---------------------------------------- -------------------------------- ,transcriber's ,note3 ,this version of the text cannot represent certain typographical effects4 ,italics are delimited with the #8' character as italic4 ,this is a very visual work1 with numerous diagrams and meticulous discussion of them4 ,the ,,html or e,book versions would be more suitable if those aspects of the work are of interest4 ,there is a single footnote following the ,translator's note4 ,if an article spans several pages1 marginal paragraph notes were repeated4 ,these redundant notes have been removed4 ,minor errors1 attributable to the printer1 have been corrected4 ,please see the transcriber's note at the end of this text for details regarding the handling of any textual issues encountered during its preparation4 ,the ,english ,works of ,thomas ,hobbes3 ,volume #a ,illustration3 ,,thomas ,,hobbes4 ,w ,humphrys sc4 ,london4 ,published ,march #ast #ahci1 by ,john ,bohn4 #ag1 ,henrietta ,street1 ,covent ,garden4 ,,the ,,english ,,works ,,of ,,thomas ,,hobbes ,,of ,,malmesbury2 ,,now ,,first ,,collected ,,and ,,edited ,,by ,,sir ,,william ,,molesworth1 ,,bart4 --- ,,vol4 ,i4 --- ,,london3 ,,john ,,bohn1 ,,henrietta ,,street1 ,,covent ,,garden4 --- ,,mdcccxxxix4 ,,london3 ,c4 ,,richards1 ,,printer1 ,,st4 ,,martin',s ,,lane4 ,,to ,,george ,,grote1 ,,esq4 ,m4,p4 ,,for ,,the ,,city ,,of ,,london4 -------------- ,,dear ,,grote1 ,i dedicate to you this edition of the ,works of ,hobbes2 first1 because ,i know you will be well pleased to see a complete collection of all the writings of an ,author for whom you have so high an admiration4 ,secondly1 because ,i am indebted to you for my first acquaintance with the speculations of one of the greatest and most original thinkers in the ,english language1 whose works1 ,i have often heard you regret1 were so scarce1 and so much less read and studied than they deserved to be4 ,it now1 therefore1 gives me great satisfaction to be able to gratify a wish1 you have frequently expressed1 that some person1 who had time and due reverence for that illustrious man1 would undertake to edite his works1 and bring his views again before his countrymen1 who have so long and so unjustly neglected him4 ,and likewise1 ,i am desirous1 in some way1 to express the sincere regard and respect that ,i feel for you1 and the gratitude that ,i owe you for the valuable instruction1 that ,i have obtained from your society1 and from the friendship with which you have honoured me1 during the many years we have been companions in political life4 ,yours1 truly1 ,,william ,,molesworth4 ,february #beth1 #ahci4 #gi1 ,eaton ,square1 ,london4 ---------------------------------------- -------------------------------- ,,elements ,,of ,,philosophy4 ,,the ,,first ,,section1 ,,concerning ,,body1 ,,written ,,in ,,latin ,,by ,,thomas ,,hobbes ,,of ,,malmesbury ,,and ,,translated ,,into ,,english4 ,,the ,,translator ,,to ,,the ,,reader4 ,if1 when ,i had finished my translation of this first section of the ,elements of ,philosophy1 ,i had presently committed the same to the press1 it might have come to your hands sooner than now it doth4 ,but as ,i undertook it with much diffidence of my own ability to perform it well2 so ,i thought fit1 before ,i published it1 to pray ,mr4 ,hobbes to view1 correct1 and order it according to his own mind and pleasure4 ,wherefore1 though you find some places enlarged1 others altered1 and two chapters1 ,,xviii and ,,xx1 almost wholly changed1 you may nevertheless remain assured1 that as now ,i present it to you1 it doth not at all vary from the author's own sense and meaning4 ,as for the ,six ,lessons to the ,savilian ,professors at ,oxford1 they are not of my translation1 but were written1 as here you have them in ,english1 by ,mr4 ,hobbes himself2 and are joined to this book1 because they are chiefly in defence of the same4,a ,footnote ,a3 ,they will be published in a separate volume1 with other works of a similar description4 ,w4 ,m4 ,,the ,,author',s ,,epistle ,,dedicatory1 ,,to ,,the ,,right ,,honorable1 ,,my ,,most ,,honored ,,lord1 ,,william1 ,,earl ,,of ,,devonshire4 ------- ,this first section of the ,elements of ,philosophy1 the monument of my service and your ,lordship's bounty1 though1 after the ,third ,section published1 long deferred1 yet at last finished1 ,i now present1 my most excellent ,lord1 and dedicate to your ,lordship4 ,a little book1 but full2 and great enough1 if men count well for great2 and to an attentive reader versed in the demonstrations of mathematicians1 that is1 to your ,lordship1 clear and easy to understand1 and almost new throughout1 without any offensive novelty4 ,i know that that part of philosophy1 wherein are considered lines and figures1 has been delivered to us notably improved by the ancients2 and withal a most perfect pattern of the logic by which they were enabled to find out and demonstrate such excellent theorems as they have done4 ,i know also that the hypothesis of the earth's diurnal motion was the invention of the ancients2 but that both it1 and astronomy1 that is1 celestial physics1 springing up together with it1 were by succeeding philosophers strangled with the snares of words4 ,and therefore the beginning of astronomy1 except observations1 ,i think is not to be derived from farther time than from ,nicolaus ,copernicus2 who in the age next preceding the present revived the opinion of ,pythagoras1 ,aristarchus1 and ,philolaus4 ,after him1 the doctrine of the motion of the earth being now received1 and a difficult question thereupon arising concerning the descent of heavy bodies1 ,galileus in our time1 striving with that difficulty1 was the first that opened to us the gate of natural philosophy universal1 which is the knowledge of the nature of motion4 ,so that neither can the age of natural philosophy be reckoned higher than to him4 ,lastly1 the science of man's body1 the most profitable part of natural science1 was first discovered with admirable sagacity by our countryman ,doctor ,harvey1 principal ,physician to ,king ,james and ,king ,charles1 in his books of the ,motion of the ,blood1 and of the ,generation of ,living ,creatures2 who is the only man ,i know1 that conquering envy1 hath established a new doctrine in his life-time4 ,before these1 there was nothing certain in natural philosophy but every man's experiments to himself1 and the natural histories1 if they may be called certain1 that are no certainer than civil histories4 ,but since these1 astronomy and natural philosophy in general have1 for so little time1 been extraordinarily advanced by ,joannes ,keplerus1 ,petrus ,gassendus1 and ,marinus ,mersennus2 and the science of human bodies in special by the wit and industry of physicians1 the only true natural philosophers1 especially of our most learned men of the ,college of ,physicians in ,london4 ,natural ,philosophy is therefore but young2 but ,civil ,philosophy yet much younger1 as being no older 7,i say it provoked1 and that my detractors may know how little they have wrought upon me7 than my own book ,de ,cive4 ,but what8 were there no philosophers natural nor civil among the ancient ,greeks8 ,there were men so called2 witness ,lucian1 by whom they are derided2 witness divers cities1 from which they have been often by public edicts banished4 ,but it follows not that there was philosophy4 ,there walked in old ,greece a certain phantasm1 for superficial gravity1 though full within of fraud and filth1 a little like philosophy2 which unwary men1 thinking to be it1 adhered to the professors of it1 some to one1 some to another1 though they disagreed among themselves1 and with great salary put their children to them to be taught1 instead of wisdom1 nothing but to dispute1 and1 neglecting the laws1 to determine every question according to their own fancies4 ,the first doctors of the ,church1 next the ,apostles1 born in those times1 whilst they endeavoured to defend the ,christian faith against the ,gentiles by natural reason1 began also to make use of philosophy1 and with the decrees of ,holy ,scripture to mingle the sentences of heathen philosophers2 and first some harmless ones of ,plato1 but afterwards also many foolish and false ones out of the physics and metaphysics of ,aristotle2 and bringing in the enemies1 betrayed unto them the citadel of ,christianity4 ,from that time1 instead of the worship of ,god1 there entered a thing called school divinity1 walking on one foot firmly1 which is the ,holy ,scripture1 but halted on the other rotten foot1 which the ,apostle ,paul called vain1 and might have called pernicious philosophy2 for it hath raised an infinite number of controversies in the ,christian world concerning religion1 and from those controversies1 wars4 ,it is like that ,empusa in the ,athenian comic poet1 which was taken in ,athens for a ghost that changed shapes1 having one brazen leg1 but the other was the leg of an ass1 and was sent1 as was believed1 by ,hecate1 as a sign of some approaching evil fortune4 ,against this ,empusa ,i think there cannot be invented a better exorcism1 than to distinguish between the rules of religion1 that is1 the rules of honouring ,god1 which we have from the laws1 and the rules of philosophy1 that is1 the opinions of private men2 and to yield what is due to religion to the ,holy ,scripture1 and what is due to philosophy to natural reason4 ,and this ,i shall do1 if ,i but handle the ,elements of ,philosophy truly and clearly1 as ,i endeavour to do4 ,therefore having in the ,third ,section1 which ,i have published and dedicated to your ,lordship1 long since reduced all power ecclesiastical and civil by strong arguments of reason1 without repugnance to ,god's word1 to one and the same sovereign authority2 ,i intend now1 by putting into a clear method the true foundations of natural philosophy1 to fright and drive away this metaphysical ,empusa2 not by skirmish1 but by letting in the light upon her4 ,for ,i am confident1 if any confidence of a writing can proceed from the writer's fear1 circumspection1 and diffidence1 that in the three former parts of this book all that ,i have said is sufficiently demonstrated from definitions2 and all in the fourth part from suppositions not absurd4 ,but if there appear to your ,lordship anything less fully demonstrated than to satisfy every reader1 the cause was this1 that ,i professed to write not all to all1 but some things to geometricians only4 ,but that your ,lordship will be satisfied1 ,i cannot doubt4 ,there remains the second section1 which is concerning ,man4 ,that part thereof1 where ,i handle the ,optics1 containing six chapters1 together with the tables of the figures belonging to them1 ,i have already written and engraven lying by me above these six years4 ,the rest shall1 as soon as ,i can1 be added to it2 though by the contumelies and petty injuries of some unskilful men1 ,i know already1 by experience1 how much greater thanks will be due than paid me1 for telling men the truth of what men are4 ,but the burthen ,i have taken on me ,i mean to carry through2 not striving to appease1 but rather to revenge myself of envy1 by encreasing it4 ,for it contents me that ,i have your ,lordship's favour1 which1 being all you require1 ,i acknowledge2 and for which1 with my prayers to ,almighty ,god for your ,lordship's safety1 ,i shall1 to my power1 be always thankful4 ,your ,lordship's most humble servant1 ,,thomas ,,hobbes4 ,london1 ,april #bc1 #afee4 ,,the ,,author',s ,,epistle ,,to ,,the ,,reader4 ,think not1 ,courteous ,reader1 that the philosophy1 the elements whereof ,i am going to set in order1 is that which makes philosophers' stones1 nor that which is found in the metaphysic codes2 but that it is the natural reason of man1 busily flying up and down among the creatures1 and bringing back a true report of their order1 causes and effects4 ,philosophy1 therefore1 the child of the world and your own mind1 is within yourself2 perhaps not fashioned yet1 but like the world its father1 as it was in the beginning1 a thing confused4 ,do1 therefore1 as the statuaries do1 who1 by hewing off that which is superfluous1 do not make but find the image4 ,or imitate the creation3 if you will be a philosopher in good earnest1 let your reason move upon the deep of your own cogitations and experience2 those things that lie in confusion must be set asunder1 distinguished1 and every one stamped with its own name set in order2 that is to say1 your method must resemble that of the creation4 ,the order of the creation was1 light1 distinction of day and night1 the firmament1 the luminaries1 sensible creatures1 man2 and1 after the creation1 the commandment4 ,therefore the order of contemplation will be1 reason1 definition1 space1 the stars1 sensible quality1 man2 and after man is grown up1 subjection to command4 ,in the first part of this section1 which is entitled ,logic1 ,i set up the light of reason4 ,in the second1 which hath for title the ,grounds of ,philosophy1 ,i distinguish the most common notions by accurate definition1 for the avoiding of confusion and obscurity4 ,the third part concerns the expansion of space1 that is ,geometry4 ,the fourth contains the ,motion of the ,stars1 together with the doctrine of sensible qualities4 ,in the second section1 if it please ,god1 shall be handled ,man4 ,in the third section1 the doctrine of ,subjection is handled already4 ,this is the method ,i followed2 and if it like you1 you may use the same2 for ,i do but propound1 not commend to you anything of mine4 ,but whatsoever shall be the method you will like1 ,i would very fain commend philosophy to you1 that is to say1 the study of wisdom1 for want of which we have all suffered much damage lately4 ,for even they1 that study wealth1 do it out of love to wisdom2 for their treasures serve them but for a looking-glass1 wherein to behold and contemplate their own wisdom4 ,nor do they1 that love to be employed in public business1 aim at anything but place wherein to show their wisdom4 ,neither do voluptuous men neglect philosophy1 but only because they know not how great a pleasure it is to the mind of man to be ravished in the vigorous and perpetual embraces of the most beauteous world4 ,lastly1 though for nothing else1 yet because the mind of man is no less impatient of empty time than nature is of empty place1 to the end you be not forced for want of what to do1 to be troublesome to men that have business1 or take hurt by falling into idle company1 but have somewhat of your own wherewith to fill up your time1 ,i recommend unto you to study philosophy4 ,farewell4 ,t4 ,h4 ,,titles ,,of ,,the ,,chapters4 ,,part ,,first1 ,,or ,,logic4 ,,chap4 ,,page4 #a4 ,of ,philosophy #a #b4 ,of ,names #ac #c4 ,of ,proposition #bi #d4 ,of ,syllogism #dd #e4 ,of ,erring1 ,falsity1 and ,captions #ee #f4 ,of ,method #fe ,,part ,,second1 ,,or ,,the ,,first ,,grounds ,,of ,,philosophy4 #g4 ,of ,place and ,time #ia #h4 ,of ,body and ,accident #aja #i4 ,of ,cause and ,effect #abj #aj4 ,of ,power and ,act #abg #aa4 ,of ,identity and ,difference #acb #ab4 ,of ,quantity #ach #ac4 ,of ,analogism1 or the ,same ,proportion #add #ad4 ,of ,strait and ,crooked1 ,angle and ,figure #agf ,,part ,,third1 ,,of ,,the ,,proportions ,,of ,,motions ,,and ,,magnitudes4 #ae4 ,of the ,nature1 ,properties1 and divers #bjc ,considerations of ,motion and ,endeavour #af4 ,of ,motion ,accelerated and ,uniform1 and of ,motion #bah by ,concourse #ag4 ,of ,figures ,deficient #bdf #ah4 ,of the ,equation of ,strait ,lines with the ,crooked #bfh ,lines of ,parabolas1 and other ,figures made in imitation of ,parabolas #ai4 ,of ,angles of ,incidence and ,reflection1 equal by #bgc ,supposition #bj4 ,of the ,dimension of a ,circle1 and the ,division of #bhg ,angles or ,arches #ba4 ,of ,circular ,motion #cag #bb4 ,of other ,variety of ,motions #ccc #bc4 ,of the ,centre of ,equiponderation of ,bodies #cej pressing downwards in ,strait ,parallel ,lines #bd4 ,of ,refraction and ,reflection #cgd ,,part ,,fourth1 ,,of ,,physics1 ,,or ,,the ,,phenomena ,,of ,,nature4 #be4 ,of ,sense and ,animal ,motion #chg #bf4 ,of the ,world and of the ,stars #daj #bg4 ,of ,light1 ,heat1 and of ,colours #dde #bh4 ,of ,cold1 ,wind1 ,hard1 ,ice1 ,restitution of ,bodies #dff bent1 ,diaphanous1 ,lightning and ,thunder1 and of the ,heads of ,rivers #bi4 ,of ,sound1 ,odour1 ,savour1 and ,touch #dhe #cj4 ,of ,gravity #ejh ---------------------------------------- -------------------------------- ,,computation ,,or ,,logic4 ------- ,,chapter ,i4 ,,of ,,philosophy4 #a4 ,the ,introduction4--#b4 ,the ,definition of ,philosophy explained4--#c4 ,ratiocination of the ,mind4--#d4 ,properties1 what they are4--#e4 ,how ,properties are known by ,generation1 and contrarily4--#f4 ,the ,scope of ,philosophy4--#g4 ,the ,utility of it4--#h4 ,the ,subject4--#i4 ,the ,parts of it4--#aj4 ,the ,epilogue4 ,sidenote3 ,introduction4 ,philosophy seems to me to be amongst men now1 in the same manner as corn and wine are said to have been in the world in ancient time4 ,for from the beginning there were vines and ears of corn growing here and there in the fields2 but no care was taken for the planting and sowing of them4 ,men lived therefore upon acorns2 or if any were so bold as to venture upon the eating of those unknown and doubtful fruits1 they did it with danger of their health4 ,in like manner1 every man brought ,philosophy1 that is1 ,natural ,reason1 into the world with him2 for all men can reason to some degree1 and concerning some things3 but where there is need of a long series of reasons1 there most men wander out of the way1 and fall into error for want of method1 as it were for want of sowing and planting1 that is1 of improving their reason4 ,and from hence it comes to pass1 that they who content themselves with daily experience1 which may be likened to feeding upon acorns1 and either reject1 or not much regard philosophy1 are commonly esteemed1 and are1 indeed1 men of sounder judgment than those who1 from opinions1 though not vulgar1 yet full of uncertainty1 and carelessly received1 do nothing but dispute and wrangle1 like men that are not well in their wits4 ,i confess1 indeed1 that that part of philosophy by which magnitudes and figures are computed1 is highly improved4 ,but because ,i have not observed the like advancement in the other parts of it1 my purpose is1 as far forth as ,i am able1 to lay open the few and first ,elements of ,philosophy in general1 as so many seeds from which pure and true ,philosophy may hereafter spring up by little and little4 ,i am not ignorant how hard a thing it is to weed out of men's minds such inveterate opinions as have taken root there1 and been confirmed in them by the authority of most eloquent writers2 especially seeing true 7that is1 accurate7 ,philosophy professedly rejects not only the paint and false colours of language1 but even the very ornaments and graces of the same2 and the first grounds of all science are not only not beautiful1 but poor1 arid1 and1 in appearance1 deformed4 ,nevertheless1 there being certainly some men1 though but few1 who are delighted with truth and strength of reason in all things1 ,i thought ,i might do well to take this pains for the sake even of those few4 ,i proceed therefore to the matter1 and take my beginning from the very definition of philosophy1 which is this4 ,sidenote3 ,definition of ,philosophy explained4 #b4 ,,philosophy is such knowledge of effects or appearances1 as we acquire by true ratiocination from the knowledge we have first of their causes or generation2 ,and again1 of such causes or generations as may be from knowing first their effects4 ,for the better understanding of which definition1 we must consider1 first1 that although ,sense and ,memory of things1 which are common to man and all living creatures1 be knowledge1 yet because they are given us immediately by nature1 and not gotten by ratiocination1 they are not philosophy4 ,secondly1 seeing ,experience is nothing but memory2 and ,prudence1 or prospect into the future time1 nothing but expectation of such things as we have already had experience of1 ,prudence also is not to be esteemed philosophy4 ,by ,,ratiocination1 ,i mean computation4 ,now to compute1 is either to collect the sum of many things that are added together1 or to know what remains when one thing is taken out of another4 ,ratiocination1 therefore1 is the same with addition and substraction2 and if any man add multiplication and division1 ,i will not be against it1 seeing multiplication is nothing but addition of equals one to another1 and division nothing but a substraction of equals one from another1 as often as is possible4 ,so that all ratiocination is comprehended in these two operations of the mind1 addition and substraction4 ,sidenote3 ,ratiocination of the ,mind4 #c4 ,but how by the ratiocination of our mind1 we add and substract in our silent thoughts1 without the use of words1 it will be necessary for me to make intelligible by an example or two4 ,if therefore a man see something afar off and obscurely1 although no appellation had yet been given to anything1 he will1 notwithstanding1 have the same idea of that thing for which now1 by imposing a name on it1 we call it body4 ,again1 when1 by coming nearer1 he sees the same thing thus and thus1 now in one place and now in another1 he will have a new idea thereof1 namely1 that for which we now call such a thing animated4 ,thirdly1 when standing nearer1 he perceives the figure1 hears the voice1 and sees other things which are signs of a rational mind1 he has a third idea1 though it have yet no appellation1 namely1 that for which we now call anything rational4 ,lastly1 when1 by looking fully and distinctly upon it1 he conceives all that he has seen as one thing1 the idea he has now is compounded of his former ideas1 which are put together in the mind in the same order in which these three single names1 body1 animated1 rational1 are in speech compounded into this one name1 body-animated-rational1 or man4 ,in like manner1 of the several conceptions of four sides1 equality of sides1 and right angles1 is compounded the conception of a square4 ,for the mind may conceive a figure of four sides without any conception of their equality1 and of that equality without conceiving a right angle2 and may join together all these single conceptions into one conception or one idea of a square4 ,and thus we see how the conceptions of the mind are compounded4 ,again1 whosoever sees a man standing near him1 conceives the whole idea of that man2 and if1 as he goes away1 he follow him with his eyes only1 he will lose the idea of those things which were signs of his being rational1 whilst1 nevertheless1 the idea of a body-animated remains still before his eyes1 so that the idea of rational is substracted from the whole idea of man1 that is to say1 of body-animated-rational1 and there remains that of body-animated2 and a while after1 at a greater distance1 the idea of animated will be lost1 and that of body only will remain2 so that at last1 when nothing at all can be seen1 the whole idea will vanish out of sight4 ,by which examples1 ,i think1 it is manifest enough what is the internal ratiocination of the mind without words4 ,we must not therefore think that computation1 that is1 ratiocination1 has place only in numbers1 as if man were distinguished from other living creatures 7which is said to have been the opinion of ,pythagoras7 by nothing but the faculty of numbering2 for magnitude1 body1 motion1 time1 degrees of quality1 action1 conception1 proportion1 speech and names 7in which all the kinds of philosophy consist7 are capable of addition and substraction4 ,now such things as we add or substract1 that is1 which we put into an account1 we are said to consider1 in ,greek 1 in which language also signifies to compute1 reason1 or reckon4 ,sidenote3 ,properties1 what they are4 #d4 ,but effects and the appearances of things to sense1 are faculties or powers of bodies1 which make us distinguish them from one another2 that is to say1 conceive one body to be equal or unequal1 like or unlike to another body2 as in the example above1 when by coming near enough to any body1 we perceive the motion and going of the same1 we distinguish it thereby from a tree1 a column1 and other fixed bodies2 and so that motion or going is the property thereof1 as being proper to living creatures1 and a faculty by which they make us distinguish them from other bodies4 ,sidenote3 ,how ,properties are known by ,generation1 and contrarily4 #e4 ,how the knowledge of any effect may be gotten from the knowledge of the generation thereof1 may easily be understood by the example of a circle3 for if there be set before us a plain figure1 having1 as near as may be1 the figure of a circle1 we cannot possibly perceive by sense whether it be a true circle or no2 than which1 nevertheless1 nothing is more easy to be known to him that knows first the generation of the propounded figure4 ,for let it be known that the figure was made by the circumduction of a body whereof one end remained unmoved1 and we may reason thus2 a body carried about1 retaining always the same length1 applies itself first to one radius1 then to another1 to a third1 a fourth1 and successively to all2 and1 therefore1 the same length1 from the same point1 toucheth the circumference in every part thereof1 which is as much as to say1 as all the radii are equal4 ,we know1 therefore1 that from such generation proceeds a figure1 from whose one middle point all the extreme points are reached unto by equal radii4 ,and in like manner1 by knowing first what figure is set before us1 we may come by ratiocination to some generation of the same1 though perhaps not that by which it was made1 yet that by which it might have been made2 for he that knows that a circle has the property above declared1 will easily know whether a body carried about1 as is said1 will generate a circle or no4 ,sidenote3 ,scope of ,philosophy4 #f4 ,the end or scope of philosophy is1 that we may make use to our benefit of effects formerly seen2 or that1 by application of bodies to one another1 we may produce the like effects of those we conceive in our mind1 as far forth as matter1 strength1 and industry1 will permit1 for the commodity of human life4 ,for the inward glory and triumph of mind that a man may have for the mastering of some difficult and doubtful matter1 or for the discovery of some hidden truth1 is not worth so much pains as the study of ,philosophy requires2 nor need any man care much to teach another what he knows himself1 if he think that will be the only benefit of his labour4 ,the end of knowledge is power2 and the use of theorems 7which1 among geometricians1 serve for the finding out of properties7 is for the construction of problems2 and1 lastly1 the scope of all speculation is the performing of some action1 or thing to be done4 ,sidenote3 ,utility of ,philosophy4 #g4 ,but what the utility of philosophy is1 especially of natural philosophy and geometry1 will be best understood by reckoning up the chief commodities of which mankind is capable1 and by comparing the manner of life of such as enjoy them1 with that of others which want the same4 ,now1 the greatest commodities of mankind are the arts2 namely1 of measuring matter and motion2 of moving ponderous bodies2 of architecture2 of navigation2 of making instruments for all uses2 of calculating the celestial motions1 the aspects of the stars1 and the parts of time2 of geography1 c4 ,by which sciences1 how great benefits men receive is more easily understood than expressed4 ,these benefits are enjoyed by almost all the people of ,europe1 by most of those of ,asia1 and by some of ,africa3 but the ,americans1 and they that live near the ,poles1 do totally want them4 ,but why8 ,have they sharper wits than these8 ,have not all men one kind of soul1 and the same faculties of mind8 ,what1 then1 makes this difference1 except philosophy8 ,philosophy1 therefore1 is the cause of all these benefits4 ,but the utility of moral and civil philosophy is to be estimated1 not so much by the commodities we have by knowing these sciences1 as by the calamities we receive from not knowing them4 ,now1 all such calamities as may be avoided by human industry1 arise from war1 but chiefly from civil war2 for from this proceed slaughter1 solitude1 and the want of all things4 ,but the cause of war is not that men are willing to have it2 for the will has nothing for object but good1 at least that which seemeth good4 ,nor is it from this1 that men know not that the effects of war are evil2 for who is there that thinks not poverty and loss of life to be great evils8 ,the cause1 therefore1 of civil war is1 that men know not the causes neither of war nor peace1 there being but few in the world that have learned those duties which unite and keep men in peace1 that is to say1 that have learned the rules of civil life sufficiently4 ,now1 the knowledge of these rules is moral philosophy4 ,but why have they not learned them1 unless for this reason1 that none hitherto have taught them in a clear and exact method8 ,for what shall we say8 ,could the ancient masters of ,greece1 ,egypt1 ,rome1 and others1 persuade the unskilful multitude to their innumerable opinions concerning the nature of their gods1 which they themselves knew not whether they were true or false1 and which were indeed manifestly false and absurd2 and could they not persuade the same multitude to civil duty1 if they themselves had understood it8 ,or shall those few writings of geometricians which are extant1 be thought sufficient for the taking away of all controversy in the matters they treat of1 and shall those innumerable and huge volumes of ethics be thought unsufficient1 if what they teach had been certain and well demonstrated8 ,what1 then1 can be imagined to be the cause that the writings of those men have increased science1 and the writings of these have increased nothing but words1 saving that the former were written by men that knew1 and the latter by such as knew not1 the doctrine they taught only for ostentation of their wit and eloquence8 ,nevertheless1 ,i deny not but the reading of some such books is very delightful2 for they are most eloquently written1 and contain many clear1 wholesome and choice sentences1 which yet are not universally true1 though by them universally pronounced4 ,from whence it comes to pass1 that the circumstances of times1 places1 and persons being changed1 they are no less frequently made use of to confirm wicked men in their purposes1 than to make them understand the precepts of civil duties4 ,now that which is chiefly wanting in them1 is a true and certain rule of our actions1 by which we might know whether that we undertake be just or unjust4 ,for it is to no purpose to be bidden in every thing to do right1 before there be a certain rule and measure of right established1 which no man hitherto hath established4 ,seeing1 therefore1 from the not knowing of civil duties1 that is1 from the want of moral science1 proceed civil wars1 and the greatest calamities of mankind1 we may very well attribute to such science the production of the contrary commodities4 ,and thus much is sufficient1 to say nothing of the praises and other contentment proceeding from philosophy1 to let you see the utility of the same in every kind thereof4 ,sidenote3 ,subject of ,philosophy4 #h4 ,the subject of ,philosophy1 or the matter it treats of1 is every body of which we can conceive any generation1 and which we may1 by any consideration thereof1 compare with other bodies1 or which is capable of composition and resolution2 that is to say1 every body of whose generation or properties we can have any knowledge4 ,and this may be deduced from the definition of philosophy1 whose profession it is to search out the properties of bodies from their generation1 or their generation from their properties2 and1 therefore1 where there is no generation or property1 there is no philosophy4 ,therefore it excludes ,theology1 ,i mean the doctrine of ,god1 eternal1 ingenerable1 incomprehensible1 and in whom there is nothing neither to divide nor compound1 nor any generation to be conceived4 ,it excludes the doctrine of angels1 and all such things as are thought to be neither bodies nor properties of bodies2 there being in them no place neither for composition nor division1 nor any capacity of more and less1 that is to say1 no place for ratiocination4 ,it excludes history1 as well natural as political1 though most useful 7nay necessary7 to philosophy2 because such knowledge is but experience1 or authority1 and not ratiocination4 ,it excludes all such knowledge as is acquired by ,divine inspiration1 or revelation1 as not derived to us by reason1 but by ,divine grace in an instant1 and1 as it were1 by some sense supernatural4 ,it excludes not only all doctrines which are false1 but such also as are not well-grounded2 for whatsoever we know by right ratiocination1 can neither be false nor doubtful2 and1 therefore1 astrology1 as it is now held forth1 and all such divinations rather than sciences1 are excluded4 ,lastly1 the doctrine of ,god's worship is excluded from philosophy1 as being not to be known by natural reason1 but by the authority of the ,church2 and as being the object of faith1 and not of knowledge4 ,sidenote3 ,parts of ,philosophy4 #i4 ,the principal parts of philosophy are two4 ,for two chief kinds of bodies1 and very different from one another1 offer themselves to such as search after their generation and properties2 one whereof being the work of nature1 is called a natural body1 the other is called a commonwealth1 and is made by the wills and agreement of men4 ,and from these spring the two parts of philosophy1 called natural and civil4 ,but seeing that1 for the knowledge of the properties of a commonwealth1 it is necessary first to know the dispositions1 affections1 and manners of men1 civil philosophy is again commonly divided into two parts1 whereof one1 which treats of men's dispositions and manners1 is called ethics2 and the other1 which takes cognizance of their civil duties1 is called politics1 or simply civil philosophy4 ,in the first place1 therefore 7after ,i have set down such premises as appertain to the nature of philosophy in general71 ,i will discourse of bodies natural2 in the second1 of the dispositions and manners of men2 and in the third1 of the civil duties of subjects4 ,sidenote3 ,epilogue4 #aj4 ,to conclude2 seeing there may be many who will not like this my definition of philosophy1 and will say1 that1 from the liberty which a man may take of so defining as seems best to himself1 he may conclude any thing from any thing 7though ,i think it no hard matter to demonstrate that this definition of mine agrees with the sense of all men72 yet1 lest in this point there should be any cause of dispute betwixt me and them1 ,i here undertake no more than to deliver the elements of that science by which the effects of anything may be found out from the known generation of the same1 or contrarily1 the generation from the effects2 to the end that they who search after other philosophy1 may be admonished to seek it from other principles4 ------- ,,chapter ,,ii4 ,,of ,,names4 #a4 ,the necessity of sensible ,moniments or ,marks for the help of ,memory3 a ,mark defined4--#b4 ,the necessity of ,marks for the signification of the conceptions of the ,mind4--#c4 ,names supply both those necessities4--#d4 ,the ,definition of a ,name4--#e4 ,names are ,signs not of ,things1 but of our ,cogitations4--#f4 ,what it is we give ,names to4--#g4 ,names ,positive and ,negative4--#h4 ,contradictory ,names4--#i4 ,a ,common ,name4--#aj4 ,names of the ,first and ,second ,intention4--#aa4 ,universal1 ,particular1 ,individual1 and ,indefinite ,names4--#ab4 ,names ,univocal and ,equivocal4--#ac4 ,absolute and ,relative ,names4--#ad4 ,simple and ,compounded ,names4--#ae4 ,a ,predicament described4--#af4 ,some things to be noted concerning ,predicaments4 ,sidenote3 ,necessity of sensible ,moniments or ,marks for the help of ,memory4 #a4 ,how unconstant and fading men's thoughts are1 and how much the recovery of them depends upon chance1 there is none but knows by infallible experience in himself4 ,for no man is able to remember quantities without sensible and present measures1 nor colours without sensible and present patterns1 nor number without the names of numbers disposed in order and learned by heart4 ,so that whatsoever a man has put together in his mind by ratiocination without such helps1 will presently slip from him1 and not be revocable but by beginning his ratiocination anew4 ,from which it follows1 that1 for the acquiring of philosophy1 some sensible moniments are necessary1 by which our past thoughts may be not only reduced1 but also registered every one in its own order4,sidenote3 ,a ,mark defined ,these moniments ,i call ,,marks1 namely1 sensible things taken at pleasure1 that1 by the sense of them1 such thoughts may be recalled to our mind as are like those thoughts for which we took them4 ,sidenote3 ,necessity of ,marks for the signification of the conceptions of the ,mind4 #b4 ,again1 though some one man1 of how excellent a wit soever1 should spend all his time partly in reasoning1 and partly in inventing marks for the help of his memory1 and advancing himself in learning2 who sees not that the benefit he reaps to himself will not be much1 and to others none at all8 ,for unless he communicate his notes with others1 his science will perish with him4 ,but if the same notes be made common to many1 and so one man's inventions be taught to others1 sciences will thereby be increased to the general good of mankind4 ,it is therefore necessary1 for the acquiring of philosophy1 that there be certain signs1 by which what one man finds out may be manifested and made known to others4 ,now1 those things we call ,,signs are the antecedents of their consequents1 and the consequents of their antecedents1 as often as we observe them to go before or follow after in the same manner4 ,for example1 a thick cloud is a sign of rain to follow1 and rain a sign that a cloud has gone before1 for this reason only1 that we seldom see clouds without the consequence of rain1 nor rain at any time but when a cloud has gone before4 ,and of signs1 some are natural1 whereof ,i have already given an example1 others are arbitrary1 namely1 those we make choice of at our own pleasure1 as a bush hung up1 signifies that wine is to be sold there2 a stone set in the ground signifies the bound of a field2 and words so and so connected1 signify the cogitations and motions of our mind4 ,the difference1 therefore1 betwixt marks and signs is this1 that we make those for our own use1 but these for the use of others4 ,sidenote3 ,names supply both those necessities4 #c4 ,words so connected as that they become signs of our thoughts1 are called ,,speech1 of which every part is a name4 ,but seeing 7as is said7 both marks and signs are necessary for the acquiring of philosophy1 7marks by which we may remember our own thoughts1 and signs by which we may make our thoughts known to others71 names do both these offices2 but they serve for marks before they be used as signs4 ,for though a man were alone in the world1 they would be useful to him in helping him to remember2 but to teach others1 7unless there were some others to be taught7 of no use at all4 ,again1 names1 though standing singly by themselves1 are marks1 because they serve to recal our own thoughts to mind2 but they cannot be signs1 otherwise than by being disposed and ordered in speech as parts of the same4 ,for example1 a man may begin with a word1 whereby the hearer may frame an idea of something in his mind1 which1 nevertheless1 he cannot conceive to be the idea which was in the mind of him that spake1 but that he would say something which began with that word1 though perhaps not as by itself1 but as part of another word4 ,so that the nature of a name consists principally in this1 that it is a mark taken for memory's sake2 but it serves also by accident to signify and make known to others what we remember ourselves1 and1 therefore1 ,i will define it thus3 ,sidenote3 ,definition of a ,name4 #d4 ,a ,,name is a word taken at pleasure to serve for a mark1 which may raise in our mind a thought like to some thought we had before1 and which being pronounced to others1 may be to them a sign of what thought the speaker had1 or had not before in his mind4 ,and it is for brevity's sake that ,i suppose the original of names to be arbitrary1 judging it a thing that may be assumed as unquestionable4 ,for considering that new names are daily made1 and old ones laid aside2 that diverse nations use different names1 and how impossible it is either to observe similitude1 or make any comparison betwixt a name and a thing1 how can any man imagine that the names of things were imposed from their natures8 ,for though some names of living creatures and other things1 which our first parents used1 were taught by ,god himself2 yet they were by him arbitrarily imposed1 and afterwards1 both at the ,tower of ,babel1 and since1 in process of time1 growing everywhere out of use1 are quite forgotten1 and in their room have succeeded others1 invented and received by men at pleasure4 ,moreover1 whatsoever the common use of words be1 yet philosophers1 who were to teach their knowledge to others1 had always the liberty1 and sometimes they both had and will have a necessity1 of taking to themselves such names as they please for the signifying of their meaning1 if they would have it understood4 ,nor had mathematicians need to ask leave of any but themselves to name the figures they invented1 parabolas1 hyperboles1 cissoeides1 quadratices1 c4 or to call one magnitude ,a1 another ,b4 ,sidenote3 ,names are signs not of things1 but of our cogitations4 #e4 ,but seeing names ordered in speech 7as is defined7 are signs of our conceptions1 it is manifest they are not signs of the things themselves2 for that the sound of this word stone should be the sign of a stone1 cannot be understood in any sense but this1 that he that hears it collects that he that pronounces it thinks of a stone4 ,and1 therefore1 that disputation1 whether names signify the matter or form1 or something compounded of both1 and other like subtleties of the metaphysics1 is kept up by erring men1 and such as understand not the words they dispute about4 ,sidenote3 ,what it is we give names to4 #f4 ,nor1 indeed1 is it at all necessary that every name should be the name of something4 ,for as these1 a man1 a tree1 a stone1 are the names of the things themselves1 so the images of a man1 of a tree1 and of a stone1 which are represented to men sleeping1 have their names also1 though they be not things1 but only fictions and phantasms of things4 ,for we can remember these2 and1 therefore1 it is no less necessary that they have names to mark and signify them1 than the things themselves4 ,also this word future is a name1 but no future thing has yet any being1 nor do we know whether that which we call future1 shall ever have a being or no4 ,nevertheless1 seeing we use in our mind to knit together things past with those that are present1 the name future serves to signify such knitting together4 ,moreover1 that which neither is1 nor has been1 nor ever shall1 or ever can be1 has a name1 namely1 that which neither is nor has been1 c42 or more briefly this1 impossible4 ,to conclude2 this word nothing is a name1 which yet cannot be the name of any thing3 for when1 for example1 we substract #b and #c from #e1 and so nothing remaining1 we would call that substraction to mind1 this speech nothing remains1 and in it the word nothing is not unuseful4 ,and for the same reason we say truly1 less than nothing remains1 when we substract more from less2 for the mind feigns such remains as these for doctrine's sake1 and desires1 as often as is necessary1 to call the same to memory4 ,but seeing every name has some relation to that which is named1 though that which we name be not always a thing that has a being in nature1 yet it is lawful for doctrine's sake to apply the word thing to whatsoever we name2 as if it were all one whether that thing be truly existent1 or be only feigned4 ,sidenote3 ,names ,positive and ,negative4 #g4 ,the first distinction of names is1 that some are positive1 or affirmative1 others negative1 which are also called privative and indefinite4 ,positive are such as we impose for the likeness1 equality1 or identity of the things we consider2 negative1 for the diversity1 unlikeness1 or inequality of the same4 ,examples of the former are1 a man1 a philosopher2 for a man denotes any one of a multitude of men1 and a philosopher1 any one of many philosophers1 by reason of their similitude2 also1 ,socrates is a positive name1 because it signifies always one and the same man4 ,examples of negatives are such positives as have the negative particle not added to them1 as not-man1 not-philosopher4 ,but positives were before negatives2 for otherwise there could have been no use at all of these4 ,for when the name of white was imposed upon certain things1 and afterwards upon other things1 the names of black1 blue1 transparent1 c4 the infinite dissimilitudes of these with white could not be comprehended in any one name1 save that which had in it the negation of white1 that is to say1 the name not-white1 or some other equivalent to it1 in which the word white is repeated1 such as unlike to white1 c4 ,and by these negative names1 we take notice ourselves1 and signify to others what we have not thought of4 ,sidenote3 ,contradictory names4 #h4 ,positive and negative names are contradictory to one another1 so that they cannot both be the name of the same thing4 ,besides1 of contradictory names1 one is the name of anything whatsoever2 for whatsoever is1 is either man1 or not-man1 white or not-white1 and so of the rest4 ,and this is so manifest1 that it needs no farther proof or explication2 for they that say the same thing cannot both be1 and not be1 speak obscurely2 but they that say1 whatsoever is1 either is1 or is not1 speak also absurdly and ridiculously4 ,the certainty of this axiom1 viz4 of two contradictory names1 one is the name of anything whatsoever1 the other not1 is the original and foundation of all ratiocination1 that is1 of all philosophy2 and therefore it ought to be so exactly propounded1 that it may be of itself clear and perspicuous to all men2 as indeed it is1 saving to such1 as reading the long discourses made upon this subject by the writers of metaphysics 7which they believe to be some egregious learning7 think they understand not1 when they do4 ,sidenote3 ,a common name4 #i4 ,secondly1 of names1 some are common to many things1 as a man1 a tree2 others proper to one thing1 as he that writ the ,iliad1 ,homer1 this man1 that man4 ,and a common name1 being the name of many things severally taken1 but not collectively of all together 7as man is not the name of all mankind1 but of every one1 as of ,peter1 ,john1 and the rest severally7 is therefore called an universal name2 and therefore this word universal is never the name of any thing existent in nature1 nor of any idea or phantasm formed in the mind1 but always the name of some word or name2 so that when a living creature1 a stone1 a spirit1 or any other thing1 is said to be universal1 it is not to be understood1 that any man1 stone1 c4 ever was or can be universal1 but only that these words1 living creature1 stone1 c4 are universal names1 that is1 names common to many things2 and the conceptions answering them in our mind1 are the images and phantasms of several living creatures1 or other things4 ,and therefore1 for the understanding of the extent of an universal name1 we need no other faculty but that of our imagination1 by which we remember that such names bring sometimes one thing1 sometimes another1 into our mind4 ,also of common names1 some are more1 some less common4 ,more common1 is that which is the name of more things2 less common1 the name of fewer things2 as living creature is more common than man1 or horse1 or lion1 because it comprehends them all3 and therefore a more common name1 in respect of a less common1 is called the genus1 or a general name2 and this in respect of that1 the species1 or a special name4 ,sidenote3 ,names of the first and second intention4 #aj4 ,and from hence proceeds the third distinction of names1 which is1 that some are called names of the first1 others of the second intention4 ,of the first intention are the names of things1 a man1 stone1 c43 of the second are the names of names and speeches1 as universal1 particular1 genus1 species1 syllogism1 and the like4 ,but it is hard to say why those are called names of the first1 and these of the second intention1 unless perhaps it was first intended by us to give names to those things which are of daily use in this life1 and afterwards to such things as appertain to science1 that is1 that our second intention was to give names to names4 ,but whatsoever the cause hereof may be1 yet this is manifest1 that genus1 species1 definition1 c4 are names of words and names only2 and therefore to put genus and species for things1 and definition for the nature of any thing1 as the writers of metaphysics have done1 is not right1 seeing they be only significations of what we think of the nature of things4 ,sidenote3 ,universal1 particular1 individual1 and indefinite names4 #aa4 ,fourthly1 some names are of certain and determined1 others of uncertain and undetermined signification4 ,of determined and certain signification is1 first1 that name which is given to any one thing by itself1 and is called an individual name2 as ,homer1 this tree1 that living creature1 c4 ,secondly that which has any of these words1 all1 every1 both1 either1 or the like added to it2 and it is therefore called an universal name1 because it signifies every one of those things to which it is common2 and of certain signification for this reason1 that he which hears1 conceives in his mind the same thing that he which speaks would have him conceive4 ,of indefinite signification is1 first1 that name which has the word some1 or the like added to it1 and is called a particular name2 secondly1 a common name set by itself without any note either of universality or particularity1 as man1 stone1 and is called an indefinite name2 but both particular and indefinite names are of uncertain signification1 because the hearer knows not what thing it is the speaker would have him conceive2 and therefore in speech1 particular and indefinite names are to be esteemed equivalent to one another4 ,but these words1 all1 every1 some1 c4 which denote universality and particularity1 are not names1 but parts only of names2 so that every man1 and that man which the hearer conceives in his mind1 are all one2 and some man1 and that man which the speaker thought of1 signify the same4 ,from whence it is evident1 that the use of signs of this kind1 is not for a man's own sake1 or for his getting of knowledge by his own private meditation 7for every man has his own thoughts sufficiently determined without such helps as these7 but for the sake of others2 that is1 for the teaching and signifying of our conceptions to others2 nor were they invented only to make us remember1 but to make us able to discourse with others4 ,sidenote3 ,names univocal and equivocal4 #ab4 ,fifthly1 names are usually distinguished into univocal and equivocal4 ,univocal are those which in the same train of discourse signify always the same thing2 but equivocal those which mean sometimes one thing and sometimes another4 ,thus1 the name triangle is said to be univocal1 because it is always taken in the same sense2 and parabola to be equivocal1 for the signification it has sometimes of allegory or similitude1 and sometimes of a certain geometrical figure4 ,also every metaphor is by profession equivocal4 ,but this distinction belongs not so much to names1 as to those that use names1 for some use them properly and accurately for the finding out of truth2 others draw them from their proper sense1 for ornament or deceit4 ,sidenote3 ,absolute and relative names4 #ac4 ,sixthly1 of names1 some are absolute1 others relative4 ,relative are such as are imposed for some comparison1 as father1 son1 cause1 effect1 like1 unlike1 equal1 unequal1 master1 servant1 c4 ,and those that signify no comparison at all are absolute names4 ,but1 as it was noted above1 that universality is to be attributed to words and names only1 and not to things1 so the same is to be said of other distinctions of names2 for no things are either univocal or equivocal1 or relative or absolute4 ,there is also another distinction of names into concrete and abstract2 but because abstract names proceed from proposition1 and can have no place where there is no affirmation1 ,i shall speak of them hereafter4 ,sidenote3 ,simple and compounded names4 #ad4 ,lastly1 there are simple and compounded names4 ,but here it is to be noted1 that a name is not taken in philosophy1 as in grammar1 for one single word1 but for any number of words put together to signify one thing2 for among philosophers sentient animated body passes but for one name1 being the name of every living creature1 which yet1 among grammarians1 is accounted three names4 ,also a simple name is not here distinguished from a compounded name by a preposition1 as in grammar4 ,but ,i call a simple name1 that which in every kind is the most common or most universal2 and that a compounded name1 which by the joining of another name to it1 is made less universal1 and signifies that more conceptions than one were in the mind1 for which that latter name was added4 ,for example1 in the conception of man 7as is shown in the former chapter47 ,first1 he is conceived to be something that has extension1 which is marked by the word body4 ,body1 therefore1 is a simple name1 being put for that first single conception2 afterwards1 upon the sight of such and such motion1 another conception arises1 for which he is called an animated body2 and this ,i here call a compounded name1 as ,i do also the name animal1 which is equivalent to an animated body4 ,and1 in the same manner1 an animated rational body1 as also a man1 which is equivalent to it1 is a more compounded name4 ,and by this we see how the composition of conceptions in the mind is answerable to the composition of names2 for1 as in the mind one idea or phantasm succeeds to another1 and to this a third2 so to one name is added another and another successively1 and of them all is made one compounded name4 ,nevertheless we must not think bodies which are without the mind1 are compounded in the same manner1 namely1 that there is in nature a body1 or any other imaginable thing existent1 which at the first has no magnitude1 and then1 by the addition of magnitude1 comes to have quantity1 and by more or less quantity to have density or rarity2 and again1 by the addition of figure1 to be figurate1 and after this1 by the injection of light or colour1 to become lucid or coloured2 though such has been the philosophy of many4 ,sidenote3 ,a predicament described4 #ae4 ,the writers of logic have endeavoured to digest the names of all the kinds of things into certain scales or degrees1 by the continual subordination of names less common1 to names more common4 ,in the scale of bodies they put in the first and highest place body simply1 and in the next place under it less common names1 by which it may be more limited and determined1 namely animated and inanimated1 and so on till they come to individuals4 ,in like manner1 in the scale of quantities1 they assign the first place to quantity1 and the next to line1 superficies1 and solid1 which are names of less latitude2 and these orders or scales of names they usually call predicaments and categories4 ,and of this ordination not only positive1 but negative names also are capable2 which may be exemplified by such forms of the predicaments as follow3 ,,the ,,form ,,of ,,the ,,predicament ,,of ,,body4 ,not-,body1 or ,accident4 ,not animated4 ,body ,not living ,creature4 ,animated ,living ,not ,man4 ,creature ,not ,peter4 ,man ,peter4 ,quantity1 or so much4 ,both ,accident and ,body ,absolutely1 as are considered or ,quality1 or such4 ,comparatively1 which is called their ,relation4 ,,the ,,form ,,of ,,the ,,predicament ,,of ,,quantity4 ,not continual1 as ,number4 ,line4 ,quantity4 ,of itself1 as ,superficies4 ,solid4 ,continual ,time1 by ,line4 ,motion1 by ,line and ,by accident1 as ,time4 ,force1 by ,motion and ,solid4 ,where1 it is to be noted1 that line1 superficies1 and solid1 may be said to be of such and such quantity1 that is1 to be originally and of their own nature capable of equality and inequality2 but we cannot say there is either majority or minority1 or equality1 or indeed any quantity at all1 in time1 without the help of line and motion2 nor in motion1 without line and time2 nor in force1 otherwise than by motion and solid4 ,,the ,,form ,,of ,,the ,,predicament ,,of ,,quality4 ,seeing4 ,hearing4 ,primary ,smelling4 ,tasting4 ,perception ,touching4 by ,sense ,imagination4 ,secondary ,affection pleasant4 ,quality unpleasant4 ,by ,seeing1 as ,light and ,colour4 ,by ,hearing1 as ,sound4 ,sensible ,by ,smelling1 as ,odours4 ,quality ,by ,tasting1 as ,savours4 ,by ,touching1 as ,hardness1 ,heat1 ,cold1 c4 ,,the ,,form ,,of ,,the ,,predicament ,,of ,,relation4 ,magnitudes1 as ,equality and ,inequality4 ,qualities1 as ,likeness and ,unlikeness4 ,relation of ,together ,in ,place4 ,in ,time4 ,order ,in ,place ,former4 ,not together ,later ,in ,time ,former4 ,later ,sidenote3 ,some things to be noted concerning predicaments4 #af4 ,concerning which predicaments it is to be noted1 in the first place1 that as the division is made in the first predicament into contradictory names1 so it might have been done in the rest4 ,for1 as there1 body is divided into animated and not-animated1 so1 in the second predicament1 continual quantity may be divided into line and not-line1 and again1 not-line into superficies and not-superficies1 and so in the rest2 but it was not necessary4 ,secondly1 it is to be observed1 that of positive names the former comprehends the latter2 but of negatives the former is comprehended by the latter4 ,for example1 living-creature is the name of every man1 and therefore it comprehends the name man2 but1 on the contrary1 not-man is the name of everything which is not-living-creature1 and therefore the name not-living-creature1 which is put first1 is comprehended by the latter name1 not-man4 ,thirdly1 we must take heed that we do not think1 that as names1 so the diversities of things themselves may be searched out and determined by such distinctions as these2 or that arguments may be taken from hence 7as some have done ridiculously7 to prove that the kinds of things are not infinite4 ,fourthly1 ,i would not have any man think ,i deliver the forms above for a true and exact ordination of names2 for this cannot be performed as long as philosophy remains imperfect2 nor that by placing 7for example7 light in the predicament of qualities1 while another places the same in the predicament of bodies1 ,i pretend that either of us ought for this to be drawn from his opinion2 for this is to be done only by arguments and ratiocination1 and not by disposing of words into classes4 ,lastly1 ,i confess ,i have not yet seen any great use of the predicaments in philosophy4 ,i believe ,aristotle when he saw he could not digest the things themselves into such orders1 might nevertheless desire out of his own authority to reduce words to such forms1 as ,i have done2 but ,i do it only for this end1 that it may be understood what this ordination of words is1 and not to have it received for true1 till it be demonstrated by good reason to be so4 ------- ,,chapter ,,iii4 ,,of ,,proposition4 #a4 ,divers kinds of speech4--#b4 ,proposition defined4--#c4 ,subject1 predicate1 and copula1 what they are2 and abstract and concrete what4 #d4 ,the use and abuse of names abstract4--#e4 ,proposition1 universal and particular4--#f4 ,affirmative and negative4--#g4 ,true and false4--#h4 ,true and false belongs to speech1 and not to things4--#i4 ,proposition1 primary1 not primary1 definition1 axiom1 petition4--#aj4 ,proposition1 necessary and contingent4--#aa4 ,categorical and hypothetical4--#ab4 ,the same proposition diversely pronounced4--#ac4 ,propositions that may be reduced to the same categorical proposition1 are equipollent4--#ad4 ,universal propositions converted by contradictory names1 are equipollent4--#ae4 ,negative propositions are the same1 whether negation be before or after the copula4--#af4 ,particular propositions simply converted1 are equipollent4--#ag4 ,what are subaltern1 contrary1 subcontrary1 and contradictory propositions4--#ah4 ,consequence1 what it is4--#ai4 ,falsity cannot follow from truth4--#bj4 ,how one proposition is the cause of another4 ,sidenote3 ,divers kinds of speech4 #a4 ,from the connexion or contexture of names arise divers kinds of speech1 whereof some signify the desires and affections of men2 such are1 first1 interrogations1 which denote the desire of knowing3 as1 ,who is a good man8 ,in which speech there is one name expressed1 and another desired and expected from him of whom we ask the same4 ,then prayers1 which signify the desire of having something2 promises1 threats1 wishes1 commands1 complaints1 and other significations of other affections4 ,speech may also be absurd and insignificant2 as when there is a succession of words1 to which there can be no succession of thoughts in mind to answer them2 and this happens often to such1 as1 understanding nothing in some subtle matter1 do1 nevertheless1 to make others believe they understand1 speak of the same incoherently2 for the connection of incoherent words1 though it want the end of speech 7which is signification7 yet it is speech2 and is used by writers of metaphysics almost as frequently as speech significative4 ,in philosophy1 there is but one kind of speech useful1 which some call in ,latin dictum1 others enuntiatum et pronunciatum2 but most men call it proposition1 and is the speech of those that affirm or deny1 and expresseth truth or falsity4 ,sidenote3 ,proposition defined4 #b4 ,a ,,proposition is a speech consisting of two names copulated1 by which he that speaketh signifies he conceives the latter name to be the name of the same thing whereof the former is the name2 or 7which is all one7 that the former name is comprehended by the latter4 ,for example1 this speech1 man is a living creature1 in which two names are copulated by the verb is1 is a proposition1 for this reason1 that he that speaks it conceives both living creature and man to be names of the same thing1 or that the former name1 man1 is comprehended by the latter name1 living creature4 ,now the former name is commonly called the subject1 or antecedent1 or the contained name1 and the latter predicate1 consequent1 or containing name4 ,the sign of connection amongst most nations is either some word1 as the word is in the proposition man is a living creature1 or some case or termination of a word1 as in this proposition1 man walketh 7which is equivalent to this1 man is walking72 the termination by which it is said he walketh1 rather than he is walking1 signifieth that those two are understood to be copulated1 or to be names of the same thing4 ,but there are1 or certainly may be1 some nations that have no word which answers to our verb is1 who nevertheless form propositions by the position only of one name after another1 as if instead of man is a living creature1 it should be said man a living creature2 for the very order of the names may sufficiently show their connection2 and they are as apt and useful in philosophy1 as if they were copulated by the verb is4 ,sidenote3 ,subject1 predicate1 and copula1 what they are1 and abstract and concrete what4 #c4 ,wherefore1 in every proposition three things are to be considered1 viz4 the two names1 which are the subject1 and the predicate1 and their copulation2 both which names raise in our mind the thought of one and the same thing2 but the copulation makes us think of the cause for which those names were imposed on that thing4 ,as1 for example1 when we say a body is moveable1 though we conceive the same thing to be designed by both those names1 yet our mind rests not there1 but searches farther what it is to be a body1 or to be moveable1 that is1 wherein consists the difference betwixt these and other things1 for which these are so called1 others are not so called4 ,they1 therefore1 that seek what it is to be any thing1 as to be moveable1 to be hot1 c4 seek in things the causes of their names4 ,and from hence arises that distinction of names 7touched in the last chapter7 into concrete and abstract4 ,for concrete is the name of any thing which we suppose to have a being1 and is therefore called the subject1 in ,latin suppositum1 and in ,greek 2 as body1 moveable1 moved1 figurate1 a cubit high1 hot1 cold1 like1 equal1 ,appius1 ,lentulus1 and the like2 and1 abstract is that which in any subject denotes the cause of the concrete name1 as to be a body1 to be moveable1 to be moved1 to be figurate1 to be of such quantity1 to be hot1 to be cold1 to be like1 to be equal1 to be ,appius1 to be ,lentulus1 c4 ,or names equivalent to these1 which are most commonly called abstract names1 as corporiety1 mobility1 motion1 figure1 quantity1 heat1 cold1 likeness1 equality1 and 7as ,cicero has it7 ,appiety and ,lentulity4 ,of the same kind also are infinitives2 for to live and to move are the same with life and motion1 or to be living and to be moved4 ,but abstract names denote only the causes of concrete names1 and not the things themselves4 ,for example1 when we see any thing1 or conceive in our mind any visible thing1 that thing appears to us1 or is conceived by us1 not in one point1 but as having parts distant from one another1 that is1 as being extended and filling some space4 ,seeing therefore we call the thing so conceived body1 the cause of that name is1 that that thing is extended1 or the extension or corporiety of it4 ,so when we see a thing appear sometimes here1 sometimes there1 and call it moved or removed1 the cause of that name is that it is moved or the motion of the same4 ,and these causes of names are the same with the causes of our conceptions1 namely1 some power of action1 or affection of the thing conceived1 which some call the manner by which any thing works upon our senses1 but by most men they are called accidents2 ,i say accidents1 not in that sense in which accident is opposed to necessary2 but so1 as being neither the things themselves1 nor parts thereof1 do nevertheless accompany the things in such manner1 that 7saving extension7 they may all perish1 and be destroyed1 but can never be abstracted4 ,sidenote3 ,the use and abuse of names abstract4 #d4 ,there is also this difference betwixt concrete and abstract names1 that those were invented before propositions1 but these after2 for these could have no being till there were propositions1 from whose copula they proceed4 ,now in all matters that concern this life1 but chiefly in philosophy1 there is both great use and great abuse of abstract names2 and the use consists in this1 that without them we cannot1 for the most part1 either reason1 or compute the properties of bodies2 for when we would multiply1 divide1 add1 or substract heat1 light1 or motion1 if we should double or add them together by concrete names1 saying 7for example7 hot is double to hot1 light double to light1 or moved double to moved1 we should not double the properties1 but the bodies themselves that are hot1 light1 moved1 c4 which we would not do4 ,but the abuse proceeds from this1 that some men seeing they can consider1 that is 7as ,i said before7 bring into account the increasings and decreasings of quantity1 heat and other accidents1 without considering their bodies or subjects 7which they call abstracting1 or making to exist apart by themselves7 they speak of accidents1 as if they might be separated from all bodies4 ,and from hence proceed the gross errors of writers of metaphysics2 for1 because they can consider thought without the consideration of body1 they infer there is no need of a thinking-body2 and because quantity may be considered without considering body1 they think also that quantity may be without body1 and body without quantity2 and that a body has quantity by the addition of quantity to it4 ,from the same fountain spring those insignificant words1 abstract substance1 separated essence1 and the like2 as also that confusion of words derived from the ,latin verb est1 as essence1 essentiality1 entity1 entitative2 besides reality1 aliquiddity1 quiddity1 c4 which could never have been heard of among such nations as do not copulate their names by the verb is1 but by adjective verbs1 as runneth1 readeth1 c4 or by the mere placing of one name after another2 and yet seeing such nations compute and reason1 it is evident that philosophy has no need of those words essence1 entity1 and other the like barbarous terms4 ,sidenote3 ,proposition1 universal and particular4 #e4 ,there are many distinctions of propositions1 whereof the first is1 that some are universal1 others particular1 others indefinite1 and others singular2 and this is commonly called the distinction of quantity4 ,an universal proposition is that whose subject is affected with the sign of an universal name1 as every man is a living creature4 ,particular1 that whose subject is affected with the sign of a particular name1 as some man is learned4 ,an indefinite proposition has for its subject a common name1 and put without any sign1 as man is a living creature1 man is learned4 ,and a singular proposition is that whose subject is a singular name1 as ,socrates is a philosopher1 this man is black4 ,sidenote3 ,affirmative and negative4 #f4 ,the second distinction is into affirmative and negative1 and is called the distinction of quality4 ,an affirmative proposition is that whose predicate is a positive name1 as man is a living creature4 ,negative1 that whose predicate is a negative name1 as man is not a stone4 ,sidenote3 ,true false4 #g4 ,the third distinction is1 that one is true1 another false4 ,a true proposition is that1 whose predicate contains1 or comprehends its subject1 or whose predicate is the name of every thing1 of which the subject is the name2 as man is a living creature is therefore a true proposition1 because whatsoever is called man1 the same is also called living creature2 and some man is sick1 is true1 because sick is the name of some man4 ,that which is not true1 or that whose predicate does not contain its subject1 is called a false proposition1 as man is a stone4 ,now these words true1 truth1 and true proposition1 are equivalent to one another2 for truth consists in speech1 and not in the things spoken of2 and though true be sometimes opposed to apparent or feigned1 yet it is always to be referred to the truth of proposition2 for the image of a man in a glass1 or a ghost1 is therefore denied to be a very man1 because this proposition1 a ghost is a man1 is not true2 for it cannot be denied but that a ghost is a very ghost4 ,and therefore truth or verity is not any affection of the thing1 but of the proposition concerning it4 ,as for that which the writers of metaphysics say1 that a thing1 one thing1 and a very thing1 are equivalent to one another1 it is but trifling and childish2 for who does not know1 that a man1 one man1 and a very man1 signify the same4 ,sidenote3 ,true false belongs to speech1 and not to things4 #h4 ,and from hence it is evident1 that truth and falsity have no place but amongst such living creatures as use speech4 ,for though some brute creatures1 looking upon the image of a man in a glass1 may be affected with it1 as if it were the man himself1 and for this reason fear it or fawn upon it in vain2 yet they do not apprehend it as true or false1 but only as like2 and in this they are not deceived4 ,wherefore1 as men owe all their true ratiocination to the right understanding of speech2 so also they owe their errors to the misunderstanding of the same2 and as all the ornaments of philosophy proceed only from man1 so from man also is derived the ugly absurdity of false opinions4 ,for speech has something in it like to a spider's web1 7as it was said of old of ,solon's laws7 for by contexture of words tender and delicate wits are ensnared and stopped2 but strong wits break easily through them4 ,from hence also this may be deduced1 that the first truths were arbitrarily made by those that first of all imposed names upon things1 or received them from the imposition of others4 ,for it is true 7for example7 that man is a living creature1 but it is for this reason1 that it pleased men to impose both those names on the same thing4 ,sidenote3 ,proposition1 primary1 not primary1 definition1 axiom1 petition4 #i4 ,fourthly1 propositions are distinguished into primary and not primary4 ,primary is that wherein the subject is explicated by a predicate of many names1 as man is a body1 animated1 rational2 for that which is comprehended in the name man1 is more largely expressed in the names body1 animated1 and rational1 joined together2 and it is called primary1 because it is first in ratiocination2 for nothing can be proved1 without understanding first the name of the thing in question4 ,now primary propositions are nothing but definitions1 or parts of definitions1 and these only are the principles of demonstration1 being truths constituted arbitrarily by the inventors of speech1 and therefore not to be demonstrated4 ,to these propositions1 some have added others1 which they call primary and principles1 namely1 axioms1 and common notions2 which1 7though they be so evident that they need no proof7 yet1 because they may be proved1 are not truly principles2 and the less to be received for such1 in regard propositions not intelligible1 and sometimes manifestly false1 are thrust on us under the name of principles by the clamour of men1 who obtrude for evident to others1 all that they themselves think true4 ,also certain petitions are commonly received into the number of principles2 as1 for example1 that a straight line may be drawn between two points1 and other petitions of the writers of geometry2 and these are indeed the principles of art or construction1 but not of science and demonstration4 ,sidenote3 ,proposition necessary contingent4 #aj4 ,fifthly1 propositions are distinguished into necessary1 that is1 necessarily true2 and true1 but not necessarily1 which they call contingent4 ,a necessary proposition is when nothing can at any time be conceived or feigned1 whereof the subject is the name1 but the predicate also is the name of the same thing2 as man is a living creature is a necessary proposition1 because at what time soever we suppose the name man agrees with any thing1 at that time the name living-creature also agrees with the same4 ,but a contingent proposition is that1 which at one time may be true1 at another time false2 as every crow is black2 which may perhaps be true now1 but false hereafter4 ,again1 in every necessary proposition1 the predicate is either equivalent to the subject1 as in this1 man is a rational living creature2 or part of an equivalent name1 as in this1 man is a living creature1 for the name rational-living-creature1 or man1 is compounded of these two1 rational and living-creature4 ,but in a contingent proposition this cannot be2 for though this were true1 every man is a liar1 yet because the word liar is no part of a compounded name equivalent to the name man1 that proposition is not to be called necessary1 but contingent1 though it should happen to be true always4 ,and therefore those propositions only are necessary1 which are of sempiternal truth1 that is1 true at all times4 ,from hence also it is manifest1 that truth adheres not to things1 but to speech only1 for some truths are eternal2 for it will be eternally true1 if man1 then living-creature2 but that any man1 or living-creature1 should exist eternally1 is not necessary4 ,sidenote3 ,categorical hypothetical4 #aa4 ,a sixth distinction of propositions is into categorical and hypothetical4 ,a categorical proposition is that which is simply or absolutely pronounced1 as every man is a living-creature1 no man is a tree2 and hypothetical is that which is pronounced conditionally1 as1 if any thing be a man1 the same is also a living-creature1 if anything be a man1 the same is also not-a-stone4 ,a categorical proposition1 and an hypothetical answering it1 do both signify the same1 if the propositions be necessary2 but not if they be contingent4 ,for example1 if this1 every man is a living-creature1 be true1 this also will be true1 if any thing be a man1 the same is also a living-creature2 but in contingent propositions1 though this be true1 every crow is black1 yet this1 if any thing be a crow1 the same is black1 is false4 ,but an hypothetical proposition is then rightly said to be true1 when the consequence is true1 as every man is a living-creature1 is rightly said to be a true proposition1 because of whatsoever it is truly said that is a man1 it cannot but be truly said also1 the same is a living creature4 ,and therefore whensoever an hypothetical proposition is true1 the categorical answering it1 is not only true1 but also necessary2 which ,i thought worth the noting1 as an argument1 that philosophers may in most things reason more solidly by hypothetical than categorical propositions4 ,sidenote3 ,the same proposition diversely pronounced4 #ab4 ,but seeing every proposition may be1 and uses to be1 pronounced and written in many forms1 and we are obliged to speak in the same manner as most men speak1 yet they that learn philosophy from masters1 had need to take heed they be not deceived by the variety of expressions4 ,and therefore1 whensoever they meet with any obscure proposition1 they ought to reduce it to its most simple and categorical form2 in which the copulative word is must be expressed by itself1 and not mingled in any manner either with the subject or predicate1 both which must be separated and clearly distinguished one from another4 ,for example1 if this proposition1 man can not sin1 be compared with this1 man cannot sin1 their difference will easily appear if they be reduced to these1 man is able not to sin1 and1 man is not able to sin1 where the predicates are manifestly different4 ,but they ought to do this silently by themselves1 or betwixt them and their masters only2 for it will be thought both ridiculous and absurd1 for a man to use such language publicly4 ,being therefore to speak of equipollent propositions1 ,i put in the first place all those for equipollent1 that may be reduced purely to one and the same categorical proposition4 ,sidenote3 ,propositions that may be reduced to the same categorical proposition1 are equipollent4 #ac4 ,secondly1 that which is categorical and necessary1 is equipollent to its hypothetical proposition2 as this categorical1 a right-lined triangle has its three angles equal to two right angles1 to this hypothetical1 if any figure be a right-lined triangle1 the three angles of it are equal to two right angles4 ,sidenote3 ,universal propositions converted by contradictory names1 are equipollent4 #ad4 ,also1 any two universal propositions1 of which the terms of the one 7that is1 the subject and predicate7 are to the terms of the other1 and their order inverted1 as these1 every man is a living creature1 and every thing that is not a living-creature is not a man1 are equipollent4 ,for seeing every man is a living creature is a true proposition1 the name living creature contains the name man2 but they are both positive names1 and therefore 7by the last article of the precedent chapter7 the negative name not man1 contains the negative name not living-creature2 wherefore every thing that is not a living-creature1 is not a man1 is a true proposition4 ,likewise these1 no man is a tree1 no tree is a man1 are equipollent4 ,for if it be true that tree is not the name of any man1 then no one thing can be signified by the two names man and tree1 wherefore no tree is a man is a true proposition4 ,also to this1 whatsoever is not a living-creature is not a man1 where both the terms are negative1 this other proposition is equipollent1 only a living creature is a man4 ,sidenote3 ,negative propositions are the same1 whether the negation be before or after the copula4 #ae4 ,fourthly1 negative propositions1 whether the particle of negation be set after the copula as some nations do1 or before it1 as it is in ,latin and ,greek1 if the terms be the same1 are equipollent3 as1 for example1 man is not a tree1 and1 man is not-a-tree1 are equipollent1 though ,aristotle deny it4 ,also these1 every man is not a tree1 and no man is a tree1 are equipollent1 and that so manifestly1 as it needs not be demonstrated4 ,sidenote3 ,particular propositions simply converted1 are equipollent4 #af4 ,lastly1 all particular propositions that have their terms inverted1 as these1 some man is blind1 some blind thing is a man1 are equipollent2 for either of the two names1 is the name of some one and the same man2 and therefore in which soever of the two orders they be connected1 they signify the same truth4 ,sidenote3 ,what are subaltern1 contrary1 subcontrary1 and contradictory propositions4 #ag4 ,of propositions that have the same terms1 and are placed in the same order1 but varied either by quantity or quality1 some are called subaltern1 others contrary1 others subcontrary1 and others contradictory4 ,subaltern1 are universal and particular propositions of the same quality2 as1 every man is a living creature1 some man is a living creature2 or1 no man is wise1 some man is not wise4 ,of these1 if the universal be true1 the particular will be true also4 ,contrary1 are universal propositions of different quality2 as1 every man is happy1 no man is happy4 ,and of these1 if one be true1 the other is false3 also1 they may both be false1 as in the example given4 ,subcontrary1 are particular propositions of different quality2 as1 some man is learned1 some man is not learned2 which cannot be both false1 but they may be both true4 ,contradictory are those that differ both in quantity and quality2 as1 every man is a living creature1 some man is not a living-creature2 which can neither be both true1 nor both false4 ,sidenote3 ,consequence1 what it is4 #ah4 ,a proposition is said to follow from two other propositions1 when these being granted to be true1 it cannot be denied but the other is true also4 ,for example1 let these two propositions1 every man is a living creature1 and1 every living creature is a body1 be supposed true1 that is1 that body is the name of every living creature1 and living creature the name of every man4 ,seeing therefore1 if these be understood to be true1 it cannot be understood that body is not the name of every man1 that is1 that every man is a body is false1 this proposition will be said to follow from those two1 or to be necessarily inferred from them4 ,sidenote3 ,falsity cannot follow from truth4 #ai4 ,that a true proposition may follow from false propositions1 may happen sometimes2 but false from true1 never4 ,for if these1 every man is a stone1 and every stone is a living creature1 7which are both false7 be granted to be true1 it is granted also that living creature is the name of every stone1 and stone of every man1 that is1 that living creature is the name of every man2 that is to say1 this proposition every man is a living creature1 is true1 as it is indeed true4 ,wherefore a true proposition may sometimes follow from false4 ,but if any two propositions be true1 a false one can never follow from them4 ,for if true follow from false1 for this reason only1 that the false are granted to be true1 then truth from two truths granted will follow in the same manner4 ,sidenote3 ,how one proposition is the cause of another4 #bj4 ,now1 seeing none but a true proposition will follow from true1 and that the understanding of two propositions to be true1 is the cause of understanding that also to be true which is deduced from them2 the two antecedent propositions are commonly called the causes of the inferred proposition1 or conclusion4 ,and from hence it is that logicians say1 the premises are causes of the conclusion2 which may pass1 though it be not properly spoken2 for though understanding be the cause of understanding1 yet speech is not the cause of speech4 ,but when they say1 the cause of the properties of any thing1 is the thing itself1 they speak absurdly4 ,for example1 if a figure be propounded which is triangular2 seeing every triangle has all its angles together equal to two right angles1 from whence it follows that all the angles of that figure are equal to two right angles1 they say1 for this reason1 that that figure is the cause of that equality4 ,but seeing the figure does not itself make its angles1 and therefore cannot be said to be the efficient-cause1 they call it the formal-cause2 whereas indeed it is no cause at all2 nor does the property of any figure follow the figure1 but has its being at the same time with it2 only the knowledge of the figure goes before the knowledge of the properties2 and one knowledge is truly the cause of another knowledge1 namely the efficient cause4 ,and thus much concerning proposition2 which in the progress of philosophy is the first step1 like the moving towards of one foot4 ,by the due addition of another step ,i shall proceed to syllogism1 and make a complete pace4 ,of which in the next chapter4 ------- ,,chapter ,,iv4 ,,of ,,syllogism4 #a4 ,the definition of syllogism4--#b4 ,in a syllogism there are but three terms4--#c4 ,major1 minor1 and middle term2 also major and minor proposition1 what they are4--#d4 ,the middle term in every syllogism ought to be determined in both the propositions to one and the same thing4--#e4 ,from two particular propositions nothing can be concluded4--#f4 ,a syllogism is the collection of two propositions into one sum4--#g4 ,the figure of a syllogism1 what it is4--#h4 ,what is in the mind answering to a syllogism4--#i4 ,the first indirect figure1 how it is made4--#aj4 ,the second indirect figure1 how made4--#aa4 ,how the third indirect figure is made4--#ab4 ,there are many moods in every figure1 but most of them useless in philosophy4--#ac4 ,an hypothetical syllogism when equipollent to a categorical4 ,sidenote3 ,definition of syllogism4 #a4 ,a ,,speech1 consisting of three propositions1 from two of which the third follows1 is called a ,,syllogism2 and that which follows is called the conclusion2 the other two premises4 ,for example1 this speech1 every man is a living creature1 every living creature is a body1 therefore1 every man is a body1 is a syllogism1 because the third proposition follows from the two first2 that is1 if those be granted to be true1 this must also be granted to be true4 ,sidenote3 ,in a syllogism there are but three terms4 #b4 ,from two propositions which have not one term common1 no conclusion can follow2 and therefore no syllogism can be made of them4 ,for let any two premises1 a man is a living creature1 a tree is a plant1 be both of them true1 yet because it cannot be collected from them that plant is the name of a man1 or man the name of a plant1 it is not necessary that this conclusion1 a man is a plant1 should be true4 ,corollary3 therefore1 in the premises of a syllogism there can be but three terms4 ,besides1 there can be no term in the conclusion1 which was not in the premises4 ,for let any two premises be1 a man is a living creature1 a living creature is a body1 yet if any other term be put in the conclusion1 as man is two-footed2 though it be true1 it cannot follow from the premises1 because from them it cannot be collected1 that the name two-footed belongs to a man2 and therefore1 again1 in every syllogism there can be but three terms4 ,sidenote3 ,major1 minor and middle term2 also major and minor proposition1 what they are4 #c4 ,of these terms1 that which is the predicate in the conclusion1 is commonly called the major2 that which is the subject in the conclusion1 the minor1 and the other is the middle term2 as in this syllogism1 a man is a living creature1 a living creature is a body1 therefore1 a man is a body1 body is the major1 man the minor1 and living creature the middle term4 ,also of the premises1 that in which the major term is found1 is called the major proposition1 and that which has the minor term1 the minor proposition4 ,sidenote3 ,the middle term in every syllogism to be determined in both propositions to one and the same thing4 #d4 ,if the middle term be not in both the premises determined to one and the same singular thing1 no conclusion will follow1 nor syllogism be made4 ,for let the minor term be man1 the middle term living creature1 and the major term lion2 and let the premises be1 man is a living creature1 some living creature is a lion1 yet it will not follow that every or any man is a lion4 ,by which it is manifest1 that in every syllogism1 that proposition which has the middle term for its subject1 ought to be either universal or singular1 but not particular nor indefinite4 ,for example1 this syllogism1 every man is a living creature1 some living creature is four-footed1 therefore some man is four-footed1 is therefore faulty1 because the middle term1 living creature1 is in the first of the premises determined only to man1 for there the name of living creature is given to man only1 but in the latter premise it may be understood of some other living creature besides man4 ,but if the latter premise had been universal1 as here1 every man is a living creature1 every living creature is a body1 therefore every man is a body1 the syllogism had been true2 for it would have followed that body had been the name of every living creature1 that is of man2 that is to say1 the conclusion every man is a body had been true4 ,likewise1 when the middle term is a singular name1 a syllogism may be made1 ,i say a true syllogism1 though useless in philosophy1 as this1 some man is ,socrates1 ,socrates is a philosopher1 therefore1 some man is a philosopher2 for the premises being granted1 the conclusion cannot be denied4 ,sidenote3 ,from two particular propositions nothing can be concluded4 #e4 ,and therefore of two premises1 in both which the middle term is particular1 a syllogism cannot be made2 for whether the middle term be the subject in both the premises1 or the predicate in both1 or the subject in one1 and the predicate in the other1 it will not be necessarily determined to the same thing4 ,for let the premises be1 ,some man is blind1 ,in both which the middle ,some man is learned1 term is the subject1 it will not follow that blind is the name of any learned man1 or learned the name of any blind man1 seeing the name learned does not contain the name blind1 nor this that2 and therefore it is not necessary that both should be names of the same man4 ,so from these premises1 ,every man is a living-creature1 ,in both which the middle ,every horse is a living-creature1 term is the predicate1 nothing will follow4 ,for seeing living creature is in both of them indefinite1 which is equivalent to particular1 and that man may be one kind of living creature1 and horse another kind1 it is not necessary that man should be the name of horse1 or horse of man4 ,or if the premises be1 ,every man is a ,in one of which the living-creature1 middle-term ,some living creature is four-footed1 is the subject1 and in the other the predicate1 the conclusion will not follow1 because the name living creature being not determined1 it may in one of them be understood of man1 in the other of not-man4 ,sidenote3 ,a syllogism is the collection of two propositions into one sum4 #f4 ,now it is manifest from what has been said1 that a syllogism is nothing but a collection of the sum of two propositions1 joined together by a common term1 which is called the middle term4 ,and as proposition is the addition of two names1 so syllogism is the adding together of three4 ,sidenote3 ,the figure of a syllogism what it is4 #g4 ,syllogisms are usually distinguished according to their diversity of figures1 that is1 by the diverse position of the middle term4 ,and again in figure there is a distinction of certain moods1 which consist of the differences of propositions in quantity and quality4 ,the first figure is that1 in which the terms are placed one after another according to their latitude of signification2 in which order the minor term is first1 the middle term next1 and the major last2 as1 if the minor term be man1 the middle term1 living creature1 and the major term1 body1 then1 man is a living-creature1 is a body1 will be a syllogism in the first figure3 in which1 man is a living creature is the minor proposition2 the major1 living creature is a body1 and the conclusion1 or sum of both1 man is a body4 ,now this figure is called direct1 because the terms stand in direct order2 and it is varied by quantity and quality into four moods3 of which the first is that wherein all the terms are positive1 and the minor term universal1 as every man is a living creature1 every living creature is a body3 in which all the propositions are affirmative1 and universal4 ,but if the major term be a negative name1 and the minor an universal name1 the figure will be in the second mood1 as1 every man is a living creature1 every living creature is not a tree1 in which the major proposition and conclusion are both universal and negative4 ,to these two1 are commonly added two more1 by making the minor term particular4 ,also it may happen that both the major and middle terms are negative terms1 and then there arises another mood1 in which all the propositions are negative1 and yet the syllogism will be good2 as1 if the minor term be man1 the middle term not a stone1 and the major term not a flint1 this syllogism1 no man is a stone1 whatsoever is not a stone is not a flint1 therefore1 no man is a flint1 is true1 though it consist of three negatives4 ,but in philosophy1 the profession whereof is to establish universal rules concerning the properties of things1 seeing the difference betwixt negatives and affirmatives is only this1 that in the former the subject is affirmed by a negative name1 and by a positive in the latter1 it is superfluous to consider any other mood in direct figure1 besides that1 in which all the propositions are both universal and affirmative4 ,sidenote3 ,what is in the mind answering to a syllogism4 #h4 ,the thoughts in the mind answering to a direct syllogism1 proceed in this manner2 first1 there is conceived a phantasm of the thing named1 with that accident or quality thereof1 for which it is in the minor proposition called by that name which is the subject2 next1 the mind has a phantasm of the same thing with that accident1 or quality1 for which it hath the name1 that in the same proposition is the predicate2 thirdly1 the thought returns of the same thing as having that accident in it1 for which it is called by the name1 that is the predicate of the major proposition2 and lastly1 remembering that all those are the accidents of one and the same thing1 it concludes that those three names are also names of one and the same thing2 that is to say1 the conclusion is true4 ,for example1 when this syllogism is made1 man is a living creature1 a living creature is a body1 therefore1 man is a body1 the mind conceives first an image of a man speaking or discoursing1 and remembers that that1 which so appears1 is called man2 then it has the image of the same man moving1 and remembers that that1 which appears so1 is called living creature2 thirdly1 it conceives an image of the same man1 as filling some place or space1 and remembers that what appears so is called body2 and lastly1 when it remembers that that thing1 which was extended1 and moved and spake1 was one and the same thing1 it concludes that the three names1 man1 living creature1 and body1 are names of the same thing1 and that therefore man is a living creature is a true proposition4 ,from whence it is manifest1 that living creatures that have not the use of speech1 have no conception or thought in the mind1 answering to a syllogism made of universal propositions2 seeing it is necessary to think not only of the thing1 but also by turns to remember the divers names1 which for divers considerations thereof are applied to the same4 ,sidenote3 ,the first indirect figure how made4 #i4 ,the rest of the figures arise either from the inflexion1 or inversion of the first or direct figure2 which is done by changing the major1 or minor1 or both the propositions1 into converted propositions equipollent to them4 ,from whence follow three other figures2 of which1 two are inflected1 and the third inverted4 ,the first of these three is made by the conversion of the major proposition4 ,for let the minor1 middle1 and major terms stand in direct order1 thus1 man is a living creature1 is not a stone1 which is the first or direct figure2 the inflection will be by converting the major proposition in this manner1 man is a living creature1 a stone is not a living creature2 and this is the second figure1 or the first of the indirect figures2 in which the conclusion will be1 man is not a stone4 ,for 7having shown in the last chapter1 art4 #ad1 that universal propositions1 converted by contradiction of the terms1 are equipollent7 both those syllogisms conclude alike2 so that if the major be read 7like ,hebrew7 backwards1 thus1 a living creature is not a stone1 it will be direct again1 as it was before4 ,in like manner this direct syllogism1 man is not a tree1 is not a pear-tree1 will be made indirect by converting the major proposition 7by contradiction of the terms7 into another equipollent to it1 thus1 man is not a tree1 a pear-tree is a tree2 for the same conclusion will follow1 man is not a pear-tree4 ,but for the conversion of the direct figure into the first indirect figure1 the major term in the direct figure ought to be negative4 ,for though this direct1 man is a living creature1 is a body1 be made indirect1 by converting the major proposition1 thus1 ,man is a living creature1 ,not a body is not a living creature1 ,therefore1 ,every man is a body2 ,yet this conversion appears so obscure1 that this mood is of no use at all4 ,by the conversion of the major proposition1 it is manifest1 that in this figure1 the middle term is always the predicate in both the premises4 ,sidenote3 ,second indirect figure how made4 #aj4 ,the second indirect figure is made by converting the minor proposition1 so as that the middle term is the subject in both4 ,but this never concludes universally1 and therefore is of no use in philosophy4 ,nevertheless ,i will set down an example of it2 by which this direct ,every man is a living creature1 ,every living creature is a body1 by conversion of the minor proposition1 will stand thus1 ,some living creature is a man1 ,every living creature is a body1 ,therefore1 ,some man is a body4 ,for every man is a living creature cannot be converted into this1 every living creature is a man3 and therefore if this syllogism be restored to its direct form1 the minor proposition will be some man is a living creature1 and consequently the conclusion will be some man is a body1 seeing the minor term man1 which is the subject in the conclusion1 is a particular name4 ,sidenote3 ,how the third indirect figure is made4 #aa4 ,the third indirect or inverted figure1 is made by the conversion of both the premises4 ,for example1 this direct syllogism1 ,every man is a living creature1 ,every living creature is not a stone1 ,therefore1 ,every man is not a stone1 being inverted1 will stand thus1 ,every stone is not a living creature1 ,whatsoever is not a living creature1 is not a man1 ,therefore1 ,every stone is not a man2 which conclusion is the converse of the direct conclusion1 and equipollent to the same4 ,the figures1 therefore1 of syllogisms1 if they be numbered by the diverse situation of the middle term only1 are but three2 in the first whereof1 the middle term has the middle place2 in the second1 the last2 and in the third1 the first place4 ,but if they be numbered according to the situation of the terms simply1 they are four2 for the first may be distinguished again into two1 namely1 into direct and inverted4 ,from whence it is evident1 that the controversy among logicians concerning the fourth figure1 is a mere 1 or contention about the name thereof2 for1 as for the thing itself1 it is plain that the situation of the terms 7not considering the quantity or quality by which the moods are distinguished7 makes four differences of syllogisms1 which may be called figures1 or have any other name at pleasure4 ,sidenote3 ,there are many moods in every figure1 but most of them useless in philosophy4 #ab4 ,in every one of these figures there are many moods1 which are made by varying the premises according to all the differences they are capable of1 by quantity and quality2 as namely1 in the direct figure there are six moods2 in the first indirect figure1 four2 in the second1 fourteen2 and in the third1 eighteen4 ,but because from the direct figure ,i rejected as superfluous all moods besides that which consists of universal propositions1 and whose minor proposition is affirmative1 ,i do1 together with it1 reject the moods of the rest of the figures which are made by conversion of the premises in the direct figure4 ,sidenote3 ,an hypothetical syllogism when equipollent to a categorical4 #ac4 ,as it was showed before1 that in necessary propositions a categorical and hypothetical proposition are equipollent2 so likewise it is manifest that a categorical and hypothetical syllogism are equivalent4 ,for every categorical syllogism1 as this1 ,every man is a living creature1 ,every living creature is a body1 ,therefore1 ,every man is a body1 is of equal force with this hypothetical syllogism3 ,if any thing be a man1 the same is also a living creature1 ,if any thing be a living creature1 the same is a body1 ,therefore1 ,if any thing be a man1 the same is a body4 ,in like manner1 this categorical syllogism in an indirect figure1 ,no stone is a living creature1 ,every man is a living creature1 ,therefore1 ,no man is a stone1 ,or1 ,no stone is a man1 is equivalent to this hypothetical syllogism3 ,if any thing be a man1 the same is a living creature1 ,if any thing be a stone1 the same is not a living creature1 ,therefore1 ,if any thing be a stone1 the same is not a man1 ,or1 ,if any thing be a man1 the same is not a stone4 ,and thus much seems sufficient for the nature of syllogisms2 7for the doctrine of moods and figures is clearly delivered by others that have written largely and profitably of the same74 ,nor are precepts so necessary as practice for the attaining of true ratiocination2 and they that study the demonstrations of mathematicians1 will sooner learn true logic1 than they that spend time in reading the rules of syllogizing which logicians have made2 no otherwise than little children learn to go1 not by precepts1 but by exercising their feet4 ,this1 therefore1 may serve for the first pace in the way to ,philosophy4 ,in the next place ,i shall speak of the faults and errors into which men that reason unwarily are apt to fall2 and of their kinds and causes4 ------- ,,chapter ,v4 ,,of ,,erring1 ,,falsity1 ,,and ,,captions4 #a4 ,erring and falsity how they differ4 ,error of the mind by itself without the use of words1 how it happens4--#b4 ,a sevenfold incoherency of names1 every one of which makes always a false proposition4--#c4 ,examples of the first manner of incoherency4--#d4 ,of the second4--#e4 ,of the third4--#f4 ,of the fourth4--#g4 ,of the fifth4--#h4 ,of the sixth4--#i4 ,of the seventh4--#aj4 ,falsity of propositions detected by resolving the terms with definitions continued till they come to simple names1 or names that are the most general of their kind4--#aa4 ,of the fault of a syllogism consisting in the implication of the terms with the copula4--#ab4 ,of the fault which consists in equivocation4--#ac4 ,sophistical captions are oftener faulty in the matter than in the form of syllogisms4 ,sidenote3 ,erring falsity how they differ4 ,error of the mind by itself1 without the use of words1 how it happens4 #a4 ,men are subject to err not only in affirming and denying1 but also in perception1 and in silent cogitation4 ,in affirming and denying1 when they call any thing by a name1 which is not the name thereof2 as if from seeing the sun first by reflection in water1 and afterwards again directly in the firmament1 we should to both those appearances give the name of sun1 and say there are two suns2 which none but men can do1 for no other living creatures have the use of names4 ,this kind of error only deserves the name of falsity1 as arising1 not from sense1 nor from the things themselves1 but from pronouncing rashly2 for names have their constitution1 not from the species of things1 but from the will and consent of men4 ,and hence it comes to pass1 that men pronounce falsely1 by their own negligence1 in departing from such appellations of things as are agreed upon1 and are not deceived neither by the things1 nor by the sense2 for they do not perceive that the thing they see is called sun1 but they give it that name from their own will and agreement4 ,tacit errors1 or the errors of sense and cogitation1 are made1 by passing from one imagination to the imagination of another different thing2 or by feigning that to be past1 or future1 which never was1 nor ever shall be2 as when1 by seeing the image of the sun in water1 we imagine the sun itself to be there2 or by seeing swords1 that there has been or shall be fighting1 because it uses to be so for the most part2 or when from promises we feign the mind of the promiser to be such and such2 or lastly1 when from any sign we vainly imagine something to be signified1 which is not4 ,and errors of this sort are common to all things that have sense2 and yet the deception proceeds neither from our senses1 nor from the things we perceive2 but from ourselves while we feign such things as are but mere images to be something more than images4 ,but neither things1 nor imaginations of things1 can be said to be false1 seeing they are truly what they are2 nor do they1 as signs1 promise any thing which they do not perform2 for they indeed do not promise at all1 but we from them2 nor do the clouds1 but we1 from seeing the clouds1 say it shall rain4 ,the best way1 therefore1 to free ourselves from such errors as arise from natural signs1 is first of all1 before we begin to reason concerning such conjectural things1 to suppose ourselves ignorant1 and then to make use of our ratiocination2 for these errors proceed from the want of ratiocination2 whereas1 errors which consist in affirmation and negation1 7that is1 the falsity of propositions7 proceed only from reasoning amiss4 ,of these1 therefore1 as repugnant to philosophy1 ,i will speak principally4 ,sidenote3 ,a sevenfold incoherency of names1 all of which make always a false proposition4 #b4 ,errors which happen in reasoning1 that is1 in syllogizing1 consist either in the falsity of the premises1 or of the inference4 ,in the first of these cases1 a syllogism is said to be faulty in the matter of it2 and in the second case1 in the form4 ,i will first consider the matter1 namely1 how many ways a proposition may be false2 and next the form1 and how it comes to pass1 that when the premises are true1 the inference is1 notwithstanding1 false4 ,seeing1 therefore1 that proposition only is true1 7chap1 ,,iii1 art4 #g7 in which are copulated two names of one and the same thing2 and that always false1 in which names of different things are copulated1 look how many ways names of different things may be copulated1 and so many ways a false proposition may be made4 ,now1 all things to which we give names1 may be reduced to these four kinds1 namely1 bodies1 accidents1 phantasms1 and names themselves2 and therefore1 in every true proposition1 it is necessary that the names copulated1 be both of them names of bodies1 or both names of accidents1 or both names of phantasms1 or both names of names4 ,for names otherwise copulated are incoherent1 and constitute a false proposition4 ,it may happen1 also1 that the name of a body1 of an accident1 or of a phantasm1 may be copulated with the name of a speech4 ,so that copulated names may be incoherent seven manner of ways4 #a4 ,if the name of a ,body the name of an ,accident4 #b4 ,if the name of a ,body the name of a ,phantasm4 #c4 ,if the name of a ,body be the name of a ,name4 #d4 ,if the name of an ,accident copulated the name of a ,phantasm4 #e4 ,if the name of an ,accident with the name of a ,name4 #f4 ,if the name of a ,phantasm the name of a ,name4 #g4 ,if the name of a ,body1 the name of a ,speech4 ,accident1 or ,phantasm ,of all which ,i will give some examples4 ,sidenote3 ,examples of the first manner of incoherency4 #c4 ,after the first of these ways propositions are false1 when abstract names are copulated with concrete names2 as 7in ,latin and ,greek7 esse est ens1 essentia est ens1 7i472 quidditas est ens1 and many the like1 which are found in ,aristotle's ,metaphysics4 ,also1 the understanding worketh1 the understanding understandeth1 the sight seeth2 a body is magnitude1 a body is quantity1 a body is extension2 to be a man is a man1 whiteness is a white thing1 c42 which is as if one should say1 the runner is the running1 or the walk walketh4 ,moreover1 essence is separated1 substance is abstracted3 and others like these1 or derived from these1 7with which common philosophy abounds47 ,for seeing no subject of an accident 7that is1 no body7 is an accident3 no name of an accident ought to be given to a body1 nor of a body to an accident4 ,sidenote3 ,the second4 #d4 ,false1 in the second manner1 are such propositions as these2 a ghost is a body1 or a spirit1 that is1 a thin body2 sensible species fly up and down in the air1 or are moved hither and thither1 which is proper to bodies2 also1 a shadow is moved1 or is a body2 light is moved1 or is a body2 colour is the object of sight1 sound of hearing2 space or place is extended2 and innumerable others of this kind4 ,for seeing ghosts1 sensible species1 a shadow1 light1 colour1 sound1 space1 c4 appear to us no less sleeping than waking1 they cannot be things without us1 but only phantasms of the mind that imagines them2 and therefore the names of these1 copulated with the names of bodies1 cannot constitute a true proposition4 ,sidenote3 ,the third4 #e4 ,false propositions of the third kind1 are such as these2 genus est ens1 universale est ens1 ens de ente prdicatur4 ,for genus1 and universale1 and predicare1 are names of names1 and not of things4 ,also1 number is infinite1 is a false proposition2 for no number can be infinite1 but only the word number is then called an indefinite name when there is no determined number answering to it in the mind4 ,sidenote3 ,the fourth4 #f4 ,to the fourth kind belong such false propositions as these1 an object is of such magnitude or figure as appears to the beholders2 colour1 light1 sound1 are in the object2 and the like4 ,for the same object appears sometimes greater1 sometimes lesser1 sometimes square1 sometimes round1 according to the diversity of the distance and medium2 but the true magnitude and figure of the thing seen is always one and the same2 so that the magnitude and figure which appears1 is not the true magnitude and figure of the object1 nor anything but phantasm2 and therefore1 in such propositions as these1 the names of accidents are copulated with the names of phantasms4 ,sidenote3 ,the fifth4 #g4 ,propositions are false in the fifth manner1 when it is said that the definition is the essence of a thing2 whiteness1 or some other accident1 is the genus1 or universal4 ,for definition is not the essence of any thing1 but a speech signifying what we conceive of the essence thereof2 and so also not whiteness itself1 but the word whiteness1 is a genus1 or an universal name4 ,sidenote3 ,the sixth4 #h4 ,in the sixth manner they err1 that say the idea of anything is universal2 as if there could be in the mind an image of a man1 which were not the image of some one man1 but a man simply1 which is impossible2 for every idea is one1 and of one thing2 but they are deceived in this1 that they put the name of the thing for the idea thereof4 ,sidenote3 ,the seventh4 #i4 ,they err in the seventh manner1 that make this distinction between things that have being1 that some of them exist by themselves1 others by accident2 namely1 because ,socrates is a man is a necessary proposition1 and ,socrates is a musician a contingent proposition1 therefore they say some things exist necessarily or by themselves1 others contingently or by accident2 whereby1 seeing necessary1 contingent1 by itself1 by accident1 are not names of things1 but of propositions1 they that say any thing that has being1 exists by accident1 copulate the name of a proposition with the name of a thing4 ,in the same manner also1 they err1 which place some ideas in the understanding1 others in the fancy2 as if from the understanding of this proposition1 man is a living creature1 we had one idea or image of a man derived from sense to the memory1 and another to the understanding2 wherein that which deceives them is this1 that they think one idea should be answerable to a name1 another to a proposition1 which is false2 for proposition signifies only the order of those things one after another1 which we observe in the same idea of man2 so that this proposition1 man is a living creature raises but one idea in us1 though in that idea we consider that first1 for which he is called man1 and next that1 for which he is called living creature4 ,the falsities of propositions in all these several manners1 is to be discovered by the definitions of the copulated names4 ,sidenote3 ,falsity of propositions detected by resolving the terms with definitions4 #aj4 ,but when names of bodies are copulated with names of bodies1 names of accidents with names of accidents1 names of names with names of names1 and names of phantasms1 with names of phantasms1 if we1 nevertheless1 remain still doubtful whether such propositions are true1 we ought then in the first place to find out the definition of both those names1 and again the definitions of such names as are in the former definition1 and so proceed by a continual resolution till we come to a simple name1 that is1 to the most general or most universal name of that kind2 and if after all this1 the truth or falsity thereof be not evident1 we must search it out by philosophy1 and ratiocination1 beginning from definitions4 ,for every proposition1 universally true1 is either a definition1 or part of a definition1 or the evidence of it depends upon definitions4 ,sidenote3 ,of the fault of a syllogism consisting in implication of the terms with the copula4 #aa4 ,that fault of a syllogism which lies hid in the form thereof1 will always be found either in the implication of the copula with one of the terms1 or in the equivocation of some word2 and in either of these ways there will be four terms1 which 7as ,i have shewn7 cannot stand in a true syllogism4 ,now the implication of the copula with either term1 is easily detected by reducing the propositions to plain and clear predication2 as 7for example7 if any man should argue thus1 ,the hand toucheth the pen1 ,the pen toucheth the paper1 ,therefore1 ,the hand toucheth the paper2 the fallacy will easily appear by reducing it1 thus3 ,the hand1 is1 touching the pen1 ,the pen1 is1 touching the paper1 ,therefore1 ,the hand1 is1 touching the paper2 where there are manifestly these four terms1 the hand1 touching the pen1 the pen1 and touching the the paper4 ,but the danger of being deceived by sophisms of this kind1 does not seem to be so great1 as that ,i need insist longer upon them4 ,sidenote3 ,of the fault which consists in equivocation4 #ab4 ,and though there may be fallacy in equivocal terms1 yet in those that be manifestly such1 there is none at all2 nor in metaphors1 for they profess the transferring of names from one thing to another4 ,nevertheless1 sometimes equivocals 7and those not very obscure7 may deceive2 as in this argumentation3--,it belongs to metaphysics to treat of principles2 but the first principle of all1 is1 that the same thing cannot both exist and not exist at the same time2 and therefore it belongs to metaphysics to treat whether the same thing may both exist and not exist at the same time2 where the fallacy lies in the equivocation of the word principle2 for whereas ,aristotle in the beginning of his ,metaphysics1 says1 that the treating of principles belongs to primary science1 he understands by principles1 causes of things1 and certain existences which he calls primary2 but where he says a primary proposition is a principle1 by principle1 there1 he means the beginning and cause of knowledge1 that is1 the understanding of words1 which1 if any man want1 he is incapable of learning4 ,sidenote3 ,sophistical captions are oftener faulty in the matter than in the form of syllogisms4 #ac4 ,but the captions of sophists and sceptics1 by which they were wont1 of old1 to deride and oppose truth1 were faulty for the most part1 not in the form1 but in the matter of syllogism2 and they deceived not others oftener than they were themselves deceived4 ,for the force of that famous argument of ,zeno against motion1 consisted in this proposition1 whatsoever may be divided into parts1 infinite in number1 the same is infinite2 which he1 without doubt1 thought to be true1 yet nevertheless is false4 ,for to be divided into infinite parts1 is nothing else but to be divided into as many parts as any man will4 ,but it is not necessary that a line should have parts infinite in number1 or be infinite1 because ,i can divide and subdivide it as often as ,i please2 for how many parts soever ,i make1 yet their number is finite2 but because he that says parts1 simply1 without adding how many1 does not limit any number1 but leaves it to the determination of the hearer1 therefore we say commonly1 a line may be divided infinitely2 which cannot be true in any other sense4 ,sidenote3 ,conclusion4 ,and thus much may suffice concerning syllogism1 which is1 as it were1 the first pace towards philosophy2 in which ,i have said as much as is necessary to teach any man from whence all true argumentation has its force4 ,and to enlarge this treatise with all that may be heaped together1 would be as superfluous1 as if one should 7as ,i said before7 give a young child precepts for the teaching of him to go2 for the art of reasoning is not so well learned by precepts as by practice1 and by the reading of those books in which the conclusions are all made by severe demonstration4 ,and so ,i pass on to the way of philosophy1 that is1 to the method of study4 ------- ,,chapter ,,vi4 ,,of ,,method4 #a4 ,method and science defined4--#b4 ,it is more easily known concerning singular1 than universal things1 that they are2 and contrarily1 it is more easily known concerning universal1 than singular things1 why they are1 or what are their causes4--#c4 ,what it is philosophers seek to know4--#d4 ,the first part1 by which principles are found out1 is purely analytical4--#e4 ,the highest causes1 and most universal in every kind1 are known by themselves4--#f4 ,method from principles found out1 tending to science simply1 what it is4--#g4 ,that method of civil and natural science1 which proceeds from sense to principles1 is analytical2 and again1 that1 which begins at principles1 is synthetical4--#h4 ,the method of searching out1 whether any thing propounded be matter or accident4--#i4 ,the method of seeking whether any accident be in this1 or in that subject4--#aj4 ,the method of searching after the cause of any effect propounded4--#aa4 ,words serve to invention1 as marks2 to demonstration1 as signs4--#ab4 ,the method of demonstration is synthetical4--#ac4 ,definitions only are primary and universal propositions4--#ad4 ,the nature and definition of a definition4--#ae4 ,the properties of a definition4--#af4 ,the nature of a demonstration4--#ag4 ,the properties of a demonstration1 and order of things to be demonstrated4--#ah4 ,the faults of a demonstration4--#ai4 ,why the analytical method of geometricians cannot be treated of in this place4 ,sidenote3 ,method and science defined4 #a4 ,for the understanding of method1 it will be necessary for me to repeat the definition of philosophy1 delivered above 7,chap4 ,i1 art4 #b47 in this manner1 ,philosophy is the knowledge we acquire1 by true ratiocination1 of appearances1 or apparent effects1 from the knowledge we have of some possible production or generation of the same2 and of such production1 as has been or may be1 from the knowledge we have of the effects4 ,,method1 therefore1 in the study of philosophy1 is the shortest way of finding out effects by their known causes1 or of causes by their known effects4 ,but we are then said to know any effect1 when we know that there be causes of the same1 and in what subject those causes are1 and in what subject they produce that effect1 and in what manner they work the same4 ,and this is the science of causes1 or1 as they call it1 of the 4 ,all other science1 which is called the 1 is either perception by sense1 or the imagination1 or memory remaining after such perception4 ,the first beginnings1 therefore1 of knowledge1 are the phantasms of sense and imagination2 and that there be such phantasms we know well enough by nature2 but to know why they be1 or from what causes they proceed1 is the work of ratiocination2 which consists 7as is said above1 in the #ast ,chapter1 ,art4 #b7 in composition1 and division or resolution4 ,there is therefore no method1 by which we find out the causes of things1 but is either compositive or resolutive1 or partly compositive1 and partly resolutive4 ,and the resolutive is commonly called analytical method1 as the compositive is called synthetical4 ,sidenote3 ,it is easier known concerning singular than universal things1 that they are2 and contrarily it is easier known concerning universal than singular things1 why they are1 or what are their causes4 #b4 ,it is common to all sorts of method1 to proceed from known things to unknown2 and this is manifest from the cited definition of philosophy4 ,but in knowledge by sense1 the whole object is more known1 than any part thereof2 as when we see a man1 the conception or whole idea of that man is first or more known1 than the particular ideas of his being figurate1 animate1 and rational2 that is1 we first see the whole man1 and take notice of his being1 before we observe in him those other particulars4 ,and therefore in any knowledge of the 1 or that any thing is1 the beginning of our search is from the whole idea2 and contrarily1 in our knowledge of the 1 or of the causes of any thing1 that is1 in the sciences1 we have more knowledge of the causes of the parts than of the whole4 ,for the cause of the whole is compounded of the causes of the parts2 but it is necessary that we know the things that are to be compounded1 before we can know the whole compound4 ,now1 by parts1 ,i do not here mean parts of the thing itself1 but parts of its nature2 as1 by the parts of man1 ,i do not understand his head1 his shoulders1 his arms1 c4 but his figure1 quantity1 motion1 sense1 reason1 and the like2 which accidents being compounded or put together1 constitute the whole nature of man1 but not the man himself4 ,and this is the meaning of that common saying1 namely1 that some things are more known to us1 others more known to nature2 for ,i do not think that they1 which so distinguish1 mean that something is known to nature1 which is known to no man2 and therefore1 by those things1 that are more known to us1 we are to understand things we take notice of by our senses1 and1 by more known to nature1 those we acquire the knowledge of by reason2 for in this sense it is1 that the whole1 that is1 those things that have universal names1 7which1 for brevity's sake1 ,i call universal7 are more known to us than the parts1 that is1 such things as have names less universal1 7which ,i therefore call singular72 and the causes of the parts are more known to nature than the cause of the whole2 that is1 universals than singulars4 ,sidenote3 ,what it is philosophers seek to know4 #c4 ,in the study of philosophy1 men search after science either simply or indefinitely2 that is1 to know as much as they can1 without propounding to themselves any limited question2 or they enquire into the cause of some determined appearance1 or endeavour to find out the certainty of something in question1 as what is the cause of light1 of heat1 of gravity1 of a figure propounded1 and the like2 or in what subject any propounded accident is inherent2 or what may conduce most to the generation of some propounded effect from many accidents2 or in what manner particular causes ought to be compounded for the production of some certain effect4 ,now1 according to this variety of things in question1 sometimes the analytical method is to be used1 and sometimes the synthetical4 ,sidenote3 ,the first part1 by which principles are discovered1 is purely analytical4 #d4 ,but to those that search after science indefinitely1 which consists in the knowledge of the causes of all things1 as far forth as it may be attained1 7and the causes of singular things are compounded of the causes of universal or simple things7 it is necessary that they know the causes of universal things1 or of such accidents as are common to all bodies1 that is1 to all matter1 before they can know the causes of singular things1 that is1 of those accidents by which one thing is distinguished from another4 ,and1 again1 they must know what those universal things are1 before they can know their causes4 ,moreover1 seeing universal things are contained in the nature of singular things1 the knowledge of them is to be acquired by reason1 that is1 by resolution4 ,for example1 if there be propounded a conception or idea of some singular thing1 as of a square1 this square is to be resolved into a plain1 terminated with a certain number of equal and straight lines and right angles4 ,for by this resolution we have these things universal or agreeable to all matter1 namely1 line1 plain1 7which contains superficies7 terminated1 angle1 straightness1 rectitude1 and equality2 and if we can find out the causes of these1 we may compound them altogether into the cause of a square4 ,again1 if any man propound to himself the conception of gold1 he may1 by resolving1 come to the ideas of solid1 visible1 heavy1 7that is1 tending to the centre of the earth1 or downwards7 and many other more universal than gold itself2 and these he may resolve again1 till he come to such things as are most universal4 ,and in this manner1 by resolving continually1 we may come to know what those things are1 whose causes being first known severally1 and afterwards compounded1 bring us to the knowledge of singular things4 ,i conclude1 therefore1 that the method of attaining to the universal knowledge of things1 is purely analytical4 ,sidenote3 ,the highest causes1 and most universal in every kind1 are known by themselves4 #e4 ,but the causes of universal things 7of those1 at least1 that have any cause7 are manifest of themselves1 or 7as they say commonly7 known to nature2 so that they need no method at all2 for they have all but one universal cause1 which is motion4 ,for the variety of all figures arises out of the variety of those motions by which they are made2 and motion cannot be understood to have any other cause besides motion2 nor has the variety of those things we perceive by sense1 as of colours1 sounds1 savours1 c4 any other cause than motion1 residing partly in the objects that work upon our senses1 and partly in ourselves1 in such manner1 as that it is manifestly some kind of motion1 though we cannot1 without ratiocination1 come to know what kind4 ,for though many cannot understand till it be in some sort demonstrated to them1 that all mutation consists in motion2 yet this happens not from any obscurity in the thing itself1 7for it is not intelligible that anything can depart either from rest1 or from the motion it has1 except by motion71 but either by having their natural discourse corrupted with former opinions received from their masters1 or else for this1 that they do not at all bend their mind to the enquiring out of truth4 ,sidenote3 ,method from principles found out1 tending to science simply1 what it is4 #f4 ,by the knowledge therefore of universals1 and of their causes 7which are the first principles by which we know the of things7 we have in the first place their definitions1 7which are nothing but the explication of our simple conceptions47 ,for example1 he that has a true conception of place1 cannot be ignorant of this definition1 place is that space which is possessed or filled adequately by some body2 and so1 he that conceives motion aright1 cannot but know that motion is the privation of one place1 and the acquisition of another4 ,in the next place1 we have their generations or descriptions2 as 7for example7 that a line is made by the motion of a point1 superficies by the motion of a line1 and one motion by another motion1 c4 ,it remains1 that we enquire what motion begets such and such effects2 as1 what motion makes a straight line1 and what a circular2 what motion thrusts1 what draws1 and by what way2 what makes a thing which is seen or heard1 to be seen or heard sometimes in one manner1 sometimes in another4 ,now the method of this kind of enquiry1 is compositive4 ,for first we are to observe what effect a body moved produceth1 when we consider nothing in it besides its motion2 and we see presently that this makes a line1 or length2 next1 what the motion of a long body produces1 which we find to be superficies2 and so forwards1 till we see what the effects of simple motion are2 and then1 in like manner1 we are to observe what proceeds from the addition1 multiplication1 subtraction1 and division1 of these motions1 and what effects1 what figures1 and what properties1 they produce2 from which kind of contemplation sprung that part of philosophy which is called geometry4 ,from this consideration of what is produced by simple motion1 we are to pass to the consideration of what effects one body moved worketh upon another2 and because there may be motion in all the several parts of a body1 yet so as that the whole body remain still in the same place1 we must enquire first1 what motion causeth such and such motion in the whole1 that is1 when one body invades another body which is either at rest or in motion1 what way1 and with what swiftness1 the invaded body shall move2 and1 again1 what motion this second body will generate in a third1 and so forwards4 ,from which contemplation shall be drawn that part of philosophy which treats of motion4 ,in the third place we must proceed to the enquiry of such effects as are made by the motion of the parts of any body1 as1 how it comes to pass1 that things when they are the same1 yet seem not to be the same1 but changed4 ,and here the things we search after are sensible qualities1 such as light1 colour1 transparency1 opacity1 sound1 odour1 savour1 heat1 cold1 and the like2 which because they cannot be known till we know the causes of sense itself1 therefore the consideration of the causes of seeing1 hearing1 smelling1 tasting1 and touching1 belongs to this third place2 and all those qualities and changes1 above mentioned1 are to be referred to the fourth place2 which two considerations comprehend that part of philosophy which is called physics4 ,and in these four parts is contained whatsoever in natural philosophy may be explicated by demonstration1 properly so called4 ,for if a cause were to be rendered of natural appearances in special1 as1 what are the motions and influences of the heavenly bodies1 and of their parts1 the reason hereof must either be drawn from the parts of the sciences above mentioned1 or no reason at all will be given1 but all left to uncertain conjecture4 ,after physics we must come to moral philosophy2 in which we are to consider the motions of the mind1 namely1 appetite1 aversion1 love1 benevolence1 hope1 fear1 anger1 emulation1 envy1 c42 what causes they have1 and of what they be causes4 ,and the reason why these are to be considered after physics is1 that they have their causes in sense and imagination1 which are the subject of physical contemplation4 ,also the reason1 why all these things are to be searched after in the order above-said1 is1 that physics cannot be understood1 except we know first what motions are in the smallest parts of bodies2 nor such motion of parts1 till we know what it is that makes another body move2 nor this1 till we know what simple motion will effect4 ,and because all appearance of things to sense is determined1 and made to be of such and such quality and quantity by compounded motions1 every one of which has a certain degree of velocity1 and a certain and determined way2 therefore1 in the first place1 we are to search out the ways of motion simply 7in which geometry consists72 next the ways of such generated motions as are manifest2 and1 lastly1 the ways of internal and invisible motions 7which is the enquiry of natural philosophers74 ,and1 therefore1 they that study natural philosophy1 study in vain1 except they begin at geometry2 and such writers or disputers thereof1 as are ignorant of geometry1 do but make their readers and hearers lose their time4 ,sidenote3 ,that method of civil and natural science1 proceeding from sense to principles1 is analytical2 and again1 that which begins at principles is synthetical4 #g4 ,civil and moral philosophy do not so adhere to one another1 but that they may be severed4 ,for the causes of the motions of the mind are known1 not only by ratiocination1 but also by the experience of every man that takes the pains to observe those motions within himself4 ,and1 therefore1 not only they that have attained the knowledge of the passions and perturbations of the mind1 by the synthetical method1 and from the very first principles of philosophy1 may by proceeding in the same way1 come to the causes and necessity of constituting commonwealths1 and to get the knowledge of what is natural right1 and what are civil duties2 and1 in every kind of government1 what are the rights of the commonwealth1 and all other knowledge appertaining to civil philosophy2 for this reason1 that the principles of the politics consist in the knowledge of the motions of the mind1 and the knowledge of these motions from the knowledge of sense and imagination2 but even they also that have not learned the first part of philosophy1 namely1 geometry and physics1 may1 notwithstanding1 attain the principles of civil philosophy1 by the analytical method4 ,for if a question be propounded1 as1 whether such an action be just or unjust2 if that unjust be resolved into fact against law1 and that notion law into the command of him or them that have coercive power2 and that power be derived from the wills of men that constitute such power1 to the end they may live in peace1 they may at last come to this1 that the appetites of men and the passions of their minds are such1 that1 unless they be restrained by some power1 they will always be making war upon one another2 which may be known to be so by any man's experience1 that will but examine his own mind4 ,and1 therefore1 from hence he may proceed1 by compounding1 to the determination of the justice or injustice of any propounded action4 ,so that it is manifest1 by what has been said1 that the method of philosophy1 to such as seek science simply1 without propounding to themselves the solution of any particular question1 is partly analytical1 and partly synthetical2 namely1 that which proceeds from sense to the invention of principles1 analytical2 and the rest synthetical4 ,sidenote3 ,the method of searching out1 whether anything propounded be matter or accident4 #h4 ,to those that seek the cause of some certain and propounded appearance or effect1 it happens1 sometimes1 that they know not whether the thing1 whose cause is sought after1 be matter or body1 or some accident of a body4 ,for though in geometry1 when the cause is sought of magnitude1 or proportion1 or figure1 it be certainly known that these things1 namely magnitude1 proportion1 and figure1 are accidents2 yet in natural philosophy1 where all questions are concerning the causes of the phantasms of sensible things1 it is not so easy to discern between the things themselves1 from which those phantasms proceed1 and the appearances of those things to the sense2 which have deceived many1 especially when the phantasms have been made by light4 ,for example1 a man that looks upon the sun1 has a certain shining idea of the magnitude of about a foot over1 and this he calls the sun1 though he know the sun to be truly a great deal bigger2 and1 in like manner1 the phantasm of the same thing appears sometimes round1 by being seen afar off1 and sometimes square1 by being nearer4 ,whereupon it may well be doubted1 whether that phantasm be matter1 or some body natural1 or only some accident of a body2 in the examination of which doubt we may use this method4 ,the properties of matter and accidents already found out by us1 by the synthetical method1 from their definitions1 are to be compared with the idea we have before us2 and if it agree with the properties of matter or body1 then it is a body2 otherwise it is an accident4 ,seeing1 therefore1 matter cannot by any endeavour of ours be either made or destroyed1 or increased1 or diminished1 or moved out of its place1 whereas that idea appears1 vanishes1 is increased and diminished1 and moved hither and thither at pleasure2 we may certainly conclude that it is not a body1 but an accident only4 ,and this method is synthetical4 ,sidenote3 ,the method of seeking whether any accident be in this or in that subject4 #i4 ,but if there be a doubt made concerning the subject of any known accident 7for this may be doubted sometimes1 as in the precedent example1 doubt may be made in what subject that splendour and apparent magnitude of the sun is71 then our enquiry must proceed in this manner4 ,first1 matter in general must be divided into parts1 as1 into object1 medium1 and the sentient itself1 or such other parts as seem most conformable to the thing propounded4 ,next1 these parts are severally to be examined how they agree with the definition of the subject2 and such of them as are not capable of that accident are to be rejected4 ,for example1 if by any true ratiocination the sun be found to be greater than its apparent magnitude1 then that magnitude is not in the sun2 if the sun be in one determined straight line1 and one determined distance1 and the magnitude and splendour be seen in more lines and distances than one1 as it is in reflection or refraction1 then neither that splendour nor apparent magnitude are in the sun itself1 and1 therefore1 the body of the sun cannot be the subject of that splendour and magnitude4 ,and for the same reasons the air and other parts will be rejected1 till at last nothing remain which can be the subject of that splendour and magnitude but the sentient itself4 ,and this method1 in regard the subject is divided into parts1 is analytical2 and in regard the properties1 both of the subject and accident1 are compared with the accident concerning whose subject the enquiry is made1 it is synthetical4 ,sidenote3 ,method of searching for the cause of any effect1 propounded4 #aj4 ,but when we seek after the cause of any propounded effect1 we must in the first place get into our mind an exact notion or idea of that which we call cause1 namely1 that a cause is the sum or aggregate of all such accidents1 both in the agents and the patient1 as concur to the producing of the effect propounded2 all which existing together1 it cannot be understood but that the effect existeth with them2 or that it can possibly exist if any one of them be absent4 ,this being known1 in the next place we must examine singly every accident that accompanies or precedes the effect1 as far forth as it seems to conduce in any manner to the production of the same1 and see whether the propounded effect may be conceived to exist1 without the existence of any of those accidents2 and by this means separate such accidents1 as do not concur1 from such as concur to produce the said effect2 which being done1 we are to put together the concurring accidents1 and consider whether we can possibly conceive1 that when these are all present1 the effect propounded will not follow2 and if it be evident that the effect will follow1 then that aggregate of accidents is the entire cause1 otherwise not2 but we must still search out and put together other accidents4 ,for example1 if the cause of light be propounded to be sought out2 first1 we examine things without us1 and find that whensoever light appears1 there is some principal object1 as it were the fountain of light1 without which we cannot have any perception of light2 and1 therefore1 the concurrence of that object is necessary to the generation of light4 ,next we consider the medium1 and find1 that unless it be disposed in a certain manner1 namely1 that it be transparent1 though the object remain the same1 yet the effect will not follow2 and1 therefore1 the concurrence of transparency is also necessary to the generation of light4 ,thirdly1 we observe our own body1 and find that by the indisposition of the eyes1 the brain1 the nerves1 and the heart1 that is1 by obstructions1 stupidity1 and debility1 we are deprived of light1 so that a fitting disposition of the organs to receive impressions from without is likewise a necessary part of the cause of light4 ,again1 of all the accidents inherent in the object1 there is none that can conduce to the effecting of light1 but only action 7or a certain motion71 which cannot be conceived to be wanting1 whensoever the effect is present2 for1 that anything may shine1 it is not requisite that it be of such or such magnitude or figure1 or that the whole body of it be moved out of the place it is in 7unless it may perhaps be said1 that in the sun1 or other body1 that which causes light is the light it hath in itself2 which yet is but a trifling exception1 seeing nothing is meant thereby but the cause of light2 as if any man should say that the cause of light is that in the sun which produceth it72 it remains1 therefore1 that the action1 by which light is generated1 is motion only in the parts of the object4 ,which being understood1 we may easily conceive what it is the medium contributes1 namely1 the continuation of that motion to the eye2 and1 lastly1 what the eye and the rest of the organs of the sentient contribute1 namely1 the continuation of the same motion to the last organ of sense1 the heart4 ,and in this manner the cause of light may be made up of motion continued from the original of the same motion1 to the original of vital motion1 light being nothing but the alteration of vital motion1 made by the impression upon it of motion continued from the object4 ,but ,i give this only for an example1 for ,i shall speak more at large of light1 and the generation of it1 in its proper place4 ,in the mean time it is manifest1 that in the searching out of causes1 there is need partly of the analytical1 and partly of the synthetical method2 of the analytical1 to conceive how circumstances conduce severally to the production of effects2 and of the synthetical1 for the adding together and compounding of what they can effect singly by themselves4 ,and thus much may serve for the method of invention4 ,it remains that ,i speak of the method of teaching1 that is1 of demonstration1 and of the means by which we demonstrate4 ,sidenote3 ,words serve to invention as marks2 to demonstration as signs4 #aa4 ,in the method of invention1 the use of words consists in this1 that they may serve for marks1 by which1 whatsoever we have found out may be recalled to memory2 for without this all our inventions perish1 nor will it be possible for us to go on from principles beyond a syllogism or two1 by reason of the weakness of memory4 ,for example1 if any man1 by considering a triangle set before him1 should find that all its angles together taken are equal to two right angles1 and that by thinking of the same tacitly1 without any use of words either understood or expressed2 and it should happen afterwards that another triangle1 unlike the former1 or the same in different situation1 should be offered to his consideration1 he would not know readily whether the same property were in this last or no1 but would be forced1 as often as a different triangle were brought before him 7and the difference of triangles is infinite7 to begin his contemplation anew2 which he would have no need to do if he had the use of names1 for every universal name denotes the conceptions we have of infinite singular things4 ,nevertheless1 as ,i said above1 they serve as marks for the help of our memory1 whereby we register to ourselves our own inventions2 but not as signs by which we declare the same to others2 so that a man may be a philosopher alone by himself1 without any master2 ,adam had this capacity4 ,but to teach1 that is1 to demonstrate1 supposes two at the least1 and syllogistical speech4 ,sidenote3 ,the method of demonstration is synthetical4 #ab4 ,and seeing teaching is nothing but leading the mind of him we teach1 to the knowledge of our inventions1 in that track by which we attained the same with our own mind2 therefore1 the same method that served for our invention1 will serve also for demonstration to others1 saving that we omit the first part of method which proceeded from the sense of things to universal principles1 which1 because they are principles1 cannot be demonstrated2 and seeing they are known by nature1 7as was said above in the #eth article7 they need no demonstration1 though they need explication4 ,the whole method1 therefore1 of demonstration1 is synthetical1 consisting of that order of speech which begins from primary or most universal propositions1 which are manifest of themselves1 and proceeds by a perpetual composition of propositions into syllogisms1 till at last the learner understand the truth of the conclusion sought after4 ,sidenote3 ,definitions only are primary1 universal propositions4 #ac4 ,now1 such principles are nothing but definitions1 whereof there are two sorts2 one of names1 that signify such things as have some conceivable cause1 and another of such names as signify things of which we can conceive no cause at all4 ,names of the former kind are1 body1 or matter1 quantity1 or extension1 motion1 and whatsoever is common to all matter4 ,of the second kind1 are such a body1 such and so great motion1 so great magnitude1 such figure1 and whatsoever we can distinguish one body from another by4 ,and names of the former kind are well enough defined1 when1 by speech as short as may be1 we raise in the mind of the hearer perfect and clear ideas or conceptions of the things named1 as when we define motion to be the leaving of one place1 and the acquiring of another continually2 for though no thing moved1 nor any cause of motion be in that definition1 yet1 at the hearing of that speech1 there will come into the mind of the hearer an idea of motion clear enough4 ,but definitions of things1 which may be understood to have some cause1 must consist of such names as express the cause or manner of their generation1 as when we define a circle to be a figure made by the circumduction of a straight line in a plane1 c4 ,besides definitions1 there is no other proposition that ought to be called primary1 or 7according to severe truth7 be received into the number of principles4 ,for those axioms of ,euclid1 seeing they may be demonstrated1 are no principles of demonstration1 though they have by the consent of all men gotten the authority of principles1 because they need not be demonstrated4 ,also1 those petitions1 or postulata1 7as they call them7 though they be principles1 yet they are not principles of demonstration1 but of construction only2 that is1 not of science1 but of power2 or 7which is all one7 not of theorems1 which are speculations1 but of problems1 which belong to practice1 or the doing of something4 ,but as for those common received opinions1 ,nature abhors vacuity1 ,nature doth nothing in vain1 and the like1 which are neither evident in themselves1 nor at all to be demonstrated1 and which are oftener false than true1 they are much less to be acknowledged for principles4 ,to return1 therefore1 to definitions2 the reason why ,i say that the cause and generation of such things1 as have any cause or generation1 ought to enter into their definitions1 is this4 ,the end of science is the demonstration of the causes and generations of things2 which if they be not in the definitions1 they cannot be found in the conclusion of the first syllogism1 that is made from those definitions2 and if they be not in the first conclusion1 they will not be found in any further conclusion deduced from that2 and1 therefore1 by proceeding in this manner1 we shall never come to science2 which is against the scope and intention of demonstration4 ,sidenote3 ,the nature definition of a definition4 #ad4 ,now1 seeing definitions 7as ,i have said7 are principles1 or primary propositions1 they are therefore speeches2 and seeing they are used for the raising of an idea of some thing in the mind of the learner1 whensoever that thing has a name1 the definition of it can be nothing but the explication of that name by speech2 and if that name be given it for some compounded conception1 the definition is nothing but a resolution of that name into its most universal parts4 ,as when we define man1 saying man is a body animated1 sentient1 rational1 those names1 body animated1 c4 are parts of that whole name man2 so that definitions of this kind always consist of genus and difference2 the former names being all1 till the last1 general2 and the last of all1 difference4 ,but if any name be the most universal in its kind1 then the definition of it cannot consist of genus and difference1 but is to be made by such circumlocution1 as best explicateth the force of that name4 ,again1 it is possible1 and happens often1 that the genus and difference are put together1 and yet make no definition2 as these words1 a straight line1 contain both the genus and difference2 but are not a definition1 unless we should think a straight line may be thus defined1 a straight line is a straight line3 and yet if there were added another name1 consisting of different words1 but signifying the same thing which these signify1 then these might be the definition of that name4 ,from what has been said1 it may be understood how a definition ought to be defined1 namely1 that it is a proposition1 whose predicate resolves the subject1 when it may2 and when it may not1 it exemplifies the same4 ,sidenote3 ,properties of a definition4 #ae4 ,the properties of a definition are3 ,first1 that it takes away equivocation1 as also all that multitude of distinctions1 which are used by such as think they may learn philosophy by disputation4 ,for the nature of a definition is to define1 that is1 to determine the signification of the defined name1 and to pare from it all other signification besides what is contained in the definition itself2 and therefore one definition does as much1 as all the distinctions 7how many soever7 that can be used about the name defined4 ,secondly1 that it gives an universal notion of the thing defined1 representing a certain universal picture thereof1 not to the eye1 but to the mind4 ,for as when one paints a man1 he paints the image of some man2 so he1 that defines the name man1 makes a representation of some man to the mind4 ,thirdly1 that it is not necessary to dispute whether definitions are to be admitted or no4 ,for when a master is instructing his scholar1 if the scholar understand all the parts of the thing defined1 which are resolved in the definition1 and yet will not admit of the definition1 there needs no further controversy betwixt them1 it being all one as if he refused to be taught4 ,but if he understand nothing1 then certainly the definition is faulty2 for the nature of a definition consists in this1 that it exhibit a clear idea of the thing defined2 and principles are either known by themselves1 or else they are not principles4 ,fourthly1 that1 in philosophy1 definitions are before defined names4 ,for in teaching philosophy1 the first beginning is from definitions2 and all progression in the same1 till we come to the knowledge of the thing compounded1 is compositive4 ,seeing1 therefore1 definition is the explication of a compounded name by resolution1 and the progression is from the parts to the compound1 definitions must be understood before compounded names2 nay1 when the names of the parts of any speech be explicated1 it is not necessary that the definition should be a name compounded of them4 ,for example1 when these names1 equilateral1 quadrilateral1 right-angled1 are sufficiently understood1 it is not necessary in geometry that there should be at all such a name as square2 for defined names are received in philosophy for brevity's sake only4 ,fifthly1 that compounded names1 which are defined one way in some one part of philosophy1 may in another part of the same be otherwise defined2 as a parabola and an hyperbole have one definition in geometry1 and another in rhetoric2 for definitions are instituted and serve for the understanding of the doctrine which is treated of4 ,and1 therefore1 as in one part of philosophy1 a definition may have in it some one fit name for the more brief explanation of some proposition in geometry2 so it may have the same liberty in other parts of philosophy2 for the use of names is particular 7even where many agree to the settling of them7 and arbitrary4 ,sixthly1 that no name can be defined by any one word2 because no one word is sufficient for the resolving of one or more words4 ,seventhly1 that a defined name ought not to be repeated in the definition4 ,for a defined name is the whole compound1 and a definition is the resolution of that compound into parts2 but no total can be part of itself4 ,sidenote3 ,nature of a demonstration4 #af4 ,any two definitions1 that may be compounded into a syllogism1 produce a conclusion2 which1 because it is derived from principles1 that is1 from definitions1 is said to be demonstrated2 and the derivation or composition itself is called a demonstration4 ,in like manner1 if a syllogism be made of two propositions1 whereof one is a definition1 the other a demonstrated conclusion1 or neither of them is a definition1 but both formerly demonstrated1 that syllogism is also called a demonstration1 and so successively4 ,the definition therefore of a demonstration is this1 a demonstration is a syllogism1 or series of syllogisms1 derived and continued1 from the definitions of names1 to the last conclusion4 ,and from hence it may be understood1 that all true ratiocination1 which taketh its beginning from true principles1 produceth science1 and is true demonstration4 ,for as for the original of the name1 although that1 which the ,greeks called 1 and the ,latins demonstratio1 was understood by them for that sort only of ratiocination1 in which1 by the describing of certain lines and figures1 they placed the thing they were to prove1 as it were before men's eyes1 which is properly 1 or to shew by the figure2 yet they seem to have done it for this reason1 that unless it were in geometry1 7in which only there is place for such figures7 there was no ratiocination certain1 and ending in science1 their doctrines concerning all other things being nothing but controversy and clamour2 which1 nevertheless1 happened1 not because the truth to which they pretended could not be made evident without figures1 but because they wanted true principles1 from which they might derive their ratiocination2 and1 therefore1 there is no reason but that if true definitions were premised in all sorts of doctrines1 the demonstrations also would be true4 ,sidenote3 ,properties of a demonstration1 and order of things to be demonstrated4 #ag4 ,it4 is proper to methodical demonstration1 ,first1 that there be a true succession of one reason to another1 according to the rules of syllogizing delivered above4 ,secondly1 that the premises of all syllogisms be demonstrated from the first definitions4 ,thirdly1 that after definitions1 he that teaches or demonstrates any thing1 proceed in the same method by which he found it out2 namely1 that in the first place those things be demonstrated1 which immediately succeed to universal definitions 7in which is contained that part of philosophy which is called philosophia prima74 ,next1 those things which may be demonstrated by simple motion 7in which geometry consists74 ,after geometry1 such things as may be taught or shewed by manifest action1 that is1 by thrusting from1 or pulling towards4 ,and after these1 the motion or mutation of the invisible parts of things1 and the doctrine of sense and imaginations1 and of the internal passions1 especially those of men1 in which are comprehended the grounds of civil duties1 or civil philosophy2 which takes up the last place4 ,and that this method ought to be kept in all sorts of philosophy1 is evident from hence1 that such things as ,i have said are to be taught last1 cannot be demonstrated1 till such as are propounded to be first treated of1 be fully understood4 ,of which method no other example can be given1 but that treatise of the elements of philosophy1 which ,i shall begin in the next chapter1 and continue to the end of the work4 ,sidenote3 ,faults of a demonstration4 #ah4 ,besides those paralogisms1 whose fault lies either in the falsity of the premises1 or the want of true composition1 of which ,i have spoken in the precedent chapter1 there are two more1 which are frequent in demonstration2 one whereof is commonly called petitio principii2 the other is the supposing of a false cause2 and these do not only deceive unskilful learners1 but sometimes masters themselves1 by making them take that for well demonstrated1 which is not demonstrated at all4 ,petitio principii is1 when the conclusion to be proved is disguised in other words1 and put for the definition or principle from whence it is to be demonstrated2 and thus1 by putting for the cause of the thing sought1 either the thing itself or some effect of it1 they make a circle in their demonstration4 ,as for example1 he that would demonstrate that the earth stands still in the centre of the world1 and should suppose the earth's gravity to be the cause thereof1 and define gravity to be a quality by which every heavy body tends towards the centre of the world1 would lose his labour2 for the question is1 what is the cause of that quality in the earth8 and1 therefore1 he that supposes gravity to be the cause1 puts the thing itself for its own cause4 ,of a false cause ,i find this example in a certain treatise where the thing to be demonstrated is the motion of the earth4 ,he begins1 therefore1 with this1 that seeing the earth and the sun are not always in the same situation1 it must needs be that one of them be locally moved1 which is true2 next1 he affirms that the vapours1 which the sun raises from the earth and sea1 are1 by reason of this motion1 necessarily moved1 which also is true2 from whence he infers the winds are made1 and this may pass for granted2 and by these winds he says1 the waters of the sea are moved1 and by their motion the bottom of the sea1 as if it were beaten forwards1 moves round2 and let this also be granted2 wherefore1 he concludes1 the earth is moved2 which is1 nevertheless1 a paralogism4 ,for1 if that wind were the cause why the earth was1 from the beginning1 moved round1 and the motion either of the sun or the earth were the cause of that wind1 then the motion of the sun or the earth was before the wind itself2 and if the earth were moved1 before the wind was made1 then the wind could not be the cause of the earth's revolution2 but1 if the sun were moved1 and the earth stand still1 then it is manifest the earth might remain unmoved1 notwithstanding that wind2 and therefore that motion was not made by the cause which he allegeth4 ,but paralogisms of this kind are very frequent among the writers of physics1 though none can be more elaborate than this in the example given4 ,sidenote3 ,why the analytical method of geometricians cannot be treated of in this place4 #ai4 ,it may to some men seem pertinent to treat in this place of that art of the geometricians1 which they call logistica1 that is1 the art1 by which1 from supposing the thing in question to be true1 they proceed by ratiocination1 till either they come to something known1 by which they may demonstrate the truth of the thing sought for2 or to something which is impossible1 from whence they collect that to be false1 which they supposed true4 ,but this art cannot be explicated here1 for this reason1 that the method of it can neither be practised1 nor understood1 unless by such as are well versed in geometry2 and among geometricians themselves1 they1 that have most theorems in readiness1 are the most ready in the use of this logistica2 so that1 indeed1 it is not a distinct thing from geometry itself2 for there are1 in the method of it1 three parts2 the first whereof consists in the finding out of equality betwixt known and unknown things1 which they call equation2 and this equation cannot be found out1 but by such as know perfectly the nature1 properties1 and transpositions of proportion1 as also the addition1 subtraction1 multiplication1 and division of lines and superficies1 and the extraction of roots2 which are the parts of no mean geometrician4 ,the second is1 when an equation is found1 to be able to judge whether the truth or falsity of the question may be deduced from it1 or no2 which yet requires greater knowledge4 ,and the third is1 when such an equation is found1 as is fit for the solution of the question1 to know how to resolve the same in such manner1 that the truth or falsity may thereby manifestly appear2 which1 in hard questions1 cannot be done without the knowledge of the nature of crooked-lined figures2 but he that understands readily the nature and properties of these1 is a complete geometrician4 ,it happens besides1 that for the finding out of equations1 there is no certain method1 but he is best able to do it1 that has the best natural wit4 ,,part ,,ii4 ,,the ,,first ,,grounds ,,of ,,philosophy4 ------- ,,chapter ,,vii4 ,,of ,,place ,,and ,,time4 #a4 ,things that have no existence1 may nevertheless be understood and computed4--#b4 ,what is ,space4--#c4 ,time4--#d4 ,part4--#e4 ,division4--#f4 ,one4--#g4 ,number4--#h4 ,composition4--#i4 ,the whole4--#aj4 ,spaces and times contiguous1 and continual4--#aa4 ,beginning1 end1 way1 finite1 infinite4--#ab4 ,what is infinite in power4 ,nothing infinite can be truly said to be either whole1 or one2 nor infinite spaces or times1 many4--#ac4 ,division proceeds not to the least4 ,sidenote3 ,things that have no existence1 may nevertheless be understood and computed4 #a4 ,in the teaching of natural philosophy1 ,i cannot begin better 7as ,i have already shewn7 than from privation2 that is1 from feigning the world to be annihilated4 ,but1 if such annihilation of all things be supposed1 it may perhaps be asked1 what would remain for any man 7whom only ,i except from this universal annihilation of things7 to consider as the subject of philosophy1 or at all to reason upon2 or what to give names unto for ratiocination's sake4 ,i say1 therefore1 there would remain to that man ideas of the world1 and of all such bodies as he had1 before their annihilation1 seen with his eyes1 or perceived by any other sense2 that is to say1 the memory and imagination of magnitudes1 motions1 sounds1 colours1 c4 as also of their order and parts4 ,all which things1 though they be nothing but ideas and phantasms1 happening internally to him that imagineth2 yet they will appear as if they were external1 and not at all depending upon any power of the mind4 ,and these are the things to which he would give names1 and subtract them from1 and compound them with one another4 ,for seeing1 that after the destruction of all other things1 ,i suppose man still remaining1 and namely that he thinks1 imagines1 and remembers1 there can be nothing for him to think of but what is past2 nay1 if we do but observe diligently what it is we do when we consider and reason1 we shall find1 that though all things be still remaining in the world1 yet we compute nothing but our own phantasms4 ,for when we calculate the magnitude and motions of heaven or earth1 we do not ascend into heaven that we may divide it into parts1 or measure the motions thereof1 but we do it sitting still in our closets or in the dark4 ,now things may be considered1 that is1 be brought into account1 either as internal accidents of our mind1 in which manner we consider them when the question is about some faculty of the mind2 or as species of external things1 not as really existing1 but appearing only to exist1 or to have a being without us4 ,and in this manner we are now to consider them4 ,sidenote3 ,what is ,space4 #b4 ,if therefore we remember1 or have a phantasm of any thing that was in the world before the supposed annihilation of the same2 and consider1 not that the thing was such or such1 but only that it had a being without the mind1 we have presently a conception of that we call space3 an imaginary space indeed1 because a mere phantasm1 yet that very thing which all men call so4 ,for no man calls it space for being already filled1 but because it may be filled2 nor does any man think bodies carry their places away with them1 but that the same space contains sometimes one1 sometimes another body2 which could not be if space should always accompany the body which is once in it4 ,and this is of itself so manifest1 that ,i should not think it needed any explaining at all1 but that ,i find space to be falsely defined by certain philosophers1 who infer from thence1 one1 that the world is infinite 7for taking space to be the extension of bodies1 and thinking extension may encrease continually1 he infers that bodies may be infinitely extended72 and1 another1 from the same definition1 concludes rashly1 that it is impossible even to ,god himself to create more worlds than one2 for1 if another world were to be created1 he says1 that seeing there is nothing without this world1 and therefore 7according to his definition7 no space1 that new world must be placed in nothing2 but in nothing nothing can be placed2 which he affirms only1 without showing any reason for the same2 whereas the contrary is the truth3 for more cannot be put into a place already filled1 so much is empty space fitter than that1 which is full1 for the receiving of new bodies4 ,having therefore spoken thus much for these men's sakes1 and for theirs that assent to them1 ,i return to my purpose1 and define space thus3 ,,space is the phantasm of a thing existing without the mind simply2 that is to say1 that phantasm1 in which we consider no other accident1 but only that it appears without us4 ,sidenote3 ,time4 #c4 ,as a body leaves a phantasm of its magnitude in the mind1 so also a moved body leaves a phantasm of its motion1 namely1 an idea of that body passing out of one space into another by continual succession4 ,and this idea1 or phantasm1 is that1 which 7without receding much from the common opinion1 or from ,aristotle's definition7 ,i call ,time4 ,for seeing all men confess a year to be time1 and yet do not think a year to be the accident or affection of any body1 they must needs confess it to be1 not in the things without us1 but only in the thought of the mind4 ,so when they speak of the times of their predecessors1 they do not think after their predecessors are gone1 that their times can be any where else than in the memory of those that remember them4 ,and as for those that say1 days1 years1 and months are the motions of the sun and moon1 seeing it is all one to say1 motion past and motion destroyed1 and that future motion is the same with motion which is not yet begun1 they say that1 which they do not mean1 that there neither is1 nor has been1 nor shall be any time3 for of whatsoever it may be said1 it has been or it shall be1 of the same also it might have been said heretofore1 or may be said hereafter1 it is4 ,what then can days1 months1 and years1 be1 but the names of such computations made in our mind8 ,time therefore is a phantasm1 but a phantasm of motion1 for if we would know by what moments time passes away1 we make use of some motion or other1 as of the sun1 of a clock1 of the sand in an hour-glass1 or we mark some line upon which we imagine something to be moved1 there being no other means by which we can take notice of any time at all4 ,and yet1 when ,i say time is a phantasm of motion1 ,i do not say this is sufficient to define it by2 for this word time comprehends the notion of former and latter1 or of succession in the motion of a body1 in as much as it is first here then there4 ,wherefore a complete definition of time is such as this1 ,,time is the phantasm of before and after in motion2 which agrees with this definition of ,aristotle1 time is the number of motion according to former and latter2 for that numbering is an act of the mind2 and therefore it is all one to say1 time is the number of motion according to former and latter2 and time is a phantasm of motion numbered4 ,but that other definition1 time is the measure of motion1 is not so exact1 for we measure time by motion and not motion by time4 ,sidenote3 ,part4 #d4 ,one space is called part of another space1 and one time part of another time1 when this contains that and something besides4 ,from whence it may be collected1 that nothing can rightly be called a ,,part1 but that which is compared with something that contains it4 ,sidenote3 ,division4 #e4 ,and therefore to make parts1 or to part or ,,divide space or time1 is nothing else but to consider one and another within the same2 so that if any man divide space or time1 the diverse conceptions he has are more1 by one1 than the parts he makes2 for his first conception is of that which is to be divided1 then of some part of it1 and again of some other part of it1 and so forwards as long as he goes on in dividing4 ,but it is to be noted1 that here1 by division1 ,i do not mean the severing or pulling asunder of one space or time from another 7for does any man think that one hemisphere may be separated from the other hemisphere1 or the first hour from the second87 but diversity of consideration2 so that division is not made by the operation of the hands but of the mind4 ,sidenote3 ,one4 #f4 ,when space or time is considered among other spaces or times1 it is said to be ,,one1 namely one of them2 for except one space might be added to another1 and subtracted from another space1 and so of time1 it would be sufficient to say space or time simply1 and superfluous to say one space or one time1 if it could not be conceived that there were another4 ,the common definition of one1 namely1 that one is that which is undivided1 is obnoxious to an absurd consequence2 for it may thence be inferred1 that whatsoever is divided is many things1 that is1 that every divided thing1 is divided things1 which is insignificant4 ,sidenote3 ,number4 #g4 ,,number is one and one1 or one one and one1 and so forwards2 namely1 one and one make the number two1 and one one and one the number three2 so are all other numbers made2 which is all one as if we should say1 number is unities4 ,sidenote3 ,composition4 #h4 ,to ,,compound space of spaces1 or time of times1 is first to consider them one after another1 and then altogether as one2 as if one should reckon first the head1 the feet1 the arms1 and the body1 severally1 and then for the account of them all together put man4 ,and that which is so put for all the severals of which it consists1 is called the ,,whole2 and those severals1 when by the division of the whole they come again to be considered singly1 are parts thereof2 and therefore the whole and all the parts taken together are the same thing4 ,and as ,i noted above1 that in division it is not necessary to pull the parts asunder2 so in composition1 it is to be understood1 that for the making up of a whole there is no need of putting the parts together1 so as to make them touch one another1 but only of collecting them into one sum in the mind4 ,for thus all men1 being considered together1 make up the whole of mankind1 though never so much dispersed by time and place2 and twelve hours1 though the hours of several days1 may be compounded into one number of twelve4 ,sidenote3 ,the whole4 #i4 ,this being well understood1 it is manifest1 that nothing can rightly be called a whole1 that is not conceived to be compounded of parts1 and that it may be divided into parts2 so that if we deny that a thing has parts1 we deny the same to be a whole4 ,for example1 if we say the soul can have no parts1 we affirm that no soul can be a whole soul4 ,also it is manifest1 that nothing has parts till it be divided2 and when a thing is divided1 the parts are only so many as the division makes them4 ,again1 that a part of a part is a part of the whole2 and thus any part of the number four1 as two1 is a part of the number eight2 for four is made of two and two2 but eight is compounded of two1 two1 and four1 and therefore two1 which is a part of the part four1 is also a part of the whole eight4 ,sidenote3 ,spaces and times contiguous and continual4 #aj4 ,two spaces are said to be ,,contiguous1 when there is no other space betwixt them4 ,but two times1 betwixt which there is no other time1 are called immediate1 as ,a ,b1 ,b ,c4,sidenote3 ,a ,b ,c ,and any two spaces1 as well as times1 are said to be ,,continual1 when they have one common part1 as ,a ,c1 ,b ,d1 where the part ,b ,c is common2 ,sidenote3 ,a ,b ,c ,d and more spaces and times are continual1 when every two which are next one another are continual4 ,a ,b ,c ,d ------------- ,a ,b ,c ------------ ,sidenote3 ,beginning1 end1 way1 finite1 infinite4 #aa4 ,that part which is between two other parts1 is called a ,,mean2 and that which is not between two other parts1 an ,,extreme4 ,and of extremes1 that which is first reckoned is the ,,beginning1 and that which last1 the ,,end2 and all the means together taken are the ,,way4 ,also1 extreme parts and limits are the same thing4 ,and from hence it is manifest1 that beginning and end depend upon the order in which we number them2 and that to terminate or limit space and time1 is the same thing with imagining their beginning and end2 as also that every thing is ,,finite or ,,infinite1 according as we imagine or not imagine it limited or terminated every way2 and that the limits of any number are unities1 and of these1 that which is the first in our numbering is the beginning1 and that which we number last1 is the end4 ,when we say number is infinite1 we mean only that no number is expressed2 for when we speak of the numbers two1 three1 a thousand1 c4 they are always finite4 ,but when no more is said but this1 number is infinite1 it is to be understood as if it were said1 this name number is an indefinite name4 ,sidenote3 ,what is infinite in power4 ,nothing infinite can be truly said to be either whole or one2 nor infinite spaces or times1 many4 #ab4 ,space or time is said to finite in power1 or terminable1 when there may be assigned a number of finite spaces or times1 as of paces or hours1 than which there can be no greater number of the same measure in that space or time2 and infinite in power is that space or time1 in which a greater number of the said paces or hours may be assigned1 than any number that can be given4 ,but we must note1 that1 although in that space or time which is infinite in power1 there may be numbered more paces or hours than any number that can be assigned1 yet their number will always be finite2 for every number is finite4 ,and therefore his ratiocination was not good1 that undertaking to prove the world to be finite1 reasoned thus2 ,if the world be infinite1 then there may be taken in it some part which is distant from us an infinite number of paces3 but no such part can be taken2 wherefore the world is not infinite2 because that consequence of the major proposition is false2 for in an infinite space1 whatsoever we take or design in our mind1 the distance of the same from us is a finite space2 for in the very designing of the place thereof1 we put an end to that space1 of which we ourselves are the beginning2 and whatsoever any man with his mind cuts off both ways from infinite1 he determines the same1 that is1 he makes it finite4 ,of infinite space or time1 it cannot be said that it is a whole or one3 not a whole1 because not compounded of parts2 for seeing parts1 how many soever they be1 are severally finite1 they will also1 when they are all put together1 make a whole finite3 nor one1 because nothing can be said to be one1 except there be another to compare it with2 but it cannot be conceived that there are two spaces1 or two times1 infinite4 ,lastly1 when we make question whether the world be finite or infinite1 we have nothing in our mind answering to the name world2 for whatsoever we imagine1 is therefore finite1 though our computation reach the fixed stars1 or the ninth or tenth1 nay1 the thousandth sphere4 ,the meaning of the question is this only1 whether ,god has actually made so great an addition of body to body1 as we are able to make of space to space4 ,sidenote3 ,division proceeds not to the least4 #ac4 ,and1 therefore1 that which is commonly said1 that space and time may be divided infinitely1 is not to be so understood1 as if there might be any infinite or eternal division2 but rather to be taken in this sense1 whatsoever is divided1 is divided into such parts as may again be divided2 or thus1 the least divisible thing is not to be given2 or1 as geometricians have it1 no quantity is so small1 but a less may be taken2 which may easily be demonstrated in this manner4 ,let any space or time1 that which was thought to be the least divisible1 be divided into two equal parts1 ,a and ,b4 ,i say either of them1 as ,a1 may be divided again4 ,for suppose the part ,a to be contiguous to the part ,b of one side1 and of the other side to some other space equal to ,b4 ,this whole space1 therefore1 being greater than the space given1 is divisible4 ,wherefore1 if it be divided into two equal parts1 the part in the middle1 which is ,a1 will be also divided into two equal parts2 and therefore ,a was divisible4 ------- ,,chapter ,,viii4 ,,of ,,body ,,and ,,accident4 #a4 ,body defined4--#b4 ,accident defined4--#c4 ,how an accident may be understood to be in its subject4--#d4 ,magnitude1 what it is4--#e4 ,place1 what it is1 and that it is immovable4--#f4 ,what is full and empty4--#g4 ,here1 there1 somewhere1 what they signify4--#h4 ,many bodies cannot be in one place1 nor one body in many places4--#i4 ,contiguous and continual1 what they are4--#aj4 ,the definition of motion4 ,no motion intelligible but with time4--#aa4 ,what it is to be at rest1 to have been moved1 and to be moved4 ,no motion to be conceived1 without the conception of past and future4--#ab4 ,a point1 a line1 superficies and solid1 what they are4--#ac4 ,equal1 greater1 and less in bodies and magnitudes1 what they are4--#ad4 ,one and the same body has always one and the same magnitude4--#ae4 ,velocity1 what it is4--#af4 ,equal1 greater1 and less in times1 what they are4--#ag4 ,equal1 greater1 and less1 in velocity1 what4--#ah4 ,equal1 greater1 and less1 in motion1 what4--#ai4 ,that which is at rest1 will always be at rest1 except it be moved by some external thing2 and that which is moved1 will always be moved1 unless it be hindered by some external thing4--#bj4 ,accidents are generated and destroyed1 but bodies not so4--#ba4 ,an accident cannot depart from its subject4--#bb4 ,nor be moved4--#bc4 ,essence1 form1 and matter1 what they are4--#bd4 ,first matter1 what4--#be4 ,that the whole is greater than any part thereof1 why demonstrated4 ,sidenote3 ,body defined4 #a4 ,having understood what imaginary space is1 in which we supposed nothing remaining without us1 but all those things to be destroyed1 that1 by existing heretofore1 left images of themselves in our minds2 let us now suppose some one of those things to be placed again in the world1 or created anew4 ,it is necessary1 therefore1 that this new-created or replaced thing do not only fill some part of the space above mentioned1 or be coincident and coextended with it1 but also that it have no dependance upon our thought4 ,and this is that which1 for the extension of it1 we commonly call body2 and because it depends not upon our thought1 we say is a thing subsisting of itself2 as also existing1 because without us2 and1 lastly1 it is called the subject1 because it is so placed in and subjected to imaginary space1 that it may be understood by reason1 as well as perceived by sense4 ,the definition1 therefore1 of body may be this1 a body is that1 which having no dependance upon our thought1 is coincident or coextended with some part of space4 ,sidenote3 ,accident defined4 #b4 ,but what an accident is cannot so easily be explained by any definition1 as by examples4 ,let us imagine1 therefore1 that a body fills any space1 or is coextended with it2 that coextension is not the coextended body3 and1 in like manner1 let us imagine that the same body is removed out of its place2 that removing is not the removed body3 or let us think the same not removed2 that not removing or rest is not the resting body4 ,what1 then1 are these things8 ,they are accidents of that body4 ,but the thing in question is1 what is an accident8 which is an enquiry after that which we know already1 and not that which we should enquire after4 ,for who does not always and in the same manner understand him that says any thing is extended1 or moved1 or not moved8 ,but most men will have it be said that an accident is something1 namely1 some part of a natural thing1 when1 indeed1 it is no part of the same4 ,to satisfy these men1 as well as may be1 they answer best that define an accident to be the manner by which any body is conceived2 which is all one as if they should say1 an accident is that faculty of any body1 by which it works in us a conception of itself4 ,which definition1 though it be not an answer to the question propounded1 yet it is an answer to that question which should have been propounded1 namely1 whence does it happen that one part of any body appears here1 another there8 ,for this is well answered thus3 it happens from the extension of that body4 ,or1 how comes it to pass that the whole body1 by succession1 is seen now here1 now there8 and the answer will be1 by reason of its motion4 ,or1 lastly1 whence is it that any body possesseth the same space for sometime8 and the answer will be1 because it is not moved4 ,for if concerning the name of a body1 that is1 concerning a concrete name1 it be asked1 what is it8 the answer must be made by definition2 for the question is concerning the signification of the name4 ,but if it be asked concerning an abstract name1 what is it8 the cause is demanded why a thing appears so or so4 ,as if it be asked1 what is hard8 ,the answer will be1 hard is that1 whereof no part gives place1 but when the whole gives place4 ,but if it be demanded1 what is hardness8 a cause must be shewn why a part does not give place1 except the whole give place4 ,wherefore1 ,i define an accident to be the manner of our conception of body4 ,sidenote3 ,how an accident may be understood to be in its subject4 #c4 ,when an accident is said to be in a body1 it is not so to be understood1 as if any thing were contained in that body2 as if1 for example1 redness were in blood1 in the same manner1 as blood is in a bloody cloth1 that is1 as a part in the whole2 for so1 an accident would be a body also4 ,but1 as magnitude1 or rest1 or motion1 is in that which is great1 or which resteth1 or which is moved1 7which1 how it is to be understood1 every man understands7 so also1 it is to be understood1 that every other accident is in its subject4 ,and this1 also1 is explicated by ,aristotle no otherwise than negatively1 namely1 that an accident is in its subject1 not as any part thereof1 but so as that it may be away1 the subject still remaining2 which is right1 saving that there are certain accidents which can never perish except the body perish also2 for no body can be conceived to be without extension1 or without figure4 ,all other accidents1 which are not common to all bodies1 but peculiar to some only1 as to be at rest1 to be moved1 colour1 hardness1 and the like1 do perish continually1 and are succeeded by others2 yet so1 as that the body never perisheth4 ,and as for the opinion that some may have1 that all other accidents are not in their bodies in the same manner that extension1 motion1 rest1 or figure1 are in the same2 for example1 that colour1 heat1 odour1 virtue1 vice1 and the like1 are otherwise in them1 and1 as they say1 inherent2 ,i desire they would suspend their judgment for the present1 and expect a little1 till it be found out by ratiocination1 whether these very accidents are not also certain motions either of the mind of the perceiver1 or of the bodies themselves which are perceived2 for in the search of this1 a great part of natural philosophy consists4 ,sidenote3 ,magnitude1 what it is4 #d4 ,the extension of a body1 is the same thing with the magnitude of it1 or that which some call real space4 ,but this magnitude does not depend upon our cogitation1 as imaginary space doth2 for this is an effect of our imagination1 but magnitude is the cause of it2 this is an accident of the mind1 that of a body existing out of the mind4 ,sidenote3 ,place1 what it is1 and that it is immovable4 #e4 ,that space1 by which word ,i here understand imaginary space1 which is coincident with the magnitude of any body1 is called the place of that body2 and the body itself is that which we call the thing placed4 ,now place1 and the magnitude of the thing placed1 differ4 ,first in this1 that a body keeps always the same magnitude1 both when it is at rest1 and when it is moved2 but when it is moved1 it does not keep the same place4 ,secondly in this1 that place is a phantasm of any body of such and such quantity and figure2 but magnitude is the peculiar accident of every body2 for one body may at several times have several places1 but has always one and the same magnitude4 ,thirdly in this1 that place is nothing out of the mind1 nor magnitude any thing within it4 ,and lastly1 place is feigned extension1 but magnitude true extension2 and a placed body is not extension1 but a thing extended4 ,besides1 place is immovable2 for1 seeing that which is moved1 is understood to be carried from place to place1 if place were moved1 it would also be carried from place to place1 so that one place must have another place1 and that place another place1 and so on infinitely1 which is ridiculous4 ,and as for those1 that1 by making place to be of the same nature with real space1 would from thence maintain it to be immovable1 they also make place1 though they do not perceive they make it so1 to be a mere phantasm4 ,for whilst one affirms that place is therefore said to be immovable1 because space in general is considered there2 if he had remembered that nothing is general or universal besides names or signs1 he would easily have seen that that space1 which he says is considered in general1 is nothing but a phantasm1 in the mind or the memory1 of a body of such magnitude and such figure4 ,and whilst another says3 real space is made immovable by the understanding2 as when1 under the superficies of running water1 we imagine other and other water to come by continual succession1 that superficies fixed there by the understanding1 is the immovable place of the river3 what else does he make it to be but a phantasm1 though he do it obscurely and in perplexed words8 ,lastly1 the nature of place does not consist in the superficies of the ambient1 but in solid space2 for the whole placed body is coextended with its whole place1 and every part of it with every answering part of the same place2 but seeing every placed body is a solid thing1 it cannot be understood to be coextended with superficies4 ,besides1 how can any whole body be moved1 unless all its parts be moved together with it8 ,or how can the internal parts of it be moved1 but by leaving their place8 ,but the internal parts of a body cannot leave the superficies of an external part contiguous to it2 and1 therefore1 it follows1 that if place be the superficies of the ambient1 then the parts of a body moved1 that is1 bodies moved1 are not moved4 ,sidenote3 ,what is full and empty4 #f4 ,space1 or place1 that is possessed by a body1 is called full1 and that which is not so possessed1 is called empty4 ,sidenote3 ,here1 there1 somewhere1 what they signify4 #g4 ,here1 there1 in the country1 in the city1 and other the like names1 by which answer is made to the question where is it8 are not properly names of place1 nor do they of themselves bring into the mind the place that is sought2 for here and there signify nothing1 unless the thing be shewn at the same time with the finger or something else2 but when the eye of him that seeks1 is1 by pointing or some other sign1 directed to the thing sought1 the place of it is not hereby defined by him that answers1 but found out by him that asks the question4 ,now such shewings as are made by words only1 as when we say1 in the country1 or in the city1 are some of greater latitude than others1 as when we say1 in the country1 in the city1 in such a street1 in a house1 in the chamber1 in bed1 c4 ,for these do1 by little and little1 direct the seeker nearer to the proper place2 and yet they do not determine the same1 but only restrain it to a lesser space1 and signify no more1 than that the place of the thing is within a certain space designed by those words1 as a part is in the whole4 ,and all such names1 by which answer is made to the question where8 have1 for their highest genus1 the name somewhere4 ,from whence it may be understood1 that whatsoever is somewhere1 is in some place properly so called1 which place is part of that greater space that is signified by some of these names1 in the country1 in the city1 or the like4 ,sidenote3 ,many bodies cannot be in one place1 nor one body in many places4 #h4 ,a body1 and the magnitude1 and the place thereof1 are divided by one and the same act of the mind2 for1 to divide an extended body1 and the extension thereof1 and the idea of that extension1 which is place1 is the same with dividing any one of them2 because they are coincident1 and it cannot be done but by the mind1 that is by the division of space4 ,from whence it is manifest1 that neither two bodies can be together in the same place1 nor one body be in two places at the same time4 ,not two bodies in the same place2 because when a body that fills its whole place is divided into two1 the place itself is divided into two also1 so that there will be two places4 ,not one body in two places2 for the place that a body fills being divided into two1 the placed body will be also divided into two2 for1 as ,i said1 a place and the body that fills that place1 are divided both together2 and so there will be two bodies4 ,sidenote3 ,contiguous and continual1 what they are4 #i4 ,two bodies are said to be contiguous to one another1 and continual1 in the same manner as spaces are2 namely1 those are contiguous1 between which there is no space4 ,now1 by space ,i understand1 here as formerly1 an idea or phantasm of a body4 ,wherefore1 though between two bodies there be put no other body1 and consequently no magnitude1 or1 as they call it1 real space1 yet if another body may be put between them1 that is1 if there intercede any imagined space which may receive another body1 then those bodies are not contiguous4 ,and this is so easy to be understood1 that ,i should wonder at some men1 who being otherwise skilful enough in philosophy1 are of a different opinion1 but that ,i find that most of those that affect metaphysical subtleties wander from truth1 as if they were led out of their way by an ignis fatuus4 ,for can any man that has his natural senses1 think that two bodies must therefore necessarily touch one another1 because no other body is between them8 ,or that there can be no vacuum1 because vacuum is nothing1 or as they call it1 non ens8 ,which is as childish1 as if one should reason thus2 no man can fast1 because to fast is to eat nothing2 but nothing cannot be eaten4 ,continual1 are any two bodies that have a common part2 and more than two are continual1 when every two1 that are next to one another1 are continual4 ,sidenote3 ,the definition of motion4 ,no motion intelligible but with time4 #aj4 ,,motion is a continual relinquishing of one place1 and acquiring of another2 and that place which is relinquished is commonly called the terminus a quo1 as that which is acquired is called the terminus ad quem2 ,i say a continual relinquishing1 because no body1 how little soever1 can totally and at once go out of its former place into another1 so1 but that some part of it will be in a part of a place which is common to both1 namely1 to the relinquished and the acquired places4 ,sidenote3 ,a ,g ,b ,i ,e ----------- 3 3 3 3 ----------- ,c ,h ,d ,k ,f ,for example1 let any body be in the place ,a ,c ,b ,d2 the same body cannot come into the place ,b ,d ,e ,f1 but it must first be in ,g ,h ,i ,k1 whose part ,g ,h ,b ,d is common to both the places ,a ,c ,b ,d1 and ,g ,h ,i ,k1 and whose part ,b ,d ,i ,k1 is common to both the places ,g ,h ,i ,k1 and ,b ,d ,e ,f4 ,now it cannot be conceived that any thing can be moved without time2 for time is1 by the definition of it1 a phantasm1 that is1 a conception of motion2 and1 therefore1 to conceive that any thing may be moved without time1 were to conceive motion without motion1 which is impossible4 ,sidenote3 ,what it is to be at rest1 to have been moved1 and to be moved4 ,no motion to be conceived without the conception of past and future4 #aa4 ,that is said to be at rest1 which1 during any time1 is in one place2 and that to be moved1 or to have been moved1 which1 whether it be now at rest or moved1 was formerly in another place than that which it is now in4 ,from which definitions it may be inferred1 first1 that whatsoever is moved1 has been moved2 for if it be still in the same place in which it was formerly1 it is at rest1 that is1 it is not moved1 by the definition of rest2 but if it be in another place1 it has been moved1 by the definition of moved4 ,secondly1 that what is moved1 will yet be moved2 for that which is moved1 leaveth the place where it is1 and therefore will be in another place1 and consequently will be moved still4 ,thirdly1 that whatsoever is moved1 is not in one place during any time1 how little soever that time be2 for by the definition of rest1 that which is in one place during any time1 is at rest4 ,there is a certain sophism against motion1 which seems to spring from the not understanding of this last proposition4 ,for they say1 that1 if any body be moved1 it is moved either in the place where it is1 or in the place where it is not2 both which are false2 and therefore nothing is moved4 ,but the falsity lies in the major proposition2 for that which is moved1 is neither moved in the place where it is1 nor in the place where is not2 but from the place where it is1 to the place where it is not4 ,indeed it cannot be denied but that whatsoever is moved1 is moved somewhere1 that is1 within some space2 but then the place of that body is not that whole space1 but a part of it1 as is said above in the seventh article4 ,from what is above demonstrated1 namely1 that whatsoever is moved1 has also been moved1 and will be moved1 this also may be collected1 that there can be no conception of motion1 without conceiving past and future time4 ,sidenote3 ,a point1 a line1 superficies1 and solid1 what they are4 #ab4 ,though there be no body which has not some magnitude1 yet if1 when any body is moved1 the magnitude of it be not at all considered1 the way it makes is called a line1 or one single dimension2 and the space1 through which it passeth1 is called length2 and the body itself1 a point2 in which sense the earth is called a point1 and the way of its yearly revolution1 the ecliptic line4 ,but if a body1 which is moved1 be considered as long1 and be supposed to be so moved1 as that all the several parts of it be understood to make several lines1 then the way of every part of that body is called breadth1 and the space which is made is called superficies1 consisting of two dimensions1 one whereof to every several part of the other is applied whole4 ,again1 if a body be considered as having superficies1 and be understood to be so moved1 that all the several parts of it describe several lines1 then the way of every part of that body is called thickness or depth1 and the space which is made is called solid1 consisting of three dimensions1 any two whereof are applied whole to every several part of the third4 ,but if a body be considered as solid1 then it is not possible that all the several parts of it should describe several lines2 for what way soever it be moved1 the way of the following part will fall into the way of the part before it1 so that the same solid will still be made which the foremost superficies would have made by itself4 ,and therefore there can be no other dimension in any body1 as it is a body1 than the three which ,i have now described2 though1 as it shall be shewed hereafter1 velocity1 which is motion according to length1 may1 by being applied to all the parts of a solid1 make a magnitude of motion1 consisting of four dimensions2 as the goodness of gold1 computed in all the parts of it1 makes the price and value thereof4 ,sidenote3 ,equal1 great1 greater1 and less1 in bodies and magnitudes1 what they are4 #ac4 ,bodies1 how many soever they be1 that can fill every one the place of every one1 are said to be equal every one to every other4 ,now1 one body may fill the same place which another body filleth1 though it be not of the same figure with that other body1 if so be that it may be understood to be reducible to the same figure1 either by flexion or transposition of the parts4 ,and one body is greater than another body1 when a part of that is equal to all this2 and less1 when all that is equal to a part of this4 ,also1 magnitudes are equal1 or greater1 or lesser1 than one another1 for the same consideration1 namely1 when the bodies1 of which they are the magnitudes1 are either equal1 or greater1 or less1 c4 ,sidenote3 ,one and the same body has always one and the same magnitude4 #ad4 ,one and the same body is always of one and the same magnitude4 ,for seeing a body and the magnitude and place thereof cannot be comprehended in the mind otherwise than as they are coincident1 if any body be understood to be at rest1 that is1 to remain in the same place during some time1 and the magnitude thereof be in one part of that time greater1 and in another part less1 that body's place1 which is one and the same1 will be coincident sometimes with greater1 sometimes with less magnitude1 that is1 the same place will be greater and less than itself1 which is impossible4 ,but there would be no need at all of demonstrating a thing that is in itself so manifest1 if there were not some1 whose opinion concerning bodies and their magnitudes is1 that a body may exist separated from its magnitude1 and have greater or less magnitude bestowed upon it1 making use of this principle for the explication of the nature of rarum and densum4 ,sidenote3 ,velocity1 what it is4 #ae4 ,motion1 in as much as a certain length may in a certain time be transmitted by it1 is called ,,velocity or swiftness3 c4 ,for though swift be very often understood with relation to slower or less swift1 as great is in respect of less1 yet nevertheless1 as magnitude is by philosophers taken absolutely for extension1 so also velocity or swiftness may be put absolutely for motion according to length4 ,sidenote3 ,equal1 greater1 and less1 in times1 what they are4 #af4 ,many motions are said to be made in equal times1 when every one of them begins and ends together with some other motion1 or if it had begun together1 would also have ended together with the same4 ,for time1 which is a phantasm of motion1 cannot be reckoned but by some exposed motion2 as in dials by the motion of the sun or of the hand2 and if two or more motions begin and end with this motion1 they are said to be made in equal times2 from whence also it is easy to understand what it is to be moved in greater or longer time1 and in less time or not so long2 namely1 that that is longer moved1 which beginning with another1 ends later2 or ending together1 began sooner4 ,sidenote3 ,equal1 greater1 and less1 in velocity1 what4 #ag4 ,motions are said to be equally swift1 when equal lengths are transmitted in equal times2 and greater swiftness is that1 wherein greater length is passed in equal time1 or equal length in less time4 ,also that swiftness by which equal lengths are passed in equal parts of time1 is called uniform swiftness or motion2 and of motions not uniform1 such as become swifter or slower by equal increasings or decreasings in equal parts of time1 are said to be accelerated or retarded uniformly4 ,sidenote3 ,equal1 greater1 and less1 in motion1 what4 #ah4 ,but motion is said to be greater1 less1 and equal1 not only in regard of the length which is transmitted in a certain time1 that is1 in regard of swiftness only1 but of swiftness applied to every smallest particle of magnitude2 for when any body is moved1 every part of it is also moved2 and supposing the parts to be halves1 the motions of those halves have their swiftness equal to one another1 and severally equal to that of the whole2 but the motion of the whole is equal to those two motions1 either of which is of equal swiftness with it2 and therefore it is one thing for two motions to be equal to one another1 and another thing for them to be equally swift4 ,and this is manifest in two horses that draw abreast1 where the motion of both the horses together is of equal swiftness with the motion of either of them singly2 but the motion of both is greater than the motion of one of them1 namely1 double4 ,wherefore motions are said to be simply equal to one another1 when the swiftness of one1 computed in every part of its magnitude1 is equal to the swiftness of the other computed also in every part of its magnitude3 and greater than one another1 when the swiftness of one computed as above1 is greater than the swiftness of the other so computed2 and less1 when less4 ,besides1 the magnitude of motion computed in this manner is that which is commonly called ,,force4 ,sidenote3 ,that which is at rest will always be at rest1 except it be moved by some external thing4 #ai4 ,whatsoever is at rest1 will always be at rest1 unless there be some other body besides it1 which1 by endeavouring to get into its place by motion1 suffers it no longer to remain at rest4 ,for suppose that some finite body exist and be at rest1 and that all space besides be empty2 if now this body begin to be moved1 it will certainly be moved some way2 seeing therefore there was nothing in that body which did not dispose it to rest1 the reason why it is moved this way is in something out of it2 and in like manner1 if it had been moved any other way1 the reason of motion that way had also been in something out of it2 but seeing it was supposed that nothing is out of it1 the reason of its motion one way would be the same with the reason of its motion every other way1 wherefore it would be moved alike all ways at once2 which is impossible4 ,sidenote3 ,that which is moved will always be moved1 unless it be hindered by some external thing4 ,in like manner1 whatsoever is moved1 will always be moved1 except there be some other body besides it1 which causeth it to rest4 ,for if we suppose nothing to be without it1 there will be no reason why it should rest now1 rather than at another time2 wherefore its motion would cease in every particle of time alike2 which is not intelligible4 ,sidenote3 ,accidents are generated and destroyed1 but bodies not so4 #bj4 ,when we say a living creature1 a tree1 or any other specified body is generated or destroyed1 it is not to be so understood as if there were made a body of that which is not-body1 or not a body of a body1 but of a living creature not a living creature1 of a tree not a tree1 c4 that is1 that those accidents for which we call one thing a living creature1 another thing a tree1 and another by some other name1 are generated and destroyed2 and that therefore the same names are not to be given to them now1 which were given them before4 ,but that magnitude for which we give to any thing the name of body is neither generated nor destroyed4 ,for though we may feign in our mind that a point may swell to a huge bulk1 and that this may again contract itself to a point2 that is1 though we may imagine something to arise where before was nothing1 and nothing to be there where before was something1 yet we cannot comprehend in our mind how this may possibly be done in nature4 ,and therefore philosophers1 who tie themselves to natural reason1 suppose that a body can neither be generated nor destroyed1 but only that it may appear otherwise than it did to us1 that is1 under different species1 and consequently be called by other and other names2 so that that which is now called man1 may at another time have the name of not-man2 but that which is once called body1 can never be called not-body4 ,but it is manifest1 that all other accidents besides magnitude or extension may be generated and destroyed2 as when a white thing is made black1 the whiteness that was in it perisheth1 and the blackness that was not in it is now generated2 and therefore bodies1 and the accidents under which they appear diversely1 have this difference1 that bodies are things1 and not generated2 accidents are generated1 and not things4 ,sidenote3 ,an accident cannot depart from its subject #ba4 ,and therefore1 when any thing appears otherwise than it did by reason of other and other accidents1 it is not to be thought that an accident goes out of one subject into another1 7for they are not1 as ,i said above1 in their subjects as a part in the whole1 or as a contained thing in that which contains it1 or as a master of a family in his house17 but that one accident perisheth1 and another is generated4 ,for example1 when the hand1 being moved1 moves the pen1 motion does not go out of the hand into the pen2 for so the writing might be continued though the hand stood still2 but a new motion is generated in the pen1 and is the pen's motion4 ,sidenote3 ,nor be moved4 #bb4 ,and therefore also it is improper to say1 an accident is moved2 as when1 instead of saying1 figure is an accident of a body carried away1 we say1 a body carries away its figure4 ,sidenote3 ,essence1 form1 and matter1 what they are4 #bc4 ,now that accident for which we give a certain name to any body1 or the accident which denominates its subject1 is commonly called the ,,essence thereof2 as rationality is the essence of a man2 whiteness1 of any white thing1 and extension the essence of a body4 ,and the same essence1 in as much as it is generated1 is called the ,,form4 ,again1 a body1 in respect of any accident1 is called the ,,subject1 and in respect of the form it is called the ,,matter4 ,also1 the production or perishing of any accident makes its subject be said to be changed2 only the production or perishing of form makes it be said it is generated or destroyed2 but in all generation and mutation1 the name of matter still remains4 ,for a table made of wood is not only wooden1 but wood2 and a statue of brass is brass as well as brazen2 though ,aristotle1 in his ,metaphysics1 says1 that whatsoever is made of any thing ought not to be called 1 but 2 as that which is made of wood1 not 1 but 1 that is1 not wood1 but wooden4 ,sidenote3 ,first matter1 what4 #bd4 ,and as for that matter which is common to all things1 and which philosophers1 following ,aristotle1 usually call materia prima1 that is1 first matter1 it is not any body distinct from all other bodies1 nor is it one of them4 ,what then is it8 ,a mere name2 yet a name which is not of vain use2 for it signifies a conception of body without the consideration of any form or other accident except only magnitude or extension1 and aptness to receive form and other accident4 ,so that whensoever we have use of the name body in general1 if we use that of materia prima1 we do well4 ,for as when a man not knowing which was first1 water or ice1 would find out which of the two were the matter of both1 he would be fain to suppose some third matter which were neither of these two2 so he that would find out what is the matter of all things1 ought to suppose such as is not the matter of anything that exists4 ,wherefore materia prima is nothing2 and therefore they do not attribute to it either form or any other accident besides quantity2 whereas all singular things have their forms and accidents certain4 ,materia prima1 therefore1 is body in general1 that is1 body considered universally1 not as having neither form nor any accident1 but in which no form nor any other accident but quantity are at all considered1 that is1 they are not drawn into argumentation4 ,sidenote3 ,that the whole is greater than any part thereof1 why demonstrated4 #be4 ,from what has been said1 those axioms may be demonstrated1 which are assumed by ,euclid in the beginning of his first element1 about the equality and inequality of magnitudes2 of which1 omitting the rest1 ,i will here demonstrate only this one1 the whole is greater than any part thereof2 to the end that the reader may know that those axioms are not indemonstrable1 and therefore not principles of demonstration2 and from hence learn to be wary how he admits any thing for a principle1 which is not at least as evident as these are4 ,greater is defined to be that1 whose part is equal to the whole of another4 ,now if we suppose any whole to be ,a1 and a part of it to be ,b2 seeing the whole ,b is equal to itself1 and the same ,b is a part of ,a2 therefore a part of ,a will be equal to the whole ,b4 ,wherefore1 by the definition above1 ,a is greater than ,b2 which was to be proved4 ------- ,,chapter ,,ix4 ,,of ,,cause ,,and ,,effect4 #a4 ,action and passion1 what they are4--#b4 ,action and passion mediate and immediate4--#c4 ,cause simply taken4 ,cause without which no effect follows1 or cause necessary by supposition4--#d4 ,cause efficient and material4--#e4 ,an entire cause is always sufficient to produce its effect4 ,at the same instant that the cause is entire1 the effect is produced4 ,every effect has a necessary cause4--#f4 ,the generation of effects is continual4 ,what is the beginning in causation4--#g4 ,no cause of motion but in a body contiguous and moved4--#h4 ,the same agents and patients1 if alike disposed1 produce like effects though at different times4--#i4 ,all mutation is motion4--#aj4 ,contingent accidents1 what they are4 ,sidenote3 ,action and passion1 what they are4 #a4 ,a body is said to work upon or act1 that is to say1 do something to another body1 when it either generates or destroys some accident in it3 and the body in which an accident is generated or destroyed is said to suffer1 that is1 to have something done to it by another body2 as when one body by putting forwards another body generates motion in it1 it is called the ,,agent2 and the body in which motion is so generated1 is called the ,,patient2 so fire that warms the hand is the agent1 and the hand1 which is warmed1 is the patient4 ,that accident1 which is generated in the patient1 is called the ,,effect4 ,sidenote3 ,action and passion1 mediate1 and immediate4 #b4 ,when an agent and patient are contiguous to one another1 their action and passion are then said to be immediate1 otherwise1 mediate2 and when another body1 lying betwixt the agent and patient1 is contiguous to them both1 it is then itself both an agent and a patient2 an agent in respect of the body next after it1 upon which it works1 and a patient in respect of the body next before it1 from which it suffers4 ,also1 if many bodies be so ordered that every two which are next to one another be contiguous1 then all those that are betwixt the first and the last are both agents and patients1 and the first is an agent only1 and the last a patient only4 ,sidenote3 ,cause simply taken4 #c4 ,an agent is understood to produce its determined or certain effect in the patient1 according to some certain accident or accidents1 with which both it and the patient are affected2 that is to say1 the agent hath its effect precisely such1 not because it is a body1 but because such a body1 or so moved4 ,for otherwise all agents1 seeing they are all bodies alike1 would produce like effects in all patients4 ,and therefore the fire1 for example1 does not warm1 because it is a body1 but because it is hot2 nor does one body put forward another body because it is a body1 but because it is moved into the place of that other body4 ,sidenote3 ,cause without which no effect follows1 or cause necessary by supposition4,the cause1 therefore1 of all effects consists in certain accidents both in the agents and in the patients2 which when they are all present1 the effect is produced2 but if any one of them be wanting1 it is not produced2 and that accident either of the agent or patient1 without which the effect cannot be produced1 is called causa sine qua non1 or cause necessary by supposition1 as also the cause requisite for the production of the effect4 ,but a ,,cause simply1 or an entire cause1 is the aggregate of all the accidents both of the agents how many soever they be1 and of the patient1 put together2 which when they are all supposed to be present1 it cannot be understood but that the effect is produced at the same instant2 and if any one of them be wanting1 it cannot be understood but that the effect is not produced4 ,sidenote3 ,cause efficient1 and material4 #d4 ,the aggregate of accidents in the agent or agents1 requisite for the production of the effect1 the effect being produced1 is called the efficient cause thereof2 and the aggregate of accidents in the patient1 the effect being produced1 is usually called the material cause2 ,i say the effect being produced2 for where there is no effect1 there can be no cause2 for nothing can be called a cause1 where there is nothing that can be called an effect4 ,but the efficient and material causes are both but partial causes1 or parts of that cause1 which in the next precedent article ,i called an entire cause4 ,and from hence it is manifest1 that the effect we expect1 though the agents be not defective on their part1 may nevertheless be frustrated by a defect in the patient2 and when the patient is sufficient1 by a defect in the agents4 ,sidenote3 ,an entire cause is always sufficient to produce its effect4 #e4 ,an entire cause is always sufficient for the production of its effect1 if the effect be at all possible4 ,for let any effect whatsoever be propounded to be produced2 if the same be produced1 it is manifest that the cause which produced it was a sufficient cause2 but if it be not produced1 and yet be possible1 it is evident that something was wanting either in some agent1 or in the patient1 without which it could not be produced2 that is1 that some accident was wanting which was requisite for its production2 and therefore1 that cause was not entire1 which is contrary to what was supposed4 ,sidenote3 ,at the same instant that the cause is entire1 the effect is produced4 ,it follows also from hence1 that in whatsoever instant the cause is entire1 in the same instant the effect is produced4 ,for if it be not produced1 something is still wanting1 which is requisite for the production of it2 and therefore the cause was not entire1 as was supposed4 ,sidenote3 ,every effect has a necessary cause4 ,and seeing a necessary cause is defined to be that1 which being supposed1 the effect cannot but follow2 this also may be collected1 that whatsoever effect is produced at any time1 the same is produced by a necessary cause4 ,for whatsoever is produced1 in as much as it is produced1 had an entire cause1 that is1 had all those things1 which being supposed1 it cannot be understood but that the effect follows2 that is1 it had a necessary cause4 ,and in the same manner it may be shewn1 that whatsoever effects are hereafter to be produced1 shall have a necessary cause2 so that all the effects that have been1 or shall be produced1 have their necessity in things antecedent4 ,sidenote3 ,the generation of effects is continual4 ,what is the beginning in causation4 #f4 ,and from this1 that whensoever the cause is entire1 the effect is produced in the same instant1 it is manifest that causation and the production of effects consist in a certain continual progress2 so that as there is a continual mutation in the agent or agents1 by the working of other agents upon them1 so also the patient1 upon which they work1 is continually altered and changed4 ,for example3 as the heat of the fire increases more and more1 so also the effects thereof1 namely1 the heat of such bodies as are next to it1 and again1 of such other bodies as are next to them1 increase more and more accordingly2 which is already no little argument that all mutation consists in motion only2 the truth whereof shall be further demonstrated in the ninth article4 ,but in this progress of causation1 that is1 of action and passion1 if any man comprehend in his imagination a part thereof1 and divide the same into parts1 the first part or beginning of it cannot be considered otherwise than as action or cause2 for1 if it should be considered as effect or passion1 then it would be necessary to consider something before it1 for its cause or action2 which cannot be1 for nothing can be before the beginning4 ,and in like manner1 the last part is considered only as effect2 for it cannot be called cause1 if nothing follow it2 but after the last1 nothing follows4 ,and from hence it is1 that in all action the beginning and cause are taken for the same thing4 ,but every one of the intermediate parts are both action and passion1 and cause and effect1 according as they are compared with the antecedent or subsequent part4 ,sidenote3 ,no cause of motion but in a body contiguous and moved4 #g4 ,there can be no cause of motion1 except in a body contiguous and moved4 ,for let there be any two bodies which are not contiguous1 and betwixt which the intermediate space is empty1 or1 if filled1 filled with another body which is at rest2 and let one of the propounded bodies be supposed to be at rest2 ,i say it shall always be at rest4 ,for if it shall be moved1 the cause of that motion1 by the #hth chapter1 article #ai1 will be some external body2 and1 therefore1 if between it and that external body there be nothing but empty space1 then whatsoever the disposition be of that external body or of the patient itself1 yet if it be supposed to be now at rest1 we may conceive it will continue so till it be touched by some other body4 ,but seeing cause1 by the definition1 is the aggregate of all such accidents1 which being supposed to be present1 it cannot be conceived but that the effect will follow1 those accidents1 which are either in external bodies1 or in the patient itself1 cannot be the cause of future motion4 ,and in like manner1 seeing we may conceive that whatsoever is at rest will still be at rest1 though it be touched by some other body1 except that other body be moved2 therefore in a contiguous body1 which is at rest1 there can be no cause of motion4 ,wherefore there is no cause of motion in any body1 except it be contiguous and moved4 ,the same reason may serve to prove that whatsoever is moved1 will always be moved on in the same way and with the same velocity1 except it be hindered by some other contiguous and moved body2 and consequently that no bodies1 either when they are at rest1 or when there is an interposition of vacuum1 can generate or extinguish or lessen motion in other bodies4 ,there is one that has written that things moved are more resisted by things at rest1 than by things contrarily moved2 for this reason1 that he conceived motion not to be so contrary to motion as rest4 ,that which deceived him was1 that the words rest and motion are but contradictory names2 whereas motion1 indeed1 is not resisted by rest1 but by contrary motion4 ,sidenote3 ,the same agents and patients1 if alike disposed1 produce like effects1 though at different times4 #h4 ,but if a body work upon another body at one time1 and afterwards the same body work upon the same body at another time1 so that both the agent and patient1 and all their parts1 be in all things as they were2 and there be no difference1 except only in time1 that is1 that one action be former1 the other later in time2 it is manifest of itself1 that the effects will be equal and like1 as not differing in anything besides time4 ,and as effects themselves proceed from their causes1 so the diversity of them depends upon the diversity of their causes also4 ,sidenote3 ,all mutation is motion4 #i4 ,this being true1 it is necessary that mutation can be nothing else but motion of the parts of that body which is changed4 ,for first1 we do not say anything is changed1 but that which appears to our senses otherwise than it appeared formerly4 ,secondly1 both those appearances are effects produced in the sentient2 and1 therefore1 if they be different1 it is necessary1 by the preceding article1 that either some part of the agent1 which was formerly at rest1 is now moved1 and so the mutation consists in this motion2 or some part1 which was formerly moved1 is now otherwise moved1 and so also the mutation consists in this new motion2 or which1 being formerly moved1 is now at rest1 which1 as ,i have shewn above1 cannot come to pass without motion2 and so again1 mutation is motion2 or lastly1 it happens in some of these manners to the patient1 or some of its parts2 so that mutation1 howsoever it be made1 will consist in the motion of the parts1 either of the body which is perceived1 or of the sentient body1 or of both4 ,mutation therefore is motion1 namely1 of the parts either of the agent or of the patient2 which was to be demonstrated4 ,and to this it is consequent1 that rest cannot be the cause of anything1 nor can any action proceed from it2 seeing neither motion nor mutation can be caused by it4 ,sidenote3 ,contingent accidents4 #aj4 ,accidents1 in respect of other accidents which precede them1 or are before them in time1 and upon which they do not depend as upon their causes1 are called contingent accidents2 ,i say1 in respect of those accidents by which they are not generated2 for1 in respect of their causes1 all things come to pass with equal necessity2 for otherwise they would have no causes at all2 which1 of things generated1 is not intelligible4 ------- ,,chapter ,x4 ,,of ,,power ,,and ,,act4 #a4 ,power and cause are the same thing4--#b4 ,an act is produced at the same instant in which the power is plenary4--#c4 ,active and passive power are parts only of plenary power4--#d4 ,an act1 when said to be possible4--#e4 ,an act necessary and contingent1 what4--#f4 ,active power consists in motion4--#g4 ,cause1 formal and final1 what they are4 ,sidenote3 ,power and cause are the same thing4 #a4 ,correspondent to cause and effect1 are ,,power and ,,act2 nay1 those and these are the same things2 though1 for divers considerations1 they have divers names4 ,for whensoever any agent has all those accidents which are necessarily requisite for the production of some effect in the patient1 then we say that agent has power to produce that effect1 if it be applied to a patient4 ,but1 as ,i have shewn in the precedent chapter1 those accidents constitute the efficient cause2 and therefore the same accidents1 which constitute the efficient cause1 constitute also the power of the agent4 ,wherefore the power of the agent and the efficient cause are the same thing4 ,but they are considered with this difference1 that cause is so called in respect of the effect already produced1 and power in respect of the same effect to be produced hereafter2 so that cause respects the past1 power the future time4 ,also1 the power of the agent is that which is commonly called active power4 ,in like manner1 whensoever any patient has all those accidents which it is requisite it should have1 for the production of some effect in it1 we say it is in the power of that patient to produce that effect1 if it be applied to a fitting agent4 ,but those accidents1 as is defined in the precedent chapter1 constitute the material cause2 and therefore the power of the patient1 commonly called passive power1 and material cause1 are the same thing2 but with this different consideration1 that in cause the past time1 and in power the future1 is respected4 ,wherefore the power of the agent and patient together1 which may be called entire or plenary power1 is the same thing with entire cause2 for they both consist in the sum or aggregate of all the accidents1 as well in the agent as in the patient1 which are requisite for the production of the effect4 ,lastly1 as the accident produced is1 in respect of the cause1 called an effect1 so in respect of the power1 it is called an act4 ,sidenote3 ,an act is produced at the same instant in which the power is plenary4 #b4 ,as therefore the effect is produced in the same instant in which the cause is entire1 so also every act that may be produced1 is produced in the same instant in which the power is plenary4 ,and as there can be no effect but from a sufficient and necessary cause1 so also no act can be produced but by sufficient power1 or that power by which it could not but be produced4 ,sidenote3 ,active and passive power are parts only of plenary power4 #c4 ,and as it is manifest1 as ,i have shewn1 that the efficient and material causes are severally and by themselves parts only of an entire cause1 and cannot produce any effect but by being joined together1 so also power1 active and passive1 are parts only of plenary and entire power2 nor1 except they be joined1 can any act proceed from them2 and therefore these powers1 as ,i said in the first article1 are but conditional1 namely1 the agent has power1 if it be applied to a patient2 and the patient has power1 if it be applied to an agent2 otherwise neither of them have power1 nor can the accidents1 which are in them severally1 be properly called powers2 nor any action be said to be possible for the power of the agent alone or of the patient alone4 ,sidenote3 ,an act1 when said to be possible4 #d4 ,for that is an impossible act1 for the production of which there is no power plenary4 ,for seeing plenary power is that in which all things concur1 which are requisite for the production of an act1 if the power shall never be plenary1 there will always be wanting some of those things1 without which the act cannot be produced2 wherefore that act shall never be produced2 that is1 that act is ,,impossible3 and every act1 which is not impossible1 is ,,possible4 ,every act1 therefore1 which is possible1 shall at some time be produced2 for if it shall never be produced1 then those things shall never concur which are requisite for the production of it2 wherefore that act is impossible1 by the definition2 which is contrary to what was supposed4 ,sidenote3 ,an act necessary and contingent1 what4 #e4 ,a necessary act is that1 the production whereof it is impossible to hinder2 and therefore every act1 that shall be produced1 shall necessarily be produced2 for1 that it shall not be produced1 is impossible2 because1 as is already demonstrated1 every possible act shall at some time be produced2 nay1 this proposition1 what shall be1 shall be1 is as necessary a proposition as this1 a man is a man4 ,but here1 perhaps1 some man may ask whether those future things1 which are commonly called contingents1 are necessary4 ,i say1 therefore1 that generally all contingents have their necessary causes1 as is shewn in the preceding chapter2 but are called contingents in respect of other events1 upon which they do not depend2 as the rain1 which shall be tomorrow1 shall be necessary1 that is1 from necessary causes2 but we think and say it happens by chance1 because we do not yet perceive the causes thereof1 though they exist now2 for men commonly call that casual or contingent1 whereof they do not perceive the necessary cause2 and in the same manner they used to speak of things past1 when not knowing whether a thing be done or no1 they say it is possible it never was done4 ,wherefore1 all propositions concerning future things1 contingent or not contingent1 as this1 it will rain tomorrow1 or this1 tomorrow the sun will rise1 are either necessarily true1 or necessarily false2 but we call them contingent1 because we do not yet know whether they be true or false2 whereas their verity depends not upon our knowledge1 but upon the foregoing of their causes4 ,but there are some1 who though they confess this whole proposition1 tomorrow it will either rain1 or not rain1 to be true1 yet they will not acknowledge the parts of it1 as1 tomorrow it will rain1 or1 tomorrow it will not rain1 to be either of them true by itself2 because they say neither this nor that is true determinately4 ,but what is this determinately true1 but true upon our knowledge1 or evidently true8 ,and therefore they say no more but that it is not yet known whether it be true or no2 but they say it more obscurely1 and darken the evidence of the truth with the same words1 with which they endeavour to hide their own ignorance4 ,sidenote3 ,active power consists in motion4 #f4 ,in the #ith article of the preceding chapter1 ,i have shewn that the efficient cause of all motion and mutation consists in the motion of the agent1 or agents2 and in the first article of this chapter1 that the power of the agent is the same thing with the efficient cause4 ,from whence it may be understood1 that all active power consists in motion also2 and that power is not a certain accident1 which differs from all acts1 but is1 indeed1 an act1 namely1 motion1 which is therefore called power1 because another act shall be produced by it afterwards4 ,for example1 if of three bodies the first put forward the second1 and this the third1 the motion of the second1 in respect of the first which produceth it1 is the act of the second body2 but1 in respect of the third1 it is the active power of the same second body4 ,sidenote3 ,cause1 formal and final1 what they are4 #g4 ,the writers of metaphysics reckon up two other causes besides the efficient and material1 namely1 the ,,essence1 which some call the formal cause1 and the ,,end1 or final cause2 both which are nevertheless efficient causes4 ,for when it is said the essence of a thing is the cause thereof1 as to be rational is the cause of man1 it is not intelligible2 for it is all one1 as if it were said1 to be a man is the cause of man2 which is not well said4 ,and yet the knowledge of the essence of anything1 is the cause of the knowledge of the thing itself2 for1 if ,i first know that a thing is rational1 ,i know from thence1 that the same is man2 but this is no other than an efficient cause4 ,a final cause has no place but in such things as have sense and will2 and this also ,i shall prove hereafter to be an efficient cause4 ------- ,,chapter ,,xi4 ,,of ,,identity ,,and ,,difference4 #a4 ,what it is for one thing to differ from another4--#b4 ,to differ in number1 magnitude1 species1 and genus1 what4--#c4 ,what is relation1 proportion1 and relatives4--#d4 ,proportionals1 what4--#e4 ,the proportion of magnitudes to one another1 wherein it consists4--#f4 ,relation is no new accident1 but one of those that were in the relative before the relation or comparison was made4 ,also the causes of accidents in the correlatives1 are the cause of relation4--#g4 ,of the beginning of individuation4 ,sidenote3 ,what it is for one thing to differ from another4 #a4 ,hitherto ,i have spoken of body simply1 and accidents common to all bodies1 as magnitude1 motion1 rest1 action1 passion1 power1 possible1 c42 and ,i should now descend to those accidents by which one body is distinguished from another1 but that it is first to be declared what it is to be distinct and not distinct1 namely1 what are the ,,same and ,,different2 for this also is common to all bodies1 that they may be distinguished and differenced from one another4 ,now1 two bodies are said to differ from one another1 when something may be said of one of them1 which cannot be said of the other at the same time4 ,sidenote3 ,to differ in number1 magnitude1 species1 and genus1 what4 #b4 ,and1 first of all1 it is manifest that no two bodies are the same2 for seeing they are two1 they are in two places at the same time2 as that1 which is the same1 is at the same time in one and the same place4 ,all bodies therefore differ from one another in number1 namely1 as one and another2 so that the same and different in number1 are names opposed to one another by contradiction4 ,in magnitude bodies differ when one is greater than another1 as a cubit long1 and two cubits long1 of two pound weight1 and of three pound weight4 ,and to these1 equals are opposed4 ,bodies1 which differ more than in magnitude1 are called unlike2 and those1 which differ only in magnitude1 like4 ,also1 of unlike bodies1 some are said to differ in the species1 others in the genus2 in the species1 when their difference is perceived by one and the same sense1 as white and black2 and in the genus1 when their difference is not perceived but by divers senses1 as white and hot4 ,sidenote3 ,what is relation1 proportion1 and relatives4 #c4 ,and the likeness1 or unlikeness1 equality1 or inequality of one body to another1 is called their ,,relation2 and the bodies themselves relatives or correlatives2 ,aristotle calls them 2 the first whereof is usually named the antecedent1 and the second the consequent2 and the relation of the antecedent to the consequent1 according to magnitude1 namely1 the equality1 the excess or defect thereof1 is called the ,,proportion of the antecedent to the consequent2 so that proportion is nothing but the equality or inequality of the magnitude of the antecedent compared to the magnitude of the consequent by their difference only1 or compared also with their difference4 ,for example1 the proportion of three to two consists only in this1 that three exceeds two by unity2 and the proportion of two to five in this1 that two1 compared with five1 is deficient of it by three1 either simply1 or compared with the numbers different2 and therefore in the proportion of unequals1 the proportion of the less to the greater1 is called ,,defect2 and that of the greater to the less1 ,,excess4 ,sidenote3 ,proportionals1 what4 #d4 ,besides1 of unequals1 some are more1 some less1 and some equally unequal2 so that there is proportion of proportions1 as well as of magnitudes2 namely1 where two unequals have relation to two other unequals2 as1 when the inequality which is between #b and #c1 is compared with the inequality which is between #d and #e4 ,in which comparison there are always four magnitudes2 or1 which is all one1 if there be but three1 the middlemost is twice numbered2 and if the proportion of the first to the second1 be equal to the proportion of the third to the fourth1 then the four are said to be proportionals2 otherwise they are not proportionals4 ,sidenote3 ,the proportion of magnitudes to one another1 wherein it consists4 #e4 ,the proportion of the antecedent to the consequent consists in their difference1 not only simply taken1 but also as compared with one of the relatives2 that is1 either in that part of the greater1 by which it exceeds the less1 or in the remainder1 after the less is taken out of the greater2 as the proportion of two to five consists in the three by which five exceeds two1 not in three simply only1 but also as compared with five or two4 ,for though there be the same difference between two and five1 which is between nine and twelve1 namely three1 yet there is not the same inequality2 and therefore the proportion of two to five is not in all relation the same with that of nine to twelve1 but only in that which is called arithmetical4 ,sidenote3 ,relation is no new accident1 but one of those that were in the relative1 before the relation or comparison was made4 ,also the causes of accidents in correlatives are the cause of relation4 #f4 ,but we must not so think of relation1 as if it were an accident differing from all the other accidents of the relative2 but one of them1 namely1 that by which the comparison is made4 ,for example1 the likeness of one white to another white1 or its unlikeness to black1 is the same accident with its whiteness2 and equality and inequality1 the same accident with the magnitude of the thing compared1 though under another name3 for that which is called white or great1 when it is not compared with something else1 the same when it is compared1 is called like or unlike1 equal or unequal4 ,and from this it follows that the causes of the accidents1 which are in relatives1 are the causes also of likeness1 unlikeness1 equality and inequality2 namely1 that he1 that makes two unequal bodies1 makes also their inequality2 and he1 that makes a rule and an action1 makes also1 if the action be congruous to the rule1 their congruity2 if incongruous1 their incongruity4 ,and thus much concerning comparison of one body with another4 ,sidenote3 ,of the beginning of individuation4 #g4 ,but the same body may at different times be compared with itself4 ,and from hence springs a great controversy among philosophers about the beginning of individuation1 namely1 in what sense it may be conceived that a body is at one time the same1 at another time not the same it was formerly4 ,for example1 whether a man grown old be the same man he was whilst he was young1 or another man2 or whether a city be in different ages the same1 or another city4 ,some place individuity in the unity of matter2 others1 in the unity of form2 and one says it consists in the unity of the aggregate of all the accidents together4 ,for matter1 it is pleaded that a lump of wax1 whether it be spherical or cubical1 is the same wax1 because the same matter4 ,for form1 that when a man is grown from an infant to be an old man1 though his matter be changed1 yet he is still the same numerical man2 for that identity which cannot be attributed to the matter1 ought probably to be ascribed to the form4 ,for the aggregate of accidents1 no instance can be made2 but because1 when any new accident is generated1 a new name is commonly imposed on the thing1 therefore he1 that assigned this cause of individuity1 thought the thing itself also was become another thing4 ,according to the first opinion1 he that sins1 and he that is punished1 should not be the same man1 by reason of the perpetual flux and change of man's body2 nor should the city1 which makes laws in one age and abrogates them in another1 be the same city2 which were to confound all civil rights4 ,according to the second opinion1 two bodies existing both at once1 would be one and the same numerical body4 ,for if1 for example1 that ship of ,theseus1 concerning the difference whereof made by continual reparation in taking out the old planks and putting in new1 the sophisters of ,athens were wont to dispute1 were1 after all the planks were changed1 the same numerical ship it was at the beginning2 and if some man had kept the old planks as they were taken out1 and by putting them afterwards together in the same order1 had again made a ship of them1 this1 without doubt1 had also been the same numerical ship with that which was at the beginning2 and so there would have been two ships numerically the same1 which is absurd4 ,but1 according to the third opinion1 nothing would be the same it was2 so that a man standing would not be the same he was sitting2 nor the water1 which is in the vessel1 the same with that which is poured out of it4 ,wherefore the beginning of individuation is not always to be taken either from matter alone1 or from form alone4 ,but we must consider by what name anything is called1 when we inquire concerning the identity of it4 ,for it is one thing to ask concerning ,socrates1 whether he be the same man1 and another to ask whether he be the same body2 for his body1 when he is old1 cannot be the same it was when he was an infant1 by reason of the difference of magnitude2 for one body has always one and the same magnitude2 yet1 nevertheless1 he may be the same man4 ,and therefore1 whensoever the name1 by which it is asked whether a thing be the same it was1 is given it for the matter only1 then1 if the matter be the same1 the thing also is individually the same2 as the water1 which was in the sea1 is the same which is afterwards in the cloud2 and any body is the same1 whether the parts of it be put together1 or dispersed2 or whether it be congealed1 or dissolved4 ,also1 if the name be given for such form as is the beginning of motion1 then1 as long as that motion remains1 it will be the same individual thing2 as that man will be always the same1 whose actions and thoughts proceed all from the same beginning of motion1 namely1 that which was in his generation2 and that will be the same river which flows from one and the same fountain1 whether the same water1 or other water1 or something else than water1 flow from thence2 and that the same city1 whose acts proceed continually from the same institution1 whether the men be the same or no4 ,lastly1 if the name be given for some accident1 then the identity of the thing will depend upon the matter2 for1 by the taking away and supplying of matter1 the accidents that were1 are destroyed1 and other new ones are generated1 which cannot be the same numerically2 so that a ship1 which signifies matter so figured1 will be the same as long as the matter remains the same2 but if no part of the matter be the same1 then it is numerically another ship2 and if part of the matter remain and part be changed1 then the ship will be partly the same1 and partly not the same4 ------- ,,chapter ,,xii4 ,,of ,,quantity4 #a4 ,the definition of quantity4--#b4 ,the exposition of quantity1 what it is4--#c4 ,how line1 superficies1 and solid1 are exposed4--#d4 ,how time is exposed4--#e4 ,how number is exposed4--#f4 ,how velocity is exposed4--#g4 ,how weight is exposed4--#h4 ,how the proportion of magnitudes is exposed4--#i4 ,how the proportion of times and velocities is exposed4 ,sidenote3 ,definition of quantity4 #a4 ,what and how manifold dimension is1 has been said in the #hth chapter1 namely1 that there are three dimensions1 line or length1 superficies1 and solid2 every one of which1 if it be determined1 that is1 if the limits of it be made known1 is commonly called quantity2 for by quantity all men understand that which is signified by that word1 by which answer is made to the question1 ,how much is it8 ,whensoever1 therefore1 it is asked1 for example1 ,how long is the journey8 it is not answered indefinitely1 length2 nor1 when it is asked1 ,how big is the field8 is it answered indefinitely1 superficies2 nor1 if a man ask1 ,how great is the bulk8 indefinitely1 solid3 but it is answered determinately1 the journey is a hundred miles2 the field is a hundred acres2 the bulk is a hundred cubical feet2 or at least in some such manner1 that the magnitude of the thing enquired after may by certain limits be comprehended in the mind4 ,,quantity1 therefore1 cannot otherwise be defined1 than to be a dimension determined1 or a dimension1 whose limits are set out1 either by their place1 or by some comparison4 ,sidenote3 ,the exposition of quantity1 what it is4 #b4 ,and quantity is determined two ways2 one1 by the sense1 when some sensible object is set before it2 as when a line1 a superficies or solid1 of a foot or cubit1 marked out in some matter1 is objected to the eyes2 which way of determining1 is called exposition1 and the quantity so known is called exposed quantity2 the other by memory1 that is1 by comparison with some exposed quantity4 ,in the first manner1 when it is asked of what quantity a thing is1 it is answered1 of such quantity as you see exposed4 ,in the second manner1 answer cannot be made but by comparison with some exposed quantity2 for if it be asked1 how long is the way8 the answer is1 so many thousand paces2 that is1 by comparing the way with a pace1 or some other measure1 determined and known by exposition2 or the quantity of it is to some other quantity known by exposition1 as the diameter of a square is to the side of the same1 or by some other the like means4 ,but it is to be understood1 that the quantity exposed must be some standing or permanent thing1 such as is marked out in consistent or durable matter2 or at least something which is revocable to sense2 for otherwise no comparison can be made by it4 ,seeing1 therefore1 by what has been said in the next preceding chapter1 comparison of one magnitude with another is the same thing with proportion2 it is manifest1 that quantity determined in the second manner is nothing else but the proportion of a dimension not exposed to another which is exposed2 that is1 the comparison of the equality or inequality thereof with an exposed quantity4 ,sidenote3 ,how line1 superficies1 and solids1 are exposed4 #c4 ,lines1 superficies1 and solids1 are exposed1 first1 by motion1 in such manner as in the #hth chapter ,i have said they are generated2 but so as that the marks of such motion be permanent2 as when they are designed upon some matter1 as a line upon paper2 or graven in some durable matter4 ,secondly1 by apposition2 as when one line or length is applied to another line or length1 one breadth to another breadth1 and one thickness to another thickness2 which is as much as to describe a line by points1 a superficies by lines1 and a solid by superficies2 saving that by points in this place are to be understood very short lines2 and1 by superficies1 very thin solids4 ,thirdly1 lines and superficies may be exposed by section1 namely1 a line may be made by cutting an exposed superficies2 and a superficies1 by the cutting of an exposed solid4 ,sidenote3 ,how time is exposed4 #d4 ,time is exposed1 not only by the exposition of a line1 but also of some moveable thing1 which is moved uniformly upon that line1 or at least is supposed so to be moved4 ,for1 seeing time is an idea of motion1 in which we consider former and latter1 that is succession1 it is not sufficient for the exposition of time that a line be described2 but we must also have in our mind an imagination of some moveable thing passing over that line2 and the motion of it must be uniform1 that time may be divided and compounded as often as there shall be need4 ,and1 therefore1 when philosophers1 in their demonstrations1 draw a line1 and say1 ,let that line be time1 it is to be understood as if they said1 ,let the conception of uniform motion upon that line1 be time4 ,for though the circles in dials be lines1 yet they are not of themselves sufficient to note time by1 except also there be1 or be supposed to be1 a motion of the shadow or the hand4 ,sidenote3 ,how number is exposed4 #e4 ,number is exposed1 either by the exposition of points1 or of the names of number1 one1 two1 three1 c42 and those points must not be contiguous1 so as that they cannot be distinguished by notes1 but they must be so placed that they may be discerned one from another2 for1 from this it is1 that number is called discreet quantity1 whereas all quantity1 which is designed by motion1 is called continual quantity4 ,but that number may be exposed by the names of number1 it is necessary that they be recited by heart and in order1 as one1 two1 three1 c42 for by saying one1 one1 one1 and so forward1 we know not what number we are at beyond two or three2 which also appear to us in this manner1 not as number1 but as figure4 ,sidenote3 ,how velocity is exposed4 #f4 ,for the exposition of velocity1 which1 by the definition thereof1 is a motion which1 in a certain time1 passeth over a certain space1 it is requisite1 not only that time be exposed1 but that there be also exposed that space which is transmitted by the body1 whose velocity we would determine2 and that a body be understood to be moved in that space also2 so that there must be exposed two lines1 upon one of which uniform motion must be understood to be made1 that the time may be determined2 and1 upon the other1 the velocity is to be computed4 ,as if we would expose the velocity of the we would expose the velocity of the body ,a1 we draw two lines ,a ,b and ,c ,d1 and place a body in ,c also2 which done1 we say the velocity of the body ,a is so great1 that it passeth over the line ,a ,b in the same time in which the body ,c passeth over the line ,c ,d with uniform motion4 ,a ,b ------ ,c ,d ------ ,sidenote3 ,how weight is exposed4 #g4 ,weight is exposed by any heavy body1 of what matter soever1 so it be always alike heavy4 ,sidenote3 ,how the proportion of magnitudes is exposed4 #h4 ,the proportion of two magnitudes is then exposed1 when the magnitudes themselves are exposed1 namely1 the proportion of equality1 when the magnitudes are equal2 and of inequality1 when they are unequal4 ,for seeing1 by the #eth article of the preceding chapter1 the proportion of two unequal magnitudes consists in their difference1 compared with either of them2 and when two unequal magnitudes are exposed1 their difference is also exposed2 it follows1 that when magnitudes1 which have proportion to one another1 are exposed1 their proportion also is exposed with them2 and1 in like manner1 the proportion of equals1 which consists in this1 that there is no difference of magnitude betwixt them1 is exposed at the same time when the equal magnitudes themselves are exposed4 ,for example1 if the exposed lines ,a ,b and ,c ,d be equal1 the proportion of equality is exposed in them2 and if the exposed lines1 ,e ,f and ,e ,g be unequal1 the proportion which ,e ,f has to ,e ,g1 and that which ,e ,g has to ,e ,f are also exposed in them2 for not only the lines themselves1 but also their difference1 ,g ,f1 is exposed4 ,the proportion of unequals is quantity2 for the difference1 ,g ,f1 in which it consists1 is quantity4 ,but the proportion of equality is not quantity2 because1 between equals1 there is no difference2 nor is one equality greater than another1 as one inequality is greater than another inequality4 ,a ,b ------- ,c ,d ------- ,e ,f ,f ------- ,sidenote3 ,how the proportion of times and velocities is exposed4 ,the proportion of two times1 or of two uniform velocities1 is then exposed1 when two lines are exposed by which two bodies are understood to be moved uniformly2 and therefore the same two lines serve to exhibit both their own proportion1 and that of the times and velocities1 according as they are considered to be exposed for the magnitudes themselves1 or for the times or velocities4 ,for let the two lines ,a and ,b be exposed2 their proportion therefore 7by the last foregoing article7 is exposed2 and if they be considered as drawn with equal and uniform velocity1 then1 seeing their times are greater1 or equal1 or less1 according as the same spaces are transmitted in greater1 or equal1 or less time1 the lines ,a and ,b will exhibit the equality or inequality1 that is1 the proportion of the times4 ,to conclude1 if the same lines1 ,a and ,b1 be considered as drawn in the same time1 then1 seeing their velocities are greater1 or equal1 or less1 according as they pass over in the same time longer1 or equal1 or shorter lines1 the same lines1 ,a and ,b1 will exhibit the equality1 or inequality1 that is1 the proportion of their velocities4 ,a --- ,b ------- ,,chapter ,,xiii4 ,,of ,,analogism1 ,,or ,,the ,,same ,,proportion4 #a1 #b1 #c1 #d4 ,the nature and definition of proportion1 arithmetical and geometrical4--#e4 ,the definition1 and some properties of the same arithmetical proportion4--#f1 #g4 ,the definition and transmutations of analogism1 or the same geometrical proportion4--#h1 #i44 ,the definitions of hyperlogism and hypologism1 that is1 of greater and less proportion1 and their transmutations4--#aj1 #aa1 #ab4 ,comparison of analogical quantities1 according to magnitude4--#ac1 #ad1 #ae4 ,composition of proportions4--#af1 #ag1 #ah1 #ai1 #bj1 #ba1 #bb1 #bc1 #bd1 #be4 ,the definition and properties of continual proportion4--#bf1 #bg1 #bh1 #bi4 ,comparison of arithmetical and geometrical proportions4 ,note1 that in this chapter the sign signifies that the quantities betwixt which it is put1 are added together2 and this sign - the remainder after the latter quantity is taken out of the former4 ,so that ,a ,b is equal to both ,a and ,b together2 and where you see ,a - ,b1 there ,a is the whole1 ,b the part taken out of it1 and ,a - ,b the remainder4 ,also1 two letters1 set together without any sign1 signify1 unless they belong to a figure1 that one of the quantities is multiplied by the other2 as ,a ,b signifies the product of ,a multiplied by ,b4 ,sidenote3 ,the nature and definition of proportion1 arithmetical geometrical4 #a4 ,great and little are not intelligible1 but by comparison4 ,now that1 to which they are compared1 is something exposed2 that is1 some magnitude either perceived by sense1 or so defined by words1 that it may be comprehended by the mind4 ,also that1 to which any magnitude is compared1 is either greater or less1 or equal to it4 ,and therefore proportion 7which1 as ,i have shewn1 is the estimation or comprehension of magnitudes by comparison17 is threefold1 namely1 proportion of equality1 that is1 of equal to equal2 or of excess1 which is of the greater to the less2 or of defect1 which is the proportion of the less to the greater4 ,again1 every one of these proportions is two-fold2 for if it be asked concerning any magnitude given1 how great it is1 the answer may be made by comparing it two ways2 first1 by saying it is greater or less than another magnitude1 by so much2 as seven is less than ten1 by three unities2 and this is called arithmetical proportion4 ,secondly1 by saying it is greater or less than another magnitude1 by such a part or parts thereof2 as seven is less than ten1 by three tenth parts of the same ten4 ,and though this proportion be not always explicable by number1 yet it is a determinate proportion1 and of a different kind from the former1 and called geometrical proportion1 and most commonly proportion simply4 #b4 ,proportion1 whether it be arithmetical or geometrical1 cannot be exposed but in two magnitudes1 7of which the former is commonly called the antecedent1 and the latter the consequent of the proportion7 as ,i have shewn in the #hth article of the preceding chapter4 ,and1 therefore1 if two proportions be to be compared1 there must be four magnitudes exposed1 namely1 two antecedents and two consequents2 for though it happen sometimes that the consequent of the former proportion be the same with the antecedent of the latter1 yet in that double comparison it must of necessity be twice numbered2 so that there will be always four terms4 #c4 ,of two proportions1 whether they be arithmetical or geometrical1 when the magnitudes compared in both 7which ,euclid1 in the fifth definition of his sixth book1 calls the quantities of proportions17 are equal1 then one of the proportions cannot be either greater or less than the other2 for one equality is neither greater nor less than another equality4 ,but of two proportions of inequality1 whether they be proportions of excess or of defect1 one of them may be either greater or less than the other1 or they may both be equal2 for though there be propounded two magnitudes that are unequal to one another1 yet there may be other two more1 unequal1 and other two equally unequal1 and other two less unequal than the two which were propounded4 ,and from hence it may be understood1 that the proportions of excess and defect are quantity1 being capable of more and less2 but the proportion of equality is not quantity1 because not capable neither of more1 nor of less4 ,and therefore proportions of inequality may be added together1 or subtracted from one another1 or be multiplied or divided by one another1 or by number2 but proportions of equality not so4 #d4 ,two equal proportions are commonly called the same proportion2 and1 it is said1 that the proportion of the first antecedent to the first consequent is the same with that of the second antecedent to the second consequent4 ,and when four magnitudes are thus to one another in geometrical proportion1 they are called proportionals2 and by some1 more briefly1 analogism4 ,and greater proportion is the proportion of a greater antecedent to the same consequent1 or of the same antecedent to a less consequent2 and when the proportion of the first antecedent to the first consequent is greater than that of the second antecedent to the second consequent1 the four magnitudes1 which are so to one another1 may be called hyperlogism4 ,less proportion is the proportion of a less antecedent to the same consequent1 or of the same antecedent to a greater consequent2 and when the proportion of the first antecedent to the first consequent is less than that of the second to the second1 the four magnitudes may be called hypologism4 ,sidenote3 ,the definition and some properties of the same arithmetical proportion4 #e4 ,one arithmetical proportion is the same with another arithmetical proportion1 when one of the antecedents exceeds its consequent1 or is exceeded by it1 as much as the other antecedent exceeds its consequent1 or is exceeded by it4 ,and therefore1 in four magnitudes1 arithmetically proportional1 the sum of the extremes is equal to the sum of the means4 ,for if ,a4 ,b 33 ,c4 ,d be arithmetically proportional1 and the difference on both sides be the same excess1 or the same defect1 ,e1 then ,b ,c 7if ,a be greater than ,b7 will be equal to ,a - ,e ,c2 and ,a ,d will be equal to ,a ,c - ,e2 but ,a - ,e ,c and ,a ,c - ,e are equal4 ,or if ,a be less than ,b1 then ,b ,c will be equal to ,a ,e ,c2 and ,a ,d will be equal to ,a ,c ,e2 but ,a ,e ,c and ,a ,c ,e are equal4 ,also1 if there be never so many magnitudes1 arithmetically proportional1 the sum of them all will be equal to the product of half the number of the terms multiplied by the sum of the extremes4 ,for if ,a4 ,b 33 ,c4 ,d 33 ,e4 ,f be arithmetically proportional1 the couples ,a ,f1 ,b ,e1 ,c ,d will be equal to one another2 and their sum will be equal to ,a ,f1 multiplied by the number of their combinations1 that is1 by half the number of the terms4 ,if1 of four unequal magnitudes1 any two1 together taken1 be equal to the other two together taken1 then the greatest and the least of them will be in the same combination4 ,let the unequal magnitudes be ,a1 ,b1 ,c1 ,d2 and let ,a ,b be equal to ,c ,d2 and let ,a be the greatest of them all2 ,i say ,b will be the least4 ,for1 if it may be1 let any of the rest1 as ,d1 be the least4 ,seeing therefore ,a is greater than ,c1 and ,b than ,d1 ,a,b will be greater than ,c ,d2 which is contrary to what was supposed4 ,if there be any four magnitudes1 the sum of the greatest and least1 the sum of the means1 the difference of the two greatest1 and the difference of the two least1 will be arithmetically proportional4 ,for1 let there be four magnitudes1 whereof ,a is the greatest1 ,d the least1 and ,b and ,c the means2 ,i say ,a ,d4 ,b ,c 33 ,a - ,b4 ,c - ,d are arithmetically proportional4 ,for the difference between the first antecedent and its consequent is this1 ,a ,d - ,b - ,c2 and the difference between the second antecedent and its consequent this1 ,a - ,b - ,c ,d2 but these two differences are equal2 and therefore1 by this #eth article1 ,a ,d4 ,b ,c 33 ,a - ,b4 ,c - ,d are arithmetically proportional4 ,if1 of four magnitudes1 two be equal to the other two1 they will be in reciprocal arithmetical proportion4 ,for let ,a ,b be equal to ,c ,d1 ,i say ,a4 ,c 33 ,d4 ,b are arithmetically proportional4 ,for if they be not1 let ,a4 ,c 33 ,d4 ,e 7supposing ,e to be greater or less than ,b7 be arithmetically proportional1 and then ,a ,e will be equal to ,c ,d2 wherefore ,a ,b and ,c ,d are not equal2 which is contrary to what was supposed4 ,sidenote3 ,the definition and transmutations of analogism1 or the same geometrical proportion4 #f4 ,one geometrical proportion is the same with another geometrical proportion2 when the same cause1 producing equal effects in equal times1 determines both the proportions4 ,if a point uniformly moved describe two lines1 either with the same1 or different velocity1 all the parts of them which are contemporary1 that is1 which are described in the same time1 will be two to two1 in geometrical proportion1 whether the antecedents be taken in the same line1 or not4 ,for1 from the point ,a 7in the #ajth figure at the end of the #adth chapter7 let the two lines1 ,a ,d1 ,a ,g1 be described with uniform motion2 and let there be taken in them two parts ,a ,b1 ,a ,e1 and again1 two other parts1 ,a ,c1 ,a ,f2 in such manner1 that ,a ,b1 ,a ,e1 be contemporary1 and likewise ,a ,c1 ,a ,f contemporary4 ,i say first 7taking the antecedents ,a ,b1 ,a ,c in the line ,a ,d1 and the conquents ,a ,e1 ,a ,f in the line ,a ,g7 that ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f are proportionals4 ,for seeing 7by the #hth chap1 and the #aeth art47 velocity is motion considered as determined by a certain length or line1 in a certain time transmitted by it1 the quantity of the line ,a ,b will be determined by the velocity and time by which the same ,a ,b is described2 and for the same reason1 the quantity of the line ,a ,c will be determined by the velocity and time1 by which the same ,a ,c is described2 and therefore the proportion of ,a ,b to ,a ,c1 whether it be proportion of equality1 or of excess or defect1 is determined by the velocities and times by which ,a ,b1 ,a ,c are described2 but seeing the motion of the point ,a upon ,a ,b and ,a ,c is uniform1 they are both described with equal velocity2 and therefore whether one of them have to the other the proportion of majority or of minority1 the sole cause of that proportion is the difference of their times2 and by the same reason it is evident1 that the proportion of ,a ,e to ,a ,f is determined by the difference of their times only4 ,seeing therefore ,a ,b1 ,a ,e1 as also ,a ,c1 ,a ,f are contemporary1 the difference of the times in which ,a ,b and ,a ,c are described1 is the same with that in which ,a ,e and ,a ,f are described4 ,wherefore the proportion of ,a ,b to ,a ,c1 and the proportion of ,a ,e to ,a ,f are both determined by the same cause4 ,but the cause1 which so determines the proportion of both1 works equally in equal times1 for it is uniform motion2 and therefore 7by the last precedent definition7 the proportion of ,a ,b to ,a ,c is the same with that of ,a ,e to ,a ,f2 and consequently ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f are proportionals2 which is the first4 ,secondly1 7taking the antecedents in different lines7 ,i say1 ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f are proportionals2 for seeing ,a ,b1 ,a ,e are described in the same time1 the difference of the velocities in which they are described is the sole cause of the proportion they have to one another4 ,and the same may be said of the proportion of ,a ,c to ,a ,f4 ,but seeing both the lines ,a ,d and ,a ,g are passed over by uniform motion1 the difference of the velocities in which ,a ,b1 ,a ,e are described1 will be the same with the difference of the velocities1 in which ,a ,c1 ,a ,f are described4 ,wherefore the cause which determines the proportion of ,a ,b to ,a ,e1 is the same with that which determines the proportion of ,a ,c to ,a ,f2 and therefore ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f1 are proportionals2 which remained to be proved4 ,coroll4 ,i4 ,if four magnitudes be in geometrical proportion1 they will also be proportionals by permutation1 that is1 by transposing the middle terms4 ,for ,i have shown1 that not only ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f1 but also that1 by permutation1 ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f are proportionals4 ,coroll4 ,,ii4 ,if there be four proportionals1 they will also be proportionals by inversion or conversion1 that is1 by turning the antecedents into consequents4 ,for if in the last analogism1 ,i had for ,a ,b1 ,a ,c1 put by inversion ,a ,c1 ,a ,b1 and in like manner converted ,a ,e1 ,a ,f into ,a ,f1 ,a ,e1 yet the same demonstration had served4 ,for as well ,a ,c1 ,a ,b1 as ,a ,b1 ,a ,c are of equal velocity2 and ,a ,c1 ,a ,f1 as well as ,a ,f1 ,a ,c are contemporary4 ,coroll4 ,,iii4 ,if proportionals be added to proportionals1 or taken from them1 the aggregates1 or remainders1 will be proportionals4 ,for contemporaries1 whether they be added to contemporaries1 or taken from them1 make the aggregates or remainders contemporary1 though the addition or subtraction be of all the terms1 or of the antecedents alone1 or of the consequents alone4 ,coroll4 ,,iv4 ,if both the antecedents of four proportionals1 or both the consequents1 or all the terms1 be multiplied or divided by the same number or quantity1 the products or quotients will be proportionals4 ,for the multiplication and division of proportionals1 is the same with the addition and subtraction of them4 ,coroll4 ,v4 ,if there be four proportionals1 they will also be proportionals by composition1 that is1 by compounding an antecedent of the antecedent and consequent put together1 and by taking for consequent either the consequent singly1 or the antecedent singly4 ,for this composition is nothing but addition of proportionals1 namely1 of consequents to their own antecedents1 which by supposition are proportionals4 ,coroll4 ,,vi4 ,in like manner1 if the antecedent singly1 or consequent singly1 be put for antecedent1 and the consequent be made of both put together1 these also will be proportionals4 ,for it is the inversion of proportion by composition4 ,coroll4 ,,vii4 ,if there be four proportionals1 they will also be proportionals by division1 that is1 by taking the remainder after the consequent is subtracted from the antecedent1 or the difference between the antecedent and consequent for antecedent1 and either the whole or the subtracted for consequent2 as if ,a4 ,b 33 ,c4 ,d be proportionals1 they will by division be ,a - ,b4 ,b 33 ,c - ,d4 ,d1 and ,a - ,b4 ,a 33 ,c - ,d4 ,c2 and when the consequent is greater than the antecedent1 ,b - ,a4 ,a 33 ,d - ,c4 ,c1 and ,b - ,a4 ,b 33 ,d - ,c4 ,d4 ,for in all these divisions1 proportionals are1 by the very supposition of the analogism ,a4 ,b 33 ,c4 ,d1 taken from ,a and ,b1 and from ,c and ,d4 ,coroll4 ,,viii4 ,if there be four proportionals1 they will also be proportionals by the conversion of proportion1 that is1 by inverting the divided proportion1 or by taking the whole for antecedent1 and the difference or remainder for consequent4 ,as1 if ,a4 ,b 33 ,c4 ,d be proportionals1 then ,a4 ,a - ,b 33 ,c4 ,c - ,d1 as also ,b4 ,a - ,b 33 ,d4 ,c - ,d will be proportionals4 ,for seeing these inverted be proportionals1 they are also themselves proportionals4 ,coroll4 ,,ix4 ,if there be two analogisms which have their quantities equal1 the second to the second1 and the fourth to the fourth1 then either the sum or difference of the first quantities will be to the second1 as the sum or difference of the third quantities is to the fourth4 ,let ,a4 ,b 33 ,c4 ,d and ,e4 ,b 33 ,f4 ,d be analogisms2 ,i say ,a ,e4 ,b 33 ,c ,f4 ,d are proportionals4 ,for the said analogisms will by permutation be ,a4 ,c 33 ,b4 ,d1 and ,e4 ,f 33 ,b4 ,d2 and therefore ,a4 ,c 33 ,e4 ,f will be proportionals1 for they have both the proportion of ,b to ,d common4 ,wherefore1 if in the permutation of the first analogism1 there be added ,e and ,f to ,a and ,c1 which ,e and ,f are proportional to ,a and ,c1 then 7by the third coroll47 ,a ,e4 ,b 33 ,c ,f4 ,d will be proportionals2 which was to be proved4 ,also in the same manner it may be shown1 that ,a - ,e4 ,b 33 ,c - ,f4 ,d are proportionals4 #g4 ,if there be two analogisms1 where four antecedents make an analogism1 their consequents also shall make an analogism2 as also the sums of their antecedents will be proportional to the sums of their consequents4 ,for if ,a4 ,b 33 ,c4 ,d and ,e4 ,f 33 ,g4 ,h be two analogisms1 and ,a4 ,e 33 ,c4 ,g be proportionals1 then by permutation ,a4 ,c 33 ,e4 ,g1 and ,e4 ,g 33 ,f4 ,h1 and ,a4 ,c 33 ,b4 ,d will be proportionals2 wherefore ,b4 ,d 33 ,e4 ,g1 that is1 ,b4 ,d 33 ,f4 ,h1 and by permutation ,b4 ,f 33 ,d4 ,h are proportionals2 which is the first4 ,secondly1 ,i say ,a ,e4 ,b ,f 33 ,c ,g4 ,d ,h will be proportionals4 ,for seeing ,a4 ,e 33 ,c4 ,g are proportionals1 ,a ,e4 ,e 33 ,c ,g4 ,g will also by composition be proportionals1 and by permutation ,a ,e4 ,c ,g 33 ,e4 ,g will be proportionals2 wherefore1 also ,a ,e4 ,c ,g 33 ,b ,f4 ,d ,h will be proportionals4 ,again1 seeing1 as is shown above1 ,b4 ,f 33 ,d4 ,h are proportionals1 ,b ,f4 ,f 33 ,d ,h4 ,h will also by composition be proportionals2 and by permutation ,b ,f4 ,d ,h 33 ,f4 ,h will also be proportionals2 wherefore ,a ,e4 ,c ,g 33 ,b ,f4 ,d ,h are proportionals2 which remained to be proved4 ,coroll4 ,by the same reason1 if there be never so many analogisms1 and the antecedents be proportional to the antecedents1 it may be demonstrated also that the consequents will be proportional to the consequents1 as also the sum of the antecedents to the sum of the consequents4 ,sidenote3 ,the definition of hyperlogism and hypologism1 that is1 of greater and less proportion1 and their transmutations4 #h4 ,in an hyperlogism1 that is1 where the proportion of the first antecedent to its consequent is greater than the proportion of the second antecedent to its consequent1 the permutation of the proportionals1 and the addition of proportionals to proportionals1 and substraction of them from one another1 as also their composition and division1 and their multiplication and division by the same number1 produce always an hyperlogism4 ,for suppose ,a4 ,b 33 ,c 4 ,d and ,a4 ,c 33 ,e4 ,f be analogisms1 ,a ,e4 ,b 33 ,c ,f 4 ,d will also be an analogism2 but ,a ,e4 ,b 33 ,c4 ,d will be an hyperlogism2 wherefore by permutation1 ,a ,e4 ,c 33 ,b4 ,d is an hyperlogism1 because ,a4 ,b 33 ,c4 ,d is an analogism4 ,secondly1 if to the hyperlogism ,a ,e4 ,b 33 ,c4 ,d the proportionals ,g and ,h be added1 ,a ,e ,g4 ,b 33 ,c ,h4 ,d will be an hyperlogism1 by reason ,a ,e ,g4 ,b 33 ,c ,f ,h4 ,d is an analogism4 ,also1 if ,g and ,h be taken away1 ,a ,e - ,g4 ,b 33 ,c - ,h4 ,d will be an hyperlogism2 for ,a ,e - ,g4 ,b 33 ,c ,f - ,h4 ,d is an analogism4 ,thirdly1 by composition ,a ,e ,b4 ,b 33 ,c ,d4 ,d will be an hyperlogism1 because ,a ,e ,b4 ,b 33 ,c ,f ,d4 ,d is ah analogism1 and so it will be in all the varieties of composition4 ,fourthly1 by division1 ,a ,e - ,b4 ,b 33 ,c - ,d4 ,d will by an hyperlogism1 by reason ,a ,e - ,b4 ,b 33 ,c ,f - ,d4 ,d is an analogism4 ,also ,a ,e - ,b4 ,a ,e 33 ,c - ,d4 ,c is an hyperlogism2 for ,a ,e - ,b4 ,a ,e 33 ,c ,f - ,d4 ,c is an analogism4 ,fifthly1 by multiplication #d ,a #d ,e4 ,b 33 #d ,c4 ,d is an hyperlogism1 because #d ,a4 ,b 33 #d ,c4 ,d is an analogism2 and by division ,a ,e4 ,b 33 ,c4,d is an hyperlogism1 because ,a4 ,b 33 ,c4 ,d is an analogism4 #i4 ,but if ,a ,e4 ,b 33 ,c4 ,d be an hyperlogism1 then by inversion ,b4 ,a ,e 33 ,d4 ,c will be an hypologism1 because ,b4 ,a 33 ,d4 ,c being an analogism1 the first consequent will be too great4 ,also1 by conversion of proportion1 ,a ,e4 ,a ,e - ,b 33 ,c4 ,c - ,d is an hypologism1 because the inversion of it1 namely ,a ,e - ,b4 ,a ,e 33 ,c - ,d4 ,c is an hyperlogism1 as ,i have shown but now4 ,so also ,b4 ,a ,e - ,b 33 ,d4 ,c - ,d is an hypologism1 because1 as ,i have newly shown1 the inversion of it1 namely ,a ,e - ,b4 ,b 33 ,c - ,d4 ,d is an hyperlogism4 ,note that this hypologism ,a ,e4 ,a ,e - ,b 33 ,c4 ,c - ,d is commonly thus expressed2 if the proportion of the whole1 7,a ,e7 to that which is taken out of it 7,b71 be greater than the proportion of the whole 7,c7 to that which is taken out of it 7,d71 then the proportion of the whole 7,a ,e7 to the remainder 7,a ,e - ,b7 will be less than the proportion of the whole 7,c7 to the remainder 7,c - ,d74 ,sidenote3 ,comparison of analogical quantities1 according to magnitude4 #aj4 ,if there be four proportionals1 the difference of the two first1 to the difference of the two last1 will be as the first antecedent is to the second antecedent1 or as the first consequent to the second consequent4 ,for if ,a4 ,b 33 ,c4 ,d be proportionals1 then by division ,a - ,b4 ,b 33 ,c - ,d4 ,d will be proportionals2 and by permutation ,a - ,b4 ,c - ,d 33 ,b4 ,d2 that is1 the differences are proportional to the consequents1 and therefore they are so also to the antecedents4 #aa4 ,of four proportionals1 if the first be greater than the second1 the third also shall be greater than the fourth4 ,for seeing the first is greater than the second1 the proportion of the first to the second is the proportion of excess2 but the proportion of the third to the fourth is the same with that of the first to the second2 and therefore also the proportion of the third to the fourth is the proportion of excess2 wherefore the third is greater than the fourth4 ,in the same manner it may be proved1 that whensoever the first is less than the second1 the third also is less than the fourth2 and when those are equal1 that these also are equal4 #ab4 ,if there be four proportionals whatsoever1 ,a4 ,b 33 ,c4,d1 and the first and third be multiplied by any one number1 as by #b2 and again the second and fourth be multiplied by any one number1 as by #c2 and the product of the first #b ,a1 be greater than the product of the second #c ,b2 the product also of the third #b ,c1 will be greater than the product of the fourth #c ,d4 ,but if the product of the first be less than the product of the second1 then the product of the third will be less than that of the fourth4 ,and lastly1 if the products of the first and second be equal1 the products of the third and fourth shall also be equal4 ,now this theorem is all one with ,euclid's definition of the same proportion2 and it may be demonstrated thus4 ,seeing ,a4 ,b 33 ,c4 ,d are proportionals1 by permutation also 7art4 #f1 coroll4 ,i47 ,a4 ,c 33 ,b 4 ,d will be proportionals2 wherefore 7by coroll4 ,,iv4 art4 #f7 #b ,a4 #b ,c 33 #c ,b4 #c ,d will be proportionals2 and again1 by permutation1 #b ,a4 #c ,b 33 #b ,c4 #c ,d will be proportionals2 and therefore1 by the last article1 if #b ,a be greater than #c ,b1 then #b ,c will be greater than #c ,d2 if less1 less2 and if equal1 equal2 which was to be demonstrated4 ,sidenote3 ,composition of proportions4 ,if any three magnitudes be propounded1 or three things whatsoever that have any proportion one to another1 as three numbers1 three times1 three degrees1 c42 the proportions of the first to the second1 and of the second to the third1 together taken1 are equal to the proportion of the first to the third4 ,let there be three lines1 for any proportion may be reduced to the proportion of lines1 ,a ,b1 ,a ,c1 ,a ,d2 and in the first place1 let the proportion as well of the first ,a ,b to the second ,a ,c1 as of the second ,a ,c to the third ,,ad1 be the proportion of defect1 or of less to greater2 ,i say the proportions together taken of ,a ,b to ,a ,c1 and of ,a ,c to ,a ,d1 are equal to the proportion of ,a ,b to ,a ,d4 ,suppose the point ,a to be moved over the whole line ,a ,d with uniform motion2 then the proportions as well of ,a ,b to ,a ,c1 as of ,a ,c to ,a ,d1 are determined by the difference of the times in which they are described2 that is1 ,a ,b has to ,a ,c such proportion as is determined by the different times of their description2 and ,a ,c to ,,ad such proportion as is determined by their times4 ,but the proportion of ,a ,b to ,a ,d is such as is determined by the difference of the times in which ,a ,b and ,a ,d are described2 and the difference of the times in which ,a ,b and ,a ,c are described1 together with the difference of the times in which ,a ,c and ,a ,d are described1 is the same with the difference of the times in which ,a ,b and ,a ,d are described4 ,and therefore1 the same cause which determines the two proportions of ,a ,b to ,a ,c and of ,a ,c to ,a ,d1 determines also the proportion of ,a ,b to ,a ,d4 ,wherefore1 by the definition of the same proportion1 delivered above in the #fth article1 the proportion of ,a ,b to ,a ,c together with the proportion of ,a ,c to ,a ,d1 is the same with the proportion of ,a ,b to ,a ,d4 ,a ,b ,c ,d ------------- ,in the second place1 let ,a ,d be the first1 ,a ,c the second1 and ,a ,b the third1 and let their proportion be the proportion of excess1 or the greater to less2 then1 as before1 the proportions of ,a ,d to ,a ,c1 and of ,a ,c to ,a ,b1 and of ,a ,d to ,a ,b1 will be determined by the difference of their times2 which in the description of ,a ,d and ,a ,c1 and of ,a ,c and ,a ,b together taken1 is the same with the difference of the times in the description of ,a ,d and ,a ,b4 ,wherefore the proportion of ,a ,d to ,a ,b is equal to the two proportions of ,a ,d to ,a ,c and of ,a ,c to ,a ,b4 ,in the last place4 ,if one of the proportions1 namely of ,a ,d to ,a ,b1 be the proportion of excess1 and another of them1 as of ,a ,b to ,a ,c be the proportion of defect1 thus also the proportion of ,a ,d to ,a ,c will be equal to the two proportions together taken of ,a ,d to ,a ,b1 and of ,a ,b to ,a ,c4 ,for the difference of the times in which ,,ad and ,,ab are described1 is excess of time2 for there goes more time to the description of ,a ,d than of ,a ,b2 and the difference of the times in which ,a ,b and ,a ,c are described1 is defect of time1 for less time goes to the description of ,a ,b than of ,a ,c2 but this excess and defect being added together1 make ,d ,b - ,b ,c1 which is equal to ,d ,c1 by which the first ,a ,d exceeds the third ,a ,c2 and therefore the proportions of the first ,a ,d to the second ,a ,b1 and of the second ,a ,b to the third ,a ,c1 are determined by the same cause which determines the proportion of the first ,a ,d to the third ,a ,c4 ,wherefore1 if any three magnitudes1 c4 ,coroll4 ,i4 ,if there be never so many magnitudes having proportion to one another1 the proportion of the first to the last is compounded of the proportions of the first to the second1 of the second to the third1 and so on till you come to the last2 or1 the proportion of the first to the last is the same with the sum of all the intermediate proportions4 ,for any number of magnitudes having proportion to one another1 as ,a1 ,b1 ,c1 ,d1 ,e being propounded1 the proportion of ,a to ,e1 as is newly shown1 is compounded of the proportions of ,a to ,d and of ,d to ,e2 and again1 the proportion of ,a to ,d1 of the proportions of ,a to ,c1 and of ,c to ,d2 and lastly1 the proportion of ,a to ,c1 of the proportions of ,a to ,b1 and of ,b to ,c4 ,coroll4 ,,ii4 ,from hence it may be understood how any two proportions may be compounded4 ,for if the proportions of ,a to ,b1 and of ,c to ,d1 be propounded to be added together1 let ,b have to something else1 as to ,e1 the same proportion which ,c has to ,d1 and let them be set in this order1 ,a1 ,b1 ,e2 for so the proportion of ,a to ,e will evidently be the sum of the two proportions of ,a to ,b1 and of ,b to ,e1 that is1 of ,c to ,d4 ,or let it be as ,d to ,c1 so ,a to something else1 as to ,e1 and let them be ordered thus1 ,e1 ,a1 ,b2 for the proportion of ,e to ,b will be compounded of the proportions of ,e to ,a1 that is1 of ,c to ,d1 and of ,a to ,b4 ,also1 it may be understood how one proportion may be taken out of another4 ,for if the proportion of ,c to ,d be to be subtracted out of the proportion of ,a to ,b1 let it be as ,c to ,d1 so ,a to something else1 as ,e1 and setting them in this order1 ,a1 ,e1 ,b1 and taking away the proportion of ,a to ,e1 that is1 of ,c to ,d1 there will remain the proportion of ,e to ,b4 ,coroll4 ,,iii4 ,if there be two orders of magnitudes which have proportion to one another1 and the several proportions of the first order be the same and equal in number with the proportions of the second order2 then1 whether the proportions in both orders be successively answerable to one another1 which is called ordinate proportion1 or not successively answerable1 which is called perturbed proportion1 the first and the last in both will be proportionals4 ,for the proportion of the first to the last is equal to all the intermediate proportions2 which being in both orders the same1 and equal in number1 the aggregates of those proportions will also be equal to one another2 but to their aggregates1 the proportions of the first to the last are equal2 and therefore the proportion of the first to the last in one order1 is the same with the proportion of the first to the last in the other order4 ,wherefore the first and the last in both are proportionals4 ,sidenote3 ,composition of proportions4 #ad4 ,if any two quantities be made of the mutual multiplication of many quantities1 which have proportion to one another1 and the efficient quantities on both sides be equal in number1 the proportion of the products will be compounded of the several proportions1 which the efficient quantities have to one another4 ,a ,b4 ,a4 ,c ,b4 ,c4 ,c ,d4 ,e4 ,first1 let the two products be ,a ,b and ,c ,d1 whereof one is made of the multiplication of ,a into ,b1 and the other of the multiplication of ,c into ,d4 ,i say the proportion of ,a ,b to ,c ,d is compounded of the proportions of the efficient ,a to the efficient ,c1 and of the efficient ,b to the efficient ,d4 ,for let ,a ,b1 ,c ,b and ,c ,d be set in order2 and as ,b is to ,d1 so let ,c be to another quantity as ,e2 and let ,a1 ,c1 ,e be set also in order4 ,then 7by coroll4 ,,iv4 of the #fth art47 it will be as ,a ,b the first quantity to ,c ,b the second quantity in the first order1 so ,a to ,c in the second order2 and again1 as ,c ,b to ,c ,d in the first order1 so ,b to ,d1 that is1 by construction1 so ,c to ,e in the second order2 and therefore 7by the last corollary7 ,a ,b4 ,c ,d 33 ,a4 ,e will be proportionals4 ,but the proportion of ,a to ,e is compounded of the proportions of ,a to ,c1 and of ,b to ,d2 wherefore also the proportion of ,a ,b to ,c ,d is compounded of the same4 ,secondly1 let the two products be ,a ,b ,f1 and ,c ,d ,g1 each of them made of three efficients1 the first of ,a1 ,b and ,f1 and the second of ,c1 ,d and ,g2 ,i say1 the proportion of ,a ,b ,f to ,c ,d ,g is compounded of the proportions of ,a to ,c1 of ,b to ,d1 and of ,f to ,g4 ,for let them be set in order as before2 and as ,b is to ,d1 so let ,c be to another quantity ,e2 and again1 as ,f is to ,g1 so let ,e be to another1 ,h2 and let the first order stand thus1 ,,abf1 ,,cbf1 ,,cdf and ,,cdg2 and the second order thus1 ,a1 ,c1 ,e1 ,h4 ,then the proportion of ,a ,b ,f to ,c ,b ,f in the first order1 will be as ,a to ,c in the second2 and the proportion of ,,cbf to ,,cdf in the first order1 as ,b to ,d1 that is1 as ,c to ,e 7by construction7 in the second order2 and the proportion of ,,cdf to ,,cdg in the first1 as ,f to ,g1 that is1 as ,e to ,h 7by construction7 in the second order2 and therefore ,a ,b ,f4 ,c ,d ,g33 ,a4 ,h will be proportionals4 ,but the proportion of ,a to ,h is compounded of the proportions of ,a to ,c1 ,b to ,d1 and ,f to ,g4 ,wherefore the proportion of the product ,a ,b ,f to ,c ,d ,g is also compounded of the same4 ,and this operation serves1 how many soever the efficients be that make the quantities given4 ,,abf4 ,a4 ,,cbf4 ,c4 ,,cdf4 ,e4 ,,cdg4 ,h4 ,from hence ariseth another way of compounding many proportions into one1 namely1 that which is supposed in the #eth definition of the #fth book of ,euclid2 which is1 by multiplying all the antecedents of the proportions into one another1 and in like manner all the consequents into one another4 ,and from hence also it is evident1 in the first place1 that the cause why parallelograms1 which are made by the duction of two straight lines into one another1 and all solids which are equal to figures so made1 have their proportions compounded of the proportions of the efficients2 and in the second place1 why the multiplication of two or more fractions into one another is the same thing with the composition of the proportions of their several numerators to their several denominators4 ,for example1 if these fractions #ab1 #bc1 #cd be to be multiplied into one another1 the numerators #a1 #b1 #c1 are first to be multiplied into one another1 which make #f2 and next the denominators #b1 #c1 #d1 which make #bd2 and these two products make the fraction #fbd4 ,in like manner1 if the proportions of #a to #b1 of #b to #c1 and of #c to #d1 be to be compounded1 by working as ,i have shown above1 the same proportion of #f to #bd will be produced4 #ae4 ,if any proportion be compounded with itself inverted1 the compound will be the proportion of equality4 ,for let any proportion be given1 as of ,a to ,b1 and let the inverse of it be that of ,c to ,d2 and as ,c to ,d1 so let ,b be to another quantity2 for thus they will be compounded 7by the second coroll4 of the #abth art47 ,now seeing the proportion of ,c to ,d is the inverse of the proportion of ,a to ,b1 it will be as ,c to ,d1 so ,b to ,a2 and therefore if they be placed in order1 ,a1 ,b1 ,a1 the proportion compounded of the proportions of ,a to ,b1 and of ,c to ,d1 will be the proportion of ,a to ,a1 that is1 the proportion of equality4 ,and from hence the cause is evident why two equal products have their efficients reciprocally proportional4 ,for1 for the making of two products equal1 the proportions of their efficients must be such1 as being compounded may make the proportion of equality1 which cannot be except one be the inverse of the other2 for if betwixt ,a and ,a any other quantity1 as ,c1 be interposed1 their order will be ,a1 ,c1 ,a1 and the later proportion of ,c to ,a will be the inverse of the former proportion of ,a to ,c4 ,sidenote3 ,the definition and properties of continual proportion4 #af4 ,a proportion is said to be multiplied by a number1 when it is so often taken as there be unities in that number2 and if the proportion be of the greater to the less1 then shall also the quantity of the proportion be increased by the multiplication2 but when the proportion is of the less to the greater1 then as the number increaseth1 the quantity of the proportion diminisheth2 as in these three numbers1 #d1 #b1 #a1 the proportion of #d to #a is not only the duplicate of #d to #b1 but also twice as great2 but inverting the order of those numbers thus1 #a1 #b1 #d1 the proportion of #a to #b is greater than that of #a to #d2 and therefore though the proportion of #a to #d be the duplicate of #a to #b1 yet it is not twice so great as that of #a to #b1 but contrarily the half of it4 ,in like manner1 a proportion is said to be divided1 when between two quantities are interposed one or more means in continual proportion1 and then the proportion of the first to the second is said to be subduplicate of that of the first to the third1 and subtriplicate of that of the first to the fourth1 c4 ,this mixture of proportions1 where some are proportions of excess1 others of defect1 as in a merchant's account of debtor and creditor1 is not so easily reckoned as some think2 but maketh the composition of proportions sometimes to be addition1 sometimes substraction2 which soundeth absurdly to such as have always by composition understood addition1 and by diminution substraction4 ,therefore to make this account a little clearer1 we are to consider 7that which is commonly assumed1 and truly7 that if there be never so many quantities1 the proportion of the first to the last is compounded of the proportions of the first to the second1 and of the second to the third1 and so on to the last1 without regarding their equality1 excess1 or defect2 so that if two proportions1 one of inequality1 the other of equality1 be added together1 the proportion is not thereby made greater nor less2 as for example1 if the proportions of ,a to ,b and of ,b to ,b be compounded1 the proportion of the first to the second is as much as the sum of both1 because proportion of equality1 being not quantity1 neither augmenteth quantity nor lesseneth it4 ,but if there be three quantities1 ,a1 ,b1 ,c1 unequal1 and the first be the greatest1 the last least1 then the proportion of ,b to ,c is an addition to that of ,a to ,b1 and makes it greater2 and on the contrary1 if ,a be the least1 and ,c the greatest quantity1 then doth the addition of the proportion of ,b to ,c make the compounded proportion of ,a to ,c less than the proportion of ,a to ,b1 that is1 the whole less than the part4 ,the composition therefore of proportions is not in this case the augmentation of them1 but the diminution2 for the same quantity 7,euclid ,v4 #h7 compared with two other quantities1 hath a greater proportion to the lesser of them than to the greater4 ,likewise1 when the proportions compounded are one of excess1 the other of defect1 if the first be of excess1 as in these numbers1 #h1 #f1 #i1 the proportion compounded1 namely1 of #h to #i1 is less than the proportion of one of the parts of it1 namely1 of #h to #f2 but if the proportion of the first to the second be of defect1 and that of the second to the third be of excess1 as in these numbers1 #f1 #h1 #d1 then shall the proportion of the first to the third be greater than that of the first to the second1 as #f hath a greater proportion to #d than to #h2 the reason whereof is manifestly this1 that the less any quantity is deficient of another1 or the more one exceedeth another1 the proportion of it to that other is the greater4 ,suppose now three quantities in continual proportion1 ,a ,b #d1 ,a ,c #f1 ,a ,d #i4 ,because therefore ,a ,d is greater than ,a ,c1 but not greater than ,a ,d1 the proportion of ,a ,d to ,,ac will be 7by ,euclid1 ,v4 #h7 greater than that of ,a ,d to ,a ,d2 and likewise1 because the proportions of ,a ,d to ,a ,c1 and of ,a ,c to ,a ,b are the same1 the proportions of ,a ,d to ,a ,c and of ,a ,c to ,a ,b1 being both proportions of excess1 make the whole proportion of ,a ,d to ,a ,b1 or of #i to #d1 not only the duplicate of ,a ,d to ,a ,c1 that is1 of #i to #f1 but also the double1 or twice so great4 ,on the other side1 because the proportion of ,a ,d to ,a ,d1 or #i to #i1 being proportion of equality1 is no quantity1 and yet greater than that of ,a ,c to ,a ,d1 or #f to #i1 it will be as #j - #i to #j - #f1 so ,a ,c to ,a ,d1 and again1 as #j - #i to #j - #f1 so #j - #f to #j - #d2 but #j - #d1 #j - #f1 #j - #i are in continual proportion2 and because #j - #d is greater than #j - #f1 the proportion of #j - #d to #j - #f will be double to the proportion of #j - #d to #j - #i1 double ,i say1 and yet not duplicate1 but subduplicate4 ,if any be unsatisfied with this ratiocination1 let him first consider that 7by ,euclid ,v4 #h7 the proportion of ,a ,b to ,a ,c is greater than that of ,a ,b to ,a ,d1 wheresoever ,d be placed in the line ,a ,c prolonged2 and the further off the point ,d is from ,c1 so much the greater is the proportion of ,a ,b to ,a ,c than that of ,a ,b to ,a ,d4 ,there is therefore some point 7which suppose be ,e7 in such distance from ,c1 as that the proportion of ,a ,b to ,a ,c will be twice as great as that of ,a ,b to ,a ,e4 ,that considered1 let him determine the length of the line ,a ,e1 and demonstrate1 if he can1 that ,a ,e is greater or less than ,a ,d4 ,c ,a----------,e ,by the same method1 if there be more quantities than three1 as ,a1 ,b1 ,c1 ,d1 in continual proportion1 and ,a be the least1 it may be made appear that the proportion of ,a to ,b is triple magnitude1 though subtriple in multitude1 to the proportion of ,a to ,d4 #ag4 ,if there be never so many quantities1 the number whereof is odd1 and their order such1 that from the middlemost quantity both ways they proceed in continual proportion1 the proportion of the two which are next on either side to the middlemost is subduplicate to the proportion of the two which are next to these on both sides1 and subtriplicate of the proportion of the two which are yet one place more remote1 c4 ,for let the magnitudes be ,c1 ,b1 ,a1 ,d1 ,e1 and let ,a1 ,b1 ,c1 as also ,a1 ,d1 ,e be in continual proportion2 ,i say the proportion of ,d to ,b is subduplicate of the proportion of ,e to ,c4 ,for the proportion of ,d to ,b is compounded of the proportions of ,d to ,a1 and of ,a to ,b once taken2 but the proportion of ,e to ,c is compounded of the same twice taken2 and therefore the proportion of ,d to ,b is subduplicate of the proportion of ,e to ,c4 ,and in the same manner1 if there were three terms on either side1 it might be demonstrated that the proportion of ,d to ,b would be subtriplicate of that of the extremes1 c4 #ah4 ,if there be never so many continual proportionals1 as the first1 second1 third1 c4 their differences will be proportional to them4 ,for the second1 third1 c4 are severally consequents of the preceding1 and antecedents of the following proportion4 ,but 7by art4 #aj7 the difference of the first antecedent and consequent1 to difference of the second antecedent and consequent1 is as the first antecedent to the second antecedent1 that is1 as the first term to the second1 or as the second to the third1 c4 in continual proportionals4 #ai4 ,if there be three continual proportionals1 the sum of the extremes1 together with the mean twice taken1 the sum of the mean and either of the extremes1 and the same extreme1 are continual proportionals4 ,for let ,a4 ,b4 ,c be continual proportionals4 ,seeing1 therefore1 ,a4 ,b 33 ,b4 ,c are proportionals1 by composition also ,a ,b4 ,b 33 ,b ,c4 ,c will be proportionals2 and by permutation ,a ,b4 ,b ,c 33 ,b4 ,c will also be proportionals2 and again1 by composition ,a #b;,b ,c4 ,b ,c 33 ,b ,c4 ,c2 which was to be proved4 #bj4 ,in four continual proportionals1 the greatest and the least put together is a greater quantity than the other two put together4 ,let ,a4 ,b 33 ,c4 ,d be continual proportionals2 whereof let the greatest be ,a1 and the least be ,d2 ,i say ,a ,d is greater than ,b ,c4 ,for by art4 #aj1 ,a - ,b4 ,c - ,d 33 ,a4 ,c are proportionals2 and therefore ,a - ,b is1 by art4 #aa1 greater than ,c - ,d4 ,add ,b on both sides1 and ,a will be greater than ,c ,b - ,d4 ,and again1 add ,d on both sides1 and ,a ,d will be greater than ,b ,c2 which was to be proved4 ,sidenote3 ,the definition and properties of continual proportion4 #ba4 ,if there be four proportionals1 the extremes multiplied into one another1 and the means multiplied into one another1 will make equal products4 ,let ,a4 ,b 33 ,c4 ,d be proportionals2 ,i say ,a ,d is equal to ,b ,c4 ,for the proportion of ,a ,d to ,b ,c is compounded1 by art4 #ac1 of the proportions of ,a to ,b1 and ,d to ,c1 that is1 its inverse ,b to ,a2 and therefore1 by art4 #ad1 this compounded proportion is the proportion of equality2 and therefore also1 the proportion of ,a ,d to ,b ,c is the proportion of equality4 ,wherefore they are equal4 #bb4 ,if there be four quantities1 and the proportion of the first to the second be duplicate of the proportion of the third to the fourth1 the product of the extremes to the product of the means1 will be as the third to the fourth4 ,let the four quantities be ,a1 ,b1 ,c and ,d2 and let the proportion of ,a to ,b be duplicate of the proportion of ,c to ,d1 ,i say ,a ,d1 that is1 the product of ,a into ,d is to ,b ,c1 that is1 to the product of the means1 as ,c to ,d4 ,for seeing the proportion of ,a to ,b is duplicate of the proportion of ,c to ,d1 if it be as ,c to ,d1 so ,d to another1 ,e1 then ,a4 ,b 33 ,c4 ,e will be proportionals2 for the proportion of ,a to ,b is by supposition duplicate of the proportion of ,c to ,d2 and ,c to ,e duplicate also of that of ,c to ,d by the definition1 art4 #ae4 ,wherefore1 by the last article1 ,a ,e or ,a into ,e is equal to ,b ,c or ,b into ,c2 but1 by coroll4 ,,iv4 art4 #f1 ,a ,d is to ,a ,e as ,d to ,e1 that is1 as ,c to ,d2 and therefore ,a ,d is to ,b ,c1 which as ,i have shown is equal to ,a ,e1 as ,c to ,d2 which was to be proved4 ,moreover1 if the proportion of the first ,a to the second ,b be triplicate of the proportion of the third ,c to the fourth ,d1 the product of the extremes to the product of the means will be duplicate of the proportion of the third to the fourth4 ,for if it be as ,c to ,d so ,d to ,e1 and again1 as ,d to ,e so ,e to another1 ,f1 then the proportion of ,c to ,f will be triplicate of the proportion of ,c to ,d2 and consequently1 ,a4 ,b 33 ,c4 ,f will be proportionals1 and ,a ,f equal to ,b ,c4 ,but as ,a ,d to ,a ,f1 so is ,d to ,f2 and therefore1 also1 as ,a ,d to ,b ,c1 so ,d to ,f1 that is1 so ,c to ,e2 but the proportion of ,c to ,e is duplicate of the proportion of ,c to ,d2 wherefore1 also1 the proportion of ,a ,d to ,b ,c is duplicate of that of ,c to ,d1 as was propounded4 #bc4 ,if there be four proportionals1 and a mean be interposed betwixt the first and second1 and another betwixt the third and fourth1 the first of these means will be to the second1 as the first of the proportionals is to the third1 or as the second of them is to the fourth4 ,for let ,a4 ,b 33 ,c4 ,d be proportionals1 and let ,e be a mean betwixt ,a and ,b1 and ,f a mean betwixt ,c and ,d2 ,i say ,a4 ,c 33 ,e4 ,f are proportionals4 ,for the proportion of ,a to ,e is subduplicate of the proportion of ,a to ,b1 or of ,c to ,d4 ,also1 the proportion of ,c to ,f is subduplicate of that of ,c to ,d2 and therefore ,a4 ,e 33 ,c4 ,f are proportionals2 and by permutation ,a4 ,c 33 ,e4 ,f are also proportionals2 which was to be proved4 #bd4 ,any thing is said to be divided into extreme and mean proportion1 when the whole and the parts are in continual proportion4 ,as for example1 when ,a ,b4 ,a4 ,b are continual proportionals2 or when the straight line ,a ,c is so divided in ,b1 that ,a ,c4 ,a ,b4 ,b ,c are in continual proportion4 ,and if the same line ,a ,c be again divided in ,d1 so as that ,a ,c4 ,c ,d4 ,a ,d be continual proportionals2 then also ,a ,c4 ,a ,b4 ,a ,d will be continual proportionals2 and in like manner1 though in contrary order1 ,c ,a4 ,c ,d4 ,c ,b will be continual proportionals2 which cannot happen in any line otherwise divided4 ,a ,b ,c ------------ ,d #be4 ,if there be three continual proportionals1 and again1 three other continual proportions1 which have the same middle term1 their extremes will be in reciprocal proportion4 ,for let ,a4 ,b4 ,c and ,d4 ,b4 ,e be continual proportionals1 ,i say ,a4 ,d 33 ,e4 ,c shall be proportionals4 ,for the proportion of ,a to ,d is compounded of the proportions of ,a to ,b1 and of ,b to ,d2 and the proportion of ,e to ,c is compounded of those of ,e to ,b1 that is1 of ,b to ,d1 and of ,b to ,c1 that is1 of ,a to ,b4 ,wherefore1 by equality1 ,a4 ,d 33 ,e4 ,c are proportionals4 ,sidenote3 ,comparison of arithmetical and geometrical proportion4 ,if any two unequal quantities be made extremes1 and there be interposed betwixt them any number of means in geometrical proportion1 and the same number of means in arithmetical proportion1 the several means in geometrical proportion will be less than the several means in arithmetical proportion4 ,for betwixt ,a the lesser1 and ,e the greater extreme1 let there be interposed three means1 ,b1 ,c1 ,d1 in geometrical proportion1 and as many more1 ,f1 ,g1 ,h1 in arithmetical proportion2 ,i say ,b will be less than ,f1 ,c than ,g1 and ,d than ,h4 ,for first1 the difference between ,a and ,f is the same with that between ,f and ,g1 and with that between ,g and ,h1 by the definition of arithmetical proportion2 and therefore1 the difference of the proportionals which stand next to one another1 to the difference of the extremes1 is1 when there is but one mean1 half their difference2 when two1 a third part of it2 when three1 a quarter1 c42 so that in this example it is a quarter4 ,but the difference between ,d and ,e1 by art4 #ag1 is more than a quarter of the difference between the extremes1 because the proportion is geometrical1 and therefore the difference between ,a and ,d is less than three quarters of the same difference of the extremes4 ,in like manner1 if the difference between ,a and ,d be understood to be divided into three equal parts1 it may be proved1 that the difference between ,a and ,c is less than two quarters of the difference of the extremes ,a and ,e4 ,and lastly1 if the difference between ,a and ,c be divided into two equal parts1 that the difference between ,a and ,b is less than a quarter of the difference of the extremes ,a and ,e4 ,a ,a - - ,b ,f --- --- ,c ,g ----- ----- ,d ,h ------- ------- ,e ,e --------- --------- ,from the consideration hereof1 it is manifest1 that ,b1 that is ,a together with something else which is less than a fourth part of the difference of the extremes ,a and ,e1 is less than ,f1 that is1 than the same ,a with something else which is equal to the said fourth part4 ,also1 that ,c1 that is ,a with something else which is less than two fourth parts of the said difference1 is less than ,g1 that is1 than ,a together with the said two-fourths4 ,and lastly1 that ,d1 which exceeds ,a by less than three-fourths of the said difference1 is less than ,h1 which exceeds the same ,a by three entire fourths of the said difference4 ,and in the same manner it would be if there were four means1 saving that instead of fourths of the difference of the extremes we are to take fifth parts2 and so on4 #bg4 ,,lemma4 ,if a quantity being given1 first one quantity be both added to it and subtracted from it1 and then another greater or less1 the proportion of the remainder to the aggregate1 is greater where the less quantity is added and substracted1 than where the greater quantity is added and substracted4 ,let ,b be added to and substracted from the quantity ,a2 so that ,a - ,b be the remainder1 and ,a ,b the aggregate2 and again1 let ,c1 a greater quantity than ,b1 be added to and substracted from the same ,a1 so that ,a - ,c be the remainder and ,a ,c the aggregate2 ,i say ,a - ,b4 ,a ,b 33 ,a - ,c4 ,a ,c will be an hyperlogism4 ,for ,a - ,b4 ,a 33 ,a - ,c4 ,a is an hyperlogism of a greater antecedent to the same consequent2 and therefore ,a - ,b4 ,a ,b 33 ,a - ,c4 ,a ,c is a much greater hyperlogism1 being made of a greater antecedent to a less consequent4 #bh4 ,if unequal parts be taken from two equal quantities1 and betwixt the whole and the part of each there be interposed two means1 one in geometrical1 the other in arithmetical proportion2 the difference betwixt the two means will be greatest1 where the difference betwixt the whole and its part is greatest4 ,for let ,a ,b and ,a ,b be two equal quantities1 from which let two unequal parts be taken1 namely1 ,a ,e the less1 and ,a ,f the greater2 and betwixt ,a ,b and ,a ,e let ,a ,g be a mean in geometrical proportion1 and ,a ,h a mean in arithmetical proportion4 ,also betwixt ,a ,b and ,a ,f let ,a ,i be a mean in geometrical proportion1 and ,a ,k a mean in arithmetical proportion2 ,i say ,h ,g is greater than ,k ,i4 ,a ,e ,g ,h ,b ------------- ------------- ,a ,f ,i ,k ,b ,for in the first place we have this ,a ,b4 ,a ,g 33 ,b ,g4 ,g ,e1 by analogism article #ah4 ,then by composition we have this ,a ,b ,a ,g4 ,a ,b 33 ,b ,g ,g ,e that is1 ,b ,e4 ,b ,g4 ,and by taking the halves of the ,a ,b ,a ,g4 ,a ,b 33 ,b ,g ,g ,e1 antecedents this third that is1 ,b ,h4 ,b ,g4 ,and by conversion a fourth ,a ,b4 ,a ,b ,a ,g 33 ,b ,g4 ,b ,h4 ,and by division this fifth ,a ,b - ,a ,g4 ,a ,b ,a ,g 33 ,h ,g4 ,b ,h4 ,and by doubling the first antecedent and the first consequent ,a ,b - ,a ,g4 ,a ,b ,a ,g 33 ,h ,g4 ,b ,h4 ,also by the same method may be found out this analogism ,a ,b - ,a ,i4 ,a ,b ,,ai 33 ,k ,i4 ,b ,k4 ,now seeing the proportion of ,a ,b to ,a ,e is greater than that of ,a ,b to ,a ,f1 the proportion of ,a ,b to ,a ,g1 which is half the greater proportion1 is greater than the proportion of ,a ,b to ,a ,i the half of the less proportion2 and therefore ,a ,i is greater than ,a ,g4 ,wherefore the proportion of ,a ,b - ,a ,g to ,a ,b ,a ,g1 by the precedent lemma1 will be greater than the proportion of ,a ,b - ,a ,i to ,a ,b ,a ,i2 and therefore also the proportion of ,h ,g to ,b ,h will be greater than that of ,k ,i to ,b ,k1 and much greater than the proportion of ,k ,i to ,b ,h1 which is greater than ,b ,k2 for ,b ,h is the half of ,b ,e1 as ,b ,k is the half of ,b ,f1 which1 by supposition1 is less than ,b ,e4 ,wherefore ,h ,g is greater than ,k ,i2 which was to be proved4 ,coroll4 ,it is manifest from hence1 that if any quantity be supposed to be divided into equal parts infinite in number1 the difference between the arithmetical and geometrical means will be infinitely little1 that is1 none at all4 ,and upon this foundation1 chiefly1 the art of making those numbers1 which are called ,logarithms1 seems to have been built4 #bi4 ,if any number of quantities be propounded1 whether they be unequal1 or equal to one another2 and there be another quantity1 which multiplied by the number of the propounded quantities1 is equal to them all2 that other quantity is a mean in arithmetical proportion to all those propounded quantities4 ------- ,,chap4 ,,xiv4 ,,of ,,strait ,,and ,,crooked1 ,,angle ,,and ,,figure4 #a4 ,the definition and properties of a strait line4--#b4 ,the definition and properties of a plane superficies4--#c4 ,several sorts of crooked lines4--#d4 ,the definition and properties of a circular line4--#e4 ,the properties of a strait line taken in a plane4--#f44 ,the definition of tangent lines4--#g4 ,the definition of an angle1 and the kinds thereof4--#h4 ,in concentric circles1 arches of the same angle are to one another1 as the whole circumferences are4--#i4 ,the quantity of an angle1 in what it consists4 --#aj4 ,the distinction of angles1 simply so called4--#aa4 ,of strait lines from the centre of a circle to a tangent of the same4 --#ab4 ,the general definition of parallels1 and the properties of strait parallels4--#ac4 ,the circumferences of circles are to one another1 as their diameters are4--#ad4 ,in triangles1 strait lines parallel to the bases are to one another1 as the parts of the sides which they cut off from the vertex4--#ae4 ,by what fraction of a strait line the circumference of a circle is made4--#af4 ,that an angle of contingence is quantity1 but of a different kind from that of an angle simply so called2 and that it can neither add nor take away any thing from the same4--#ag4 ,that the inclination of planes is angle simply so called4--#ah4 ,a solid angle what it is4--#ai4 ,what is the nature of asymptotes4--#bj4 ,situation1 by what it is determined4--#ba4 ,what is like situation2 what is figure2 and what are like figures4 ,sidenote3 ,the definition end properties of a strait line4 #a4 ,between two points given1 the shortest line is that1 whose extreme points cannot be drawn farther asunder without altering the quantity1 that is1 without altering the proportion of that line to any other line given4 ,for the magnitude of a line is computed by the greatest distance which may be between its extreme points2 so that any one line1 whether it be extended or bowed1 has always one and the same length1 because it can have but one greatest distance between its extreme points4 ,and seeing the action1 by which a strait line is made crooked1 or contrarily a crooked line is made strait1 is nothing but the bringing of its extreme points nearer to one another1 or the setting of them further asunder1 a crooked line may rightly be defined to be that1 whose extreme points may be understood to be drawn further asunder2 and a strait line to be that1 whose extreme points cannot be drawn further asunder2 and comparatively1 a more crooked1 to be that line whose extreme points are nearer to one another than those of the other1 supposing both the lines to be of equal length4 ,now1 howsoever a line be bowed1 it makes always a sinus or cavity1 sometimes on one side1 sometimes on another2 so that the same crooked line may either have its whole cavity on one side only1 or it may have it part on one side and part on the other side4 ,which being well understood1 it will be easy to understand the following comparisons of strait and crooked lines4 ,first1 if a strait and a crooked line have their extreme points common1 the crooked line is longer than the strait line4 ,for if the extreme points of the crooked line be drawn out to their greatest distance1 it will be made a strait line1 of which that1 which was a strait line from the beginning1 will be but a part2 and therefore the strait line was shorter than the crooked line1 which had the same extreme points4 ,and for the same reason1 if two crooked lines have their extreme points common1 and both of them have all their cavity on one and the same side1 the outermost of the two will be the longest line4 ,secondly1 a strait line and a perpetually crooked line cannot be coincident1 no1 not in the least part4 ,for if they should1 then not only some strait line would have its extreme points common with some crooked line1 but also they would1 by reason of their coincidence1 be equal to one another2 which1 as ,i have newly shown1 cannot be4 ,thirdly1 between two points given1 there can be understood but one strait line2 because there cannot be more than one least interval or length between the same points4 ,for if there may be two1 they will either be coincident1 and so both of them will be one strait line2 or if they be not coincident1 then the application of one to the other by extension will make the extended line have its extreme points at greater distance than the other2 and consequently1 it was crooked from the beginning4 ,fourthly1 from this last it follows1 that two strait lines cannot include a superficies4 ,for if they have both their extreme points common1 they are coincident2 and if they have but one or neither of them common1 then at one or both ends the extreme points will be disjoined1 and include no superficies1 but leave all open and undetermined4 ,fifthly1 every part of a strait line is a strait line4 ,for seeing every part of a strait line is the least that can be drawn between its own extreme points1 if all the parts should not constitute a strait line1 they would altogether be longer than the whole line4 ,sidenote3 ,the definition and properties of a plane superficies4 #b4 ,a plane or a plane superficies1 is that which is described by a strait line so moved1 that all the several points thereof describe several strait lines4 ,a strait line1 therefore1 is necessarily all of it in the same plane which it describes4 ,also the strait lines1 which are made by the points that describe a plane1 are all of them in the same plane4 ,moreover1 if any line whatsoever be moved in a plane1 the lines1 which are described by it1 are all of them in the same plane4 ,all other superficies1 which are not plane1 are crooked1 that is1 are either concave or convex4 ,and the same comparisons1 which were made of strait and crooked lines1 may also be made of plane and crooked superficies4 ,for1 first1 if a plane and crooked superficies be terminated with the same lines1 the crooked superficies is greater than the plane superficies4 ,for if the lines1 of which the crooked superficies consists1 be extended1 they will be found to be longer than those of which the plane superficies consists1 which cannot be extended1 because they are strait4 ,secondly1 two superficies1 whereof the one is plane1 and the other continually crooked1 cannot be coincident1 no1 not in the least part4 ,for if they were coincident1 they would be equal2 nay1 the same superficies would be both plane and crooked1 which is impossible4 ,thirdly1 within the same terminating lines there can be no more than one plane superficies2 because there can be but one least superficies within the same4 ,fourthly1 no number of plane superficies can include a solid1 unless more than two of them end in a common vertex4 ,for if two planes have both the same terminating lines1 they are coincident1 that is1 they are but one superficies2 and if their terminating lines be not the same1 they leave one or more sides open4 ,fifthly1 every part of a plane superficies is a plane superficies4 ,for seeing the whole plane superficies is the least of all those1 that have the same terminating lines2 and also every part of the same superficies is the least of all those1 that are terminated with the same lines2 if every part should not constitute a plane superficies1 all the parts put together would not be equal to the whole4 ,sidenote3 ,several sorts of crooked lines4 #c4 ,of straitness1 whether it be in lines or in superficies1 there is but one kind2 but of crookedness there are many kinds2 for of crooked magnitudes1 some are congruous1 that is1 are coincident when they are applied to one other2 others are incongruous4 ,again1 some are or uniform1 that is1 have their parts1 howsoever taken1 congruous to one another2 others are or of several forms4 ,moreover1 of such as are crooked1 some are continually crooked1 others have parts which are not crooked4 ,sidenote3 ,definition and properties of a circular line4 #d4 ,if a strait line be moved in a plane1 in such manner1 that while one end of it stands still1 the whole line be carried round about till it come again into the same place from whence it was first moved1 it will describe a plane superficies1 which will be terminated every way by that crooked line1 which is made by that end of the strait line which was carried round4 ,now this superficies is called a ,,circle2 and of this circle1 the unmoved point is the centre2 the crooked line which terminates it1 the perimeter2 and every part of that crooked line1 a circumference or arch2 the strait line1 which generated the circle1 is the semidiameter or radius2 and any strait line1 which passeth through the centre and is terminated on both sides in the circumference1 is called the diameter4 ,moreover1 every point of the radius1 which describes the circle1 describes in the same time its own perimeter1 terminating its own circle1 which is said to be concentric to all the other circles1 because this and all those have one common centre4 ,wherefore in every circle1 all strait lines from the centre to the circumference are equal4 ,for they are all coincident with the radius which generates the circle4 ,also the diameter divides both the perimeter and the circle itself into two equal parts4 ,for if those two parts be applied to one another1 and the semiperimeters be coincident1 then1 seeing they have one common diameter1 they will be equal2 and the semicircles will be equal also2 for these also will be coincident4 ,but if the semiperimeters be not coincident1 then some one strait line1 which passes through the centre1 which centre is in the diameter1 will be cut by them in two points4 ,wherefore1 seeing all the strait lines from the centre to the circumference are equal1 a part of the same strait line will be equal to the whole2 which is impossible4 ,for the same reason the perimeter of a circle will be uniform1 that is1 any one part of it will be coincident with any other equal part of the same4 ,sidenote3 ,the properties of a strait line taken in a plane4 #e4 ,from hence may be collected this property of a strait line1 namely1 that it is all contained in that plane which contains both its extreme points4 ,for seeing both its extreme points are in the plane1 that strait line1 which describes the plane1 will pass through them both2 and if one of them be made a centre1 and at the distance between both a circumference be described1 whose radius is the strait line which describes the plane1 that circumference will pass through the other point4 ,wherefore between the two propounded points1 there is one strait line1 by the definition of a circle1 contained wholly in the propounded plane2 and therefore if another strait line might be drawn between the same points1 and yet not be contained in the same plane1 it would follow1 that between two points two strait lines may be drawn2 which has been demonstrated to be impossible4 ,it may also be collected1 that if two planes cut one another1 their common section will be a strait line4 ,for the two extreme points of the intersection are in both the intersecting planes2 and between those points a strait line may be drawn2 but a strait line between any two points is in the same plane1 in which the points are2 and seeing these are in both the planes1 the strait line which connects them will also be in both the same planes1 and therefore it is the common section of both4 ,and every other line1 that can be drawn between those points1 will be either coincident with that line1 that is1 it will be the same line2 or it will not be coincident1 and then it will be in neither1 or but in one of those planes4 ,as a strait line may be understood to be moved round about whilst one end thereof remains fixed1 as the centre2 so in like manner it is easy to understand1 that a plane may be circumduced about a strait line1 whilst the strait line remains still in one and the same place1 as the axis of that motion4 ,now from hence it is manifest1 that any three points are in some one plane4 ,for as any two points1 if they be connected by a strait line1 are understood to be in the same plane in which the strait line is2 so1 if that plane be circumduced about the same strait line1 it will in its revolution take in any third point1 howsoever it be situate2 and then the three points will be all in that plane2 and consequently the three strait lines which connect those points1 will also be in the same plane4 ,sidenote3 ,definition of tangent lines4 #f4 ,two lines are said to touch one another1 which being both drawn to one and the same point1 will not cut one another1 though they be produced1 produced1 ,i say1 in the same manner in which they were generated4 ,and therefore if two strait lines touch one another in any one point1 they will be contiguous through their whole length4 ,also two lines continually crooked will do the same1 if they be congruous and be applied to one another according to their congruity2 otherwise1 if they be incongruously applied1 they will1 as all other crooked lines1 touch one another1 where they touch1 but in one point only4 ,which is manifest from this1 that there can be no congruity between a strait line and a line that is continually crooked2 for otherwise the same line might be both strait and crooked4 ,besides1 when a strait line touches a crooked line1 if the strait line be never so little moved about upon the point of contact1 it will cut the crooked line2 for seeing it touches it but in one point1 if it incline any way1 it will do more than touch it2 that is1 it will either be congruous to it1 or it will cut it2 but it cannot be congruous to it2 and therefore it will cut it4 ,sidenote3 ,the definition of an angle1 and the kinds thereof4 #g4 ,an angle1 according to the most general acceptation of the word1 may be thus defined2 when two lines1 or many superficies1 concur in one sole point1 and diverge every where else1 the quantity of that divergence is an ,,angle4 ,and an angle is of two sorts2 for1 first1 it may be made by the concurrence of lines1 and then it is a superficial angle2 or by the concurrence of superficies1 and then it is called a solid angle4 ,again1 from the two ways by which two lines may diverge from one another1 superficial angles are divided into two kinds4 ,for two strait lines1 which are applied to one another1 and are contiguous in their whole length1 may be separated or pulled open in such manner1 that their concurrence in one point will still remain2 and this separation or opening may be either by circular motion1 the centre whereof is their point of concurrence1 and the lines will still retain their straitness1 the quantity of which separation or divergence is an angle simply so called2 or they may be separated by continual flexion or curvation in every imaginable point2 and the quantity of this separation is that1 which is called an angle of contingence4 ,besides1 of superficial angles simply so called1 those1 which are in a plane superficies1 are plane2 and those1 which are not plane1 are denominated from the superficies in which they are4 ,lastly1 those are strait-lined angles1 which are made by strait lines2 as those which are made by crooked lines are crooked-lined2 and those which are made both of strait and crooked lines1 are mixed angles4 ,sidenote3 ,in concentric circles1 arches of the same angle are to one another1 as the whole circumferences are4 #h4 ,two arches intercepted between two radii of concentric circles1 have the same proportion to one another1 which their whole perimeters have to one another4 ,for let the point ,a 7in the first figure7 be the centre of the two circles ,b ,c ,d and ,e ,f ,g1 in which the radii ,a ,e ,b and ,a ,f ,c intercept the arches ,b ,c and ,e ,f2 ,i say the proportion of the arch ,b ,c to the arch ,e ,f is the same with that of the perimeter ,b ,c ,d to the perimeter ,e ,f ,g4 ,for if the radius ,a ,f ,c be understood to be moved about the centre ,a with circular and uniform motion1 that is1 with equal swiftness everywhere1 the point ,c will in a certain time describe the perimeter ,b ,c ,d1 and in a part of that time the arch ,b ,c2 and because the velocities are equal by which both the arch and the whole perimeter are described1 the proportion of the magnitude of the perimeter ,b ,c ,d to the magnitude of the arch ,,bc is determined by nothing but the difference of the times in which the perimeter and the arch are described4 ,but both the perimeters are described in one and the same time1 and both the arches in one and the same time2 and therefore the proportions of the perimeter ,b ,c ,d to the arch ,b ,c1 and of the perimeter ,e ,f ,g to the arch ,e ,f1 are both determined by the same cause4 ,wherefore ,b ,c ,d4 ,b ,c 33 ,e ,f ,g4 ,e ,f are proportionals 7by the #fth art4 of the last chapter71 and by permutation ,b ,c ,d4 ,e ,f ,g 33 ,b ,c4 ,e ,f will also be proportionals2 which was to be demonstrated4 ,sidenote3 ,the quantity of an angle1 in what it consists4 #i4 ,nothing is contributed towards the quantity of an angle1 neither by the length1 nor by the equality1 nor by the inequality of the lines which comprehend it4 ,for the lines ,a ,b and ,a ,c comprehend the same angle which is comprehended by the lines ,a ,e and ,a ,f1 or ,a ,b and ,a ,f4 ,nor is an angle either increased or diminished by the absolute quantity of the arch1 which subtends the same2 for both the greater arch ,b ,c and the lesser arch ,e ,f are subtended to the same angle4 ,but the quantity of an angle is estimated by the quantity of the subtending arch compared with the quantity of the whole perimeter4 ,and therefore the quantity of an angle simply so called may be thus defined3 the quantity of an angle is an arch or circumference of a circle1 determined by its proportion to the whole perimeter4 ,so that when an arch is intercepted between two strait lines drawn from the centre1 look how great a portion that arch is of the whole perimeter1 so great is the angle4 ,from whence it may be understood1 that when the lines which contain an angle are strait lines1 the quantity of that angle may be taken at any distance from the centre4 ,but if one or both of the containing lines be crooked1 then the quantity of the angle is to be taken in the least distance from the centre1 or from their concurrence2 for the least distance is to be considered as a strait line1 seeing no crooked line can be imagined so little1 but that there may be a less strait line4 ,and although the least strait line cannot be given1 because the least given line may still be divided1 yet we may come to a part so small1 as is not at all considerable2 which we call a point4 ,and this point may be understood to be in a strait line which touches a crooked line2 for an angle is generated by separating1 by circular motion1 one strait line from another which touches it1 as has been said above in the #gth article4 ,wherefore an angle1 which two crooked lines make1 is the same with that which is made by two strait lines which touch them4 ,sidenote3 ,the distinction of angles1 simply so called4 #aj4 ,from hence it follows1 that vertical angles1 such as are ,a ,b ,c1 ,d ,b ,f in the second figure1 are equal to one another4 ,for if1 from the two semiperimeters ,d ,a ,c1 ,f ,d ,a1 which are equal to one another1 the common arch ,d ,a be taken away1 the remaining arches ,a ,c1 ,d ,f will be equal to one another4 ,another distinction of angles is into right and oblique4 ,a right angle is that1 whose quantity is the fourth part of the perimeter4 ,and the lines1 which make a right angle1 are said to be perpendicular to one another4 ,also1 of oblique angles1 that which is greater than a right1 is called an obtuse angle2 and that which is less1 an acute angle4 ,from whence it follows1 that all the angles that can possibly be made at one and the same point1 together taken1 are equal to four right angles2 because the quantities of them all put together make the whole perimeter4 ,also1 that all the angles1 which are made on one side of a strait line1 from any one point taken in the same1 are equal to two right angles2 for if that point be made the centre1 that strait line will be the diameter of a circle1 by whose circumference the quantity of an angle is determined2 and that diameter will divide the perimeter into two equal parts4 ,sidenote3 ,of strait lines from the centre of a circle to a tangent of the same4 #aa4 ,if a tangent be made the diameter of a circle1 whose centre is the point of contact1 a strait line drawn from the centre of the former circle to the centre of the latter circle1 will make two angles with the tangent1 that is1 with the diameter of the latter circle1 equal to two right angles1 by the last article4 ,and because1 by the #fth article1 the tangent has on both sides equal inclination to the circle1 each of them will be a right angle2 as also the semidiameter will be perpendicular to the same tangent4 ,moreover1 the semidiameter1 inasmuch as it is the semidiameter1 is the least strait line which can be drawn from the centre to the tangent2 and every other strait line1 that reaches the tangent1 will pass out of the circle1 and will therefore be greater than the semidiameter4 ,in like manner1 of all the strait lines1 which may be drawn from the centre to the tangent1 that is the greatest which makes the greatest angle with the perpendicular2 which will be manifest1 if about the same centre another circle be described1 whose semidiameter is a strait line taken nearer to the perpendicular1 and there be drawn a perpendicular1 that is1 a tangent1 to the same4 ,from whence it is also manifest1 that if two strait lines1 which make equal angles on either side of the perpendicular1 be produced to the tangent1 they will be equal4 ,sidenote3 ,the general definition of parallels2 the properties of strait parallels4 #ab4 ,there is in ,euclid a definition of strait-lined parallels2 but ,i do not find that parallels in general are anywhere defined2 and therefore for an universal definition of them1 ,i say that any two lines whatsoever1 strait or crooked1 as also any two superficies1 are ,,parallel2 when two equal strait lines1 wheresoever they fall upon them1 make always equal angles with each of them4 ,from which definition it follows2 first1 that any two strait lines1 not inclined opposite ways1 falling upon two other strait lines1 which are parallel1 and intercepting equal parts in both of them1 are themselves also equal and parallel4 ,as if ,a ,b and ,c ,d 7in the third figure71 inclined both the same way1 fall upon the parallels ,a ,c and ,b ,d1 and ,a ,c and ,b ,d be equal1 ,a ,b and ,c ,d will also be equal and parallel4 ,for the perpendiculars ,b ,e and ,d ,f being drawn1 the right angles ,e ,b ,d and ,f ,d ,h will be equal4 ,wherefore1 seeing ,e ,f and ,b ,d are parallel1 the angles ,e ,b ,a and ,f ,d ,c will be equal4 ,now if ,d ,c be not equal to ,b ,a1 let any other strait line equal to ,b ,a be drawn from the point ,d2 which1 seeing it cannot fall upon the point ,c1 let it fall upon ,g4 ,wherefore ,a ,g will be either greater or less than ,b ,d2 and therefore the angles ,e ,b ,a and ,f ,d ,c are not equal1 as was supposed4 ,wherefore ,a ,b and ,c ,d are equal2 which is the first4 ,again1 because they make equal angles with the perpendiculars ,b ,e and ,d ,f2 therefore the angle ,c ,d ,h will be equal to the angle ,a ,b ,d1 and1 by the definition of parallels1 ,a ,b and ,c ,d will be parallel2 which is the second4 ,that plane1 which is included both ways within parallel lines1 is called a ,,parallelogram4 ,coroll4 ,i4 ,from this last it follows1 that the angles ,a ,b ,d and ,c ,d ,h are equal1 that is1 that a strait line1 as ,b ,h1 falling upon two parallels1 as ,a ,b and ,c ,d1 makes the internal angle ,a ,b ,d equal to the external and opposite angle ,c ,d ,h4 ,coroll4 ,,ii4 ,and from hence again it follows1 that a strait line falling upon two parallels1 makes the alternate angles equal1 that is1 the angle ,a ,g ,f1 in the fourth figure1 equal to the angle ,g ,f ,d4 ,for seeing ,g ,f ,d is equal to the external opposite angle ,e ,g ,b1 it will be also equal to its vertical angle ,a ,g ,f1 which is alternate to ,g ,f ,d4 ,coroll4 ,,iii4 ,that the internal angles on the same side of the line ,f ,g are equal to two right angles4 ,for the angles at ,f1 namely1 ,g ,f ,c and ,g ,f ,d1 are equal to two right angles4 ,but ,g ,f ,d is equal to its alternate angle ,a ,g ,f4 ,wherefore both the angles ,g ,f ,c and ,a ,g ,f1 which are internal on the same side of the line ,f ,g1 are equal to two right angles4 ,coroll4 ,,iv4 ,that the three angles of a strait-lined plain triangle are equal to two right angles2 and any side being produced1 the external angle will be equal to the two opposite internal angles4 ,for if there be drawn by the vertex of the plain triangle ,a ,b ,c 7fig4 #e7 a parallel to any of the sides1 as to ,a ,b1 the angles ,a and ,b will be equal to their alternate angles ,e and ,f1 and the angle ,c is common4 ,but1 by the #ajth article1 the three angles ,e1 ,c and ,f1 are equal to two right angles2 and therefore the three angles of the triangle are equal to the same2 which is the first4 ,again1 the two angles ,b and ,d are equal to two right angles1 by the #ajth article4 ,wherefore taking away ,b1 there will remain the angles ,a and ,c1 equal to the angle ,d2 which is the second4 ,coroll4 ,v4 ,if the angles ,a and ,b be equal1 the sides ,a ,c and ,c ,b will also be equal1 because ,a ,b and ,e ,f are parallel2 and1 on the contrary1 if the sides ,a ,c and ,c ,b be equal1 the angles ,a and ,b will also be equal4 ,for if they be not equal1 let the angles ,b and ,g be equal4 ,wherefore1 seeing ,g ,b and ,e ,f are parallels1 and the angles ,g and ,b equal1 the sides ,g ,c and ,c ,b will also be equal2 and because ,c ,b and ,a ,c are equal by supposition1 ,c ,g and ,c ,a will also be equal2 which cannot be1 by the #aath article4 ,coroll4 ,,vi4 ,from hence it is manifest1 that if two radii of a circle be connected by a strait line1 the angles they make with that connecting line will be equal to one another2 and if there be added that segment of the circle1 which is subtended by the same line which connects the radii1 then the angles1 which those radii make with the circumference1 will also be equal to one another4 ,for a strait line1 which subtends any arch1 makes equal angles with the same2 because1 if the arch and the subtense be divided in the middle1 the two halves of the segment will be congruous to one another1 by reason of the uniformity both of the circumference of the circle1 and of the strait line4 ,sidenote3 ,the circumferences of circles are to one another as their diameters are4 #ac4 ,perimeters of circles are to one another1 as their semidiameters are4 ,for let there be any two circles1 as1 in the first figure1 ,b ,c ,d the greater1 and ,e ,f ,g the lesser1 having their common centre at ,a2 and let their semidiameters be ,a ,c and ,a ,e4 ,i say1 ,a ,c has the same proportion to ,a ,e1 which the perimeter ,b ,c ,d has to the perimeter ,e ,f ,g4 ,for the magnitude of the semidiameters ,a ,c and ,a ,e is determined by the distance of the points ,c and ,e from the centre ,a2 and the same distances are acquired by the uniform motion of a point from ,a to ,c1 in such manner1 that in equal times the distances acquired be equal4 ,but the perimeters ,b ,c ,d and ,e ,f ,g are also determined by the same distances of the points ,c and ,e from the centre ,a2 and therefore the perimeters ,b ,c ,d and ,e ,f ,g1 as well as the semidiameters ,a ,c and ,a ,e1 have their magnitudes determined by the same cause1 which cause makes1 in equal times1 equal spaces4 ,wherefore1 by the #acth chapter and #fth article1 the perimeters of circles and their semidiameters are proportionals2 which was to be proved4 ,sidenote3 ,in triangles strait lines parallel to the bases are to one another1 as the parts of the sides which they cut off from the vertex4 #ad4 ,if two strait lines1 which constitute an angle1 be cut by strait-lined parallels1 the intercepted parallels will be to one another1 as the parts which they cut off from the vertex4 ,let the strait lines ,a ,b and ,a ,c1 in the #fth figure1 make an angle at ,a1 and be cut by the two strait-lined parallels ,b ,c and ,d ,e1 so that the parts cut off from the vertex in either of those lines1 as in ,a ,b1 may be ,a ,b and ,a ,d4 ,i say1 the parallels ,b ,c and ,d ,e are to one another1 as the parts ,a ,b and ,a ,d4 ,for let ,a ,b be divided into any number of equal parts1 as into ,a ,f1 ,f ,d1 ,d ,b2 and by the points ,f and ,d1 let ,f ,g and ,d ,e be drawn parallel to the base ,b ,c1 and cut ,a ,c in ,g and ,e2 and again1 by the points ,g and ,e1 let other strait lines be drawn parallel to ,a ,b1 and cut ,b ,c in ,h and ,i4 ,if now the point ,a be understood to be moved uniformly over ,a ,b1 and in the same time ,b be moved to ,c1 and all the points ,f1 ,d1 and ,b be moved uniformly and with equal swiftness over ,f ,g1 ,d ,e1 and ,b ,c2 then shall ,b pass over ,b ,h1 equal to ,f ,g1 in the same time that ,a passes over ,a ,f2 and ,a ,f and ,f ,g will be to one another1 as their velocities are2 and when ,a is in ,f1 ,d will be in ,k2 when ,a is in ,d1 ,d will be in ,e2 and in what manner the point ,a passes by the points ,f1 ,d1 and ,b1 in the same manner the point ,b will pass by the points ,h1 ,i1 and ,c2 and the strait lines ,f ,g1 ,d ,k1 ,k ,e1 ,b ,h1 ,h ,i1 and ,i ,c1 are equal1 by reason of their parallelism2 and therefore1 as the velocity in ,a ,b is to the velocity in ,b ,c1 so is ,a ,d to ,d ,e2 but as the velocity in ,a ,b is to the velocity in ,b ,c1 so is ,a ,b to ,b ,c2 that is to say1 all the parallels will be severally to all the parts cut off from the vertex1 as ,a ,f is to ,f ,g4 ,wherefore1 ,a ,f4 ,g ,f 33 ,a ,d4 ,d ,e 33 ,a ,b4 ,b ,c are proportionals4 ,the subtenses of equal angles in different circles1 as the strait lines ,b ,c and ,f ,e 7in fig4 #a71 are to one another as the arches which they subtend4 ,for 7by art4 #h7 the arches of equal angles are to one another as their perimeters are2 and 7by art4 #ac7 the perimeters as their semidiameters2 but the subtenses ,b ,c and ,f ,e are parallel to one another by reason of the equality of the angles which they make with the semidiameters2 and therefore the same subtenses1 by the last precedent article1 will be proportional to the semidiameters1 that is1 to the perimeters1 that is1 to the arches which they subtend4 ,sidenote3 ,by what fraction of a strait line the circumference of a circle is made4 #ae4 ,if in a circle any number of equal subtenses be placed immediately after one another1 and strait lines be drawn from the extreme point of the first subtense to the extreme points of all the rest1 the first subtense being produced will make with the second subtense an external angle double to that1 which is made by the same first subtense1 and a tangent to the circle touching it in the extreme points thereof2 and if a strait line which subtends two of those arches be produced1 it will make an external angle with the third subtense1 triple to the angle which is made by the tangent with the first subtense2 and so continually4 ,for with the radius ,a ,b 7in fig4 #g7 let a circle be described1 and in it let any number of equal subtenses1 ,b ,c1 ,c ,d1 and ,d ,e1 be placed2 also let ,b ,d and ,b ,e be drawn2 and by producing ,b ,c1 ,b ,d and ,b ,e to any distance in ,g1 ,h and ,i1 let them make angles with the subtenses which succeed one another1 namely1 the external angles ,g ,c ,d1 and ,h ,d ,e4 ,lastly1 let the tangent ,k ,b be drawn1 making with the first subtense the angle ,k ,b ,c4 ,i say the angle ,g ,c ,d is double to the angle ,k ,b ,c1 and the angle ,h ,d ,e triple to the same angle ,k ,b ,c4 ,for if ,a ,c be drawn cutting ,b ,d in ,m1 and from the point ,c there be drawn ,l ,c perpendicular to the same ,a ,c1 then ,c ,l and ,m ,d will be parallel1 by reason of the right angles at ,c and ,m2 and therefore the alterne angles ,l ,c ,d and ,b ,d ,c will be equal3 as also the angles ,b ,d ,c and ,c ,b ,d will be equal1 because of the equality of the strait lines ,b ,c and ,c ,d4 ,wherefore the angle ,g ,c ,d is double to either of the angles ,c ,b ,d or ,c ,d ,b2 and therefore also the angle ,g ,c ,d is double to the angle ,l ,c ,d1 that is1 to the angle ,k ,b ,c4 ,again1 ,c ,d is parallel to ,b ,e1 by reason of the equality of the angles ,c ,b ,e and ,d ,e ,b1 and of the strait lines ,c ,b and ,d ,e2 and therefore the angles ,g ,c ,d and ,g ,b ,e are equal2 and consequently ,g ,b ,e1 as also ,d ,e ,b is double to the angle ,k ,b ,c4 ,but the external angle ,h ,d ,e is equal to the two internal ,d ,e ,b and ,d ,b ,e2 and therefore the angle ,h ,d ,e is triple to the angle ,k ,b ,c1 c42 which was to be proved4 ,coroll4 ,i4 ,from hence it is manifest1 that the angles ,k ,b ,c and ,c ,b ,d1 as also1 that all the angles that are comprehended by two strait lines meeting in the circumference of a circle and insisting upon equal arches1 are equal to one another4 ,coroll4 ,,ii4 ,if the tangent ,b ,k be moved in the circumference with uniform motion about the centre ,b1 it will in equal times cut off equal arches2 and will pass over the whole perimeter in the same time in which itself describes a semiperimeter about the centre ,b4 ,coroll4 ,,iii4 ,from hence also we may understand1 what it is that determines the bending or curvation of a strait line into the circumference of a circle2 namely1 that it is fraction continually increasing in the same manner1 as numbers1 from one upwards1 increase by the continual addition of unity4 ,for the indefinite strait line ,k ,b being broken in ,b according to any angle1 as that of ,k ,b ,c1 and again in ,c according to a double angle1 and in ,d according to an angle which is triple1 and in ,e according to an angle which is quadruple to the first angle1 and so continually1 there will be described a figure which will indeed be rectilineal1 if the broken parts be considered as having magnitude2 but if they be understood to be the least that can be1 that is1 as so many points1 then the figure described will not be rectilineal1 but a circle1 whose circumference will be the broken line4 ,coroll4 ,,iv4 ,from what has been said in this present article1 it may also be demonstrated1 that an angle in the centre is double to an angle in the circumference of the same circle1 if the intercepted arches be equal4 ,for seeing that strait line1 by whose motion an angle is determined1 passes over equal arches in equal times1 as well from the centre as from the circumference2 and while that1 which is from the circumference1 is passing over half its own perimeter1 it passes in the same time over the whole perimeter of that which is from the centre1 the arches1 which it cuts off in the perimeter whose centre is ,a1 will be double to those1 which it makes in its own semiperimeter1 whose centre is ,b4 ,but in equal circles1 as arches are to one another1 so also are angles4 ,it may also be demonstrated1 that the external angle made by a subtense produced and the next equal subtense is equal to an angle from the centre insisting upon the same arch2 as in the last diagram1 the angle ,g ,c ,d is equal to the angle ,c ,a ,d2 for the external angle ,g ,c ,d is double to the angle ,c ,b ,d2 and the angle ,c ,a ,d insisting upon the same arch ,c ,d is also double to the same angle ,c ,b ,d or ,k ,b ,c4 ,sidenote3 ,that an angle of contingence is quantity1 but of a different kind from that of an angle simply so called2 and that it can neither add nor take away anything from the same4 #af4 ,an angle of contingence1 if it be compared with an angle simply so called1 how little soever1 has such proportion to it as a point has to a line2 that is1 no proportion at all1 nor any quantity4 ,for first1 an angle of contingence is made by continual flexion2 so that in the generation of it there is no circular motion at all1 in which consists the nature of an angle simply so called2 and therefore it cannot be compared with it according to quantity4 ,secondly1 seeing the external angle made by a subtense produced and the next subtense is equal to an angle from the centre insisting upon the same arch1 as in the last figure the angle ,g ,c ,d is equal to the angle ,c ,a ,d1 the angle of contingence will be equal to that angle from the centre1 which is made by ,a ,b and the same ,a ,b2 for no part of a tangent can subtend any arch2 but as the point of contact is to be taken for the subtense1 so the angle of contingence is to be accounted for the external angle1 and equal to that angle whose arch is the same point ,b4 ,now1 seeing an angle in general is defined to be the opening or divergence of two lines1 which concur in one sole point2 and seeing one opening is greater than another1 it cannot be denied1 but that by the very generation of it1 an angle of contingence is quantity2 for wheresoever there is greater and less1 there is also quantity2 but this quantity consists in greater and less flexion2 for how much the greater a circle is1 so much the nearer comes the circumference of it to the nature of a strait line2 for the circumference of a circle being made by the curvation of a strait line1 the less that strait line is1 the greater is the curvation2 and therefore1 when one strait line is a tangent to many circles1 the angle of contingence1 which it makes with a less circle1 is greater than that which it makes with a greater circle4 ,nothing therefore is added to or taken from an angle simply so called1 by the addition to it or taking from it of never so many angles of contingence4 ,and as an angle of one sort can never be equal to an angle of the other sort1 so they cannot be either greater or less than one another4 ,from whence it follows1 that an angle of a segment1 that is1 the angle1 which any strait line makes with any arch1 is equal to the angle which is made by the same strait line1 and another which touches the circle in the point of their concurrence2 as in the last figure1 the angle which is made between ,g ,b and ,b ,k is equal to that which is made between ,g ,b and the arch ,b ,c4 ,sidenote3 ,that the inclination of planes is angle simply so called4 #ag4 ,an angle1 which is made by two planes1 is commonly called the inclination of those planes2 and because planes have equal inclination in all their parts1 instead of their inclination an angle is taken1 which is made by two strait lines1 one of which is in one1 the other in the other of those planes1 but both perpendicular to the common section4 ,sidenote3 ,a solid angle what it is4 #ah4 ,a solid angle may be conceived two ways4 ,first1 for the aggregate of all the angles1 which are made by the motion of a strait line1 while one extreme point thereof remaining fixed1 it is carried about any plain figure1 in which the fixed point of the strait line is not contained4 ,and in this sense1 it seems to be understood by ,euclid4 ,now it is manifest1 that the quantity of a solid angle so conceived is no other1 than the aggregate of all the angles in a superficies so described1 that is1 in the superficies of a pyramidal solid4 ,secondly1 when a pyramis or cone has its vertex in the centre of a sphere1 a solid angle may be understood to be the proportion of a spherical superficies subtending that vertex to the whole superficies of the sphere4 ,in which sense1 solid angles are to one another as the spherical bases of solids1 which have their vertex in the centre of the same sphere4 ,sidenote3 ,what is the nature of asymptotes4 #ai4 ,all the ways1 by which two lines respect one another1 or all the variety of their position1 may be comprehended under four kinds2 for any two lines whatsoever are either parallels1 or being produced1 if need be1 or moved one of them to the other parallelly to itself1 they make an angle2 or else1 by the like production and motion1 they touch one another2 or lastly1 they are asymptotes4 ,the nature of parallels1 angles1 and tangents1 has been already declared4 ,it remains that ,i speak briefly of the nature of asymptotes4 ,asymptosy depends upon this1 that quantity is infinitely divisible4 ,and from hence it follows1 that any line being given1 and a body supposed to be moved from one extreme thereof towards the other1 it is possible1 by taking degrees of velocity always less and less1 in such proportion as the parts of the line are made less by continual division1 that the same body may be always moved forwards in that line1 and yet never reach the end of it4 ,for it is manifest1 that if any strait line1 as ,a ,f1 7in the #hth figure7 be cut anywhere in ,b1 and again ,b ,f be cut in ,c1 and ,c ,f in ,d1 and ,d ,f in ,e1 and so eternally1 and there be drawn from the point ,f1 the strait line ,f ,f at any angle ,a ,f ,f2 and lastly1 if the strait lines ,a ,f1 ,b ,f1 ,c ,f1 ,d ,f1 ,e ,f1 c41 having the same proportion to one another with the segments of the line ,a ,f1 be set in order and parallel to the same ,a ,f1 the crooked line ,a ,b ,c ,d ,e1 and the strait line ,f ,f1 will be asymptotes1 that is1 they will always come nearer and nearer together1 but never touch one another4 ,now1 because any line may be cut eternally according to the proportions which the segments have to one another1 therefore the divers kinds of asymptotes are infinite in number1 and not necessary to be further spoken of in this place4 ,in the nature of asymptotes in general there is no more1 than that they come still nearer and nearer1 but never touch4 ,but in special in the asymptosy of hyperbolic lines1 it is understood they should approach to a distance less than any given quantity4 ,sidenote3 ,situation1 by what it is determined4 #bj4 ,,situation is the relation of one place to another2 and where there are many places1 their situation is determined by four things2 by their distances from one another2 by several distances from a place assigned2 by the order of strait lines drawn from a place assigned to the places of them all2 and by the angles which are made by the lines so drawn4 ,for if their distances1 order1 and angles1 be given1 that is1 be certainly known1 their several places will also be so certainly known1 as that they can be no other4 ,sidenote3 ,what is like situation2 what is figure2 and what are like figures4 #ba4 ,points1 how many soever they be1 have like situation with an equal number of other points1 when all the strait lines1 that are drawn from some one point to all these1 have severally the same proportion to those1 that are drawn in the same order and at equal angles from some one point to all those4 ,for let there be any number of points as ,a1 ,b1 and ,c1 7in the #ith figure7 to which from some one point ,d let the strait lines ,d ,a1 ,d ,b1 and ,d ,c be drawn2 and let there be an equal number of other points1 as ,e1 ,f1 and ,g1 and from some point ,h let the strait lines ,h ,e1 ,h ,f1 and ,h ,g be drawn1 so that the angles ,a ,d ,b and ,b ,d ,c be severally and in the same order equal to the angles ,e ,h ,f and ,f ,h ,g1 and the strait lines ,d ,a1 ,d ,b1 and ,d ,c proportional to the strait lines ,h ,e1 ,h ,f1 and ,h ,g2 ,i say1 the three points ,a1 ,b1 and ,c1 have like situation with the three points ,e1 ,f1 and ,g1 or are placed alike4 ,for if ,h ,e be understood to be laid upon ,d ,a1 so that the point ,h be in ,d1 the point ,f will be in the strait line ,d ,b1 by reason of the equality of the angles ,a ,d ,b and ,e ,h ,f2 and the point ,g will be in the strait line ,d ,c1 by reason of the equality of the angles ,b ,d ,c and ,f ,h ,g2 and the strait lines ,a ,b and ,e ,f1 as also ,b ,c and ,f ,g1 will be parallel1 because ,a ,d4 ,e ,h 33 ,b ,d4 ,f ,h 33 ,c ,d4 ,g ,h are proportionals by construction2 and therefore the distances between the points ,a and ,b1 and the points ,b and ,c1 will be proportional to the distances between the points ,e and ,f1 and the points ,f and ,g4 ,wherefore1 in the situation of the points ,a1 ,b1 and ,c1 and the situation of the points ,e1 ,f and ,g1 the angles in the same order are equal2 so that their situations differ in nothing but the inequality of their distances from one another1 and of their distances from the points ,d and ,h4 ,now1 in both the orders of points1 those inequalities are equal2 for ,a ,b4 ,b ,c 33 ,e ,f4 ,f ,g1 which are their distances from one another1 as also ,d ,a4 ,d ,b4 ,d ,c 33 ,h ,e4 ,h ,f4 ,h ,g1 which are their distances from the assumed points ,d and ,h1 are proportionals4 ,their difference1 therefore1 consists solely in the magnitude of their distances4 ,but1 by the definition of like1 7chapter ,i4 article #b7 those things1 which differ only in magnitude1 are like4 ,wherefore the points ,a1 ,b1 and ,c1 have to one another like situation with the points ,e1 ,f1 and ,g1 or are placed alike2 which was to be proved4 ,,figure is quantity1 determined by the situation or placing of all its extreme points4 ,now ,i call those points extreme1 which are contiguous to the place which is without the figure4 ,in lines therefore and superficies1 all points may be called extreme2 but in solids only those which are in the superficies that includes them4 ,like figures are those1 whose extreme points in one of them are all placed like all the extreme points in the other2 for such figures differ in nothing but magnitude4 ,and like figures are alike placed1 when in both of them the homologal strait lines1 that is1 the strait lines which connect the points which answer one another1 are parallel1 and have their proportional sides inclined the same way4 ,and seeing every strait line is like every other strait line1 and every plane like every other plane1 when nothing but planeness is considered2 if the lines1 which include planes1 or the superficies1 which include solids1 have their proportions known1 it will not be hard to know whether any figure be like or unlike to another propounded figure4 ,and thus much concerning the first grounds of philosophy4 ,the next place belongs to geometry2 in which the quantities of figures are sought out from the proportions of lines and angles4 ,wherefore it is necessary for him1 that would study geometry1 to know first what is the nature of quantity1 proportion1 angle and figure4 ,having therefore explained these in the three last chapters1 ,i thought fit to add them to this part2 and so pass to the next4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xiv4 ,fig4 #a-aj ---------------------------------------- -------------------------------- ,,part ,,iii ,,proportions ,,of ,,motions ,,and ,,magnitudes4 ------- ,,chapter ,,xv4 ,,of ,,the ,,nature1 ,,properties1 ,,and ,,divers ,,considerations ,,of ,,motion ,,and ,,endeavour4 #a44 ,repetition of some principles of the doctrine of motion formerly set down4--#b4 ,other principles added to them4--#c4 ,certain theorems concerning the nature of motion4--#d4 ,divers considerations of motion4--#e4 ,the way by which the first endeavour of bodies moved tendeth4--#f4 ,in motion which is made by concourse1 one of the movents ceasing1 the endeavour is made by the way by which the rest tend4--#g4 ,all endeavour is propagated in infinitum4--#h4 ,how much greater the velocity or magnitude is of a movent1 so much the greater is the efficacy thereof upon any other body in its way4 ,sidenote3 ,repetition of some principles of the doctrine of motion formerly set down4 #a4 ,the next things in order to be treated of are ,,motion and ,,magnitude1 which are the most common accidents of all bodies4 ,this place therefore most properly belongs to the elements of geometry4 ,but because this part of philosophy1 having been improved by the best wits of all ages1 has afforded greater plenty of matter than can well be thrust together within the narrow limits of this discourse1 ,i thought fit to admonish the reader1 that before he proceed further1 he take into his hands the works of ,euclid1 ,archimedes1 ,apollonius1 and other as well ancient as modern writers4 ,for to what end is it1 to do over again that which is already done8 ,the little therefore that ,i shall say concerning geometry in some of the following chapters1 shall be such only as is new1 and conducing to natural philosophy4 ,i have already delivered some of the principles of this doctrine in the eighth and ninth chapters2 which ,i shall briefly put together here1 that the reader in going on may have their light nearer at hand4 ,first1 therefore1 in chap4 ,,viii4 art4 #aj1 motion is defined to be the continual privation of one place1 and acquisition of another4 ,secondly1 it is there shown1 that whatsoever is moved is moved in time4 ,thirdly1 in the same chapter1 art4 #aa1 ,i have defined rest to be when a body remains for some time in one place4 ,fourthly1 it is there shown1 that whatsoever is moved is not in any determined place2 as also that the same has been moved1 is still moved1 and will yet be moved2 so that in every part of that space1 in which motion is made1 we may consider three times1 namely1 the past1 the present1 and future time4 ,fifthly1 in art4 #ae of the same chapter1 ,i have defined velocity or swiftness to be motion considered as power1 namely1 that power by which a body moved may in a certain time transmit a certain length2 which also may more briefly be enunciated thus1 velocity is the quantity of motion determined by time and line4 ,sixthly1 in the same chapter1 art4 #af1 ,i have shown that motion is the measure of time4 ,seventhly1 in the same chapter1 art4 #ag1 ,i have defined motions to be equally swift1 when in equal times equal lengths are transmitted by them4 ,eighthly1 in art4 #ah of the same chapter1 motions are defined to be equal1 when the swiftness of one moved body1 computed in every part of its magnitude1 is equal to the swiftness of another1 computed also in every part of its magnitude4 ,from whence it is to be noted1 that motions equal to one another1 and motions equally swift1 do not signify the same thing2 for when two horses draw abreast1 the motion of both is greater than the motion of either of them singly2 but the swiftness of both together is but equal to that of either4 ,ninthly1 in art4 #ai of the same chapter1 ,i have shown1 that whatsoever is at rest will always be at rest1 unless there be some other body besides it1 which by getting into its place suffers it no longer to remain at rest4 ,and that whatsoever is moved1 will always be moved1 unless there be some other body besides it1 which hinders its motion4 ,tenthly1 in chap4 ,,ix4 art4 #g1 ,i have demonstrated1 that when any body is moved which was formerly at rest1 the immediate efficient cause of that motion is in some other moved and contiguous body4 ,eleventhly1 ,i have shown in the same place1 that whatsoever is moved1 will always be moved in the same way1 and with the same swiftness1 if it be not hindered by some other moved and contiguous body4 ,sidenote3 ,other principles added to them4 #b4 ,to which principles ,i shall here add those that follow4 ,first1 ,i define ,,endeavour to be motion made in less space and time than can be given2 that is1 less than can be determined or assigned by exposition or number2 that is1 motion made through the length of a point1 and in an instant or point of time4 ,for the explaining of which definition it must be remembered1 that by a point is not to be understood that which has no quantity1 or which cannot by any means be divided2 for there is no such thing in nature2 but that1 whose quantity is not at all considered1 that is1 whereof neither quantity nor any part is computed in demonstration2 so that a point is not to be taken for an indivisible1 but for an undivided thing2 as also an instant is to be taken for an undivided1 and not for an indivisible time4 ,in like manner1 endeavour is to be conceived as motion2 but so as that neither the quantity of the time in which1 nor of the line in which it is made1 may in demonstration be at all brought into comparison with the quantity of that time1 or of that line of which it is a part4 ,and yet1 as a point may be compared with a point1 so one endeavour may be compared with another endeavour1 and one may be found to be greater or less than another4 ,for if the vertical points of two angles be compared1 they will be equal or unequal in the same proportion which the angles themselves have to one another4 ,or if a strait line cut many circumferences of concentric circles1 the inequality of the points of intersection will be in the same proportion which the perimeters have to one another4 ,and in the same manner1 if two motions begin and end both together1 their endeavours will be equal or unequal1 according to the proportion of their velocities2 as we see a bullet of lead descend with greater endeavour than a ball of wool4 ,secondly1 ,i define ,,impetus1 or quickness of motion1 to be the swiftness or velocity of the body moved1 but considered in the several points of that time in which it is moved4 ,in which sense impetus is nothing else but the quantity or velocity of endeavour4 ,but considered with the whole time1 it is the whole velocity of the body moved taken together throughout all the time1 and equal to the product of a line representing the time1 multiplied into a line representing the arithmetically mean impetus or quickness4 ,which arithmetical mean1 what it is1 is defined in the #bith article of chapter ,,xiii4 ,and because in equal times the ways that are passed are as the velocities1 and the impetus is the velocity they go withal1 reckoned in all the several points of the times1 it followeth that during any time whatsoever1 howsoever the impetus be increased or decreased1 the length of the way passed over shall be increased or decreased in the same proportion2 and the same line shall represent both the way of the body moved1 and the several impetus or degrees of swiftness wherewith the way is passed over4 ,and if the body moved be not a point1 but a strait line moved so as that every point thereof make a several strait line1 the plane described by its motion1 whether uniform1 accelerated1 or retarded1 shall be greater or less1 the time being the same1 in the same proportion with that of the impetus reckoned in one motion to the impetus reckoned in the other4 ,for the reason is the same in parallelograms and their sides4 ,for the same cause also1 if the body moved be a plane1 the solid described shall be still greater or less in the proportions of the several impetus or quicknesses reckoned through one line1 to the several impetus reckoned through another4 ,this understood1 let ,a ,b ,c ,d1 7in figure #a1 chap4 ,,xvii47 be a parallelogram2 in which suppose the side ,a ,b to be moved parallelly to the opposite side ,c ,d1 decreasing all the way till it vanish in the point ,c1 and so describing the figure ,a ,b ,e ,f ,c2 the point ,b1 as ,a ,b decreaseth1 will therefore describe the line ,b ,e ,f ,c2 and suppose the time of this motion designed by the line ,c ,d2 and in the same time ,c ,d1 suppose the side ,a ,c to be moved parallel and uniformly to ,b ,d4 ,from the point ,o taken at adventure in the line ,c ,d1 draw ,o ,r parallel to ,b ,d1 cutting the line ,b ,e ,f ,c in ,e1 and the side ,a ,b in ,r4 ,and again1 from the point ,q taken also at adventure in the line ,c ,d1 draw ,q ,s parallel to ,b ,d1 cutting the line ,b ,e ,f ,c in ,f1 and the side ,a ,b in ,s2 and draw ,e ,g and ,f ,h parallel to ,c ,d1 cutting ,a ,c in ,g and ,h4 ,lastly1 suppose the same construction done in all the points possible of the line ,b ,e ,f ,c4 ,i say1 that as the proportions of the swiftness wherewith ,q ,f1 ,o ,e1 ,d ,b1 and all the rest supposed to be drawn parallel to ,d ,b and terminated in the line ,b ,e ,f ,c1 are to the proportions of their several times designed by the several parallels ,h ,f1 ,g ,e1 ,a ,b1 and all the rest supposed to be drawn parallel to the line of time ,c ,d and terminated in the line ,b ,e ,f ,c1 the aggregate to the aggregate1 so is the area or plane ,d ,b ,e ,f ,c to the area or plane ,a ,c ,f ,e ,b4 ,for as ,a ,b decreasing continually by the line ,b ,e ,f ,c vanisheth in the time ,c ,d into the point ,c1 so in the same time the line ,d ,c continually decreasing vanisheth by the same line ,c ,f ,e ,b into the point ,b2 and the point ,d describeth in that decreasing motion the line ,d ,b equal to the line ,a ,c described by the point ,a in the decreasing motion of ,a ,b2 and their swiftnesses are therefore equal4 ,again1 because in the time ,g ,e the point ,o describeth the line ,,oe1 and in the same time the point ,s describeth the line ,s ,e1 the line ,o ,e shall be to the line ,,se1 as the swiftness wherewith ,,oe is described to the swiftness wherewith ,,se is described4 ,in like manner1 because in the same time ,h ,f the point ,q describeth the line ,q ,f1 and the point ,r the line ,r ,f1 it shall be as the swiftness by which ,q ,f is described to the swiftness by which ,r ,f is described1 so the line itself ,q ,f to the line itself ,r ,f2 and so in all the lines that can possibly be drawn parallel to ,b ,d in the points where they cut the line ,b ,e ,f ,c4 ,but all the parallels to ,b ,d1 as ,s ,e1 ,r ,f1 ,a ,c1 and the rest that can possibly be drawn from the line ,a ,b to the line ,b ,e ,f ,c1 make the area of the plane ,a ,b ,e ,f ,c2 and all the parallels to the same ,b ,d1 as ,q ,f1 ,o ,e1 ,d ,b and the rest drawn to the points where they cut the same line ,b ,e ,f ,c1 make the area of the plane ,b ,e ,f ,c ,d4 ,as therefore the aggregate of the swiftnesses wherewith the plane ,b ,e ,f ,c ,d is described1 is to the aggregate of the swiftnesses wherewith the plane ,a ,c ,f ,e ,b is described1 so is the plane itself ,b ,e ,f ,c ,d to the plane itself ,a ,c ,f ,e ,b4 ,but the aggregate of the times represented by the parallels ,a ,b1 ,g ,e1 ,h ,f and the rest1 maketh also the area ,a ,c ,f ,e ,b4 ,and therefore1 as the aggregate of all the lines ,q ,f1 ,o ,e1 ,d ,b and all the rest of the lines parallel to ,b ,d and terminated in the line ,b ,e ,f ,c1 is to the aggregate of all the lines ,h ,f1 ,g ,e1 ,a ,b and all the rest of the lines parallel to ,c ,d and terminated in the same line ,b ,e ,f ,c2 that is1 as the aggregate of the lines of swiftness to the aggregate of the lines of time1 or as the whole swiftness in the parallels to ,d ,b to the whole time in the parallels to ,c ,d1 so is the plane ,b ,e ,f ,c ,d to the plane ,a ,c ,f ,e ,b4 ,and the proportions of ,q ,f to ,f ,h1 and of ,o ,e to ,e ,g1 and of ,d ,b to ,b ,a1 and so of all the rest taken together1 are the proportions of the plane ,d ,b ,e ,f ,c to the plane ,a ,b ,e ,f ,c4 ,but the lines ,q ,f1 ,o ,e1 ,d ,b and the rest are the lines that design the swiftness2 and the lines ,h ,f1 ,g ,e1 ,a ,b and the rest are the lines that design the times of the motions2 and therefore the proportion of the plane ,d ,b ,e ,f ,c to the plane ,a ,b ,e ,f ,c is the proportion of all the velocities taken together to all the times taken together4 ,wherefore1 as the proportions of the swiftnesses1 c42 which was to be demonstrated4 ,the same holds also in the diminution of the circles1 whereof the lines of time are the semidiameters1 as may easily be conceived by imagining the whole plane ,a ,b ,c ,d turned round upon the axis ,b ,d2 for the line ,b ,e ,f ,c will be everywhere in the superficies so made1 and the lines ,h ,f1 ,g ,e1 ,a ,b1 which are here parallelograms1 will be there cylinders1 the diameters of whose bases are the lines ,h ,f1 ,g ,e1 ,a ,b1 c4 and the altitude a point1 that is to say1 a quantity less than any quantity that can possibly be named2 and the lines ,q ,f1 ,o ,e1 ,d ,b1 c4 small solids whose lengths and breadths are less than any quantity that can be named4 ,but this is to be noted1 that unless the proportion of the sum of the swiftnesses to the proportion of the sum of the times be determined1 the proportion of the figure ,d ,b ,e ,f ,c to the figure ,a ,b ,e ,f ,c cannot be determined4 ,thirdly1 ,i define ,,resistance to be the endeavour of one moved body either wholly or in part contrary to the endeavour of another moved body1 which toucheth the same4 ,i say1 wholly contrary1 when the endeavour of two bodies proceeds in the same strait line from the opposite extremes1 and contrary in part1 when two bodies have their endeavour in two lines1 which1 proceeding from the extreme points of a strait line1 meet without the same4 ,fourthly1 that ,i may define what it is to ,,press1 ,i say1 that of two moved bodies one presses the other1 when with its endeavour it makes either all or part of the other body to go out of its place4 ,fifthly1 a body1 which is pressed and not wholly removed1 is said to ,,restore itself1 when1 the pressing body being taken away1 the parts which were moved do1 by reason of the internal constitution of the pressed body1 return every one into its own place4 ,and this we may observe in springs1 in blown bladders1 and in many other bodies1 whose parts yield more or less to the endeavour which the pressing body makes at the first arrival2 but afterwards1 when the pressing body is removed1 they do1 by some force within them1 restore themselves1 and give their whole body the same figure it had before4 ,sixthly1 ,i define ,,force to be the impetus or quickness of motion multiplied either into itself1 or into the magnitude of the movent1 by means whereof the said movent works more or less upon the body that resists it4 ,sidenote3 ,certain theorems concerning the nature of motion4 #c4 ,having premised thus much1 ,i shall now demonstrate1 first1 that if a point moved come to touch another point which is at rest1 how little soever the impetus or quickness of its motion be1 it shall move that other point4 ,for if by that impetus it do not at all move it out of its place1 neither shall it move it with double the same impetus4 ,for nothing doubled is still nothing2 and for the same reason it shall never move it with that impetus1 how many times soever it be multiplied1 because nothing1 however it be multiplied1 will for ever be nothing4 ,wherefore1 when a point is at rest1 if it do not yield to the least impetus1 it will yield to none2 and consequently it will be impossible that that1 which is at rest1 should ever be moved4 ,secondly1 that when a point moved1 how little soever the impetus thereof be1 falls upon a point of any body at rest1 how hard soever that body be1 it will at the first touch make it yield a little4 ,for if it do not yield to the impetus which is in that point1 neither will it yield to the impetus of never so many points1 which have all their impetus severally equal to the impetus of that point4 ,for seeing all those points together work equally1 if any one of them have no effect1 the aggregate of them all together shall have no effect as many times told as there are points in the whole body1 that is1 still no effect at all2 and by consequent there would be some bodies so hard that it would be impossible to break them2 that is1 a finite hardness1 or a finite force1 would not yield to that which is infinite2 which is absurd4 ,coroll4 ,it is therefore manifest1 that rest does nothing at all1 nor is of any efficacy2 and that nothing but motion gives motion to such things as be at rest1 and takes it from things moved4 ,thirdly1 that cessation in the movent does not cause cessation in that which was moved by it4 ,for 7by number #aa of art4 #a of this chapter7 whatsoever is moved perseveres in the same way and with the same swiftness1 as long as it is not hindered by something that is moved against it4 ,now it is manifest1 that cessation is not contrary motion2 and therefore it follows that the standing still of the movent does not make it necessary that the thing moved should also stand still4 ,coroll4 ,they are therefore deceived1 that reckon the taking away of the impediment or resistance for one of the causes of motion4 ,sidenote3 ,divers considerations of motions4 #d4 ,motion is brought into account for divers respects2 first1 as in a body undivided1 that is1 considered as a point2 or1 as in a divided body4 ,in an undivided body1 when we suppose the way1 by which the motion is made1 to be a line2 and in a divided body1 when we compute the motion of the several parts of that body1 as of parts4 ,secondly1 from the diversity of the regulation of motion1 it is in body1 considered as undivided4 sometimes uniform and sometimes multiform4 ,uniform is that by which equal lines are always transmitted in equal times2 and multiform1 when in one time more1 in another time less space is transmitted4 ,again1 of multiform motions1 there are some in which the degrees of acceleration and retardation proceed in the same proportions1 which the spaces transmitted have1 whether duplicate1 or triplicate1 or by whatsoever number multiplied2 and others in which it is otherwise4 ,thirdly1 from the number of the movents2 that is1 one motion is made by one movent only1 and another by the concourse of many movents4 ,fourthly1 from the position of that line in which a body is moved1 in respect of some other line2 and from hence one motion is called perpendicular1 another oblique1 another parallel4 ,fifthly1 from the position of the movent in respect of the moved body2 from whence one motion is pulsion or driving1 another traction or drawing4 ,pulsion1 when the movent makes the moved body go before it2 and traction1 when it makes it follow4 ,again1 there are two sorts of pulsion2 one1 when the motions of the movent and moved body begin both together1 which may be called trusion or thrusting and vection2 the other1 when the movent is first moved1 and afterwards the moved body1 which motion is called percussion or stroke4 ,sixthly1 motion is considered sometimes from the effect only which the movent works in the moved body1 which is usually called moment4 ,now moment is the excess of motion which the movent has above the motion or endeavour of the resisting body4 ,seventhly1 it may be considered from the diversity of the medium2 as one motion may be made in vacuity or empty place2 another in a fluid2 another in a consistent medium1 that is1 a medium whose parts are by some power so consistent and cohering1 that no part of the same will yield to the movent1 unless the whole yield also4 ,eighthly1 when a moved body is considered as having parts1 there arises another distinction of motion into simple and compound4 ,simple1 when all the several parts describe several equal lines2 compounded1 when the lines described are unequal4 ,sidenote3 ,the way by which the first endeavour of bodies moved tendeth4 #e4 ,all endeavour tends towards that part1 that is to say1 in that way which is determined by the motion of the movent1 if the movent be but one2 or1 if there be many movents1 in that way which their concourse determines4 ,for example1 if a moved body have direct motion1 its first endeavour will be in a strait line2 if it have circular motion1 its first endeavour will be in the circumference of a circle4 ,sidenote3 ,in motion1 which it made by concourse1 one of the movents ceasing1 the endeavour is made by the way by which the rest tend4 #f4 ,and whatsoever the line be1 in which a body has its motion from the concourse of two movents1 as soon as in any point thereof the force of one of the movents ceases1 there immediately the former endeavour of that body will be changed into an endeavour in the line of the other movent4 ,wherefore1 when any body is carried on by the concourse of two winds1 one of those winds ceasing1 the endeavour and motion of that body will be in that line1 in which it would have been carried by that wind alone which blows still4 ,and in the describing of a circle1 where that which is moved has its motion determined by a movent in a tangent1 and by the radius which keeps it in a certain distance from the centre1 if the retention of the radius cease1 that endeavour1 which was in the circumference of the circle1 will now be in the tangent1 that is1 in a strait line4 ,for1 seeing endeavour is computed in a less part of the circumference than can be given1 that is1 in a point1 the way by which a body is moved in the circumference is compounded of innumerable strait lines1 of which every one is less than can be given2 which are therefore called points4 ,wherefore when any body1 which is moved in the circumference of a circle1 is freed from the retention of the radius1 it will proceed in one of those strait lines1 that is1 in a tangent4 ,sidenote3 ,all endeavour is propagated in infinitum4 #g4 ,all endeavour1 whether strong or weak1 is propagated to infinite distance2 for it is motion4 ,if therefore the first endeavour of a body be made in space which is empty1 it will always proceed with the same velocity2 for it cannot be supposed that it can receive any resistance at all from empty space2 and therefore1 7by art4 #g1 chap1 ,,ix7 it will always proceed in the same way and with the same swiftness4 ,and if its endeavour be in space which is filled1 yet1 seeing endeavour is motion1 that which stands next in its way shall be removed1 and endeavour further1 and again remove that which stands next1 and so infinitely4 ,wherefore the propagation of endeavour1 from one part of full space to another1 proceeds infinitely4 ,besides1 it reaches in any instant to any distance1 how great soever4 ,for in the same instant in which the first part of the full medium removes that which is next it1 the second also removes that part which is next to it2 and therefore all endeavour1 whether it be in empty or in full space1 proceeds not only to any distance1 how great soever1 but also in any time1 how little soever1 that is1 in an instant4 ,nor makes it any matter1 that endeavour1 by proceeding1 grows weaker and weaker1 till at last it can no longer be perceived by sense2 for motion may be insensible2 and ,i do not here examine things by sense and experience1 but by reason4 ,sidenote3 ,how much greater the velocity or magnitude is of a movent1 so much the greater is the efficacy thereof upon any other body in its way4 #h4 ,when two movents are of equal magnitude1 the swifter of them works with greater force than the slower1 upon a body that resists their motion4 ,also1 if two movents have equal velocity1 the greater of them works with more force than the less4 ,for where the magnitude is equal1 the movent of greater velocity makes the greater impression upon that body upon which it falls2 and where the velocity is equal1 the movent of greater magnitude falling upon the same point1 or an equal part of another body1 loses less of its velocity1 because the resisting body works only upon that part of the movent which it touches1 and therefore abates the impetus of that part only2 whereas in the mean time the parts1 which are not touched1 proceed1 and retain their whole force1 till they also come to be touched2 and their force has some effect4 ,wherefore1 for example1 in batteries a longer than a shorter piece of timber of the same thickness and velocity1 and a thicker than a slenderer piece of the same length and velocity1 work a greater effect upon the wall4 ------- ,,chapter ,,xvi4 ,,of ,,motion ,,accelerated ,,and ,,uniform1 ,,and ,,of ,,motion ,,by ,,concourse4 #a4 ,the velocity of any body1 in what time soever it be computed1 is that which is made of the multiplication of the impetus1 or quickness of its motion into the time4--#b-e4 ,in all motion1 the lengths which are passed through are to one another1 as the products made by the impetus multiplied into the time4--#f4 ,if two bodies be moved with uniform motion through two lengths1 the proportion of those lengths to one another will be compounded of the proportions of time to time1 and impetus to impetus1 directly taken4--#g4 ,if two bodies pass through two lengths with uniform motion1 the proportion of their times to one another will be compounded of the proportions of length to length1 and impetus to impetus reciprocally taken2 also the proportion of their impetus to one another will be compounded of the proportions of length to length1 and time to time reciprocally taken4--#h4 ,if a body be carried on with uniform motion by two movents together1 which meet in an angle1 the line by which it passes will be a strait line1 subtending the complement of that angle to two right angles4--#i1 c4 ,if a body be carried by two movents together1 one of them being moved with uniform1 the other with accelerated motion1 and the proportion of their lengths to their times being explicable in numbers1 how to find out what line that body describes4 ,sidenote3 ,the velocity of any body1 in what time soever it be computed1 is that which is made of the multiplication of the impetus or quickness of its motion into the time4 #a4 ,the velocity of any body1 in whatsoever time it be moved1 has its quantity determined by the sum of all the several quicknesses or impetus1 which it hath in the several points of the time of the body's motion4 ,for seeing velocity1 7by the definition of it1 chap1 ,,viii1 art4 #ae7 is that power by which a body can in a certain time pass through a certain length2 and quickness of motion or impetus1 7by chap4 ,,xv1 art4 #b1 num4 #b7 is velocity taken in one point of time only1 all the impetus1 together taken in all the points of time1 will be the same thing with the mean impetus multiplied into the whole time1 or which is all one1 will be the velocity of the whole motion4 ,coroll4 ,if the impetus be the same in every point1 any strait line representing it may be taken for the measure of time3 and the quicknesses or impetus applied ordinately to any strait line making an angle with it1 and representing the way of the body's motion1 will design a parallelogram which shall represent the velocity of the whole motion4 ,but if the impetus or quickness of motion begin from rest and increase uniformly1 that is1 in the same proportion continually with the times which are passed1 the whole velocity of the motion shall be represented by a triangle1 one side whereof is the whole time1 and the other the greatest impetus acquired in that time2 or else by a parallelogram1 one of whose sides is the whole time of motion1 and the other1 half the greatest impetus2 or lastly1 by a parallelogram having for one side a mean proportional between the whole time and the half of that time1 and for the other side the half of the greatest impetus4 ,for both these parallelograms are equal to one another1 and severally equal to the triangle which is made of the whole line of time1 and of the greatest acquired impetus2 as is demonstrated in the elements of geometry4 ,sidenote3 ,in all motion1 the lengths which are passed through are to one another1 as the products made by the impetus multiplied into time4 #b4 ,in all uniform motions the lengths which are transmitted are to one another1 as the product of the mean impetus multiplied into its time1 to the product of the mean impetus multiplied also into its time4 ,for let ,a ,b 7in fig4 #a7 be the time1 and ,a ,c the impetus by which any body passes with uniform motion through the length ,d ,e2 and in any part of the time ,a ,b1 as in the time ,a ,f1 let another body be moved with uniform motion1 first1 with the same impetus ,a ,c4 ,this body1 therefore1 in the time ,a ,f with the impetus ,a ,c will pass through the length ,a ,f4 ,seeing1 therefore1 when bodies are moved in the same time1 and with the same velocity and impetus in every part of their motion1 the proportion of one length transmitted to another length transmitted1 is the same with that of time to time1 it followeth1 that the length transmitted in the time ,a ,b with the impetus ,a ,c will be to the length transmitted in the time ,a ,f with the same impetus ,a ,c1 as ,a ,b itself is to ,a ,f1 that is1 as the parallelogram ,a ,i is to the parallelogram ,a ,h1 that is1 as the product of the time ,a ,b into the mean impetus ,a ,c is to the product of the time ,a ,f into the same impetus ,a ,c4 ,again1 let it be supposed that a body be moved in the time ,a ,f1 not with the same but with some other uniform impetus1 as ,a ,l4 ,seeing therefore1 one of the bodies has in all the parts of its motion the impetus ,a ,c1 and the other in like manner the impetus ,a ,l1 the length transmitted by the body moved with the impetus ,a ,c will be to the length transmitted by the body moved with the impetus ,a ,l1 as ,a ,c itself is to ,a ,l1 that is1 as the parallelogram ,a ,h is to the parallelogram ,f ,l4 ,wherefore1 by ordinate proportion it will be1 as the parallelogram ,a ,i to the parallelogram ,f ,l1 that is1 as the product of the mean impetus into the time is to the product of the mean impetus into the time1 so the length transmitted in the time ,a ,b with the impetus ,a ,c1 to the length transmitted in the time ,a ,f with the impetus ,a ,l2 which was to be demonstrated4 ,coroll4 ,seeing1 therefore1 in uniform motion1 as has been shown1 the lengths transmitted are to one another as the parallelograms which are made by the multiplication of the mean impetus into the times1 that is1 by reason of the equality of the impetus all the way1 as the times themselves1 it will also be1 by permutation1 as time to length1 so time to length2 and in general1 to this place are applicable all the properties and transmutations of analogisms1 which ,i have set down and demonstrated in chapter ,,xiii4 #c4 ,in motion begun from rest and uniformly accelerated1 that is1 where the impetus increaseth continually according to the proportion of the times1 it will also be1 as one product made by the mean impetus multiplied into the time1 to another product made likewise by the mean impetus multiplied into the time1 so the length transmitted in the one time to the length transmitted in the other time4 ,for let ,a ,b 7in fig4 #a7 represent a time2 in the beginning of which time ,a1 let the impetus be as the point ,a2 but as the time goes on1 so let the impetus increase uniformly1 till in the last point of that time ,a ,b1 namely in ,b1 the impetus acquired be ,b ,i4 ,again1 let ,a ,f represent another time1 in whose beginning ,a1 let the impetus be as the point itself ,a2 but as the time proceeds1 so let the impetus increase uniformly1 till in the last point ,f of the time ,a ,f the impetus acquired be ,f ,k2 and let ,d ,e be the length passed through in the time ,a ,b with impetus uniformly increased4 ,i say1 the length ,d ,e is to the length transmitted in the time ,a ,f1 as the time ,a ,b multiplied into the mean of the impetus increasing through the time ,a ,b1 is to the time ,a ,f multiplied into the mean of the impetus increasing through the time ,a ,f4 ,for seeing the triangle ,a ,b ,i is the whole velocity of the body moved in the time ,a ,b1 till the impetus acquired be ,b ,i2 and the triangle ,a ,f ,k the whole velocity of the body moved in the time ,a ,f with impetus increasing till there be acquired the impetus ,f ,k2 the length ,d ,e to the length acquired in the time ,a ,f with impetus increasing from rest in ,a till there be acquired the impetus ,f ,k1 will be as the triangle ,a ,b ,i to the triangle ,a ,f ,k1 that is1 if the triangles ,a ,b ,i and ,a ,f ,k be like1 in duplicate proportion of the time ,a ,b to the time ,a ,f2 but if unlike1 in the proportion compounded of the proportions of ,a ,b to ,a ,f and of ,b ,i to ,f ,k4 ,wherefore1 as ,a ,b ,i is to ,a ,f ,k1 so let ,d ,e be to ,d ,p2 for so1 the length transmitted in the time ,a ,b with impetus increasing to ,b ,i1 will be to the length transmitted in the time ,a ,f with impetus increasing to ,f ,k1 as the triangle ,a ,b ,i is to the triangle ,a ,f ,k2 but the triangle ,a ,b ,i is made by the multiplication of the time ,a ,b into the mean of the impetus increasing to ,b ,i2 and the triangle ,a ,f ,k is made by the multiplication of the time ,a ,f into the mean of the impetus increasing to ,f ,k2 and therefore the length ,d ,e which is transmitted in the time ,a ,b with impetus increasing to ,b ,i1 to the length ,d ,p which is transmitted in the time ,a ,f with impetus increasing to ,f ,k1 is as the product which is made of the time ,a ,b multiplied into its mean impetus1 to the product of the time ,a ,f multiplied also into its mean impetus2 which was to be proved4 ,coroll4 ,i4 ,in motion uniformly accelerated1 the proportion of the lengths transmitted to that of their times1 is compounded of the proportions of their times to their times1 and impetus to impetus4 ,coroll4 ,,ii4 ,in motion uniformly accelerated1 the lengths transmitted in equal times1 taken in continual succession from the beginning of motion1 are as the differences of square numbers beginning from unity1 namely1 as #c1 #e1 #g1 c4 ,for if in the first time the length transmitted be as #a1 in the first and second times the length transmitted will be as #d1 which is the square of #b1 and in the three first times it will be as #i1 which is the square of #c1 and in the four first times as #af1 and so on4 ,now the differences of these squares are #c1 #e1 #g1 c4 ,coroll4 ,,iii4 ,in motion uniformly accelerated from rest1 the length transmitted is to another length transmitted uniformly in the same time1 but with such impetus as was acquired by the accelerated motion in the last point of that time1 as a triangle to a parallelogram1 which have their altitude and base common4 ,for seeing the length ,d ,e 7in fig4 #a7 is passed through with velocity as the triangle ,a ,b ,i1 it is necessary that for the passing through of a length which is double to ,d ,e1 the velocity be as the parallelogram ,a ,i2 for the parallelogram ,a ,i is double to the triangle ,a ,b ,i4 #d4 ,in motion1 which beginning from rest is so aclerated1 that the impetus thereof increases continually in proportion duplicate to the proportion of the times in which it is made1 a length transmitted in one time will be to a length transmitted in another time1 as the product made by the mean impetus multiplied into the time of one of those motions1 to the product of the mean impetus multiplied into the time of the other motion4 ,for let ,a ,b 7in fig4 #b7 represent a time1 in whose first instant ,a let the impetus be as the point ,a2 but as the time proceeds1 so let the impetus increase continually in duplicate proportion to that of the times1 till in the last point of time ,b the impetus acquired be ,b ,i2 then taking the point ,f anywhere in the time ,a ,b1 let the impetus ,f ,k acquired in the time ,a ,f be ordinately applied to that point ,f4 ,seeing therefore the proportion of ,f ,k to ,b ,i is supposed to be duplicate to that of ,a ,f to ,a ,b1 the proportion of ,a ,f to ,a ,b will be subduplicate to that of ,f ,k to ,b ,i2 and that of ,a ,b to ,a ,f will be 7by chap4 ,,xiii4 art4 #af7 duplicate to that of ,b ,i to ,f ,k2 and consequently the point ,k will be in a parabolical line1 whose diameter is ,a ,b and base ,b ,i2 and for the same reason1 to what point soever of the time ,a ,b the impetus acquired in that time be ordinately applied1 the strait line designing that impetus will be in the same parabolical line ,a ,k ,i4 ,wherefore the mean impetus multiplied into the whole time ,a ,b will be the parabola ,a ,k ,i ,b1 equal to the parallelogram ,a ,m1 which parallelogram has for one side the line of time ,a ,b and for the other the line of the impetus ,a ,l1 which is two-thirds of the impetus ,b ,i2 for every parabola is equal to two-thirds of that parallelogram with which it has its altitude and base common4 ,wherefore the whole velocity in ,a ,b will be the parallelogram ,a ,m1 as being made by the multiplication of the impetus ,a ,l into the time ,a ,b4 ,and in like manner1 if ,f ,n be taken1 which is two-thirds of the impetus ,f ,k1 and the parallelogram ,f ,o be completed1 ,f ,o will be the whole velocity in the time ,a ,f1 as being made by the uniform impetus ,a ,o or ,f ,n multiplied into the time ,a ,f4 ,let now the length transmitted in the time ,a ,b and with the velocity ,a ,m be the strait line ,d ,e2 and lastly1 let the length transmitted in the time ,a ,f with the velocity ,a ,n be ,d ,p2 ,i say that as ,a ,m is to ,a ,n1 or as the parabola ,a ,k ,i ,b to the parabola ,a ,k ,f1 so is ,d ,e to ,d ,p4 ,for as ,a ,m is to ,f ,l1 that is1 as ,a ,b is to ,a ,f1 so let ,d ,e be to ,d ,g4 ,now the proportion of ,a ,m to ,a ,n is compounded of the proportions of ,a ,m to ,f ,l1 and of ,f ,l to ,a ,n4 ,but as ,a ,m to ,f ,l1 so by construction is ,d ,e to ,d ,g2 and as ,f ,l is to ,a ,n 7seeing the time in both is the same1 namely1 ,a ,f71 so is the length ,d ,g to the length ,d ,p2 for lengths transmitted in the same time are to one another as their velocities are4 ,wherefore by ordinate proportion1 as ,a ,m is to ,a ,n1 that is1 as the mean impetus ,a ,l multiplied into its time ,a ,b1 is to the mean impetus ,a ,o multiplied into ,a ,f1 so is ,d ,e to ,d ,p2 which was to be proved4 ,coroll4 ,i4 ,lengths transmitted with motion so accelerated1 that the impetus increase continually in duplicate proportion to that of their times1 if the base represent the impetus1 are in triplicate proportion of their impetus acquired in the last point of their times4 ,for as the length ,d ,e is to the length ,d ,p1 so is the parallelogram ,a ,m to the parallelogram ,a ,n1 and so the parabola ,a ,k ,i ,b to the parabola ,a ,k ,f4 ,but the proportion of the parabola ,a ,k ,i ,b to the parabola ,a ,k ,f is triplicate to the proportion which the base ,b ,i has to the base ,f ,k4 ,wherefore also the proportion of ,d ,e to ,d ,p is triplicate to that of ,b ,i to ,f ,k4 ,coroll4 ,,ii4 ,lengths transmitted in equal times succeeding one another from the beginning1 by motion so accelerated1 that the proportion of the impetus be duplicate to the proportion of the times1 are to one another as the differences of cubic numbers beginning at unity1 that is as #g1 #ai1 #cg1 c4 ,for if in the first time the length transmitted be as #a1 the length at the end of the second time will be as #h1 at the end of the third time as #bg1 and at the end of the fourth time as #fd1 c42 which are cubic numbers1 whose differences are #g1 #ai1 #cg1 c4 ,coroll4 ,,iii4 ,in motion so accelerated1 as that the length transmitted be always to the length transmitted in duplicate proportion to their times1 the length uniformly transmitted in the whole time1 and with impetus all the way equal to that which is last acquired1 is as a parabola to a parallelogram of the same altitude and base1 that is1 as #b to #c4 ,for the parabola ,a ,k ,i ,b is the impetus increasing in the time ,a ,b2 and the parallelogram ,a ,i is the greatest uniform impetus multiplied into the same time ,a ,b4 ,wherefore the lengths transmitted will be as a parabola to a parallelogram1 c41 that is1 as #b to #c4 #e4 ,if ,i should proceed to the explication of such motions as are made by impetus increasing in proportion triplicate1 quadruplicate1 quintuplicate1 c41 to that of their times1 it would be a labour infinite and unnecessary4 ,for by the same method by which ,i have computed such lengths1 as are transmitted with impetus increasing in single and duplicate proportion1 any man may compute such as are transmitted with impetus increasing in triplicate1 quadruplicate1 or what other proportion he pleases4 ,in making which computation he shall find1 that where the impetus increase in proportion triplicate to that of the times1 there the whole velocity will be designed by the first parabolaster 7of which see the next chapter72 and the lengths transmitted will be in proportion quadruplicate to that of the times4 ,and in like manner1 where the impetus increase in quadruplicate proportion to that of the times1 that there the whole velocity will be designed by the second parabolaster1 and the lengths transmitted will be in quintuplicate proportion to that of the times2 and so on continually4 ,sidenote3 ,if two bodies be moved with uniform motion through two lengths1 the proportion of those lengths to one another1 will be compounded of the proportions of time to time1 and impetus to impetus1 directly taken4 #f4 ,if two bodies with uniform motion transmit two lengths1 each with its own impetus and time1 the proportion of the lengths transmitted will be compounded of the proportions of time to time1 and impetus to impetus1 directly taken4 ,let two bodies be moved uniformly 7as in fig4 #c71 one in the time ,a ,b with the impetus ,a ,c1 the other in the time ,a ,d with the impetus ,a ,e4 ,i say the lengths transmitted have their proportion to one another compounded of the proportions of ,a ,b to ,a ,d1 and of ,a ,c to ,a ,e4 ,for let any length whatsoever1 as ,z1 be transmitted by one of the bodies in the time ,a ,b with the impetus ,a ,c2 and any other length1 as ,x1 be transmitted by the other body in the time ,a ,d with the impetus ,a ,e2 and let the parallelograms ,a ,f and ,a ,g be completed4 ,seeing now ,z is to ,x 7by art4 #b7 as the impetus ,a ,c multiplied into the time ,a ,b is to the impetus ,a ,e multiplied into the time ,a ,d1 that is1 as ,a ,f to ,a ,g2 the proportion of ,z to ,x will be compounded of the same proportions1 of which the proportion of ,a ,f to ,a ,g is compounded2 but the proportion of ,a ,f to ,a ,g is compounded of the proportions of the side ,a ,b to the side ,a ,d1 and of the side ,a ,c to the side ,a ,e 7as is evident by the ,elements of ,euclid71 that is1 of the proportions of the time ,a ,b to the time ,a ,d1 and of the impetus ,a ,c to the impetus ,a ,e4 ,wherefore also the proportion of ,z to ,x is compounded of the same proportions of the time ,a ,b to the time ,a ,d1 and of the impetus ,a ,c to the impetus ,a ,e2 which was to be demonstrated4 ,coroll4 ,i4 ,when two bodies are moved with uniform motion1 if the times and impetus be in reciprocal proportion1 the lengths transmitted shall be equal4 ,for if it were as ,a ,b to ,a ,d 7in the same fig4 #c7 so reciprocally ,a ,e to ,a ,c1 the proportion of ,a ,f to ,a ,g would be compounded of the proportions of ,a ,b to ,a ,d1 and of ,a ,c to ,a ,e1 that is1 of the proportions of ,a ,b to ,a ,d1 and of ,a ,d to ,a ,b4 ,wherefore1 ,a ,f would be to ,a ,g as ,a ,b to ,a ,b1 that is1 equal2 and so the two products made by the multiplication of impetus into time would be equal2 and by consequent1 ,z would be equal to ,x4 ,coroll4 ,,ii4 ,if two bodies be moved in the same time1 but with different impetus1 the lengths transmitted will be as impetus to impetus4 ,for if the time of both of them be ,a ,d1 and their different impetus be ,a ,e and ,a ,c1 the proportion of ,a ,g to ,d ,c will be compounded of the proportions of ,a ,e to ,a ,c and of ,a ,d to ,a ,d1 that is1 of the proportions of ,a ,e to ,a ,c and of ,a ,c to ,a ,c2 and so the proportion of ,a ,g to ,d ,c1 that is1 the proportion of length to length1 will be as ,a ,e to ,a ,c1 that is1 as that of impetus to impetus4 ,in like manner1 if two bodies be moved uniformly1 and both of them with the same impetus1 but in different times1 the proportion of the lengths transmitted by them will be as that of their times4 ,for if they have both the same impetus ,a ,c1 and their different times be ,a ,b and ,a ,d1 the proportion of ,a ,f to ,d ,c will be compounded of the proportions of ,a ,b to ,a ,d and of ,a ,c to ,a ,c2 that is1 of the proportions of ,a ,b to ,a ,d and of ,a ,d to ,a ,d2 and therefore the proportion of ,a ,f to ,d ,c1 that is1 of length to length1 will be the same with that of ,a ,b to ,a ,d1 which is the proportion of time to time4 ,sidenote3 ,if two bodies pass through two lengths with uniform motion1 the proportion of their times to one another1 will be compounded of the proportions of length to length1 and impetus to impetus reciprocally taken2 also the proportion of their impetus to one another1 will be compounded of the proportions of length to length1 and time to time reciprocally taken4 #g4 ,if two bodies pass through two lengths with uniform motion1 the proportion of the times in which they are moved will be compounded of the proportions of length to length and impetus to impetus reciprocally taken4 ,for let any two lengths be given1 as 7in the same fig4 #c7 ,z and ,x1 and let one of them be transmitted with the impetus ,a ,c1 the other with the impetus ,a ,e4 ,i say the proportion of the times in which they are transmitted1 will be compounded of the proportions of ,z to ,x1 and of ,a ,e1 which is the impetus with which ,x is transmitted1 to ,a ,c1 the impetus with which ,z is transmitted4 ,for seeing ,a ,f is the product of the impetus ,a ,c multiplied into the time ,a ,b1 the time of motion through ,z will be a line1 which is made by the application of the parallelogram ,a ,f to the strait line ,a ,c1 which line is ,a ,b2 and therefore ,a ,b is the time of motion through ,z4 ,in like manner1 seeing ,a ,g is the product of the impetus ,a ,e multiplied into the time ,a ,d1 the time of motion through ,x will be a line which is made by the application of ,a ,g to the strait line ,a ,d2 but ,a ,d is the time of motion through ,x4 ,now the proportion of ,a ,b to ,a ,d is compounded of the proportions of the parallelogram ,a ,f to the parallelogram ,a ,g1 and of the impetus ,a ,e to the impetus ,a ,c2 which may be demonstrated thus4 ,put the parallelograms in order ,a ,f1 ,a ,g1 ,d ,c1 and it will be manifest that the proportion of ,a ,f to ,d ,c is compounded of the proportions of ,a ,f to ,a ,g and of ,a ,g to ,d ,c2 but ,a ,f is to ,d ,c as ,a ,b to ,a ,d2 wherefore also the proportion of ,a ,b to ,a ,d is compounded of the proportions of ,a ,f to ,a ,g and of ,a ,g to ,d ,c4 ,and because the length ,z is to the length ,x as ,a ,f is to ,a ,g1 and the impetus ,a ,e to the impetus ,a ,c as ,a ,g to ,d ,c1 therefore the proportion of ,a ,b to ,a ,d will be compounded of the proportions of the length ,z to the length ,x1 and of the impetus ,a ,e to the impetus ,a ,c2 which was to be demonstrated4 ,in the same manner it may be proved1 that in two uniform motions the proportion of the impetus is compounded of the proportions of length to length and of time to time reciprocally taken4 ,for if we suppose ,a ,c 7in the same fig4 #c7 to be the time1 and ,a ,b the impetus with which the length ,z is passed through2 and ,a ,e to be the time1 and ,a ,d the impetus with which the length ,x is passed through1 the demonstration will proceed as in the last article4 ,sidenote3 ,if a body be carried on with uniform motion by two movents together1 which meet in an angle1 the line by which it passes will be a strait line1 subtending the complement of that angle to #b right angles4 #h4 ,if a body be carried by two movents together1 which move with strait and uniform motion1 and concur in any given angle1 the line by which that body passes will be a strait line4 ,let the movent ,a ,b 7in fig4 #d7 have strait and uniform motion1 and be moved till it come into the place ,c ,d2 and let another movent ,a ,c1 having likewise strait and uniform motion1 and making with the movent ,a ,b any given angle ,c ,a ,b1 be understood to be moved in the same time to ,d ,b2 and let the body be placed in the point of their concourse1 ,a4 ,i say the line which that body describes with its motion is a strait line4 ,for let the parallelogram ,a ,b ,d ,c be completed1 and its diagonal ,a ,d be drawn2 and in the strait line ,a ,b let any point ,e be taken2 and from it let ,e ,f be drawn parallel to the strait lines ,a ,c and ,b ,d1 cutting ,a ,d in ,g2 and through the point ,g let ,h ,i be drawn parallel to the strait lines ,a ,b and ,c ,d2 and lastly1 let the measure of the time be ,a ,c4 ,seeing therefore both the motions are made in the same time1 when ,a ,b is in ,c ,d1 the body also will be in ,c ,d2 and in like manner1 when ,a ,c is in ,b ,d1 the body will be in ,b ,d4 ,but ,a ,b is in ,c ,d at the same time when ,a ,c is in ,b ,d2 and therefore the body will be in ,c ,d and ,b ,d at the same time2 wherefore it will be in the common point ,d4 ,again1 seeing the motion from ,a ,c to ,b ,d is uniform1 that is1 the spaces transmitted by it are in proportion to one another as the times in which they are transmitted1 when ,a ,c is in ,e ,f1 the proportion of ,a ,b to ,a ,e will be the same with that of ,e ,f to ,e ,g1 that is1 of the time ,a ,c to the time ,a ,h4 ,wherefore ,a ,b will be in ,h ,i in the same time in which ,a ,c is in ,e ,f1 so that the body will at the same time be in ,e ,f and ,h ,i1 and therefore in their common point ,g4 ,and in the same manner it will be1 wheresoever the point ,e be taken between ,a and ,b4 ,wherefore the body will always be in the diagonal ,a ,d2 which was to be demonstrated4 ,coroll4 ,from hence it is manifest1 that the body will be carried through the same strait line ,a ,d1 though the motion be not uniform1 provided it have like acceleration2 for the proportion of ,a ,b to ,a ,e will always be the same with that of ,a ,c to ,a ,h4 ,sidenote3 ,if a body be carried by two movents together1 one of them being moved with uniform1 the other with accelerated motion1 and the proportion of their lengths to their times being explicable in numbers1 how to find out what line that body describes4 #i4 ,if a body be carried by two movents together1 which meet in any given angle1 and are moved1 the one uniformly1 the other with motion uniformly accelerated from rest1 that is1 that the proportion of their impetus be as that of their times1 that is1 that the proportion of their lengths be duplicate to that of the lines of their times1 till the line of greatest impetus acquired by acceleration be equal to that of the line of time of the uniform motion2 the line in which the body is carried will be the crooked line of a semiparabola1 whose base is the impetus last acquired1 and vertex the point of rest4 ,let the straight line ,a ,b 7in fig4 #e7 be understood to be moved with uniform motion to ,c ,d2 and let another movent in the strait line ,a ,c be supposed to be moved in the same time to ,b ,d1 but with motion uniformly accelerated1 that is1 with such motion1 that the proportion of the spaces which are transmitted be always duplicate to that of the times1 till the impetus acquired be ,b ,d equal to the strait line ,a ,c2 and let the semiparabola ,a ,g ,d ,b be described4 ,i say that by the concourse of those two movents1 the body will be carried through the semiparabolical crooked line ,a ,g ,d4 ,for let the parallelogram ,a ,b ,d ,c be completed2 and from the point ,e1 taken anywhere in the strait line ,a ,b1 let ,e ,f be drawn parallel to ,a ,c and cutting the crooked line in ,g2 and lastly1 through the point ,g let ,h ,i be drawn parallel to the strait lines ,a ,b and ,c ,d4 ,seeing therefore the proportion of ,a ,b to ,a ,e is by supposition duplicate to the proportion of ,e ,f to ,e ,g1 that is1 of the time ,a ,c to the time ,a ,h1 at the same time when ,a ,c is in ,e ,f1 ,a ,b will be in ,h ,i2 and therefore the moved body will be in the common point ,g4 ,and so it will always be1 in what part soever of ,a ,b the point ,e be taken4 ,wherefore the moved body will always be found in the parabolical line ,a ,g ,d2 which was to be demonstrated4 #aj4 ,if a body be carried by two movents together1 which meet in any given angle1 and are moved the one uniformly1 the other with impetus increasing from rest1 till it be equal to that of the uniform motion1 and with such acceleration1 that the proportion of the lengths transmitted be every where triplicate to that of the times in which they are transmitted2 the line1 in which that body is moved1 will be the crooked line of the first semiparabolaster of two means1 whose base is the impetus last acquired4 ,let the strait line ,a ,b 7in the #fth figure7 be moved uniformly to ,c ,d2 and let another movent ,a ,c be moved at the same time to ,b ,d with motion so accelerated1 that the proportion of the lengths transmitted be everywhere triplicate to the proportion of their times2 and let the impetus acquired in the end of that motion be ,b ,d1 equal to the strait line ,a ,c2 and lastly1 let ,a ,g ,d be the crooked line of the first semiparabolaster of two means4 ,i say1 that by the concourse of the two movents together1 the body will be always in that crooked line ,a ,g ,d4 ,for let the parallelogram ,a ,b ,d ,c be completed2 and from the point ,e1 taken anywhere in the strait line ,a ,b1 let ,e ,f be drawn parallel to ,a ,c1 and cutting the crooked line in ,g2 and through the point ,g let ,h ,i be drawn parallel to the strait lines ,a ,b and ,c ,d4 ,seeing therefore the proportion of ,a ,b to ,a ,e is1 by supposition1 triplicate to the proportion of ,e ,f to ,e ,g1 that is1 of the time ,a ,c to the time ,a ,h1 at the same time when ,a ,c is in ,e ,f1 ,a ,b will be in ,h ,i2 and therefore the moved body will be in the common point ,g4 ,and so it will always be1 in what part soever of ,a ,b the point ,e be taken2 and by consequent1 the body will always be in the crooked line ,a ,g ,d2 which was to be demonstrated4 #aa4 ,by the same method it may be shown1 what line it is that is made by the motion of a body carried by the concourse of any two movents1 which are moved one of them uniformly1 the other with acceleration1 but in such proportions of spaces and times as are explicable by numbers1 as duplicate1 triplicate1 c41 or such as may be designed by any broken number whatsoever4 ,for which this is the rule4 ,let the two numbers of the length and time be added together2 and let their sum be the denominator of a fraction1 whose numerator must be the number of the length4 ,seek this fraction in the table of the third article of the ,x,v,i,ith chapter2 and the line sought will be that1 which denominates the three-sided figure noted on the left hand2 and the kind of it will be that1 which is numbered above over the fraction4 ,for example1 let there be a concourse of two movents1 whereof one is moved uniformly1 the other with motion so accelerated1 that the spaces are to the times as #e to #c4 ,let a fraction be made whose denominator is the sum of #e and #c1 and the numerator #e1 namely the fraction #eh4 ,seek in the table1 and you will find #eh to be the third in that row1 which belongs to the three-sided figure of four means4 ,wherefore the line of motion made by the concourse of two such movents1 as are last of all described1 will be the crooked line of the third parabolaster of four means4 #ab4 ,if motion be made by the concourse of two movents1 whereof one is moved uniformly1 the other beginning from rest in the angle of concourse with any acceleration whatsoever2 the movent1 which is moved uniformly1 shall put forward the moved body in the several parallel spaces1 less than if both the movents had uniform motion2 and still less and less1 as the motion of the other movent is more and more accelerated4 ,let the body be placed in ,a1 7in the #gth figure7 and be moved by two movents1 by one with uniform motion from the strait line ,a ,b to the strait line ,c ,d parallel to it2 and by the other with any acceleration1 from the strait line ,a ,c to the strait line ,b ,d parallel to it2 and in the parallelogram ,a ,b ,d ,c let a space be taken between any two parallels ,e ,f and ,g ,h4 ,i say1 that whilst the movent ,a ,c passes through the latitude which is between ,e ,f and ,g ,h1 the body is less moved forwards from ,a ,b towards ,c ,d1 than it would have been1 if the motion from ,a ,c to ,b ,d had been uniform4 ,for suppose that whilst the body is made to descend to the parallel ,e ,f by the power of the movent from ,a ,c towards ,b ,d1 the same body in the same time is moved forwards to any point ,f in the line ,e ,f1 by the power of the movent from ,a ,b towards ,c ,d2 and let the strait line ,a ,f be drawn and produced indeterminately1 cutting ,g ,h in ,h4 ,seeing therefore1 it is as ,a ,e to ,a ,g1 so ,e ,f to ,g ,h2 if ,a ,c should descend towards ,b ,d with uniform motion1 the body in the time ,g ,h1 7for ,i make ,a ,c and its parallels the measure of time17 would be found in the point ,h4 ,but because ,a ,c is supposed to be moved towards ,b ,d with motion continually accelerated1 that is1 in greater proportion of space to space1 than of time to time1 in the time ,g ,h the body will be in some parallel beyond it1 as between ,g ,h and ,b ,d4 ,suppose now that in the end of the time ,g ,h it be in the parallel ,i ,k1 and in ,i ,k let ,i ,l be taken equal to ,g ,h4 ,when therefore the body is in the parallel ,i ,k1 it will be in the point ,l4 ,wherefore when it was in the parallel ,g ,h1 it was in some point between ,g and ,h1 as in the point ,m2 but if both the motions had been uniform1 it had been in the point ,h2 and therefore whilst the movent ,a ,c passes over the latitude which is between ,e ,f and ,g ,h1 the body is less moved forwards from ,a ,b towards ,c ,d1 than it would have been1 if both the motions had been uniform2 which was to be demonstrated4 #ac4 ,any length being given1 which is passed through in a given time with uniform motion1 to find out what length shall be passed through in the same time with motion uniformly accelerated1 that is1 with such motion that the proportion of the lengths passed through be continually duplicate to that of their times1 and that the line of the impetus last acquired be equal to the line of the whole time of the motion4 ,let ,a ,b 7in the #hth figure7 be a length1 transmitted with uniform motion in the time ,a ,c2 and let it be required to find another length1 which shall be transmitted in the same time with motion uniformly accelerated1 so that the line of the impetus last acquired be equal to the strait line ,a ,c4 ,let the parallelogram ,a ,b ,d ,c be completed2 and let ,b ,d be divided in the middle at ,e2 and between ,b ,e and ,b ,d let ,b ,f be a mean proportional2 and let ,a ,f be drawn and produced till it meet with ,c ,d produced in ,g2 and lastly1 let the parallelogram ,a ,c ,g ,h be completed4 ,i say1 ,a ,h is the length required4 ,for as duplicate proportion is to single proportion1 so let ,a ,h be to ,a ,i1 that is1 let ,a ,i be the half of ,a ,h2 and let ,i ,k be drawn parallel to the strait line ,a ,c1 and cutting the diagonal ,a ,d in ,k1 and the strait line ,a ,g in ,l4 ,seeing therefore ,a ,i is the half of ,a ,h1 ,i ,l will also be the half of ,b ,d1 that is1 equal to ,b ,e2 and ,i ,k equal to ,b ,f2 for ,b ,d1 that is1 ,g ,h1 ,b ,f1 and ,b ,e1 that is1 ,i ,l1 being continual proportionals1 ,a ,h1 ,a ,b and ,a ,i will also be continual proportionals4 ,but as ,a ,b is to ,a ,i1 that is1 as ,a ,h is to ,a ,b1 so is ,b ,d to ,i ,k1 and so also is ,g ,h1 that is1 ,b ,d to ,b ,f2 and therefore ,b ,f and ,i ,k are equal4 ,now the proportion of ,a ,h to ,a ,i is duplicate to the proportion of ,a ,b to ,a ,i1 that is1 to that of ,b ,d to ,i ,k1 or of ,g ,h to ,i ,k4 ,wherefore the point ,k will be in a parabola1 whose diameter is ,a ,h1 and base ,g ,h1 which ,g ,h is equal to ,a ,c4 ,the body therefore proceeding from rest in ,a1 with motion uniformly accelerated in the time ,a ,c1 when it has passed through the length ,a ,h1 will acquire the impetus ,g ,h equal to the time ,a ,c1 that is1 such impetus1 as that with it the body will pass through the length ,a ,c in the time ,a ,c4 ,wherefore any length being given1 c41 which was propounded to be done4 #ad4 ,any length being given1 which in a given time is transmitted with uniform motion1 to find out what length shall be transmitted in the same time with motion so accelerated1 that the lengths transmitted be continually in triplicate proportion to that of their times1 and the line of the impetus last of all acquired be equal to the line of time given4 ,let the given length ,a ,b 7in the #ith figure7 be transmitted with uniform motion in the time ,a ,c2 and let it be required to find what length shall be transmitted in the same time with motion so accelerated1 that the lengths transmitted be continually in triplicate proportion to that of their times1 and the impetus last acquired be equal to the time given4 ,let the parallelogram ,a ,b ,d ,c be completed2 and let ,b ,d be so divided in ,e1 that ,b ,e be a third part of the whole ,b ,d2 and let ,b ,f be a mean proportional between ,b ,d and ,b ,e2 and let ,a ,f be drawn and produced till it meet the strait line ,c ,d in ,g2 and lastly1 let the parallelogram ,a ,c ,g ,h be completed4 ,i say1 ,a ,h is the length required4 ,for as triplicate proportion is to single proportion1 so let ,a ,h be to another line1 ,a ,i1 that is1 make ,a ,i a third part of the whole ,a ,h2 and let ,i ,k be drawn parallel to the strait line ,a ,c1 cutting the diagonal ,a ,d in ,k1 and the strait line ,a ,g in ,l2 then1 as ,a ,b is to ,a ,i1 so let ,a ,i be to another1 ,a ,n2 and from the point ,n let ,n ,q be drawn parallel to ,a ,c1 cutting ,a ,g1 ,a ,d1 and ,f ,k produced in ,p1 ,m1 and ,o2 and last of all1 let ,f ,o and ,l ,m be drawn1 which will be equal and parallel to the strait lines ,b ,n and ,i ,n4 ,by this construction1 the lengths transmitted ,a ,h1 ,a ,b1 ,a ,i1 and ,a ,n1 will be continual proportionals2 and1 in like manner1 the times ,g ,h1 ,b ,f1 ,i ,l and ,n ,p1 that is1 ,n ,q1 ,n ,o1 ,n ,m and ,n ,p1 will be continual proportionals1 and in the same proportion with ,a ,h1 ,a ,b1 ,a ,i and ,a ,n4 ,wherefore the proportion of ,a ,h1 ,a ,b1 ,a ,i and ,a ,n4 ,wherefore the proportion of ,a ,h to ,a ,n is the same with that of ,b ,d1 that is1 of ,n ,q to ,n ,p2 and the proportion of ,n ,q to ,n ,p triplicate to that of ,n ,q to ,n ,o1 that is1 triplicate to that of ,b ,d to ,i ,k2 wherefore also the length ,a ,h is to the length ,a ,n in triplicate proportion to that of the time ,b ,d1 to the time ,i ,k2 and therefore the crooked line of the first three-sided figure of two means whose diameter is ,a ,h1 and base ,g ,h equal to ,a ,c1 shall pass through the point ,o2 and consequently1 ,a ,h shall be transmitted in the time ,a ,c1 and shall have its last acquired impetus ,g ,h equal to ,a ,c1 and the proportions of the lengths acquired in any of the times triplicate to the proportions of the times themselves4 ,wherefore ,a ,h is the length required to be found out4 ,by the same method1 if a length be given which is transmitted with uniform motion in any given time1 another length may be found out which shall be transmitted in the same time with motion so accelerated1 that the lengths transmitted shall be to the times in which they are transmitted1 in proportion quadruplicate1 quintuplicate1 and so on infinitely4 ,for if ,b ,d be divided in ,e1 so that ,b ,d be to ,b ,e as #d to #a2 and there be taken between ,b ,d and ,b ,e a mean proportional ,f ,b2 and as ,a ,h is to ,a ,b1 so ,a ,b be made to a third1 and again so that third to a fourth1 and that fourth to a fifth1 ,a ,n1 so that the proportion of ,a ,h to ,a ,n be quadruplicate to that of ,a ,h to ,a ,b1 and the parallelogram ,n ,b ,f ,o be completed1 the crooked line of the first three-sided figure of three means will pass through the point ,o2 and consequently1 the body moved will acquire the impetus ,g ,h equal to ,a ,c in the time ,a ,c4 ,and so of the rest4 #ae4 ,also1 if the proportion of the lengths transmitted be to that of their times1 as any number to any number1 the same method serves for the finding out of the length transmitted with such impetus1 and in such time4 ,for let ,a ,c 7in the #ajth figure7 be the time in which a body is transmitted with uniform motion from ,a to ,b2 and the parallelogram ,a ,b ,d ,c being completed1 let it be required to find out a length in which that body may be moved in the same time ,a ,c from ,a1 with motion so accelerated1 that the proportion of the lengths transmitted to that of the times be continually as #c to #b4 ,let ,b ,d be so divided in ,e1 that ,b ,d be to ,b ,e as #c to #b2 and between ,b ,d and ,b ,e let ,b ,f be a mean proportional2 and let ,a ,f be drawn and produced till it meet with ,c ,d produced in ,g2 and making ,a ,m a mean proportional between ,a ,h and ,a ,b1 let it be as ,a ,m to ,a ,b1 so ,a ,b to ,a ,i2 and so the proportion of ,a ,h to ,a ,i will be to that of ,a ,h to ,a ,b as #c to #b2 for of the proportions1 of which that of ,a ,h to ,a ,m is one1 that of ,a ,h to ,a ,b is two1 and that of ,a ,h to ,a ,i is three2 and consequently1 as #c to #b to that of ,g ,h to ,b ,f1 and 7,f ,k being drawn parallel to ,b ,i and cutting ,a ,d in ,k7 so likewise to that of ,g ,h or ,b ,d to ,i ,k4 ,wherefore the proportion of the length ,a ,h to ,a ,i is to the proportion of the time ,b ,d to ,i ,k as #c to #b2 and therefore if in the time ,a ,c the body be moved with accelerated motion1 as was propounded1 till it acquire the impetus ,h ,g equal to ,a ,c1 the length transmitted in the same time will be ,a ,h4 #af4 ,but if the proportion of the lengths to that of the times had been as #d to #c1 there should then have been taken two mean proportionals between ,a ,h and ,a ,b1 and their proportion should have been continued one term further1 so that ,a ,h to ,a ,b might have three of the same proportions1 of which ,a ,h to ,a ,i has four2 and all things else should have been done as is already shown4 ,now the way how to interpose any number of means between two lines given1 is not yet found out4 ,nevertheless this may stand for a general rule2 if there be a time given1 and a length be transmitted in that time with uniform motion2 as for example1 if the time be ,a ,c1 and the length ,a ,b1 the strait line ,a ,g1 which determines the length ,c ,g or ,a ,h1 transmitted in the same time ,a ,c with any accelerated motion1 shall so cut ,b ,d in ,f1 that ,b ,f shall be a mean proportional between ,b ,d and ,b ,e1 ,b ,e being so taken in ,b ,d1 that the proportion of length to length be everywhere to the proportion of time to time1 as the whole ,b ,d is to its part ,b ,e4 #ag4 ,if in a given time two lengths be transmitted1 one with uniform motion1 the other with motion accelerated in any proportion of the lengths to the times2 and again1 in part of the same time1 parts of the same lengths be transmitted with the same motions1 the whole length will exceed the other length in the same proportion in which one part exceeds the other part4 ,for example1 let ,a ,b 7in the #hth figure7 be a length transmitted in the time ,a ,c1 with uniform motion2 and let ,a ,h be another length transmitted in the same time with motion uniformly accelerated1 so that the impetus last acquired be ,g ,h equal to ,a ,c2 and in ,a ,h let any part ,a ,i be taken1 and transmitted in part of the time ,a ,c with uniform motion2 and let another part ,a ,b be taken and transmitted in the same part of the time ,a ,c with motion uniformly accelerated2 ,i say1 that as ,a ,h is to ,a ,b1 so will ,a ,b be to ,a ,i4 ,let ,b ,d be drawn parallel and equal to ,h ,g1 and divided in the midst at ,e1 and between ,b ,d and ,b ,e let a mean proportional be taken as ,b ,f2 and the strait line ,a ,g1 by the demonstration of art4 #ac1 shall pass through ,f4 ,and dividing ,a ,h in the midst at ,i1 ,a ,b shall be a mean proportional between ,a ,h and ,a ,i4 ,again1 because ,a ,i and ,a ,b are described by the same motions1 if ,i ,k be drawn parallel and equal to ,b ,f or ,a ,m1 and divided in the midst at ,n1 and between ,i ,k and ,i ,n be taken the mean proportional ,i ,l1 the strait line ,a ,f will1 by the demonstration of the same art4 #ac1 pass through ,l4 ,and dividing ,a ,b in the midst at ,o1 the line ,a ,i will be a mean proportional between ,a ,b and ,a ,o4 ,where ,a ,b is divided in ,i and ,o1 in like manner as ,a ,h is divided in ,b and ,i2 and as ,a ,h to ,a ,b1 so is ,a ,b to ,a ,i4 ,which was to be proved4 ,coroll4 ,also as ,a ,h to ,a ,b1 so is ,h ,b to ,b ,i2 and so also ,b ,i to ,i ,o4 ,and as this1 where one of the motions is uniformly accelerated1 is proved out of the demonstration of art4 #ac2 so1 when the accelerations are in double proportion to the times1 the same may be proved by the demonstration of art4 #ad2 and by the same method in all other accelerations1 whose proportions to the times are explicable in numbers4 #ah4 ,if two sides1 which contain an angle in any parallelogram1 be moved in the same time to the sides opposite to them1 one of them with uniform motion1 the other with motion uniformly accelerated2 that side1 which is moved uniformly1 will affect as much with its concourse through the whole length transmitted1 as it would do if the other motion were also uniform1 and the length transmitted by it in the same time were a mean proportional between the whole and the half4 ,let the side ,a ,b of the parallelogram ,a ,b ,d ,c1 7in the #aath figure7 be understood to be moved with uniform motion till it be coincident with ,c ,d2 and let the time of that motion be ,a ,c or ,b ,d4 ,also in the same time let the side ,a ,c be understood to be moved with motion uniformly accelerated1 till it be coincident with ,b ,d2 then dividing ,a ,b in the middle in ,e1 let ,a ,f be made a mean proportional between ,a ,b and ,a ,e2 and drawing ,f ,g parallel to ,a ,c1 let the side ,a ,c be understood to be moved in the same time ,a ,c with uniform motion till it be coincident with ,f ,g4 ,i say1 the whole ,a ,b confers as much to the velocity of the body placed in ,a1 when the motion of ,a ,c is uniformly accelerated till it comes to ,b ,d1 as the part ,a ,f confers to the same1 when the side ,a ,c is moved uniformly and in the same time to ,f ,g4 ,for seeing ,a ,f is a mean proportional between the whole ,a ,b and its half ,a ,e1 ,b ,d will 7by the #acth article7 be the last impetus acquired by ,a ,c1 with motion uniformly accelerated till it come to the same ,b ,d2 and consequently1 the strait line ,f ,b will be the excess1 by which the length1 transmitted by ,a ,c with motion uniformly accelerated1 will exceed the length transmitted by the same ,a ,c in the same time with uniform motion1 and with impetus every where equal to ,b ,d4 ,wherefore1 if the whole ,a ,b be moved uniformly to ,c ,d in the same time in which ,a ,c is moved uniformly to ,f ,g1 the part ,f ,b1 seeing it concurs not at all with the motion of the side ,a ,c which is supposed to be moved only to ,f ,g1 will confer nothing to its motion4 ,again1 supposing the side ,a ,c to be moved to ,b ,d with motion uniformly accelerated1 the side ,a ,b with its uniform motion to ,c ,d will less put forwards the body when it is accelerated in all the parallels1 than when it is not at all accelerated2 and by how much the greater the acceleration is1 by so much the less it will put it forwards1 as is shown in the #abth article4 ,when therefore ,a ,c is in ,f ,g with accelerated motion1 the body will not be in the side ,c ,d at the point ,g1 but at the point ,d2 so that ,g ,d will be the excess1 by which the length transmitted with accelerated motion to ,b ,d exceeds the length transmitted with uniform motion to ,f ,g2 so that the body by its acceleration avoids the action of the part ,a ,f1 and comes to the side ,c ,d in the time ,a ,c1 and makes the length ,c ,d1 which is equal to the length ,a ,b4 ,wherefore uniform motion from ,a ,b to ,c ,d in the time ,a ,c1 works no more in the whole length ,a ,b upon the body uniformly accelerated from ,a ,c to ,b ,d1 than if ,a ,c were moved in the same time with uniform motion to ,f ,g2 the difference consisting only in this1 that when ,a ,b works upon the body uniformly moved from ,a ,c to ,f ,g1 that1 by which the accelerated motion exceeds the uniform motion1 is altogether in ,f ,b or ,g ,d2 but when the same ,a ,b works upon the body accelerated1 that1 by which the accelerated motion exceeds the uniform motion1 is dispersed through the whole length ,a ,b or ,c ,d1 yet1 so that if it were collected and put together1 it would be equal to the same ,f ,b or ,g ,d4 ,wherefore1 if two sides which contain an angle1 c42 which was to be demonstrated4 #ai4 ,if two transmitted lengths have to their times any other proportion explicable by number1 and the side ,a ,b be so divided in ,e1 that ,a ,b be to ,a ,e in the same proportion which the lengths transmitted have to the times in which they are transmitted1 and between ,a ,b and ,a ,e there be taken a mean proportional ,a ,f2 it may be shown by the same method1 that the side1 which is moved with uniform motion1 works as much with its concourse through the whole length ,a ,b1 as it would do if the other motion were also uniform1 and the length transmitted in the same time ,a ,c were that mean proportional ,a ,f4 ,and thus much concerning motion by concourse4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xvi4 ,fig4 #a-aa ---------------------------------------- -------------------------------- ------- ,,chap4 ,,xvii4 ,,of ,,figures ,,deficient4 #a4 ,definitions of a deficient figure2 of a complete figure2 of the complement of a deficient figure2 and of proportions which are proportional and commensurable to one another4--#b4 ,the proportion of a deficient figure to its complement4--#c4 ,the proportions of deficient figures to the parallelograms in which they are described1 set forth in a table4--#d4 ,the description and production of the same figures4--#e4 ,the drawing of tangents to them4--#f4 ,in what proportion the same figures exceed a strait-lined triangle of the same altitude and base4--#g4 ,a table of solid deficient figures described in a cylinder4--#h4 ,in what proportion the same figures exceed a cone of the same altitude and base4--#i4 ,how a plain deficient figure may be described in a parallelogram1 so that it be to a triangle of the same base and altitude1 as another deficient figure1 plain or solid1 twice taken1 is to the same deficient figure1 together with the complete figure in which it is described4--#aj4 ,the transferring of certain properties of deficient figures described in a parallelogram to the proportions of the spaces transmitted with several degrees of velocity4--#aa4 ,of deficient figures described in a circle4--#ab4 ,the proposition demonstrated in art4 #b confirmed from the elements of philosophy4--#ac4 ,an unusual way of reasoning concerning the equality between the superficies of a portion of a sphere and a circle4--#ad4 ,how from the description of deficient figures in a parallelogram1 any number of mean proportionals may be found out between two given strait lines4 ,sidenote3 ,definition of a deficient figure4 #a4 ,i call those deficient figures which may be understood to be generated by the uniform motion of some quantity1 which decreases continually1 till at last it have no magnitude at all4 ,sidenote3 ,definitions of a complete figure2 of the complement of a deficient figure2 and of proportions which are proportional commensurable to one another4 ,and ,i call that a complete figure1 answering to a deficient figure1 which is generated with the same motion and in the same time1 by a quantity which retains always its whole magnitude4 ,the complement of a deficient figure is that which being added to the deficient figure makes it complete4 ,four proportions are said to be proportional1 when the first of them is to the second as the third is to the fourth4 ,for example1 if the first proportion be duplicate to the second1 and again1 the third be duplicate to the fourth1 those proportions are said to be proportional4 ,and commensurable proportions are those1 which are to one another as number to number4 ,as when to a proportion given1 one proportion is duplicate1 another triplicate1 the duplicate proportion will be to the triplicate proportion as #b to #c2 but to the given proportion it will be as #b to #a2 and therefore ,i call those three proportions commensurable4 ,sidenote3 ,the proportion of a deficient figure to its complement4 #b4 ,a deficient figure1 which is made by a quantity continually decreasing to nothing by proportions everywhere proportional and commensurable1 is to its complement1 as the proportion of the whole altitude to an altitude diminished in any time is to the proportion of the whole quantity1 which describes the figure1 to the same quantity diminished in the same time4 ,let the quantity ,a ,b 7in fig4 #a71 by its motion through the altitude ,a ,c1 describe the complete figure ,a ,d2 and again1 let the same quantity1 by decreasing continually to nothing in ,c1 describe the deficient figure ,a ,b ,e ,f ,c1 whose complement will be the figure ,b ,d ,c ,f ,e4 ,now let ,a ,b be supposed to be moved till it lie in ,g ,k1 so that the altitude diminished be ,g ,c1 and ,a ,b diminished be ,g ,e2 and let the proportion of the whole altitude ,a ,c to the diminished altitude ,g ,c1 be1 for example1 triplicate to the proportion of the whole quantity ,a ,b or ,g ,k to the diminished quantity ,g ,e4 ,and in like manner1 let ,h ,i be taken equal to ,g ,e1 and let it be diminished to ,h ,f2 and let the proportion of ,g ,c to ,h ,c be triplicate to that of ,h ,i to ,h ,f2 and let the same be done in as many parts of the strait line ,a ,c as is possible2 and a line be drawn through the points ,b1 ,e1 ,f and ,c4 ,i say the deficient figure ,a ,b ,e ,f ,c is to its complement ,b ,d ,c ,f ,e as #c to #a1 or as the proportion of ,a ,c to ,g ,c is to the proportion of ,a ,b1 that is1 of ,g ,k to ,g ,e4 ,for 7by art4 #b1 chapter ,,xv47 the proportion of the complement ,b ,e ,f ,c ,d to the deficient figure ,a ,b ,e ,f ,c is all the proportions of ,d ,b to ,b ,a1 ,o ,e to ,e ,g1 ,q ,f to ,f ,h1 and of all the lines parallel to ,d ,b terminated in the line ,b ,e ,f ,c1 to all the parallels to ,a ,b terminated in the same points of the line ,b ,e ,f ,c4 ,and seeing the proportions of ,d ,b to ,o ,e1 and of ,d ,b to ,q ,f c4 are everywhere triplicate of the proportions of ,a ,b to ,g ,e1 and of ,a ,b to ,h ,f c4 the proportions of ,h ,f to ,a ,b1 and of ,g ,e to ,a ,b c4 7by art4 #af1 chap4 ,,xiii471 are triplicate of the proportions of ,q ,f to ,d ,b1 and of ,o ,e to ,d ,b c4 and therefore the deficient figure ,a ,b ,e ,f ,c1 which is the aggregate of all the lines ,h ,f1 ,g ,e1 ,a ,b1 c4 is triple to the complement ,b ,e ,f ,c ,d made of all the lines ,q ,f1 ,o ,e1 ,d ,b1 c42 which was to be proved4 ,it follows from hence1 that the same complement ,b ,e ,f ,c ,d is #ad of the whole parallelogram4 ,and by the same method may be calculated in all other deficient figures1 generated as above declared1 the proportion of the parallelogram to either of its parts2 as that when the parallels increase from a point in the same proportion1 the parallelogram will be divided into two equal triangles2 when one increase is double to the other1 it will be divided into a semiparabola and its complement1 or into #b and #a4 ,the same construction standing1 the same conclusion may otherwise be demonstrated thus4 ,let the strait line ,c ,b be drawn cutting ,g ,k in ,l1 and through ,l let ,m ,n be drawn parallel to the strait line ,a ,c2 wherefore the parallelograms ,g ,m and ,l ,d will be equal4 ,then let ,l ,k be divided into three equal parts1 so that it may be to one of those parts in the same proportion which the proportion of ,a ,c to ,g ,c1 or of ,g ,k to ,g ,l1 hath to the proportion of ,g ,k to ,g ,e4 ,therefore ,l ,k will be to one of those three parts as the arithmetical proportion between ,g ,k and ,g ,l is to the arithmetical proportion between ,g ,k and the same ,g ,k wanting the third part of ,l ,k2 and ,k ,e will be somewhat greater than a third of ,l ,k4 ,seeing now the altitude ,a ,g or ,m ,l is1 by reason of the continual decrease1 to be supposed less than any quantity that can be given2 ,l ,k1 which is intercepted between the diagonal ,b ,c and the side ,b ,d1 will be also less than any quantity that can be given2 and consequently1 if ,g be put so near to ,a in g1 as that the difference between ,c g and ,c ,a be less than any quantity that can be assigned1 the difference also between ,c l 7removing ,l to l7 and ,c ,b1 will be less than any quantity that can be assigned2 and the line g l being drawn and produced to the line ,b ,d in k1 cutting the crooked line in e1 the proportion of ,g k to ,g l will still be triplicate to the proportion of ,g k to ,g e1 and the difference between k and e1 the third part of k l1 will be less than any quantity that can be given2 and therefore the parallelogram e ,d will differ from a third part of the parallelogram ,a e by a less difference than any quantity that can be assigned4 ,again1 let ,h ,i be drawn parallel and equal to ,g ,e1 cutting ,c ,b in ,p1 the crooked line in ,f1 and ,o ,e in ,i1 and the proportion of ,c g to ,c ,h will be triplicate to the proportion of ,h ,f to ,h ,p1 and ,i ,f will be greater than the third part of ,p ,i4 ,but again1 setting ,h in h so near to g1 as that the difference between ,c h and ,c g may be but as a point1 the point ,p will also in p be so near to l1 as that the difference between ,c p and ,c l will be but as a point2 and drawing h p till it meet with ,b ,d in i1 cutting the crooked line in f and having drawn e o parallel to ,b ,d1 cutting ,d ,c in o1 the parallelogram f o will differ less from the third part of the parallelogram g f1 than by any quantity that can be given4 ,and so it will be in all other spaces generated in the same manner4 ,wherefore the differences of the arithmetical and geometrical means1 which are but as so many points ,b1 e1 f1 c4 7seeing the whole figure is made up of so many indivisible spaces7 will constitute a certain line1 such as is the line ,b ,e ,f ,c1 which will divide the complete figure ,a ,d into two parts1 whereof one1 namely1 ,a ,b ,e ,f ,c1 which ,i call a deficient figure1 is triple to the other1 namely1 ,b ,d ,c ,f ,e1 which ,i call the complement thereof4 ,and whereas the proportion of the altitudes to one another is in this case everywhere triplicate to that of the decreasing quantities to one another2 in the same manner1 if the proportion of the altitudes had been everywhere quadruplicate to that of the decreasing quantities1 it might have been demonstrated that the deficient figure had been quadruple to its complement2 and so in any other proportion4 ,wherefore1 a deficient figure1 which is made1 c4 which was to be demonstrated4 ,the same rule holdeth also in the diminution of the bases of cylinders1 as is demonstrated in the second article of chapter ,,xv4 ,sidenote3 ,the proportion of deficient figures to the parallelograms in which they are described1 set forth in a table4 #c4 ,by this proposition1 the magnitudes of all deficient figures1 when the proportions by which their bases decrease continually are proportional to those by which their altitudes decrease1 may be compared with the magnitudes of their complements2 and consequently1 with the magnitudes of their complete figures4 ,and they will be found to be1 as ,i have set them down in the following tables2 in which ,i compare a parallelogram with three-sided figures2 and first1 with a strait-lined triangle1 made by the base of the parallelogram continually decreasing in such manner1 that the altitudes be always in proportion to one another as the bases are1 and so the triangle will be equal to its complement2 or the proportions of the altitudes and bases will be as #a to #a1 and then the triangle will be half the parallelogram4 ,secondly1 with that three-sided figure which is made by the continual decreasing of the bases in subduplicate proportion to that of the altitudes2 and so the deficient figure will be double to its complement1 and to the parallelogram as #b to #c4 ,then1 with that where the proportion of the altitudes is triplicate to that of the bases2 and then the deficient figure will be triple to its complement1 and to the parallelogram as #c to #d4 ,also the proportion of the altitudes to that of the bases may be as #c to #b2 and then the deficient figure will be to its complement as #c to #b1 and to the parallelogram as #c to #e2 and so forwards1 according as more mean proportionals are taken1 or as the proportions are more multiplied1 as may be seen in the following table4 ,for example1 if the bases decrease so1 that the proportion of the altitudes to that of the bases be always as #e to #b1 and it be demanded what proportion the figure made has to the parallelogram1 which is supposed to be unity2 then1 seeing that where the proportion is taken five times1 there must be four means2 look in the table amongst the three-sided figures of four means1 and seeing the proportion was as #e to #b1 look in the uppermost row for the number #b1 and descending in the second column till you meet with that three-sided figure1 you will find #eg2 which shows that the deficient figure is to the parallelogram as #eg to #a1 or as #e to #g4 #a #b #c #d #e #f #g ,parallelogram #a 3 3 3 3 3 3 ---- ,strait-sided triangle #a_/b 3 3 3 3 3 3 ---- ,three-sided figure of #a mean #b_/c 3 3 3 3 3 3 -------- ,three-sided figure of #b means #c_/d #c_/e 3 3 3 3 3 ------------ ,three-sided figure of #c means #d_/e #d_/f #d_/g 3 3 3 3 ---------------- ,three-sided figure of #d means #e_/f #e_/g #e_/he_/i 3 3 3 -------------------- ,three-sided figure of #e means #f_/g #f_/h #f_/if_/ajf_/aa 3 3 ------------------------ ,three-sided figure of #f means #g_/h #g_/ig_/aag_/aag_/abg_/ac 3 ---------------------------- ,three-sided figure of #g means #h_/ih_/ajh_/aah_/abh_/ach_/adh_/ae ---------------------------- ,sidenote3 ,description production of the same figures4 #d4 ,now for the better understanding of the nature of these three-sided figures1 ,i will show how they may be described by points2 and first1 those which are in the first column of the table4 ,any parallelogram being described1 as ,a ,b ,c ,d 7in figure #b7 let the diagonal ,b ,d be drawn2 and the strait-lined triangle ,b ,c ,d will be half the parallelogram2 then let any number of lines1 as ,e ,f1 be drawn parallel to the side ,b ,c1 and cutting the diagonal ,b ,d in ,g2 and let it be everywhere1 as ,e ,f to ,e ,g1 so ,e ,g to another1 ,e ,h2 and through all the points ,h let the line ,b ,h ,h ,d be drawn2 and the figure ,b ,h ,h ,d ,c will be that which ,i call a three-sided figure of one mean1 because in three proportionals1 as ,e ,f1 ,e ,g and ,e ,h1 there is but one mean1 namely1 ,e ,g2 and this three-sided figure will be #bc of the parallelogram1 and is called a parabola4 ,again1 let it be as ,e ,g to ,e ,h1 so ,e ,h to another1 ,e ,i1 and let the line ,b ,i ,i ,d be drawn1 making the three-sided figure ,b ,i ,i ,d ,c2 and this will be #cd of the parallelogram1 and is by many called a cubic parabola4 ,in like manner1 if the proportions be further continued in ,e ,f1 there will be made the rest of the three-sided figures of the first column2 which ,i thus demonstrate4 ,let there be drawn strait lines1 as ,h ,k and ,g ,l1 parallel to the base ,d ,c4 ,seeing therefore the proportion of ,e ,f to ,e ,h is duplicate to that of ,e ,f to ,e ,g1 or of ,b ,c to ,b ,l1 that is1 of ,c ,d to ,l ,g1 or of ,k ,m 7producing ,k ,h to ,a ,d in ,m7 to ,k ,h1 the proportion of ,b ,c to ,b ,k will be duplicate to that of ,k ,m to ,k ,h2 but as ,b ,c is to ,b ,k1 so is ,d ,c or ,k ,m to ,k ,n1 and therefore the proportion of ,k ,m to ,k ,n is duplicate to that of ,k ,m to ,k ,h2 and so it will be wheresoever the parallel ,k ,m be placed4 ,wherefore the figure ,b ,h ,h ,d ,c is double to its complement ,b ,h ,h ,d ,a1 and consequently #bc of the whole parallelogram4 ,in the same manner1 if through ,i be drawn ,o ,p ,i ,q parallel and equal to ,c ,d1 it may be demonstrated that the proportion of ,o ,q to ,o ,p1 that is1 of ,b ,c to ,b ,o1 is triplicate that of ,o ,q to ,o ,i1 and therefore that the figure ,b ,i ,i ,d ,c is triple to its complement ,b ,i ,i ,d ,a1 and consequently of the whole parallelogram1 c4 ,secondly1 such three-sided figures as are in any of the transverse rows1 may be thus described4 ,let ,a ,b ,c ,d 7in fig4 #c7 be a parallelogram1 whose diagonal is ,b ,d4 ,i would describe in it such figures1 as in the preceding table ,i call three-sided figures of three means4 ,parallel to ,d ,c1 ,i draw ,e ,f as often as is necessary1 cutting ,b ,d in ,g2 and between ,e ,f and ,e ,g1 ,i take three proportionals ,e ,h1 ,e ,i and ,e ,k4 ,if now there be drawn lines through all the points ,h1 ,i and ,k1 that through all the points ,h will make the figure ,b ,h ,d ,c1 which is the first of those three-sided figures2 and that through all the points ,i1 will make the figure ,b ,i ,d ,c1 which is the second2 and that which is drawn through all the points ,k1 will make the figure ,b ,k ,d ,c the third of those three-sided figures4 ,the first of these1 seeing the proportion of ,e ,f to ,e ,g is quadruplicate of that ,e ,f to ,e ,h1 will be to its complement as #d to #a1 and to the parallelogram as #d to #e4 ,the second1 seeing the proportion of ,e ,f to ,e ,g is to that of ,e ,f to ,e ,i as #d to #b1 will be double to its complement1 and #df or #bc of the parallelogram4 ,the third1 seeing the proportion of ,e ,f to ,e ,g is that of ,e ,f to ,e ,k as #d to #c1 will be to its complement as #d to #c1 and to the parallelogram as #d to #g4 ,any of these figures being described may be produced at pleasure1 thus2 let ,a ,b ,c ,d 7in fig4 #d7 be a parallelogram1 and in it let the figure ,b ,k ,d ,c be described1 namely1 the third three-sided figure of three means4 ,let ,b ,d be produced indefinitely to ,e1 and let ,e ,f be made parallel to the base ,d ,c1 cutting ,a ,d produced in ,g1 and ,b ,c produced in ,f2 and in ,g ,e let the point ,h be so taken1 that the proportion of ,f ,e to ,f ,g may be quadruplicate to that of ,f ,e to ,f ,h1 which may be done by making ,f ,h the greatest of three proportionals between ,f ,e and ,f ,g2 the crooked line ,b ,k ,d produced1 will pass through the point ,h4 ,for if the strait line ,b ,h be drawn1 cutting ,c ,d in ,i1 and ,h ,l be drawn parallel to ,g ,d1 and meeting ,c ,d produced in ,l2 it will be as ,f ,e to ,f ,g1 so ,c ,l to ,c ,i1 that is1 in quadruplicate proportion to that of ,f ,e to ,f ,h1 or of ,c ,d to ,c ,i4 ,wherefore if the line ,b ,k ,d be produced according to its generation1 it will fall upon the point ,h4 ,sidenote3 ,the drawing of tangents to them4 #e4 ,a strait line may be drawn so as to touch the crooked line of the said figure in any point1 in this manner4 ,let it be required to draw a tangent to the line ,b ,k ,d ,h 7in fig4 #d7 in the point ,d4 ,let the points ,b and ,d be connected1 and drawing ,d ,a equal and parallel to ,b ,c1 let ,b and ,a be connected2 and because this figure is by construction the third of three means1 let there be taken in ,a ,b three points1 so1 that by them the same ,a ,b be divided into four equal parts2 of which take three1 namely1 ,a ,m1 so that ,a ,b may be to ,a ,m1 as the figure ,b ,k ,d ,c is to its complement4 ,i say1 the strait line ,m ,d will touch the figure in the point given ,d4 ,for let there be drawn anywhere between ,a ,b and ,d ,c a parallel1 as ,r ,q1 cutting the strait line ,b ,d1 the crooked line ,b ,k ,d1 the strait line ,m ,d1 and the strait line ,a ,d1 in the points ,p1 ,k1 ,o and ,q4 ,r ,k will therefore1 by construction1 be the least of three means in geometrical proportion between ,r ,q and ,r ,p4 ,wherefore 7by coroll4 of art4 #bh1 chapter ,,xiii47 ,r ,k will be less than ,r ,o2 and therefore ,m ,d will fall without the figure4 ,now if ,m ,d be produced to ,n1 ,f ,n will be the greatest of three means in arithmetical proportion between ,f ,e and ,f ,g2 and ,f ,h will be the greatest of three means in geometrical proportion between the same ,f ,e and ,f ,g4 ,wherefore 7by the same coroll4 of art4 #bh1 chapter ,,xiii47 ,f ,h will be less than ,f ,n2 and therefore ,d ,n will fall without the figure1 and the strait line ,m ,n will touch the same figure only in the point ,d4 ,sidenote3 ,in what proportion the same figures exceed a strait-lined triangle of the same altitude and base4 #f4 ,the proportion of a deficient figure to its complement being known1 it may also be known what proportion a strait-lined triangle has to the excess of the deficient figure above the same triangle2 and these proportions ,i have set down in the following table2 where if you seek1 for example1 how much the fourth three-sided figure of five means exceeds a triangle of the same altitude and base1 you will find in the concourse of the fourth column with the three-sided figures of five means #baj2 by which is signified1 that that three-sided figure exceeds the triangle by two-tenths or by one-fifth part of the same triangle4 #a #b #c #d #e #f #g ,the triangle #a 3 3 3 3 3 3 --- ,a three-sided fig4 of #a mean #a_/c 3 3 3 3 3 3 ------- ,a three-sided fig4 of #b means#b_/d #a_/e 3 3 3 3 3 ----------- ,the ,a three-sided fig4 of #c means#c_/e #b_/f #a_/g 3 3 3 3 --------------- excess,a three-sided fig4 of #d means#d_/f #c_/g #b_/h #a_/i 3 3 3 ------------------- of ,a three-sided fig4 of #e means#e_/g #d_/h #c_/ib_/aja_/aa 3 3 ----------------------- ,a three-sided fig4 of #f means#f_/h #e_/id_/ajc_/aab_/aba_/ac 3 --------------------------- ,a three-sided fig4 of #g means#g_/if_/aje_/aad_/abc_/acb_/ada_/ae --------------------------- ,sidenote3 ,a table of solid deficient figures described in a cylinder4 #g4 ,in the next table are set down the proportion of a cone and the solids of the said three-sided figures1 namely1 the proportions between them and a cylinder4 ,as for example1 in the concourse of the second column with the three-sided figures of four means1 you have #ei2 which gives you to understand1 that the solid of the second three-sided figure of four means is to the cylinder as #ei to #a1 or as #e to #i4 #a #b #c #d #e #f #g ,a cylinder #a 3 3 3 3 3 3 ---- 3 3 3 3 3 3 ,a cone #a_/c 3 3 3 3 3 3 -------- 3 3 3 3 3 ,a three-sided fig4 of #a mean #b_/d 3 3 3 3 3 -------- 3 3 3 3 3 ,the ,a three-sided fig4 of #b means #c_/e #c_/g 3 3 3 3 3 solids ---------------- 3 3 3 of ,a three-sided fig4 of #c means #d_/f #d_/h #d_/aj 3 3 3 -------------------- 3 3 ,a three-sided fig4 of #d means #e_/g #e_/i #e_/aae_/ac 3 3 ------------------------ 3 ,a three-sided fig4 of #e means #f_/h #f_/ajf_/abf_/adf_/af 3 ---------------------------- ,a three-sided fig4 of #f means #g_/i #g_/aag_/acg_/aeg_/agg_/ai ---------------------------- ,a three-sided fig4 of #g means #h_/ajh_/abh_/adh_/afh_/ahh_/bjh_/bb ---------------------------- ,sidenote3 ,in what proportion the same figures exceed a cone of the same altitude and base4 #h4 ,lastly1 the excess of the solids of the said three-sided figures above a cone of the same altitude and base1 are set down in the table which follows3 #a #b #c #d #e #f #g ,the ,cone #a 3 3 3 3 3 3 3 ----- ,of the solid of a three- #f_/ab 3 3 3 3 3 3 sided figure of #a mean ,the ex- ---------- cesses ,ditto ditto1 #b means #ab_/ae #f_/ba 3 3 3 3 3 of the --------------- solids ,ditto ditto1 #c means #ah_/ahab_/bd #f_/cj 3 3 3 3 of these -------------------- three- ,ditto ditto1 #d means #bd_/baah_/bgab_/cc #f_/ci 3 3 3 sided ------------------------- figures ,ditto ditto1 #e means #cj_/bdbd_/cjah_/cfab_/db #f_/dh 3 3 above a ------------------------------ cone4 ,ditto ditto1 #f means #cf_/bgcj_/ccbd_/ciah_/deab_/ea #f_/eg 3 ----------------------------------- ,ditto ditto1 #g means #db_/cjcf_/cfcj_/dbbd_/dhah_/edab_/fj #f_/ff ----------------------------------- ,sidenote3 ,how a plain deficient figure may be described in a parallelogram1 so that it be to a triangle of the same base and altitude1 as another deficient figure1 plain or solid1 twice taken1 is to the same deficient figure1 together with the complete figure1 in which it is described4 #i4 ,if any of these deficient figures1 of which ,i have now spoken1 as ,a ,b ,c ,d 7in the #eth figure7 be inscribed within the complete figure ,b ,e1 having ,a ,d ,c ,e for its complement2 and there be made upon ,c ,b produced the triangle ,a ,b ,i2 and the parallelogram ,a ,b ,i ,k be completed2 and there be drawn parallel to the strait line ,c ,i1 any number of lines1 as ,m ,f1 cutting every one of them the crooked line of the deficient figure in ,d1 and the strait lines ,a ,c1 ,a ,b and ,a ,i in ,h1 ,g1 and ,l2 and as ,g ,f is to ,g ,d1 so ,g ,l be made to another1 ,g ,n2 and through all the points ,n there be drawn the line ,a ,n ,i3 there will be a deficient figure ,a ,n ,i ,b1 whose complement will be ,a ,n ,i ,k4 ,i say1 the figure ,a ,n ,i ,b is to the triangle ,a ,b ,i1 as the deficient figure ,a ,b ,c ,d twice taken is to the same deficient figure together with the complete figure ,b ,e4 ,for as the proportion of ,a ,b to ,a ,g1 that is1 of ,g ,m to ,g ,l1 is to the proportion of ,g ,m to ,g ,n1 so is the magnitude of the figure ,a ,n ,i ,b to that of its complement ,a ,n ,i ,k1 by the second article of this chapter4 ,but1 by the same article1 as the proportion of ,a ,b to ,a ,g1 that is1 of ,g ,m to ,g ,l1 is to the proportion of ,g ,f to ,g ,d1 that is1 by construction1 of ,g ,l to ,g ,n1 so is the figure ,a ,b ,c ,d to its complement ,a ,d ,c ,e4 ,and by composition1 as the proportion of ,g ,m to ,g ,l1 together with that of ,g ,l to ,g ,n1 is to the proportion of ,g ,m to ,g ,l1 so is the complete figure ,b ,e to the deficient figure ,a ,b ,c ,d4 ,and by conversion1 as the proportion of ,g ,m to ,g ,l is to both the proportions of ,g ,m to ,g ,l and of ,g ,l to ,g ,n1 that is1 to the proportion of ,g ,m to ,g ,n1 which is the proportion compounded of both1 so is the deficient figure ,a ,b ,c ,d to the complete figure ,b ,e4 ,but it was1 as the proportion of ,g ,m to ,g ,l to that of ,g ,m to ,g ,n1 so the figure ,a ,n ,i ,b to its complement ,a ,n ,i ,k4 ,and therefore1 ,a ,b ,c ,d4 ,b ,e 33 ,a ,n ,i ,b4 ,a ,n ,i ,k are proportionals4 ,and by composition1 ,a ,b ,c ,d ,b ,e4 ,a ,b ,c ,d 33 ,b ,k4 ,a ,n ,i ,b are proportionals4 ,and by doubling the consequents1 ,a ,b ,c ,d ,b ,e4 #b ,a ,b ,c ,d 33 ,b ,k4 #b ,a ,n ,i ,b are proportionals4 ,and by taking the halves of the third and the fourth1 ,a ,b ,c ,d ,b ,e4 #b ,a ,b ,c ,d 33 ,a ,b ,i4 ,a ,n ,i ,b are also proportionals2 which was to be proved4 ,sidenote3 ,the transferring of certain properties of deficient figures described in a parallelogram to the proportions of spaces transmitted with several degrees of velocity4 #aj4 ,from what has been said of deficient figures described in a parallelogram1 may be found out what proportions spaces1 transmitted with accelerated motion in determined times1 have to the times themselves1 according as the moved body is accelerated in the several times with one or more degrees of velocity4 ,for let the parallelogram ,a ,b ,c ,d1 in the #fth figure1 and in it the three-sided figure ,d ,e ,b ,c be described2 and let ,f ,g be drawn anywhere parallel to the base1 cutting the diagonal ,b ,d in ,h1 and the crooked line ,b ,e ,d in ,e2 and let the proportion of ,b ,c to ,b ,f be1 for example1 triplicate to that of ,f ,g to ,f ,e2 whereupon the figure ,d ,e ,b ,c will be triple to its complement ,b ,e ,d ,a2 and in like manner1 ,i ,e being drawn parallel to ,b ,c1 the three-sided figure ,e ,k ,b ,f will be triple to its complement ,b ,k ,e ,i4 ,wherefore the parts of the deficient figure cut off from the vertex by strait lines parallel to the base1 namely1 ,d ,e ,b ,c and ,e ,k ,b ,f1 will be to one another as the parallelograms ,a ,c and ,i ,f2 that is1 in proportion compounded of the proportions of the altitudes and bases4 ,seeing therefore the proportion of the altitude ,b ,c to the altitude ,b ,f is triplicate to the proportion of the base ,d ,c to the base ,f ,e1 the figure ,d ,e ,b ,c to the figure ,e ,k ,b ,f will be quadruplicate to the proportion of the same ,d ,c to ,f ,e4 ,and by the same method1 may be found out what proportion any of the said three-sided figures has to any part of the same1 cut off from the vertex by a strait line parallel to the base4 ,now as the said figures are understood to be described by the continual decreasing of the base1 as of ,c ,d1 for example1 till it end in a point1 as in ,b2 so also they may be understood to be described by the continual increasing of a point1 as of ,b1 till it acquire any magnitude1 as that of ,c ,d4 ,suppose now the figure ,b ,e ,d ,c to be described by the increasing of the point ,b to the magnitude ,c ,d4 ,seeing therefore the proportion of ,b ,c to ,b ,f is triplicate to that of ,c ,d to ,f ,e1 the proportion of ,f ,e to ,c ,d will1 by conversion1 as ,i shall presently demonstrate1 be triplicate to that ,b ,f to ,b ,c4 ,wherefore if the strait line ,b ,c be taken for the measure of the time in which the point ,b is moved1 the figure ,e ,k ,b ,f will represent the sum of all the increasing velocities in the time ,b ,f2 and the figure ,d ,e ,b ,c will in like manner represent the sum of all the increasing velocities in the time ,b ,c4 ,seeing therefore the proportion of the figure ,e ,k ,b ,f to the figure ,d ,e ,b ,c is compounded of the proportions of altitude to altitude1 and base to base2 and seeing the proportion of ,f ,e to ,c ,d is triplicate to that of ,b ,f to ,b ,c2 the proportion of the figure ,e ,k ,b ,f to the figure ,d ,e ,b ,c will be quadruplicate to that of ,b ,f to ,b ,c2 that is1 the proportion of the sum of the velocities in the time ,b ,f1 to the sum of the velocities in the time ,b ,c1 will be quadruplicate to the proportion of ,b ,f to ,b ,c4 ,wherefore if a body be moved from ,b with velocity so increasing1 that the velocity acquired in the time ,b ,f be to the velocity acquired in the time ,b ,c in triplicate proportion to that of the times themselves ,b ,f to ,b ,c1 and the body be carried to ,f in the time ,b ,f2 the same body in the time ,b ,c will be carried through a line equal to the fifth proportional from ,b ,f in the continual proportion of ,b ,f to ,b ,c4 ,and by the same manner of working1 we may determine what spaces are transmitted by velocities increasing according to any other proportions4 ,it remains that ,i demonstrate the proportion of ,f ,e to ,c ,d to be triplicate to that of ,b ,f to ,b ,c4 ,seeing therefore the proportion of ,c ,d1 that is1 of ,f ,g to ,f ,e is subtriplicate to that of ,b ,c to ,b ,f2 the proportion of ,f ,g to ,f ,e will also be subtriplicate to that of ,f ,g to ,f ,h4 ,wherefore the proportion of ,f ,g to ,f ,h is triplicate to that of ,f ,g1 that is1 of ,c ,d to ,f ,e4 ,but in four continual proportionals1 of which the least is the first1 the proportion of the first to the fourth1 7by the #afth article of chapter ,,xiii471 is subtriplicate to the proportion of the third to the same fourth4 ,wherefore the proportion of ,f ,h to ,g ,f is subtriplicate to that of ,f ,e to ,c ,d2 and therefore the proportion of ,f ,e to ,c ,d is triplicate to that of ,f ,h to ,f ,g1 that is1 of ,b ,f to ,b ,c2 which was to be proved4 ,it may from hence be collected1 that when the velocity of a body is increased in the same proportion with that of the times1 the degrees of velocity above one another proceed as numbers do in immediate succession from unity1 namely1 as #a1 #b1 #c1 #d1 c4 ,and when the velocity is increased in proportion duplicate to that of the times1 the degrees proceed as numbers from unity1 skipping one1 as #a1 #c1 #e1 #g1 c4 ,lastly1 when the proportions of the velocities are triplicate to those of the times1 the progression of the degrees is as that of numbers from unity1 skipping two in every place1 as #a1 #d1 #g1 #aj1 c41 and so of other proportions4 ,for geometrical proportionals1 when they are taken in every point1 are the same with arithmetical proportionals4 ,sidenote3 ,of deficient figures described in a circle4 #aa4 ,moreover1 it is to be noted that as in quantities1 which are made by any magnitudes decreasing1 the proportions of the figures to one another are as the proportions of the altitudes to those of the bases2 so also it is in those1 which are made with motion decreasing1 which motion is nothing else but that power by which the described figures are greater or less4 ,and therefore in the description of ,archimedes' spiral1 which is done by the continual diminution of the semidiameter of a circle in the same proportion in which the circumference is diminished1 the space1 which is contained within the semidiameter and the spiral line1 is a third part of the whole circle4 ,for the semidiameters of circles1 inasmuch as circles are understood to be made up of the aggregate of them1 are so many sectors2 and therefore in the description of a spiral1 the sector which describes it is diminished in duplicate proportions to the diminutions of the circumference of the circle in which it is inscribed2 so that the complement of the spiral1 that is1 that space in the circle which is without the spiral line1 is double to the space within the spiral line4 ,in the same manner1 if there be taken a mean proportional everywhere between the semidiameter of the circle1 which contains the spiral1 and that part of the semidiameter which is within the same1 there will be made another figure1 which will be half the circle4 ,and to conclude1 this rule serves for all such spaces as may be described by a line or superficies decreasing either in magnitude of power2 so that if the proportions1 in which they decrease1 be commensurable to the proportions of the times in which they decrease1 the magnitudes of the figures they describe will be known4 ,sidenote3 ,the proposition demonstrated in art4 #b confirmed from the elements of philosophy4 #ab4 ,the truth of that proposition1 which ,i demonstrated in art4 #b1 which is the foundation of all that has been said concerning deficient figures1 may be derived from the elements of philosophy1 as having its original in this2 that all equality and inequality between two effects1 that is1 all proportion1 proceeds from1 and is determined by1 the equal and unequal causes of those effects1 or from the proportion which the causes1 concurring to one effect1 have to the causes which concur to the producing of the other effect2 and that therefore the proportions of quantities are the same with the proportions of their causes4 ,seeing1 therefore1 two deficient figures1 of which one is the complement of the other1 are made1 one by motion decreasing in a certain time and proportion1 the other by the loss of motion in the same time2 the causes1 which make and determine the quantities of both the figures1 so that they can be no other than they are1 differ only in this1 that the proportions by which the quantity which generates the figure proceeds in describing of the same1 that is1 the proportions of the remainders of all the times and altitudes1 may be other proportions than those by which the same generating quantity decreases in making the complement of that figure1 that is1 the proportions of the quantity which generates the figure continually diminished4 ,wherefore1 as the proportion of the times in which motion is lost1 is to that of the decreasing quantities by which the deficient figure is generated1 so will the defect or complement be to the figure itself which is generated4 ,sidenote3 ,an unusual way of reasoning concerning the equality between the superficies of a portion of a sphere and a circle4 #ac4 ,there are also other quantities which are determinable from the knowledge of their causes1 namely1 from the comparison of the motions by which they are made2 and that more easily than from the common elements of geometry4 ,for example1 that the superficies of any portion of a sphere is equal to that circle1 whose radius is a strait line drawn from the pole of the portion to the circumference of its base1 ,i may demonstrate in this manner4 ,let ,b ,a ,c 7in fig4 #g7 be a portion of a sphere1 whose axis is ,a ,e1 and whose base is ,b ,c2 and let ,a ,b be the strait line drawn from the pole ,a to the base in ,b2 and let ,a ,d1 equal to ,a ,b1 touch the great circle ,b ,a ,c in the pole ,a4 ,it is to be proved that the circle made by the radius ,a ,d is equal to the superficies of the portion ,b ,a ,c4 ,let the plain ,a ,e ,b ,d be understood to make a revolution about the axis ,a ,e2 and it is manifest that by the strait line ,a ,d a circle will be described2 and by the arch ,a ,b the superficies of a portion of a sphere2 and lastly1 by the subtense ,a ,b the superficies of a right cone4 ,now seeing both the strait line ,a ,b and the arch ,a ,b make one and the same revolution1 and both of them have the same extreme points ,a and ,b1 the cause why the spherical superficies1 which is made by the arch1 is greater than the conical superficies1 which is made by the subtense1 is1 that ,a ,b the arch is greater than ,a ,b the subtense2 and the cause why it is greater consists in this1 that although they be both drawn from ,a to ,b1 yet the subtense is drawn strait1 but the arch angularly1 namely1 according to that angle which the arch makes with the subtense1 which angle is equal to the angle ,d ,a ,b 7for an angle of contingence adds nothing to an angle of a segment1 as has been shown in chapter ,,xiv1 article #af47 ,wherefore the magnitude of the angle ,d ,a ,b is the cause why the superficies of the portion1 described by the arch ,a ,b1 is greater than the superficies of the right cone described by the subtense ,a ,b4 ,again1 the cause why the circle described by the tangent ,a ,d is greater than the superficies of the right cone described by the subtense ,a ,b 7notwithstanding that the tangent and the subtense are equal1 and both moved round in the same time7 is this1 that ,a ,d stands at right angles to the axis1 but ,a ,b obliquely2 which obliquity consists in the same angle ,d ,a ,b4 ,seeing therefore the quantity of the angle ,d ,a ,b is that which makes the excess both of the superficies of the portion1 and of the circle made by the radius ,a ,d1 above the superficies of the right cone described by the subtense ,a ,b2 it follows1 that both the superficies of the portion and that of the circle do equally exceed the superficies of the cone4 ,wherefore the circle made by ,a ,d or ,a ,b1 and the spherical superficies made by the arch ,a ,b1 are equal to one another2 which was to be proved4 ,sidenote3 ,how from the description of deficient figures in a parallelogram1 any number of mean proportionals may be found out between two given strait lines4 #ad4 ,if these deficient figures1 which ,i have described in a parallelogram1 were capable of exact description1 then any number of mean proportionals might be found out between two strait lines given4 ,for example1 in the parallelogram ,a ,b ,c ,d1 7in figure #h7 let the three-sided figure of two means be described 7which many call a cubical parabola72 and let ,r and ,s be two given strait lines2 between which1 if it be required to find two mean proportionals1 it may be done thus4 ,let it be as ,r to ,s1 so ,b ,c to ,b ,f2 and let ,f ,e be drawn parallel to ,b ,a1 and cut the crooked line in ,e2 then through ,e let ,g ,h be drawn parallel and equal to the strait line ,a ,d1 and cut the diagonal ,b ,d in ,i2 for thus we have ,g ,i the greatest of two means between ,g ,h and ,g ,e1 as appears by the description of the figure in article #d4 ,wherefore1 if it be as ,g ,h to ,g ,i1 so ,r to another line1 ,t1 that ,t will be the greatest of two means between ,r and ,s4 ,and therefore if it be again as ,r to ,t1 so ,t to another line1 ,x1 that will be done which was required4 ,in the same manner1 four mean proportionals may be found out1 by the description of a three-sided figure of four means2 and so any other number of means1 c4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xvii4 ,fig4 #a-h ---------------------------------------- -------------------------------- ------- ,,chapter ,,xviii4 ,,of ,,the ,,equation ,,of ,,strait ,,lines ,,with ,,the ,,crooked ,,lines ,,of ,,parabolas ,,and ,,other ,,figures ,,made ,,in ,,imitation ,,of ,,parabolas4 #a4 ,to find the strait line equal to the crooked line of a semiparabola4--#b4 ,to find a strait line equal to the crooked line of the first semiparabolaster1 or to the crooked line of any other of the deficient figures of the table of the #c;d article of the precedent chapter4 ,sidenote3 ,to find a strait line equal to the crooked line of a semiparabola4 #a4 ,a parabola being given1 to find a strait line equal to the crooked line of the semiparabola4 ,let the parabolical line given be ,a ,b ,c 7in figure #a71 and the diameter found be ,a ,d1 and the base drawn ,d ,c2 and the parallelogram ,a ,d ,c ,e being completed1 draw the strait line ,a ,c4 ,then dividing ,a ,d into two equal parts in ,f1 draw ,f ,h equal and parallel to ,d ,c1 cutting ,a ,c in ,k1 and the parabolical line in ,o2 and between ,f ,h and ,f ,o take a mean proportional ,f ,p1 and draw ,a ,o1 ,a ,p and ,p ,c4 ,i say that the two lines ,a ,p and ,p ,c1 taken together as one line1 are equal to the parabolical line ,a ,b ,o ,c4 ,for the line ,a ,b ,o ,c being a parabolical line1 is generated by the concourse of two motions1 one uniform from ,a to ,e1 the other in the same time uniformly accelerated from rest in ,a to ,d4 ,and because the motion from ,a to ,e is uniform1 ,a ,e may represent the times of both those motions from the beginning to the end4 ,let therefore ,a ,e be the time2 and consequently the lines ordinately applied in the semiparabola will design the parts of time wherein the body1 that describeth the line ,a ,b ,o ,c1 is in every point of the same2 so that as at the end of the time ,a ,e or ,d ,c it is in ,c1 so at the end of the time ,f ,o it will be in ,o4 ,and because the velocity in ,a ,d is increased uniformly1 that is1 in the same proportion with the time1 the same lines ordinately applied in the semiparabola will design also the continual augmentation of the impetus1 till it be at the greatest1 designed by the base ,d ,c4 ,therefore supposing uniform motion in the line ,a ,f1 in the time ,f ,k the body in ,a by the concourse of the two uniform motions in ,a ,f and ,f ,k will be moved uniformly in the line ,a ,k2 and ,k ,o will be the increase of the impetus or swiftness gained in the time ,f ,k2 and the line ,a ,o will be uniformly described by the concourse of the two uniform motions in ,a ,f and ,f ,o in the time ,f ,o4 ,from ,o draw ,o ,l parallel to ,e ,c1 cutting ,a ,c in ,l2 and draw ,l ,n parallel to ,d ,c1 cutting ,e ,c in ,n1 and the parabolical line in ,m2 and produce it on the other side to ,a ,d in ,i2 and ,i ,n1 ,i ,m and ,i ,l will be1 by the construction of a parabola1 in continual proportion1 and equal to the three lines ,f ,h1 ,f ,p and ,f ,o2 and a strait line parallel to ,e ,c passing through ,m will fall on ,p2 and therefore ,o ,p will be the increase of impetus gained in the time ,f ,o or ,i ,l4 ,lastly1 produce ,p ,m to ,c ,d in ,q2 and ,q ,c or ,m ,n or ,p ,h will be the increase of impetus proportional to the time ,f ,p or ,i ,m or ,d ,q4 ,suppose now uniform motion from ,h to ,c in the time ,p ,h4 ,seeing therefore in the time ,f ,p with uniform motion and the impetus increased in proportion to the times1 is described the straight line ,a ,p2 and in the rest of the time and impetus1 namely1 ,p ,h1 is described the line ,c ,p uniformly2 it followeth that the whole line ,a ,p ,c is described with the whole impetus1 and in the same time wherewith is described the parabolical line ,a ,b ,c2 and therefore the line ,a ,p ,c1 made of the two strait lines ,a ,p and ,p ,c1 is equal to the parabolical line ,a ,b ,c2 which was to be proved4 ,sidenote3 ,to find a strait line equal to the crooked line of the first semiparabolaster or to the crooked line of any other of the deficient figures of the table of art4 #c of the preceding chapter4 #b4 ,to find a strait line equal to the crooked line of the first semiparabolaster4 ,sidenote3 ,discussion of ,figure #ah4b ,let ,a ,b ,c be the crooked line of the first semiparabolaster2 ,a ,d the diameter2 ,d ,c the base2 and let the parallelogram completed be ,a ,d ,c ,e1 whose diagonal is ,a ,c4 ,divide the diameter into two equal parts in ,f1 and draw ,f ,h equal and parallel to ,d ,c1 cutting ,a ,c in ,k1 the crooked line in ,o1 and ,e ,c in ,h4 ,then draw ,o ,l parallel to ,e ,c1 cutting ,a ,c in ,l2 and draw ,l ,n parallel to the base ,d ,c1 cutting the crooked line in ,m1 and the strait line ,e ,c in ,n2 and produce it on the other side to ,a ,d in ,i4 ,lastly1 through the point ,m draw ,p ,m ,q parallel and equal to ,h ,c1 cutting ,f ,h in ,p2 and join ,c ,p1 ,a ,p and ,a ,o4 ,i say1 the two strait lines ,a ,p and ,p ,c are equal to the crooked line ,a ,b ,o ,c4 ,for the line ,a ,b ,o ,c1 being the crooked line of the first semiparabolaster1 is generated by the concourse of two motions1 one uniform from ,a to ,e1 the other in the same time accelerated from rest in ,a to ,d1 so as that the impetus increaseth in proportion perpetually triplicate to that of the increase of the time1 or which is all one1 the lengths transmitted are in proportion triplicate to that of the times of their transmission2 for as the impetus or quicknesses increase1 so the lengths transmitted increase also4 ,and because the motion from ,a to ,e is uniform1 the line ,a ,e may serve to represent the time1 and consequently the lines1 ordinately drawn in the semiparabolaster1 will design the parts of time wherein the body1 beginning from rest in ,a1 describeth by its motion the crooked line ,a ,b ,o ,c4 ,and because ,d ,c1 which represents the greatest acquired impetus1 is equal to ,a ,e1 the same ordinate lines will represent the several augmentations of the impetus increasing from rest in ,a4 ,therefore1 supposing uniform motion from ,a to ,f1 in the time ,f ,k there will be described1 by the concourse of the two uniform motions ,a ,f and ,f ,k1 the line ,a ,k uniformly1 and ,k ,o will be the increase of impetus in the time ,f ,k2 and by the concourse of the two uniform motions in ,a ,f and ,f ,o will be described the line ,a ,o uniformly4 ,through the point ,l draw the strait line ,l ,m ,n parallel to ,d ,c1 cutting the strait line ,a ,d in ,i1 the crooked line ,a ,b ,c in ,m1 and the strait line ,e ,c in ,n2 and through the point ,m the strait line ,p ,m ,q parallel and equal to ,h ,c1 cutting ,d ,c in ,q and ,f ,h in ,p4 ,by the concourse therefore of the two uniform motions in ,a ,f and ,f ,p in the time ,f ,p will be uniformly described the strait line ,a ,p2 and ,l ,m or ,o ,p will be the increase of impetus to be added for the time ,f ,o4 ,and because the proportion of ,i ,n to ,i ,l is triplicate to the proportion of ,i ,n to ,i ,m1 the proportion of ,f ,h to ,f ,o will also be triplicate to the proportion of ,f ,h to ,f ,p2 and the proportional impetus gained in the time ,f ,p is ,p ,h4 ,so that ,f ,h being equal to ,d ,c1 which designed the whole impetus acquired by the acceleration1 there is no more increase of impetus to be computed4 ,now in the time ,p ,h suppose an uniform motion from ,h to ,c2 and by the two uniform motions in ,c ,h and ,h ,p will be described uniformly the strait line ,p ,c4 ,seeing therefore the two strait lines ,a ,p and ,p ,c are described in the time ,a ,e with the same increase of impetus1 wherewith the crooked line ,a ,b ,o ,c is described in the same time ,a ,e1 that is1 seeing the line ,a ,p ,c and the line ,a ,b ,o ,c are transmitted by the same body in the same time and with equal velocities1 the lines themselves are equal2 which was to be demonstrated4 ,by the same method 7if any of the semiparabolasters in the table of art4 #c of the precedent chapter be exhibited7 may be found a strait line equal to the crooked line thereof1 namely1 by dividing the diameter into two equal parts1 and proceeding as before4 ,yet no man hitherto hath compared any crooked with any strait line1 though many geometricians of every age have endeavoured it4 ,but the cause1 why they have not done it1 may be this1 that there being in ,euclid no definition of equality1 nor any mark by which to judge of it besides congruity 7which is the #hth axiom of the first ,book of his ,elements7 a thing of no use at all in the comparing of strait and crooked2 and others after ,euclid 7except ,archimedes and ,apollonius1 and in our time ,bonaventura7 thinking the industry of the ancients had reached to all that was to be done in geometry1 thought also1 that all that could be propounded was either to be deduced from what they had written1 or else that it was not at all to be done3 it was therefore disputed by some of those ancients themselves1 whether there might be any equality at all between crooked and strait lines2 which question ,archimedes1 who assumed that some strait line was equal to the circumference of a circle1 seems to have despised1 as he had reason4 ,and there is a late writer that granteth that between a strait and a crooked line there is equality2 but now1 says he1 since the fall of ,adam1 without the special assistance of ,divine ,grace it is not to be found4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xviii4 ,fig4 #a-b ---------------------------------------- -------------------------------- ------- ,,chapter ,,xix4 ,,of ,,angles ,,of ,,incidence ,,and ,,reflection1 ,,equal ,,by ,,supposition4 #a4 ,if two strait lines falling upon another strait line be parallel1 the lines reflected from them shall also be parallel4--#b4 ,if two strait lines drawn from one point fall upon another strait line1 the lines reflected from them1 if they be drawn out the other way1 will meet in an angle equal to the angle made by the lines of incidence4--#c4 ,if two strait parallel lines1 drawn not oppositely1 but from the same parts1 fall upon the circumference of a circle1 the lines reflected from them1 if produced they meet within the circle1 will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence4--#d4 ,if two strait lines drawn from the same point without a circle fall upon the circumference1 and the lines reflected from them being produced meet within the circle1 they will make an angle equal to twice that angle1 which is made by two strait lines drawn from the centre to the points of incidence1 together with the angle which the incident lines themselves make4--#e4 ,if two strait lines drawn from one point fall upon the concave circumference of a circle1 and the angle they make be less than twice the angle at the centre1 the lines reflected from them and meeting within the circle will make an angle1 which being added to the angle of the incident lines will be equal to twice the angle at the centre4--#f4 ,if through any one point two unequal chords be drawn cutting one another1 and the centre of the circle be not placed between them1 and the lines reflected from them concur wheresoever1 there cannot through the point1 through which the two former lines were drawn1 be drawn any other strait line whose reflected line shall pass through the common point of the two former lines reflected4--#g4 ,in equal chords the same is not true4--#h4 ,two points being given in the circumference of a circle1 to draw two strait lines to them1 so that their reflected lines may contain any angle given4--#i4 ,if a strait line falling upon the circumference of a circle be produced till it reach the semidiameter1 and that part of it1 which is intercepted between the circumference and the semidiameter1 be equal to that part of the semidiameter which is between the point of concourse and the centre1 the reflected line will be parallel to the semidiameter4--#aj4 ,if from a point within a circle1 two strait lines be drawn to the circumference1 and their reflected lines meet in the circumference of the same circle1 the angle made by the reflected lines will be a third part of the angle made by the incident lines4 ,sidenote3 ,angles of incidence and reflection4 ,whether a body falling upon the superficies of another body and being reflected from it1 do make equal angles at that superficies1 it belongs not to this place to dispute1 being a knowledge which depends upon the natural causes of reflection2 of which hitherto nothing has been said1 but shall be spoken of hereafter4 ,in this place1 therefore1 let it be supposed that the angle of incidence is equal to the angle of reflection2 that our present search may be applied1 not to the finding out of the causes1 but some consequences of the same4 ,i call an angle of incidence1 that which is made between a strait line and another line1 strait or crooked1 upon which it falls1 and which ,i call the line reflecting2 and an angle of reflection equal to it1 that which is made at the same point between the strait line which is reflected and the line reflecting4 ,sidenote3 ,if two strait lines falling upon another strait line be parallel1 the lines reflected from them shall also be parallel4 #a4 ,if two strait lines1 which fall upon another strait line1 be parallel1 their reflected lines shall be also parallel4 ,let the two strait lines ,a ,b and ,c ,d 7in fig4 #a71 which fall upon the strait line ,e ,f1 at the points ,b and ,d1 be parallel2 and let the lines reflected from them be ,b ,g and ,d ,h4 ,i say1 ,b ,g and ,d ,h are also parallel4 ,for the angles ,a ,b ,e and ,c ,d ,e are equal by reason of the parallelism of ,a ,b and ,c ,d2 and the angles ,g ,b ,f and ,h ,d ,f are equal to them by supposition2 for the lines ,b ,g and ,d ,h are reflected from the lines ,a ,b and ,c ,d4 ,wherefore ,b ,g and ,d ,h are parallel4 ,sidenote3 ,if two strait lines drawn from one point fall upon another strait line1 the lines reflected from them1 if they be drawn out the other way1 will meet in an angle equal to the angle made by the lines of incidence4 #b4 ,if two strait lines drawn from the same point fall upon another strait line1 the lines reflected from them1 if they be drawn out the other way1 will meet in an angle equal to the angle of the incident lines4 ,from the point ,a 7in fig4 #b7 let the two strait lines ,a ,b and ,a ,d be drawn2 and let them fall upon the strait line ,e ,k at the points ,b and ,d2 and let the lines ,b ,i and ,d ,g be reflected from them4 ,i say1 ,i ,b and ,g ,d do converge1 and that if they be produced on the other side of the line ,e ,k1 they shall meet1 as in ,f2 and that the angle ,b ,f ,d shall be equal to the angle ,b ,a ,d4 ,for the angle of reflection ,i ,b ,k is equal to the angle of incidence ,a ,b ,e2 and to the angle ,i ,b ,k its vertical angle ,e ,b ,f is equal2 and therefore the angle ,a ,b ,e is equal to the angle ,e ,b ,f4 ,again1 the angle ,a ,d ,e is equal to the angle of reflection ,g ,d ,k1 that is1 to its vertical angle ,e ,d ,f2 and therefore the two angles ,a ,b ,d and ,a ,d ,b of the triangle ,a ,b ,d are one by one equal to the two angles ,f ,b ,d and ,f ,d ,b of the triangle ,f ,b ,d2 wherefore also the third angle ,b ,a ,d is equal to the third angle ,b ,f ,d2 which was to be proved4 ,coroll4 ,i4 ,if the strait line ,a ,f be drawn1 it will be perpendicular to the strait line ,e ,k4 ,for both the angles at ,e will be equal1 by reason of the equality of the two angles ,a ,b ,e and ,f ,b ,e1 and of the two sides ,a ,b and ,f ,b4 ,coroll4 ,,ii4 ,if upon any point between ,b and ,d there fall a strait line1 as ,a ,c1 whose reflected line is ,c ,h1 this also produced beyond ,c1 will fall upon ,f2 which is evident by the demonstration above4 ,sidenote3 ,if two strait parallel lines1 drawn not oppositely1 but from the same parts1 fall upon the circumference of a circle1 the lines reflected from them1 if produced they meet within the circle1 will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence4 #c4 ,if from two points taken without a circle1 two strait parallel lines1 drawn not oppositely1 but from the same parts1 fall upon the circumference2 the lines reflected from them1 if produced they meet within the circle1 will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence4 ,let the two strait parallels ,a ,b and ,d ,c 7in fig4 #c7 fall upon the circumference ,b ,c at the points ,b and ,c2 and let the centre of the circle be ,e2 and let ,a ,b reflected be ,b ,f1 and ,d ,c reflected be ,c ,g2 and let the lines ,f ,b and ,g ,c produced meet within the circle in ,h2 and let ,e ,b and ,e ,c be connected4 ,i say the angle ,f ,h ,g is double to the angle ,b ,e ,c4 ,for seeing ,a ,b and ,d ,c are parallels1 and ,e ,b cuts ,a ,b in ,b1 the same ,e ,b produced will cut ,d ,c somewhere2 let it cut it in ,d2 and let ,d ,c be produced howsoever to ,i1 and let the intersection of ,d ,c and ,b ,f be at ,k4 ,the angle therefore ,i ,c ,h1 being external to the triangle ,c ,k ,h1 will be equal to the two opposite angles ,c ,k ,h and ,c ,h ,k4 ,again1 ,i ,c ,e being external to the triangle ,c ,d ,e1 is equal to the two angles at ,d and ,e4 ,wherefore the angle ,i ,c ,h1 being double to the angle ,i ,c ,e1 is equal to the angles at ,d and ,e twice taken2 and therefore the two angles ,c ,k ,h and ,c ,h ,k are equal to the two angles at ,d and ,e twice taken4 ,but the angle ,c ,k ,h is equal to the angles ,d and ,a ,b ,d1 that is1 ,d twice taken2 for ,a ,b and ,d ,c being parallels1 the altern angles ,d and ,a ,b ,d are equal4 ,wherefore ,c ,h ,k1 that is the angle ,f ,h ,g is also equal to the angle at ,e twice taken2 which was to be proved4 ,coroll4 ,if from two points taken within a circle two strait parallels fall upon the circumference1 the lines reflected from them shall meet in an angle1 double to that which is made by two strait lines drawn from the centre to the points of incidence4 ,for the parallels ,a ,b and ,i ,c falling upon the points ,b and ,c1 are reflected in the lines ,b ,h and ,c ,h1 and make the angle at ,h double to the angle at ,e1 as was but now demonstrated4 ,sidenote3 ,if two strait lines drawn from the same point without a circle fall upon the circumference1 and the lines reflected from them being produced meet within the circle1 they will make an angle equal to twice that angle1 which is made by two strait lines drawn from the centre to the points of incidence1 together with the angle which the incident lines themselves make4 #d4 ,if two strait lines drawn from the same point without a circle fall upon the circumference1 and the lines reflected from them being produced meet within the circle1 they will make an angle equal to twice that angle1 which is made by two strait lines drawn from the centre to the points of incidence1 together with the angle which the incident lines themselves make4 ,let the two strait lines ,a ,b and ,a ,c 7in fig4 #d7 be drawn from the point ,a to the circumference of the circle1 whose centre is ,d2 and let the lines reflected from them be ,b ,e and ,c ,g1 and1 being produced1 make within the circle the angle ,h2 also let the two strait lines ,d ,b and ,d ,c be drawn from the centre ,d to the points of incidence ,b and ,c4 ,i say1 the angle ,h is equal to twice the angle at ,d together with the angle at ,a4 ,for let ,a ,c be produced howsoever to ,i4 ,therefore the angle ,i ,c ,h1 which is external to the triangle ,c ,k ,h1 will be equal to the two angles ,c ,k ,h and ,c ,h ,k4 ,again1 the angle ,i ,c ,d1 which is external to the triangle ,c ,l ,d1 will be equal to the two angles ,c ,l ,d and ,c ,d ,l4 ,but the angle ,i ,c ,h is double to the angle ,i ,c ,d1 and is therefore equal to the angles ,c ,l ,d and ,c ,d ,l twice taken4 ,wherefore the angles ,c ,k ,h and ,c ,h ,k are equal to the angles ,c ,l ,d and ,c ,d ,l twice taken4 ,but the angle ,c ,l ,d1 being external to the triangle ,a ,l ,b1 is equal to the two angles ,l ,a ,b and ,l ,b ,a2 and consequently ,c ,l ,d twice taken is equal to ,l ,a ,b and ,l ,b ,a twice taken4 ,wherefore ,c ,k ,h and ,c ,h ,k are equal to the angle ,c ,d ,l together with ,l ,a ,b and ,l ,b ,a twice taken4 ,also the angle ,c ,k ,h is equal to the angle ,l ,a ,b once and ,a ,b ,k1 that is1 ,l ,b ,a twice taken4 ,wherefore the angle ,c ,h ,k is equal to the remaining angle ,c ,d ,l1 that is1 to the angle at ,d1 twice taken1 and the angle ,l ,a ,b1 that is1 the angle at ,a1 once taken2 which was to be proved4 ,coroll4 ,if two strait converging lines1 as ,i ,c and ,m ,b1 fall upon the concave circumference of a circle1 their reflected lines1 as ,c ,h and ,b ,h1 will meet in the angle ,h1 equal to twice the angle ,d1 together with the angle at ,a made by the incident lines produced4 ,or1 if the incident lines be ,h ,b and ,i ,c1 whose reflected lines ,c ,h and ,b ,m meet in the point ,n1 the angle ,c ,n ,b will be equal to twice the angle ,d1 together with the angle ,c ,k ,h made by the lines of incidence4 ,for the angle ,c ,n ,b is equal to the angle ,h1 that is1 to twice the angle ,d1 together with the two angles ,a1 and ,n ,b ,h1 that is1 ,k ,b ,a4 ,but the angles ,k ,b ,a and ,a are equal to the angle ,c ,k ,h4 ,wherefore the angle ,c ,n ,b is equal to twice the angle ,d1 together with the angle ,c ,k ,h made by the lines of incidence ,i ,c and ,h ,b produced to ,k4 ,sidenote3 ,if two strait lines drawn from one point fall upon the concave circumference of a circle1 and the angle they make be less than twice the angle at the centre1 the lines reflected from them and meeting within the circle will make an angle1 which being added to the angle of the incident lines will be equal to twice the angle at the centre4 #e4 ,if two strait lines drawn from one point fall upon the concave circumference of a circle1 and the angle they make be less than twice the angle at the centre1 the lines reflected from them and meeting within the circle will make an angle1 which being added to the angle of the incident lines1 will be equal to twice the angle at the centre4 ,let the two lines ,a ,b and ,a ,c 7in fig4 #e71 drawn from the point ,a1 fall upon the concave circumference of the circle whose centre is ,d2 and let their reflected lines ,b ,e and ,c ,e meet in the point ,e2 also let the angle ,a be less than twice the angle ,d4 ,i say1 the angles ,a and ,e together taken are equal to twice the angle ,d4 ,for let the strait lines ,a ,b and ,e ,c cut the strait lines ,d ,c and ,d ,b in the points ,g and ,h2 and the angle ,b ,h ,c will be equal to the two angles ,e ,b ,h and ,e2 also the same angle ,b ,h ,c will be equal to the two angles ,d and ,d ,c ,h2 and in like manner the angle ,b ,g ,c will be equal to the two angles ,a ,c ,d and ,a1 and the same angle ,b ,g ,c will be also equal to the two angles ,d ,b ,g and ,d4 ,wherefore the four angles ,e ,b ,h1 ,e1 ,a ,c ,d and ,a1 are equal to the four angles ,d1 ,d ,c ,h1 ,d ,b ,g and ,d4 ,if1 therefore1 equals be taken away on both sides1 namely1 on one side ,a ,c ,d and ,e ,b ,h1 and on the other side ,d ,c ,h and ,d ,b ,g1 7for the angle ,e ,b ,h is equal to the angle ,d ,b ,g1 and the angle ,a ,c ,d equal to the angle ,d ,c ,h71 the remainders on both sides will be equal1 namely1 on one side the angles ,a and ,e1 and on the other the angle ,d twice taken4 ,wherefore the angles ,a and ,e are equal to twice the angle ,d4 ,coroll4 ,if the angle ,a be greater than twice the angle ,d1 their reflected lines will diverge4 ,for1 by the corollary of the third proposition1 if the angle ,a be equal to twice the angle ,d1 the reflected lines ,b ,e and ,c ,e will be parallel2 and if it be less1 they will concur1 as has now been demonstrated4 ,and therefore1 if it be greater1 the reflected lines ,b ,e and ,c ,e will diverge1 and consequently1 if they be produced the other way1 they will concur and make an angle equal to the excess of the angle ,a above twice the angle ,d2 as is evident by art4 #d4 ,sidenote3 ,if through any one point two unequal chords be drawn cutting one another1 and the centre of the circle be not placed between them1 and the lines reflected from them concur wheresoever1 there cannot through the point1 through which the two former lines were drawn1 be drawn any other strait line whose reflected line shall pass through the common point of the two former lines reflected4 #f4 ,if through any one point two unequal chords be drawn cutting one another1 either within the circle1 or1 if they be produced1 without it1 and the centre of the circle be not placed between them1 and the lines reflected from them concur wheresoever2 there cannot1 through the point through which the former lines were drawn1 be drawn another strait line1 whose reflected line shall pass through the point where the two former reflected lines concur4 ,let any two unequal chords1 as ,b ,k and ,c ,h 7in fig4 #f71 be drawn through the point ,a in the circle ,b ,c2 and let their reflected lines ,b ,d and ,c ,e meet in ,f2 and let the centre not be between ,a ,b and ,a ,c2 and from the point ,a let any other strait line1 as ,a ,g1 be drawn to the circumference between ,b and ,c4 ,i say1 ,g ,n1 which passes through the point ,f1 where the reflected lines ,b ,d and ,c ,e meet1 will not be the reflected line of ,a ,g4 ,for let the arch ,b ,l be taken equal to the arch ,b ,g1 and the strait line ,b ,m equal to the strait line ,b ,a2 and ,l ,m being drawn1 let it be produced to the circumference in ,o4 ,seeing therefore ,b ,a and ,b ,m are equal1 and the arch ,b ,l equal to the arch ,b ,g1 and the angle ,m ,b ,l equal to the angle ,a ,b ,g1 ,a ,g and ,m ,l will also be equal1 and1 producing ,g ,a to the circumference in ,i1 the whole lines ,l ,o and ,g ,i will in like manner be equal4 ,but ,l ,o is greater than ,g ,f ,n1 as shall presently be demonstrated2 and therefore also ,g ,i is greater than ,g ,n4 ,wherefore the angles ,n ,g ,c and ,i ,g ,b are not equal4 ,wherefore the line ,g ,f ,n is not reflected from the line of incidence ,a ,g1 and consequently no other strait line1 besides ,a ,b and ,a ,c1 which is drawn through the point ,a1 and falls upon the circumference ,b ,c1 can be reflected to the point ,f2 which was to be demonstrated4 ,it remains that ,i prove ,l ,o to be greater than ,g ,n2 which ,i shall do in this manner4 ,l ,o and ,g ,n cut one another in ,p2 and ,p ,l is greater than ,p ,g4 ,seeing now ,l ,p4 ,p ,g 33 ,p ,n4 ,p ,o are proportionals1 therefore the two extremes ,l ,p and ,p ,o together taken1 that is ,l ,o1 are greater than ,p ,g and ,p ,n together taken1 that is1 ,g ,n2 which remained to be proved4 ,sidenote3 ,in equal chords the same is not true4 #g4 ,but if two equal chords be drawn through one point within a circle1 and the lines reflected from them meet in another point1 then another strait line may be drawn between them through the former point1 whose reflected line shall pass through the latter point4 ,let the two equal chords ,b ,c and ,e ,d 7in the #gth figure7 cut one another in the point ,a within the circle ,b ,c ,d2 and let their reflected lines ,c ,h and ,d ,i meet in the point ,f4 ,then dividing the arch ,c ,d equally in ,g1 let the two chords ,g ,k and ,g ,l be drawn through the points ,a and ,f4 ,i say1 ,g ,l will be the line reflected from the chord ,k ,g4 ,for the four chords ,b ,c1 ,c ,h1 ,e ,d and ,d ,i are by supposition all equal to one another2 and therefore the arch ,b ,c ,h is equal to the arch ,e ,d ,i2 as also the angle ,b ,c ,h to the angle ,e ,d ,i2 and the angle ,a ,m ,c to its verticle angle ,f ,m ,d2 and the strait line ,d ,m to the strait line ,g ,m2 and1 in like manner1 the strait line ,a ,c to the strait line ,f ,d2 and the chords ,c ,g and ,g ,d being drawn1 will also be equal2 and also the angles ,f ,d ,g and ,a ,c ,g1 in the equal segments ,g ,d ,i and ,g ,c ,b4 ,wherefore the strait lines ,f ,g and ,a ,g are equal2 and1 therefore1 the angle ,f ,g ,d is equal to the angle ,a ,g ,c1 that is1 the angle of incidence equal to the angle of reflection4 ,wherefore the line ,g ,l is reflected from the incident line ,c ,g2 which was to be proved4 ,coroll4 ,by the very sight of the figure it is manifest1 that if ,g be not the middle point between ,c and ,d1 the reflected line ,g ,l will not pass through the point ,f4 ,sidenote3 ,two points being given in the circumference of a circle1 to draw two strait lines to them so as that their reflected lines may contain any angle given4 #h4 ,two points in the circumference of a circle being given to draw two strait lines to them1 so as that their reflected lines may be parallel1 or contain any angle given4 ,in the circumference of the circle1 whose centre is ,a1 7in the #hth figure7 let the two points ,b and ,c be given2 and let it be required to draw to them from two points taken without the circle two incident lines1 so that their reflected lines may1 first1 be parallel4 ,let ,a ,b and ,a ,c be drawn2 as also any incident line ,d ,c1 with its reflected line ,c ,f2 and let the angle ,e ,c ,d be made double to the angle ,a2 and let ,h ,b be drawn parallel to ,e ,c1 and produced till it meet with ,d ,c produced in ,i4 ,lastly1 producing ,a ,b indefinitely to ,k1 let ,g ,b be drawn so that the angle ,g ,b ,k may be equal to the angle ,h ,b ,k1 and then ,g ,b will be the reflected line of the incident line ,h ,b4 ,i say1 ,d ,c and ,h ,b are two incident lines1 whose reflected lines ,c ,f and ,b ,g are parallel4 ,for seeing the angle ,e ,c ,d is double to the angle ,b ,a ,c1 the angle ,h ,i ,c is also1 by reason of the parallels ,e ,c and ,h ,i1 double to the same ,b ,a ,c2 therefore also ,f ,c and ,g ,b1 namely1 the lines reflected from the incident lines ,d ,c and ,h ,b1 are parallel4 ,wherefore the first thing required is done4 ,secondly1 let it be required to draw to the points ,b and ,c two strait lines of incidence1 so that the lines reflected from them may contain the given angle ,z4 ,to the angle ,e ,c ,d made at the point ,c1 let there be added on one side the angle ,d ,c ,l equal to half ,z1 and on the other side the angle ,e ,c ,m equal to the angle ,d ,c ,l2 and let the strait line ,b ,n be drawn parallel to the strait line ,c ,m2 and let the angle ,k ,b ,o be made equal to the angle ,n ,b ,k2 which being done1 ,b ,o will be the line of reflection from the line of incidence ,n ,b4 ,lastly1 from the incident line ,l ,c1 let the reflected line ,c ,o be drawn1 cutting ,b ,o at ,o1 and making the angle ,c ,o ,b4 ,i say1 the angle ,c ,o ,b is equal to the angle ,z4 ,let ,n ,b be produced till it meet with the strait line ,l ,c produced in ,p4 ,seeing1 therefore1 the angle ,l ,c ,m is1 by construction1 equal to twice the angle ,b ,a ,c1 together with the angle ,z2 the angle ,n ,p ,l1 which is equal to ,l ,c ,m by reason of the parallels ,n ,p and ,m ,c1 will also be equal to twice the same angle ,b ,a ,c1 together with the angle ,z4 ,and seeing the two strait lines ,o ,c and ,o ,b fall from the point ,o upon the points ,c and ,b2 and their reflected lines ,l ,c and ,n ,b meet in the point ,p2 the angle ,n ,p ,l will be equal to twice the angle ,b ,a ,c together with the angle ,c ,o ,b4 ,but ,i have already proved the angle ,n ,p ,l to be equal to twice the angle ,b ,a ,c together with the angle ,z4 ,therefore the angle ,c ,o ,b is equal to the angle ,z2 wherefore1 two points in the circumference of a circle being given1 ,i have drawn1 c42 which was to be done4 ,but if it be required to draw the incident lines from a point within the circle1 so that the lines reflected from them may contain an angle equal to the angle ,z1 the same method is to be used1 saving that in this case the angle ,z is not to be added to twice the angle ,b ,a ,c1 but to be taken from it4 ,sidenote3 ,if a strait line falling upon the circumference of a circle be produced till it reach the semidiameter1 and that part of it1 which is intercepted between the circumference and the semidiameter1 be equal to that part of the semidiameter which is between the point of concourse and the centre1 the reflected line will be parallel to the semidiameter4 #i4 ,if a strait line1 falling upon the circumference of a circle1 be produced till it reach the semidiameter1 and that part of it which is intercepted between the circumference and the semidiameter be equal to that part of the semidiameter which is between the point of concourse and the centre1 the reflected line will be parallel to the semidiameter4 ,let any line ,a ,b 7in the #ith figure7 be the semidiameter of the circle whose centre is ,a2 and upon the circumference ,b ,d let the strait line ,c ,d fall1 and be produced till it cut ,a ,b in ,e1 so that ,e ,d and ,e ,a may be equal2 and from the incident line ,c ,d let the line ,d ,f be reflected4 ,i say1 ,a ,b and ,d ,f will be parallel4 ,let ,a ,g be drawn through the point ,d4 ,seeing1 therefore1 ,e ,d and ,e ,a are equal1 the angles ,e ,d ,a and ,e ,a ,d will also be equal4 ,but the angles ,f ,d ,g and ,e ,d ,a are equal2 for each of them is half the angle ,e ,d ,h or ,f ,d ,c4 ,wherefore the angles ,f ,d ,g and ,e ,a ,d are equal2 and consequently ,d ,f and ,a ,b are parallel2 which was to be proved4 ,coroll4 ,if ,e ,a be greater then ,e ,d1 then ,d ,f and ,a ,b being produced will concur2 but if ,e ,a be less than ,e ,d1 then ,b ,a and ,d ,h being produced will concur4 ,sidenote3 ,if from a point within a circle two strait lines be drawn to the circumference1 and their reflected lines meet in the circumference of the same circle1 the angle made by the reflected lines will be a third part of the angle made by the incident lines4 #aj4 ,if from a point within a circle two strait lines be drawn to the circumference1 and their reflected lines meet in the circumference of the same circle1 the angle made by the lines of reflection will be a third part of the angle made by the lines of incidence4 ,from the point ,b 7in the #ajth figure7 taken within the circle whose centre is ,a1 let the two strait lines ,b ,c and ,b ,d be drawn to the circumference2 and let their reflected lines ,c ,e and ,d ,e meet in the circumference of the same circle at the point ,e4 ,i say1 the angle ,c ,e ,d will be a third part of the angle ,c ,b ,d4 ,let ,a ,c and ,a ,d be drawn4 ,seeing1 therefore1 the angles ,c ,e ,d and ,c ,b ,d together taken are equal to twice the angle ,c ,a ,d 7as has been demonstrated in the #eth article72 and the angle ,c ,a ,d twice taken is quadruple to the angle ,c ,e ,d2 the angles ,c ,e ,d and ,c ,b ,d together taken will also be equal to the angle ,c ,e ,d four times taken2 and therefore if the angle ,c ,e ,d be taken away on both sides1 there will remain the angle ,c ,b ,d on one side1 equal to the angle ,c ,e ,d thrice taken on the other side2 which was to be demonstrated4 ,coroll4 ,therefore a point being given within a circle1 there may be drawn two lines from it to the circumference1 so as their reflected lines may meet in the circumference4 ,for it is but trisecting the angle ,c ,b ,d1 which how it may be done shall be shown in the following chapter4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xix4 ,fig4 #a-aj ---------------------------------------- -------------------------------- ------- ,,chapter ,,xx4 ,,of ,,the ,,dimension ,,of ,a ,,circle1 ,,and ,,the ,,division ,,of ,,angles ,,or ,,arches4 #a4 ,the dimension of a circle never determined in numbers by ,archimedes and others4--#b4 ,the first attempt for the finding out of the dimension of a circle by lines4--#c4 ,the second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness4--#d4 ,the third attempt2 and some things propounded to be further searched into4--#e4 ,the equation of the spiral of ,archimedes with a strait line4--#f4 ,of the analysis of geometricians by the powers of lines4 ,sidenote3 ,the dimension of a circle never determined in numbers by ,archimedes and others4 #a4 ,in the comparing of an arch of a circle with a strait line1 many and great geometricians1 even from the most ancient times1 have exercised their wits2 and more had done the same1 if they had not seen their pains1 though undertaken for the common good1 if not brought to perfection1 vilified by those that envy the praises of other men4 ,amongst those ancient writers whose works are come to our hands1 ,archimedes was the first that brought the length of the perimeter of a circle within the limits of numbers very little differing from the truth2 demonstrating the same to be less than three diameters and a seventh part1 but greater than three diameters and ten seventy-one parts of the diameter4 ,so that supposing the radius to consist of #aj1jjj1jjj equal parts1 the arch of a quadrant will be between #ae1gad1bhe and #ae1gjd1bbe of the same parts4 ,in our times1 ,ludovicus ,van ,cullen and ,willebrordus ,snellius1 with joint endeavour1 have come yet nearer to the truth2 and pronounced from true principles1 that the arch of a quadrant1 putting1 as before1 #aj1jjj1jjj for radius1 differs not one whole unity from the number #ae1gjg1ifc2 which1 if they had exhibited their arithmetical operations1 and no man had discovered any error in that long work of theirs1 had been demonstrated by them4 ,this is the furthest progress that has been made by the way of numbers2 and they that have proceeded thus far deserve the praise of industry4 ,nevertheless1 if we consider the benefit1 which is the scope at which all speculation should aim1 the improvement they have made has been little or none4 ,for any ordinary man may much sooner and more accurately find a strait line equal to the perimeter of a circle1 and consequently square the circle1 by winding a small thread about a given cylinder1 than any geometrician shall do the same by dividing the radius into #aj1jjj1jjj equal parts4 ,but though the length of the circumference were exactly set out1 either by numbers1 or mechanically1 or only by chance1 yet this would contribute no help at all towards the section of angles1 unless happily these two problems1 to divide a given angle according to any proportion assigned1 and to find a strait line equal to the arch of a circle1 were reciprocal1 and followed one another4 ,seeing therefore the benefit proceeding from the knowledge of the length of the arch of a quadrant consists in this1 that we may thereby divide an angle according to any proportion1 either accurately1 or at least accurately enough for common use2 and seeing this cannot be done by arithmetic1 ,i thought fit to attempt the same by geometry1 and in this chapter to make trial whether it might not be performed by the drawing of strait and circular lines4 ,sidenote3 ,the first attempt for the finding out of the dimension of a circle by lines4 #b4 ,let the square ,a ,b ,c ,d 7in the first figure7 be described2 and with the radii ,a ,b1 ,b ,c1 and ,d ,c1 the three arches ,b ,d1 ,c ,a1 and ,a ,c2 of which let the two ,b ,d and ,c ,a cut one another in ,e1 and the two ,b ,d and ,a ,c in ,f4 ,the diagonals therefore ,b ,d and ,a ,c being drawn will cut one another in the centre of the square ,g1 and the two arches ,b ,d and ,c ,a in two equal parts in ,h and ,y2 and the arch ,b ,h ,d will be trisected in ,f and ,e4 ,through the centre ,g let the two strait lines ,k ,g ,l and ,m ,g ,n be drawn parallel and equal to the sides of the square ,a ,b and ,a ,d1 cutting the four sides of the same square in the points ,k1 ,l1 ,m1 and ,n2 which being done1 ,k ,l will pass through ,f1 and ,m ,n through ,e4 ,then let ,o ,p be drawn parallel and equal to the side ,b ,c1 cutting the arch ,b ,f ,d in ,f1 and the sides ,a ,b and ,d ,c in ,o and ,p4 ,therefore ,o ,f will be the sine of the arch ,b ,f1 which is an arch of #cj degrees2 and the same ,o ,f will be equal to half the radius4 ,lastly1 dividing the arch ,b ,f in the middle in ,q1 let ,r ,q1 the sine of the arch ,b ,q1 be drawn and produced to ,s1 so that ,q ,s be equal to ,r ,q1 and consequently ,r ,s be equal to the chord of the arch ,b ,f2 and let ,f ,s be drawn and produced to ,t in the side ,b ,c4 ,i say1 the strait line ,b ,t is equal to the arch ,b ,f2 and consequently that ,b ,v1 the triple of ,b ,t1 is equal to the arch of the quadrant ,b ,f ,e ,d4 ,let ,t ,f be produced till it meet the side ,b ,a produced in ,x2 and dividing ,o ,f in the middle in ,z1 let ,q ,z be drawn and produced till it meet with the side ,b ,a produced4 ,seeing therefore the strait lines ,r ,s and ,o ,f are parallel1 and divided in the midst in ,q and ,z1 ,q ,z produced will fall upon ,x1 and ,x ,z ,q produced to the side ,b ,c will cut ,b ,t in the midst in 4 ,upon the strait line ,f ,z1 the fourth part of the radius ,a ,b1 let the equilateral triangle a ,z ,f be constituted2 and upon the centre a1 with the radius a ,z1 let the arch ,z ,f be drawn2 which arch ,z ,f will therefore be equal to the arch ,q ,f1 the half of the arch ,b ,f4 ,again1 let the strait line ,z ,o be cut in the midst in b1 and the strait line b ,o in the midst in c2 and let the bisection be continued in this manner till the last part ,o c be the least that can possibly be taken2 and upon it1 and all the rest of the parts equal to it into which the strait line ,o ,f may be cut1 let so many equilateral triangles be understood to be constituted2 of which let the last be d ,o c4 ,if1 therefore1 upon the centre d1 with the radius d ,o1 be drawn the arch ,o c1 and upon the rest of the equal parts of the strait line ,o ,f be drawn in like manner so many equal arches1 all those arches together taken will be equal to the whole arch ,b ,f1 and the half of them1 namely1 those that are comprehended between ,o and ,z1 or between ,z and ,f1 will be equal to the arch ,b ,q or ,q ,f1 and in sum1 what part soever the strait line ,o c be of the strait line ,o ,f1 the same part will the arch ,o c be of the arch ,b ,f1 though both the arch and the chord be infinitely bisected4 ,now seeing the arch ,o c is more crooked than that part of the arch ,b ,f which is equal to it2 and seeing also that the more the strait line ,x c is produced1 the more it diverges from the strait line ,x ,o1 if the points ,o and c be understood to be moved forwards with strait motion in ,x ,o and ,x c1 the arch ,o c will thereby be extended by little and little1 till at the last it come somewhere to have the same crookedness with that part of the arch ,b ,f which is equal to it4 ,in like manner1 if the strait line ,x b be drawn1 and the point b be understood to be moved forwards at the same time1 the arch c b will also by little and little be extended1 till its crookedness come to be equal to the crookedness of that part of the arch ,b ,f which is equal to it4 ,and the same will happen in all those small equal arches which are described upon so many equal parts of the strait line ,o ,f4 ,it is also manifest1 that by strait motion in ,x ,o and ,x ,z all those small arches will lie in the arch ,b ,f1 in the points ,b1 ,q and ,f4 ,and though the same small equal arches should not be coincident with the equal parts of the arch ,b ,f in all the other points thereof1 yet certainly they will constitute two crooked lines1 not only equal to the two arches ,b ,q and ,q ,f1 and equally crooked1 but also having their cavity towards the same parts2 which how it should be1 unless all those small arches should be coincident with the arch ,b ,f in all its points1 is not imaginable4 ,they are therefore coincident1 and all the strait lines drawn from ,x1 and passing through the points of division of the strait line ,o ,f1 will also divide the arch ,b ,f into the same proportions into which ,o ,f is divided4 ,now seeing ,x b cuts off from the point ,b the fourth part of the arch ,b ,f1 let that fourth part be ,b e2 and let the sine thereof1 f e1 be produced to ,f ,t in g1 for so f e will be the fourth part of the strait line f g1 because as ,o b is to ,o ,f1 so is f e to f g4 ,but ,b ,t is greater than f g2 and therefore the same ,b ,t is greater than four sines of the fourth part of the arch ,b ,f4 ,and in like manner1 if the arch ,b ,f be subdivided into any number of equal parts whatsoever1 it may be proved that the strait line ,b ,t is greater than the sine of one of those small arches1 so many times taken as there be parts made of the whole arch ,b ,f4 ,wherefore the strait line ,b ,t is not less than the arch ,b ,f4 ,but neither can it be greater1 because if any strait line whatsoever1 less than ,b ,t1 be drawn below ,b ,t1 parallel to it1 and terminated in the strait lines ,x ,b and ,x ,t1 it would cut the arch ,b ,f2 and so the sine of some one of the parts of the arch ,b ,f1 taken so often as that small arch is found in the whole arch ,b ,f1 would be greater than so many of the same arches2 which is absurd4 ,wherefore the strait line ,b ,t is equal to the arch ,b ,f2 and the strait line ,b ,v equal to the arch of the quadrant ,b ,f ,d2 and ,b ,v four times taken1 equal to the perimeter of the circle described with the radius ,a ,b4 ,also the arch ,b ,f and the strait line ,b ,t are everywhere divided into the same proportions2 and consequently any given angle1 whether greater or less than ,b ,a ,f1 may be divided into any proportion given4 ,but the strait line ,b ,v1 though its magnitude fall within the terms assigned by ,archimedes1 is found1 if computed by the canon of signs1 to be somewhat greater than that which is exhibited by the ,rudolphine numbers4 ,nevertheless1 if in the place of ,b ,t1 another strait line1 though never so little less1 be substituted1 the division of angles is immediately lost1 as may by any man be demonstrated by this very scheme4 ,howsoever1 if any man think this my strait line ,b ,v to be too great1 yet1 seeing the arch and all the parallels are everywhere so exactly divided1 and ,b ,v comes so near to the truth1 ,i desire he would search out the reason1 why1 granting ,b ,v to be precisely true1 the arches cut off should not be equal4 ,but some man may yet ask the reason why the strait lines1 drawn from ,x through the equal parts of the arch ,b ,f1 should cut off in the tangent ,b ,v so many strait lines equal to them1 seeing the connected straight line ,x ,v passes not through the point ,d1 but cuts the strait line ,a ,d produced in l2 and consequently require some determination of this problem4 ,concerning which1 ,i will say what ,i think to be the reason1 namely1 that whilst the magnitude of the arch doth not exceed the magnitude of the radius1 that is1 the magnitude of the tangent ,b ,c1 both the arch and the tangent are cut alike by the strait lines drawn from ,x2 otherwise not4 ,for ,a ,v being connected1 cutting the arch ,b ,h ,d in ,i1 if ,x ,c being drawn should cut the same arch in the same point ,i1 it would be as true that the arch ,b ,i is equal to the radius ,b ,c1 as it is true that the arch ,b ,f is equal to the strait line ,b ,t2 and drawing ,x ,k it would cut the arch ,b ,i in the midst in i2 also drawing ,a i and producing it to the tangent ,b ,c in k1 the strait line ,b k will be the tangent of the arch ,b i1 7which arch is equal to half the radius7 and the same strait line ,b k will be equal to the strait line k ,i4 ,i say all this is true1 if the preceding demonstration be true2 and consequently the proportional section of the arch and its tangent proceeds hitherto4 ,but it is manifest by the golden rule1 that taking ,b h double to ,b ,t1 the line ,x h shall not cut off the arch ,b ,e1 which is double to the arch ,b ,f1 but a much greater4 ,for the magnitude of the straight lines ,x ,m1 ,x ,b1 and ,m ,e1 being known 7in numbers71 the magnitude of the strait line cut off in the tangent by the strait line ,x ,e produced to the tangent1 may also be known2 and it will be found to be less than ,b h2 ,wherefore the strait line ,x h being drawn1 will cut off a part of the arch of the quadrant greater than the arch ,b ,e4 ,but ,i shall speak more fully in the next article concerning the magnitude of the arch ,b ,i4 ,and let this be the first attempt for the finding out of the dimension of a circle by the section of the arch ,b ,f4 ,sidenote3 ,the second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness4 #c4 ,i shall now attempt the same by arguments drawn from the nature of the crookedness of the circle itself2 but ,i shall first set down some premises necessary for this speculation2 and ,first1 if a strait line be bowed into an arch of a circle equal to it1 as when a stretched thread1 which toucheth a right cylinder1 is so bowed in every point1 that it be everywhere coincident with the perimeter of the base of the cylinder1 the flexion of that line will be equal1 in all its points2 and consequently the crookedness of the arch of a circle is everywhere uniform2 which needs no other demonstration than this1 that the perimeter of a circle is an uniform line4 ,secondly1 and consequently3 if two unequal arches of the same circle be made by the bowing of two strait lines equal to them1 the flexion of the longer line1 whilst it is bowed into the greater arch1 is greater than the flexion of the shorter line1 whilst it is bowed into the lesser arch1 according to the proportion of the arches themselves2 and consequently1 the crookedness of the greater arch is to the crookedness of the lesser arch1 as the greater arch is to the lesser arch4 ,thirdly3 if two unequal circles and a strait line touch one another in the same point1 the crookedness of any arch taken in the lesser circle1 will be greater than the crookedness of an arch equal to it taken in the greater circle1 in reciprocal proportion to that of the radii with which the circles are described2 or1 which is all one1 any strait line being drawn from the point of contact till it cut both the circumferences1 as the part of that strait line cut off by the circumference of the greater circle to that part which is cut off by the circumference of the lesser circle4 ,for let ,a ,b and ,a ,c 7in the second figure7 be two circles1 touching one another1 and the strait line ,a ,d in the point ,a2 and let their centres be ,e and ,f2 and let it be supposed1 that as ,a ,e is to ,a ,f1 so is the arch ,a ,b to the arch ,a ,h4 ,i say the crookedness of the arch ,a ,c is to the crookedness of the arch ,a ,h1 as ,a ,e is to ,a ,f4 ,for let the strait line ,a ,d be supposed to be equal to the arch ,a ,b1 and the strait line ,a ,g to the arch ,a ,c2 and let ,a ,d1 for example1 be double to ,a ,g4 ,therefore1 by reason of the likeness of the arches ,a ,b and ,a ,c1 the strait line ,a ,b will be double to the strait line ,a ,c1 and the radius ,a ,e double to the radius ,a ,f1 and the arch ,a ,b double to the arch ,a ,h4 ,and because the strait line ,a ,d is so bowed to be coincident with the arch ,a ,b equal to it1 as the strait line ,a ,g is bowed to be coincident with the arch ,a ,c equal also to it1 the flexion of the strait line ,a ,g into the crooked line ,a ,c will be equal to the flexion of the strait line ,a ,d into the crooked line ,a ,b4 ,but the flexion of the strait line ,a ,d into the crooked line ,a ,b is double to the flexion of the strait line ,a ,g into the crooked line ,a ,h2 and therefore the flexion of the strait line ,a ,g into the crooked line ,a ,c is double to the flexion of the same strait line ,a ,g into the crooked line ,a ,h4 ,wherefore1 as the arch ,a ,b is to the arch ,a ,c or ,a ,h2 or as the radius ,a ,e is to the radius ,a ,f2 or as the chord ,a ,b is to the chord ,a ,c2 so reciprocally is the flexion or uniform crookedness of the arch ,a ,c1 to the flexion or uniform crookedness of the arch ,a ,h1 namely1 here double4 ,and this may by the same method be demonstrated in circles whose perimeters are to one another triple1 quadruple1 or in whatsoever given proportion4 ,the crookedness therefore of two equal arches taken in several circles are in proportion reciprocal to that of their radii1 or like arches1 or like chords2 which was to be demonstrated4 ,let the square ,a ,b ,c ,d be again described 7in the third figure71 and in it the quadrants ,a ,b ,d1 ,b ,c ,a and ,d ,a ,c2 and dividing each side of the square ,a ,b ,c ,d in the midst in ,e1 ,f1 ,g and ,h1 let ,e ,g and ,f ,h be connected1 which will cut one another in the centre of the square at ,i1 and divide the arch of the quadrant ,a ,b ,d into three equal parts in ,k and ,l4 ,also the diagonals ,a ,c and ,b ,d being drawn will cut one another in ,i1 and divide the arches ,b ,k ,d and ,c ,l ,a into two equal parts in ,m and ,n4 ,then with the radius ,b ,f let the arch ,f ,e be drawn1 cutting the diagonal ,b ,d in ,o2 and dividing the arch ,b ,m in the midst in ,p1 let the strait line ,e a equal to the chord ,b ,p be set off from the point ,e in the arch ,e ,f1 and let the arch a b be taken equal to the arch ,o a1 and let ,b a and ,b b be drawn and produced to the arch ,a ,n in c and d2 and lastly1 let the strait line ,a d be drawn4 ,i say the strait line ,a d is equal to the arch ,a ,n or ,b ,m4 ,i have proved in the preceding article1 that the arch ,e ,o is twice as crooked as the arch ,b ,p1 that is to say1 that the arch ,e ,o is so much more crooked than the arch ,b ,p1 as the arch ,b ,p is more crooked than the strait line ,e a4 ,the crookedness therefore of the chord ,e a1 of the arch ,b ,p1 and of the arch ,e ,o1 are as #j1 #a1 #b4 ,also the difference between the arches ,e ,o and ,e ,o1 the difference between the arches ,e ,o and ,e a1 and the difference between the arches ,e ,o and ,e b1 are as #j1 #a1 #b4 ,so also the difference between the arches ,a ,n and ,a ,n1 the difference between the arches ,a ,n and ,a c1 and the difference between the arches ,a ,n and ,a d1 are as #j1 #a1 #b2 and the strait line ,a c is double to the chord ,b ,p or ,e a1 and the strait line ,a d double to the chord ,e b4 ,again1 let the strait line ,b ,f be divided in the midst in ,q1 and the arch ,b ,p in the midst in ,r2 and describing the quadrant ,b ,q ,s 7whose arch ,q ,s is a fourth part of the arch of the quadrant ,b ,m ,d1 as the arch ,b ,r is a fourth part of the arch ,b ,m1 which is the arch of the semiquadrant ,a ,b ,m7 let the chord ,s e equal to the chord ,b ,r be set off from the point ,s in the arch ,s ,q2 and let ,b e be drawn and produced to the arch ,a ,n in f2 which being done1 the strait line ,a f will be quadruple to the chord ,b ,r or ,s e4 ,and seeing the crookedness of the arch ,s e1 or of the arch ,a c1 is double to the crookedness of the arch ,b ,r1 the excess of the crookedness of the arch ,a f above the crookedness of the arch ,a c will be subduple to the excess of the crookedness of the arch ,a c above the crookedness of the arch ,a ,n2 and therefore the arch ,n c will be double to the arch c f4 ,wherefore the arch c d is divided in the midst in f1 and the arch ,n f is of the arch ,n d4 ,and in like manner if the arch ,b ,r be bisected in ,v1 and the strait line ,b ,q in ,x1 and the quadrant ,b ,x ,y be described1 and the strait line ,y g equal to the chord ,b ,v be set off from the point ,y in the arch ,y ,x1 it may be demonstrated that the strait line ,b g being drawn and produced to the arch ,a ,n1 will cut the arch f d into two equal parts1 and that a strait line drawn from ,a to the point of that section1 will be equal to eight chords of the arch ,b ,v1 and so on perpetually2 and consequently1 that the strait line ,a d is equal to so many equal chords of equal parts of the arch ,b ,m1 as may be made by infinite bisections4 ,wherefore the strait line ,a d is equal to the arch ,b ,m or ,a ,n1 that is1 to half the arch of the quadrant ,a ,b ,d or ,b ,c ,a4 ,coroll4 ,an arch being given not greater than the arch of a quadrant 7for being made greater1 it comes again towards the radius ,b ,a produced1 from which it receded before7 if a strait line double to the chord of half the given arch be adapted from the beginning of the arch1 and by how much the arch that is subtended by it is greater than the given arch1 by so much a greater arch be subtended by another strait line1 this strait line shall be equal to the first given arch4 ,supposing the strait line ,b ,v 7in fig4 #a7 be equal to the arch of the quadrant ,b ,h ,d1 and ,a ,v be connected cutting the arch ,b ,h ,d in ,i1 it may be asked what proportion the arch ,b ,i has to the arch ,i ,d4 ,let therefore the arch ,a ,y be divided in the midst in o1 and in the strait line ,a ,d let ,a p be taken equal1 and ,a q double to the drawn chord ,a o4 ,then upon the centre ,a1 with the radius ,a q1 let an arch of a circle be drawn cutting the arch ,a ,y in r1 and let the arch ,y r be doubled at t2 which being done1 the drawn strait line ,a t 7by what has been last demonstrated7 will be equal to the arch ,a ,y4 ,again1 upon the centre ,a with the radius ,a t let the arch t u be drawn cutting ,a ,d in u2 and the strait line ,a u will be equal to the arch ,a ,y4 ,from the point u let the strait line u s be drawn equal and parallel to the strait line ,a ,b1 cutting ,m ,n in x1 and bisected by ,m ,n in the same point x4 ,therefore the strait line ,a x being drawn and produced till it meet with ,b ,c produced in ,v1 it will cut off ,b ,v double to ,b s1 that is1 equal to the arch ,b ,h ,d4 ,now let the point1 where the strait line ,a ,v cuts the arch ,b ,h ,d1 be ,i2 and let the arch ,d ,i be divided in the midst in y2 and in the strait line ,d ,c1 let ,d z be taken equal1 and ,d double to the drawn chord ,d y2 and upon the centre ,d with the radius ,d let an arch of a circle be drawn cutting the arch ,b ,h ,d in the point n2 and let the arch n m be taken equal to the arch ,i n2 which being done1 the strait line ,d m will 7by the last foregoing corollary7 be equal to the arch ,d ,i4 ,if now the strait lines ,d m and ,c ,v be equal1 the arch ,b ,i will be equal to the radius ,a ,b or ,b ,c2 and consequently ,x ,c being drawn1 will pass through the point ,i4 ,moreover1 if the semicircle ,b ,h ,d being completed1 the strait lines ,i and ,b ,i be drawn1 making a right angle 7in the semicircle7 at ,i1 and the arch ,b ,i be divided in the midst at i1 it will follow that ,a i being connected will be parallel to the strait line ,i1 and being produced to ,b ,c in k1 will cut off the strait line ,b k equal to the strait line k ,i1 and equal also to the strait line ,a cut off in ,a ,d by the strait line ,i4 ,all which is manifest1 supposing the arch ,b ,i and the radius ,b ,c to be equal4 ,but that the arch ,b ,i and the radius ,b ,c are precisely equal1 cannot 7how true soever it be7 be demonstrated1 unless that be first proved which is contained in art4 #a1 namely1 that the strait lines drawn from ,x through the equal parts of ,o ,f 7produced to a certain length7 cut off so many parts also in the tangent ,b ,c severally equal to the several arches cut off2 which they do most exactly as far as ,b ,c in the tangent1 and ,b ,i in the arch ,b ,e2 insomuch that no inequality between the arch ,b ,i and the radius ,b ,c can be discovered either by the hand or by ratiocination4 ,it is therefore to be further enquired1 whether the strait line ,a ,v cut the arch of the quadrant in ,i in the same proportion as the point ,c divides the strait line ,b ,v1 which is equal to the arch of the quadrant4 ,but however this be1 it has been demonstrated that the strait line ,b ,v is equal to the arch ,b ,h ,d4 ,sidenote3 ,the third attempt2 and some things propounded to be further searched into4 #d4 ,i shall now attempt the same dimension of a circle another way1 assuming the two following lemmas4 ,lemma ,i4 ,if to the arch of a quadrant1 and the radius1 there be taken in continual proportion a third line ,z2 then the arch of the semiquadrant1 half the chord of the quadrant1 and ,z1 will also be in continual proportion4 ,for seeing the radius is a mean proportional between the chord of a quadrant and its semichord1 and the same radius a mean proportional between the arch of the quadrant and ,z1 the square of the radius will be equal as well to the rectangle made of the chord and semichord of the quadrant1 as to the rectangle made of the arch of the quadrant and ,z2 and these two rectangles will be equal to one another4 ,wherefore1 as the arch of a quadrant is to its chord1 so reciprocally is half the chord of the quadrant to ,z4 ,but as the arch of the quadrant is to its chord1 so is half the arch of the quadrant to half the chord of the quadrant4 ,wherefore1 as half the arch of the quadrant is to half the chord of the quadrant 7or to the sine of #de degrees71 so is half the chord of the quadrant to ,z2 which was to be proved4 ,lemma ,,ii4 ,the radius1 the arch of the semiquadrant1 the sine of #de degrees1 and the semiradius1 are proportional4 ,for seeing the sine of #de degrees is a mean proportional between the radius and the semiradius2 and the same sine of #de degrees is also a mean proportional 7by the precedent lemma7 between the arch of #de degrees and ,z2 the square of the sine of #de degrees will be equal as well to the rectangle made of the radius and semiradius1 as to the rectangle made of the arch of #de degrees and ,z4 ,wherefore1 as the radius is to the arch of #de degrees1 so reciprocally is ,z to the semiradius2 which was to be demonstrated4 ,let now ,a ,b ,c ,d 7in fig4 #d7 be a square2 and with the radii ,a ,b1 ,b ,c and ,d ,a1 let the three quadrants ,a ,b ,d1 ,b ,c ,a and ,d ,a ,c1 be described2 and let the strait lines ,e ,f and ,g ,h1 drawn parallel to the sides ,b ,c and ,a ,b1 divide the square ,a ,b ,c ,d into four equal squares4 ,they will therefore cut the arch of the quadrant ,a ,b ,d into three equal parts in ,i and ,k1 and the arch of the quadrant ,b ,c ,a into three equal parts in ,k and ,l4 ,also let the diagonals ,a ,c and ,b ,d be drawn1 cutting the arches ,b ,i ,d and ,a ,l ,c in ,m and ,n4 ,then upon the centre ,h with the radius ,h ,f equal to half the chord of the arch ,b ,m ,d1 or to the sine of #de degrees1 let the arch ,f ,o be drawn cutting the arch ,c ,k in ,o2 and let ,a ,o be drawn and produced till it meet with ,b ,c produced in ,p2 also let it cut the arch ,b ,m ,d in ,q1 and the strait line ,d ,c in ,r4 ,if now the strait line ,h ,q be equal to the strait line ,d ,r1 and being produced to ,d ,c in ,s1 cut off ,d ,s equal to half the strait line ,b ,p2 ,i say then the strait line ,b ,p will be equal to the arch ,b ,m ,d4 ,for seeing ,p ,b ,a and ,a ,d ,r are like triangles1 it will be as ,p ,b to the radius ,b ,a or ,a ,d1 so ,a ,d to ,d ,r2 and therefore as well ,p ,b1 ,a ,d and ,d ,r1 as ,p ,b1 ,a ,d 7or ,a ,q7 and ,q ,h are in continual proportion2 and producing ,h ,o to ,d ,c in ,t1 ,d ,t will be equal to the sine of #de degrees1 as shall by and by be demonstrated4 ,now ,d ,s1 ,d ,t and ,d ,r are in continual proportion by the first lemma2 and by the second lemma ,d ,c4 ,d ,s33 ,d ,r4 ,d ,f are proportionals4 ,and thus it will be1 whether ,b ,p be equal or not equal to the arch of the quadrant ,b ,m ,d4 ,but if they be equal1 it will then be1 as that part of the arch ,b ,m ,d which is equal to the radius1 is to the remainder of the same arch ,b ,m ,d2 so ,a ,q to ,h ,q1 or so ,b ,c to ,c ,p4 ,and then will ,b ,p and the arch ,b ,m ,d be equal4 ,but it is not demonstrated that the strait lines ,h ,q and ,d ,r are equal2 though if from the point ,b there be drawn 7by the construction of fig4 #a7 a strait line equal to the arch ,b ,m ,d1 then ,d ,r to ,h ,q1 and also the half of the strait line ,b ,p to ,d ,s1 will always be so equal1 that no inequality can be discovered between them4 ,i will therefore leave this to be further searched into4 ,for though it be almost out of doubt1 that the strait line ,b ,p and the arch ,b ,m ,d are equal1 yet that may not be received without demonstration2 and means of demonstration the circular line admitteth none that is not grounded upon the nature of flexion1 or of angles4 ,but by that way ,i have already exhibited a strait line equal to the arch of a quadrant in the first and second aggression4 ,it remains that ,i prove ,d ,t to be equal to the sine of #de degrees4 ,in ,b ,a produced let ,a ,v be taken equal to the sine of #de degrees2 and drawing and producing ,v ,h1 it will cut the arch of the quadrant ,c ,n ,a in the midst in ,n1 and the same arch again in ,o1 and the strait line ,d ,c in ,t1 so that ,d ,t will be equal to the sine of #de degrees1 or to the strait line ,a ,v2 also the strait line ,v ,h will be equal to the strait line ,h ,i1 or the sine of #fj degrees4 ,for the square of ,a ,v is equal to two squares of the semiradius2 and consequently the square of ,v ,h is equal to three squares of the semiradius4 ,but ,h ,i is a mean proportional between the semiradius and three semiradii2 and1 therefore1 the square of ,h ,i is equal to three squares of the semiradius4 ,wherefore ,h ,i is equal to ,h ,v4 ,but because ,a ,d is cut in the midst in ,h1 therefore ,v ,h and ,h ,t are equal2 and1 therefore1 also ,d ,t is equal to the sine of #de degrees4 ,in the radius ,b ,a let ,b ,x be taken equal to the sine of #de degrees2 for so ,v ,x will be equal to the radius2 and it will be as ,v ,a to ,a ,h the semiradius1 so ,v ,x the radius to ,x ,n the sine of #de degrees4 ,wherefore ,v ,h produced passes through ,n4 ,lastly1 upon the centre ,v with the radius ,v ,a let the arch of a circle be drawn cutting ,v ,h in ,y2 which being done1 ,v ,y will be equal to ,h ,o 7for ,h ,o is1 by construction1 equal to the sine of #de degrees7 and ,y ,h will be equal to ,o ,t2 and1 therefore1 ,v ,t passes through ,o4 ,all which was to be demonstrated4 ,i will here add certain problems1 of which if any analyst can make the construction1 he will thereby be able to judge clearly of what ,i have now said concerning the dimension of a circle4 ,now these problems are nothing else 7at least to sense7 but certain symptoms accompanying the construction of the first and third figure of this chapter4 ,describing1 therefore1 again1 the square ,a ,b ,c ,d 7in fig4 #e7 and the three quadrants ,a ,b ,d1 ,b ,c ,a and ,d ,a ,c1 let the diagonals ,a ,c and ,b ,d be drawn1 cutting the arches ,b ,h ,d and ,c ,i ,a in the middle in ,h and ,i2 and the strait lines ,e ,f and ,g ,l1 dividing the square ,a ,b ,c ,d into four equal squares1 and trisecting the arches ,b ,h ,d and ,c ,i ,a1 namely1 ,b ,h ,d in ,k and ,m1 and ,c ,i ,a in ,m and ,o4 ,then dividing the arch ,b ,k in the midst in ,p1 let ,q ,p the sine of the arch ,b ,p1 be drawn and produced to ,r1 so that ,q ,r be double to ,q ,p2 and1 connecting ,k ,r1 let it be produced one way to ,b ,c in ,s1 and the other way to ,b ,a produced in ,t4 ,also let ,b ,v be made triple to ,b ,s1 and consequently1 7by the second article of this chapter7 equal to the arch ,b ,d4 ,this construction is the same with that of the first figure1 which ,i thought fit to renew discharged of all lines but such as are necessary for my present purpose4 ,in the first place1 therefore1 if ,a ,v be drawn1 cutting the arch ,b ,h ,d in ,x1 and the side ,d ,c in ,z1 ,i desire some analyst would1 if he can1 give a reason why the strait lines ,t ,e and ,t ,c should cut the arch ,b ,d1 the one in ,y1 the other in ,x1 so as to make the arch ,b ,y equal to the arch ,y ,x2 or if they be not equal1 that he would determine their difference4 ,secondly1 if in the side ,d ,a1 the strait line ,d a be taken equal to ,d ,z1 and ,v a be drawn2 why ,v a and ,v ,b should be equal2 or if they be not equal1 what is the difference4 ,thirdly1 drawing ,z b parallel and equal to the side ,c ,b1 cutting the arch ,b ,h ,d in c1 and drawing the strait line ,a c1 and producing it to ,b ,v in d2 why ,a d should be equal and parallel to the strait line a ,v1 and consequently equal also to the arch ,b ,d4 ,fourthly1 drawing e ,k the sine of the arch ,b ,k1 and taking 7in e ,a produced7 e f equal to the diagonal ,a ,c1 and connecting f ,c2 why f ,c should pass through a 7which point being given1 the length of the arch ,b ,h ,d is also given7 and c2 and why f e and f c should be equal2 or if not1 why unequal4 ,fifthly1 drawing f ,z1 ,i desire he would show1 why it is equal to ,b ,v1 or to the arch ,b ,d2 or if they be not equal1 what is their difference4 ,sixthly1 granting f ,z to be equal to the arch ,b ,d1 ,i desire he would determine whether it fall all without the arch ,b ,c ,a1 or cut the same1 or touch it1 and in what point4 ,seventhly1 the semicircle ,b ,d g being completed1 why g ,i being drawn and produced1 should pass through ,x1 by which point ,x the length of the arch ,b ,d is determined4 ,and the same g ,i being yet further produced to ,d ,c in h1 why ,a d1 which is equal to the arch ,b ,d1 should pass through that point h4 ,eighthly1 upon the centre of the square ,a ,b ,c ,d1 which let be k1 the arch of the quadrant ,e i ,l being drawn1 cutting e ,k produced in i1 why the drawn strait line i ,x should be parallel to the side ,c ,d4 ,ninthly1 in the sides ,b ,a and ,b ,c taking g l and ,b m severally equal to half ,b ,v1 or to the arch ,b ,h1 and drawing m n parallel and equal to the side ,b ,a1 cutting the arch ,b ,d in o1 why the strait line which connects ,v l should pass through the point o4 ,tenthly1 ,i would know of him why the strait line which connects a ,h should be equal to ,b m2 or if not1 how much it differs from it4 ,the analyst that can solve these problems without knowing first the length of the arch ,b ,d1 or using any other known method than that which proceeds by perpetual bisection of an angle1 or is drawn from the consideration of the nature of flexion1 shall do more than ordinary geometry is able to perform4 ,but if the dimension of a circle cannot be found by any other method1 then ,i have either found it1 or it is not at all to be found4 ,from the known length of the arch of a quadrant1 and from the proportional division of the arch and of the tangent ,b ,c1 may be deduced the section of an angle into any given proportion2 as also the squaring of the circle1 the squaring of a given sector1 and many the like propositions1 which it is not necessary here to demonstrate4 ,i will1 therefore1 only exhibit a strait line equal to the spiral of ,archimedes1 and so dismiss this speculation4 ,sidenote3 ,the equation of the spiral of ,archimedes with a strait line4 #e4 ,the length of the perimeter of a circle being found1 that strait line is also found1 which touches a spiral at the end of its first conversion4 ,for upon the centre ,a 7in fig4 #f7 let the circle ,b ,c ,d ,e be described2 and in it let ,archimedes' spiral ,a ,f ,g ,h ,b be drawn1 beginning at ,a and ending at ,b4 ,through the centre ,a let the strait line ,c ,e be drawn1 cutting the diameter ,b ,d at right angles2 and let it be produced to ,i1 so that ,a ,i be equal to the perimeter ,b ,c ,d ,e ,b4 ,therefore ,i ,b being drawn will touch the spiral ,a ,f ,g ,h ,b in ,b2 which is demonstrated by ,archimedes in his book ,de ,spiralibus4 ,and for a strait line equal to the given spiral ,a ,f ,g ,h ,b1 it may be found thus4 ,let the strait line ,a ,i1 which is equal to the perimeter ,b ,c ,d ,e1 be bisected in ,k2 and taking ,k ,l equal to the radius ,a ,b1 let the rectangle ,i ,l be completed4 ,let ,m ,l be understood to be the axis1 and ,k ,l the base of a parabola1 and let ,m ,k be the crooked line thereof4 ,now if the point ,m be conceived to be so moved by the concourse of two movents1 the one from ,i ,m to ,k ,l with velocity encreasing continually in the same proportion with the times1 the other from ,m ,l to ,i ,k uniformly1 that both those motions begin together in ,m and end in ,k2 ,galilus has demonstrated that by such motion of the point ,m1 the crooked line of a parabola will be described4 ,again1 if the point ,a be conceived to be moved uniformly in the strait line ,a ,b1 and in the same time to be carried round upon the centre ,a by the circular motion of all the points between ,a and ,b2 ,archimedes has demonstrated that by such motion will be described a spiral line4 ,and seeing the circles of all these motions are concentric in ,a2 and the interior circle is always less than the exterior in the proportion of the times in which ,a ,b is passed over with uniform motion2 the velocity also of the circular motion of the point ,a will continually increase proportionally to the times4 ,and thus far the generations of the parabolical line ,m ,k1 and of the spiral line ,a ,f ,g ,h ,b1 are like4 ,but the uniform motion in ,a ,b concurring with circular motion in the perimeters of all the concentric circles1 describes that circle1 whose centre is ,a1 and perimeter ,b ,c ,d ,e2 and1 therefore1 that circle is 7by the coroll4 of art4 #a1 chap1 ,,xvi7 the aggregate of all the velocities together taken of the point ,a whilst it describes the spiral ,a ,f ,g ,h ,b4 ,also the rectangle ,i ,k ,l ,m is the aggregate of all the velocities together taken of the point ,m1 whilst it describes the crooked line ,m ,k4 ,and1 therefore the whole velocity by which the parabolical line ,m ,k is described1 is to the whole velocity with which the spiral line ,a ,f ,g ,h ,b is described in the same time1 as the rectangle ,i ,k ,l ,m is to the circle ,b ,c ,d ,e1 that is to the triangle ,a ,i ,b4 ,but because ,a ,i is bisected in ,k1 and the strait lines ,i ,m and ,a ,b are equal1 therefore the rectangle ,i ,k ,l ,m and the triangle ,a ,i ,b are also equal4 ,wherefore the spiral line ,a ,f ,g ,h ,b1 and the parabolical line ,m ,k1 being described with equal velocity and in equal times1 are equal to one another4 ,now1 in the first article of chap4 ,,xviii1 a strait line is found out equal to any parabolical line4 ,wherefore also a strait line is found out equal to a given spiral line of the first revolution described by ,archimedes2 which was to be done4 ,sidenote3 ,of the analysis of geometricians by the powers of lines4 #f4 ,in the sixth chapter1 which is of ,method1 that which ,i should there have spoken of the analytics of geometricians ,i thought fit to defer1 because ,i could not there have been understood1 as not having then so much as named lines1 superficies1 solids1 equal and unequal1 c4 ,wherefore ,i will in this place set down my thoughts concerning it4 ,analysis is continual reasoning from the definitions of the terms of a proposition we suppose true1 and again from the definitions of the terms of those definitions1 and so on1 till we come to some things known1 the composition whereof is the demonstration of the truth or falsity of the first supposition2 and this composition or demonstration is that we call ,synthesis4 ,analytica1 therefore1 is that art1 by which our reason proceeds from something supposed1 to principles1 that is1 to prime propositions1 or to such as are known by these1 till we have so many known propositions as are sufficient for the demonstration of the truth or falsity of the thing supposed4 ,synthetica is the art itself of demonstration4 ,synthesis1 therefore1 and analysis1 differ in nothing1 but in proceeding forwards or backwards2 and ,logistica comprehends both4 ,so that in the analysis or synthesis of any question1 that is to say1 of any problem1 the terms of all the propositions ought to be convertible2 or if they be enunciated hypothetically1 the truth of the consequent ought not only to follow out of the truth of its antecedent1 but contrarily also the truth of the antecedent must necessarily be inferred from the truth of the consequent4 ,for otherwise1 when by resolution we are arrived at principles1 we cannot by composition return directly back to the thing sought for4 ,for those terms which are the first in analysis1 will be the last in synthesis2 as for example1 when in resolving1 we say1 these two rectangles are equal1 and therefore their sides are reciprocally proportional1 we must necessarily in compounding say1 the sides of these rectangles are reciprocally proportional1 and therefore the rectangles themselves are equal2 which we could not say1 unless rectangles have their sides reciprocally proportional1 and rectangles are equal1 were terms convertible4 ,now in every analysis1 that which is sought is the proportion of two quantities2 by which proportion1 a figure being described1 the quantity sought for may be exposed to sense4 ,and this exposition is the end and solution of the question1 or the construction of the problem4 ,and seeing analysis is reasoning from something supposed1 till we come to principles1 that is1 to definitions1 or to theorems formerly known2 and seeing the same reasoning tends in the last place to some equation1 we can therefore make no end of resolving1 till we come at last to the causes themselves of equality and inequality1 or to theorems formerly demonstrated from those causes2 and so have a sufficient number of those theorems for the demonstration of the thing sought for4 ,and seeing also1 that the end of the analytics is either the construction of such a problem as is possible1 or the detection of the impossibility thereof2 whensoever the problem may be solved1 the analyst must not stay1 till he come to those things which contain the efficient cause of that whereof he is to make construction4 ,but he must of necessity stay1 when he comes to prime propositions2 and these are definitions4 ,these definitions therefore must contain the efficient cause of his construction2 ,i say of his construction1 not of the conclusion which he demonstrates2 for the cause of the conclusion is contained in the premised propositions2 that is to say1 the truth of the proposition he proves is drawn from the propositions which prove the same4 ,but the cause of his construction is in the things themselves1 and consists in motion1 or in the concourse of motions4 ,wherefore those propositions1 in which analysis ends1 are definitions1 but such as signify in what manner the construction or generation of the thing proceeds4 ,for otherwise1 when he goes back by synthesis to the proof of his problem1 he will come to no demonstration at all2 there being no true demonstration but such as is scientifical2 and no demonstration is scientifical1 but that which proceeds from the knowledge of the causes from which the construction of the problem is drawn4 ,to collect therefore what has been said into few words2 ,,analysis is ratiocination from the supposed construction or generation of a thing to the efficient cause or coefficient causes of that which is constructed or generated4 ,and ,,synthesis is ratiocination from the first causes of the construction1 continued through all the middle causes till we come to the thing itself which is constructed or generated4 ,but because there are many means by which the same thing may be generated1 or the same problem be constructed1 therefore neither do all geometricians1 nor doth the same geometrician always1 use one and the same method4 ,for1 if to a certain quantity given1 it be required to construct another quantity equal1 there may be some that will inquire whether this may not be done by means of some motion4 ,for there are quantities1 whose equality and inequality may be argued from motion and time1 as well as from congruence2 and there is motion1 by which two quantities1 whether lines or superficies1 though one of them be crooked1 the other strait1 may be made congruous or coincident4 ,and this method ,archimedes made use of in his book ,de ,spiralibus4 ,also the equality or inequality of two quantities may be found out and demonstrated from the consideration of weight1 as the same ,archimedes did in his quadrature of the parabola4 ,besides1 equality and inequality are found out often by the division of the two quantities into parts which are considered as indivisable2 as ,cavallerius ,bonaventura has done in our time1 and ,archimedes often4 ,lastly1 the same is performed by the consideration of the powers of lines1 or the roots of those powers1 and by the multiplication1 division1 addition1 and subtraction1 as also by the extraction of the roots of those powers1 or by finding where strait lines of the same proportion terminate4 ,for example1 when any number of strait lines1 how many soever1 are drawn from a strait line and pass all through the same point1 look what proportion they have1 and if their parts continued from the point retain everywhere the same proportion1 they shall all terminate in a strait line4 ,and the same happens if the point be taken between two circles4 ,so that the places of all their points of termination make either strait lines1 or circumferences of circles1 and are called plane places4 ,so also when strait parallel lines are applied to one strait line1 if the parts of the strait line to which they are applied be to one another in proportion duplicate to that of the contiguous applied lines1 they will all terminate in a conical section2 which section1 being the place of their termination1 is called a solid place1 because it serves for the finding out of the quantity of any equation which consists of three dimensions4 ,there are therefore three ways of finding out the cause of equality or inequality between two given quantities2 namely1 first1 by the computation of motions2 for by equal motion1 and equal time1 equal spaces are described2 and ponderation is motion4 ,secondly1 by indivisibles3 because all the parts together taken are equal to the whole4 ,and thirdly1 by the powers3 for when they are equal1 their roots also are equal2 and contrarily1 the powers are equal1 when their roots are equal4 ,but if the question be much complicated1 there cannot by any of these ways be constituted a certain rule1 from the supposition of which of the unknown quantities the analysis may best begin2 nor out of the variety of equations1 that at first appear1 which we were best to choose2 but the success will depend upon dexterity1 upon formerly acquired science1 and many times upon fortune4 ,for no man can ever be a good analyst without being first a good geometrician2 nor do the rules of analysis make a geometrician1 as synthesis doth2 which begins at the very elements1 and proceeds by a logical use of the same4 ,for the true teaching of geometry is by synthesis1 according to ,euclid's method2 and he that hath ,euclid for his master1 may be a geometrician without ,vieta1 though ,vieta was a most admirable geometrician2 but he that has ,vieta for his master1 not so1 without ,euclid4 ,and as for that part of analysis which works by the powers1 though it be esteemed by some geometricians1 not the chiefest1 to be the best way of solving all problems1 yet it is a thing of no great extent2 it being all contained in the doctrine of rectangles1 and rectangled solids4 ,so that although they come to an equation which determines the quantity sought1 yet they cannot sometimes by art exhibit that quantity in a plane1 but in some conic section2 that is1 as geometricians say1 not geometrically1 but mechanically4 ,now such problems as these1 they call solid2 and when they cannot exhibit the quantity sought for with the help of a conic section1 they call it a lineary problem4 ,and therefore in the quantities of angles1 and of the arches of circles1 there is no use at all of the analytics which proceed by the powers2 so that the ancients pronounced it impossible to exhibit in a plane the division of angles1 except bisection1 and the bisection of the bisected parts1 otherwise than mechanically4 ,for ,pappus1 7before the #cast proposition of his fourth book7 distinguishing and defining the several kinds of problems1 says that some are plane1 others solid1 and others lineary4 ,those1 therefore1 which may be solved by strait lines and the circumferences of circles1 7that is1 which may be described with the rule and compass1 without any other instrument71 are fitly called plane2 for the lines1 by which such problems are found out1 have their generation in a plane4 ,but those which are solved by the using of some one or more conic sections in their construction1 are called solid1 because their construction cannot be made without using the superficies of solid figures1 namely1 of cones4 ,there remains the third kind1 which is called lineary1 because other lines besides those already mentioned are made use of in their construction1 c4 ,and a little after he says1 of this kind are the spiral lines1 the quadratrices1 the conchoeides1 and the cissoeides1 ,and geometricians think it no small fault1 when for the finding out of a plane problem any man makes use of conics1 or new lines4 ,now he ranks the trisection of an angle among solid problems1 and the quinquesection among lineary4 ,but what6 are the ancient geometricians to be blamed1 who made use of the quadratrix for the finding out of a strait line equal to the arch of a circle8 ,and ,pappus himself1 was he faulty1 when he found out the trisection of an angle by the help of an hyperbole8 ,or am ,i in the wrong1 who think ,i have found out the construction of both these problems by the rule and compass only8 ,neither they1 nor ,i4 ,for the ancients made use of this analysis which proceeds by the powers2 and with them it was a fault to do that by a more remote power1 which might be done by a nearer2 as being an argument that they did not sufficiently understand the nature of the thing4 ,the virtue of this kind of analysis consists in the changing and turning and tossing of rectangles and analogisms2 and the skill of analysts is mere logic1 by which they are able methodically to find out whatsoever lies hid either in the subject or predicate of the conclusion sought for4 ,but this doth not properly belong to algebra1 or the analytics specious1 symbolical1 or cossick2 which are1 as ,i may say1 the brachygraphy of the analytics1 and an art neither of teaching nor learning geometry1 but of registering with brevity and celerity the inventions of geometricians4 ,for though it be easy to discourse by symbols of very remote propositions2 yet whether such discourse deserve to be thought very profitable1 when it is made without any ideas of the things themselves1 ,i know not4 ---------------------------------------- -------------------------------- ,illustration3 ,c4 ,,xx4 of the translation4 ,fig4 #a-b ,illustration3 ,c4 ,,xx4 of the translation4 ,fig4 #c ,illustration3 ,c4 ,,xx4 of the translation4 ,fig4 #d ,illustration3 ,c4 ,,xx4 of the translation4 ,fig4 #e-f ---------------------------------------- -------------------------------- ------- ,,chapter ,,xxi4 ,,of ,,circular ,,motion4 #a4 ,in simple motion1 every strait line taken in the body moved is so carried1 that it is always parallel to the places in which it formerly was4--#b4 ,if circular motion be made about a resting centre1 and in that circle there be an epicycle1 whose revolution is made the contrary way1 in such manner that in equal times it make equal angles1 every strait line taken in that epicycle will be so carried1 that it will always be parallel to the places in which it formerly was4--#c4 ,the properties of simple motion4--#d4 ,if a fluid be moved with simple circular motion1 all the points taken in it will describe their circles in times proportional to the distances from the centre4--#e4 ,simple motion dissipates heterogeneous and congregates homogeneous bodies4--#f4 ,if a circle made by a movent moved with simple motion be commensurable to another circle made by a point which is carried about by the same movent1 all the points of both the circles will at some time return to the same situation4--#g44 ,if a sphere have simple motion1 its motion will more dissipate heterogeneous bodies by how much it is more remote from the poles4--#h4 ,if the simple circular motion of a fluid body be hindered by a body which is not fluid1 the fluid body will spread itself upon the superficies of that body4--#i4 ,circular motion about a fixed centre casteth off by the tangent such things as lie upon the circumference and stick not to it4--#aj4 ,such things1 as are moved with simple circular motion1 beget simple circular motion4--#aa4 ,if that which is so moved have one side hard and the other side fluid1 its motion will not be perfectly circular4 ,sidenote3 ,in simple motion1 every strait line taken in the body moved is so carried1 that it is always parallel to the places in which it formerly was4 #a4 ,i have already defined simple motion to be that1 in which the several points taken in a moved body do in several equal times describe several equal arches4 ,and therefore in simple circular motion it is necessary that every strait line taken in the moved body be always carried parallel to itself2 which ,i thus demonstrate4 ,first1 let ,a ,b 7in the first figure7 be any strait line taken in any solid body2 and let ,a ,d be any arch drawn upon any centre ,c and radius ,,ca4 ,let the point ,b be understood to describe towards the same parts the arch ,b ,e1 like and equal to the arch ,a ,d4 ,now in the same time in which the point ,a transmits the arch ,a ,d1 the point ,b1 which by reason of its simple motion is supposed to be carried with a velocity equal to that of ,a1 will transmit the arch ,b ,e2 and at the end of the same time the whole ,a ,b will be in ,d ,e2 and therefore ,a ,b and ,d ,e are equal4 ,and seeing the arches ,a ,d and ,b ,e are like and equal1 their subtending strait lines ,,ad and ,,be will also be equal2 and therefore the four-sided figure ,a ,b ,d ,e will be a parallelogram4 ,wherefore ,a ,b is carried parallel to itself4 ,and the same may be proved by the same method1 if any other strait line be taken in the same moved body in which the strait line ,a ,b was taken4 ,so that all strait lines1 taken in a body moved with simple circular motion1 will be carried parallel to themselves4 ,coroll4 ,i4 ,it is manifest that the same will also happen in any body which hath simple motion1 though not circular4 ,for all the points of any strait line whatsoever will describe lines1 though not circular1 yet equal2 so that though the crooked lines ,a ,d and ,b ,e were not arches of circles1 but of parabolas1 ellipses1 or of any other figures1 yet both they1 and their subtenses1 and the strait lines which join them1 would be equal and parallel4 ,coroll4 ,,ii4 ,it is also manifest1 that the radii of the equal circles ,a ,d and ,b ,e1 or the axis of a sphere1 will be so carried1 as to be always parallel to the places in which they formerly were4 ,for the strait line ,b ,f drawn to the centre of the arch ,b ,e being equal to the radius ,a ,c1 will also be equal to the strait line ,f ,e or ,c ,d2 and the angle ,b ,f ,e will be equal to the angle ,a ,c ,d4 ,now the intersection of the strait lines ,c ,a and ,b ,e being at ,g1 the angle ,c ,g ,e 7seeing ,b ,e and ,a ,d are parallel7 will be equal to the angle ,d ,a ,c4 ,but the angle ,e ,b ,f is equal to the same angle ,d ,a ,c2 and therefore the angles ,c ,g ,e and ,e ,b ,f are also equal4 ,wherefore ,a ,c and ,b ,f are parallel2 which was to be demonstrated4 ,sidenote3 ,if circular motion be made about a resting centre1 and in that circle there be an epicycle whose revolution is made the contrary way1 in such manner that in equal times it make equal angles1 every strait line taken in that epicycle will be so carried1 that it will always be parallel to the places in which it formerly was4 #b4 ,let there be a circle given 7in the second figure7 whose centre is ,a1 and radius ,a ,b2 and upon the centre ,b and any radius ,b ,c let the epicycle ,c ,d ,e be described4 ,let the centre ,b be understood to be carried about the centre ,a1 and the whole epicycle with it till it be coincident with the circle ,f ,g ,h1 whose centre is ,i2 and let ,b ,a ,i be any angle given4 ,but in the time that the centre ,b is moved to ,i1 let the epicycle ,c ,d ,e have a contrary revolution upon its own centre1 namely from ,e by ,d to ,c1 according to the same proportions2 that is1 in such manner1 that in both the circles1 equal angles be made in equal times4 ,i say ,e ,c1 the axis of the epicycle1 will be always carried parallel to itself4 ,let the angle ,f ,i ,g be made equal to the angle ,b ,a ,i2 ,i ,f and ,a ,b will therefore be parallel2 and how much the axis ,a ,g has departed from its former place ,a ,c 7the measure of which progression is the angle ,c ,a ,g1 or ,c ,b ,d1 which ,i suppose equal to it7 so much in the same time has the axis ,i ,g1 the same with ,b ,c1 departed from its own former situation4 ,wherefore1 in what time ,b ,c comes to ,i ,g by the motion from ,b to ,i upon the centre ,a1 in the same time ,g will come to ,f by the contrary motion of the epicycle2 that is1 it will be turned backwards to ,f1 and ,i ,g will lie in ,i ,f4 ,but the angles ,f ,i ,g and ,g ,a ,c are equal2 and therefore ,a ,c1 that is1 ,b ,c1 and ,i ,f1 7that is the axis1 though in different places7 will be parallel4 ,wherefore1 the axis of the epicycle ,e ,d ,c will be carried always parallel to itself2 which was to be proved4 ,coroll4 ,from hence it is manifest1 that those two annual motions which ,copernicus ascribes to the earth1 are reducible to this one circular simple motion1 by which all the points of the moved body are carried always with equal velocity1 that is1 in equal times they make equal revolutions uniformly4 ,this1 as it is the most simple1 so it is the most frequent of all circular motions2 being the same which is used by all men when they turn anything round with their arms1 as they do in grinding or sifting4 ,for all the points of the thing moved describe lines which are like and equal to one another4 ,so that if a man had a ruler1 in which many pens' points of equal length were fastened1 he might with this one motion write many lines at once4 ,sidenote3 ,properties of simple motion4 #c4 ,having shown what simple motion is1 ,i will here also set down some properties of the same4 ,first1 when a body is moved with simple motion in a fluid medium which hath no vacuity1 it changes the situation of all the parts of the fluid ambient which resist its motion2 ,i say there are no parts so small of the fluid ambient1 how far soever it be continued1 but do change their situation in such manner1 as that they leave their places continually to other small parts that come into the same4 ,for 7in the same second figure7 let any body1 as ,k ,l ,m ,n1 be understood to be moved with simple circular motion2 and let the circle1 which every point thereof describes1 have any determined quantity1 suppose that of the same ,k ,l ,m ,n4 ,wherefore the centre ,a and every other point1 and consequently the moved body itself1 will be carried sometimes towards the side where is ,k1 and sometimes towards the other side where is ,m4 ,when therefore it is carried to ,k1 the parts of the fluid medium on that side will go back2 and1 supposing all space to be full1 others on the other side will succeed4 ,and so it will be when the body is carried to the side ,m1 and to ,n1 and every way4 ,now when the nearest parts of the fluid medium go back1 it is necessary that the parts next to those nearest parts go back also2 and supposing still all space to be full1 other parts will come into their places with succession perpetual and infinite4 ,wherefore all1 even the least parts of the fluid medium1 change their places1 c4 ,which was to be proved4 ,it is evident from hence1 that simple motion1 whether circular or not circular1 of bodies which make perpetual returns to their former places1 hath greater or less force to dissipate the parts of resisting bodies1 as it is more or less swift1 and as the lines described have greater or less magnitude4 ,now the greatest velocity that can be1 may be understood to be in the least circuit1 and the least in the greatest2 and may be so supposed1 when there is need4 ,sidenote3 ,if a fluid be moved with simple circular motion1 all the points taken in it will describe their circles in times proportional to the distances from the centre4 #d4 ,secondly1 supposing the same simple motion in the air1 water1 or other fluid medium2 the parts of the medium1 which adhere to the moved body1 will be carried about with the same motion and velocity1 so that in what time soever any point of the movent finishes its circle1 in the same time every part of the medium1 which adheres to the movent1 shall also describe such a part of its circle1 as is equal to the whole circle of the movent2 ,i say1 it shall describe a part1 and not the whole circle1 because all its parts receive their motion from an interior concentric movent1 and of concentric circles the exterior are always greater than the interior2 nor can the motion imprinted by any movent be of greater velocity than that of the movent itself4 ,from whence it follows1 that the more remote parts of the fluid ambient shall finish their circles in times1 which have to one another the same proportion with their distances from the movent4 ,for every point of the fluid ambient1 as long as it toucheth the body which carries it about1 is carried about with it1 and would make the same circle1 but that it is left behind so much as the exterior circle exceeds the interior4 ,so that if we suppose some thing1 which is not fluid1 to float in that part of the fluid ambient which is nearest to the movent1 it will together with the movent be carried about4 ,now that part of the fluid ambient1 which is not the nearest but almost the nearest1 receiving its degree of velocity from the nearest1 which degree cannot be greater than it was in the giver1 doth therefore in the same time make a circular line1 not a whole circle1 yet equal to the whole circle of the nearest4 ,therefore in the same time that the movent describes its circle1 that which doth not touch it shall not describe its circle2 yet it shall describe such a part of it1 as is equal to the whole circle of the movent4 ,and after the same manner1 the more remote parts of the ambient will describe in the same time such parts of their circles1 as shall be severally equal to the whole circle of the movent2 and1 by consequent1 they shall finish their whole circles in times proportional to their distances from the movent2 which was to be proved4 ,sidenote3 ,simple motion dissipates heterogeneous and congregates homogeneous bodies4 #e4 ,thirdly1 the same simple motion of a body placed in a fluid medium1 congregates or gathers into one place such things as naturally float in that medium1 if they be homogeneous2 and if they be heterogeneous1 it separates and dissipates them4 ,but if such things as be heterogeneous do not float1 but settle1 then the same motion stirs and mingles them disorderly together4 ,for seeing bodies1 which are unlike to one another1 that is1 heterogeneous bodies1 are not unlike in that they are bodies2 for bodies1 as bodies1 have no difference2 but only from some special cause1 that is1 from some internal motion1 or motions of their smallest parts 7for ,i have shown in chap4 ,,ix1 art4 #i1 that all mutation is such motion71 it remains that heterogeneous bodies have their unlikeness or difference from one another from their internal or specifical motions4 ,now bodies which have such difference receive unlike and different motions from the same external common movent2 and therefore they will not be moved together1 that is to say1 they will be dissipated4 ,and being dissipated they will necessarily at some time or other meet with bodies like themselves1 and be moved alike and together with them2 and afterwards meeting with more bodies like themselves1 they will unite and become greater bodies4 ,wherefore homogeneous bodies are congregated1 and heterogeneous dissipated by simple motion in a medium where they naturally float4 ,again1 such as being in a fluid medium do not float1 but sink1 if the motion of the fluid medium be strong enough1 will be stirred up and carried away by that motion1 and consequently they will be hindered from returning to that place to which they sink naturally1 and in which only they would unite1 and out of which they are promiscuously carried2 that is1 they are disorderly mingled4 ,now this motion1 by which homogeneous bodies are congregated and heterogeneous are scattered1 is that which is commonly called fermentation1 from the ,latin fervere2 as the ,greeks have their 1 which signifies the same1 from ferveo4 ,for seething makes all the parts of the water change their places2 and the parts of any thing1 that is thrown into it1 will go several ways according to their several natures4 ,and yet all fervour or seething is not caused by fire2 for new wine and many other things have also their fermentation and fervour1 to which fire contributes little1 and sometimes nothing4 ,but when in fermentation we find heat1 it is made by the fermentation4 ,sidenote3 ,if a circle made by a movent moved with simple motion1 be commensurable to another circle made by a point which is carried about by the same movent1 all the points of both the circles will at some time return to the same situation4 #f4 ,fourthly1 in what time soever the movent1 whose centre is ,a 7in fig4 #b7 moved in ,k ,l ,n1 shall1 by any number of revolutions1 that is1 when the perimeters ,b ,i and ,k ,l ,n be commensurable1 have described a line equal to the circle which passes through the points ,b and ,i2 in the same time all the points of the floating body1 whose centre is ,b1 shall return to have the same situation in respect of the movent1 from which they departed4 ,for seeing it is as the distance ,b ,a1 that is1 as the radius of the circle which passes through ,b ,i is to the perimeter itself ,b ,i1 so the radius of the circle ,k ,l ,n is to the perimeter ,k ,l ,n2 and seeing the velocities of the points ,b and ,k are equal1 the time also of the revolution in ,i ,b to the time of one revolution in ,k ,l ,n1 will be as the perimeter ,b ,i to the perimeter ,k ,l ,n2 and therefore so many revolutions in ,k ,l ,n1 as together taken are equal to the perimeter ,b ,i1 will be finished in the same time in which the whole perimeter ,b ,i is finished2 and therefore also the points ,l1 ,n1 ,f and ,h1 or any of the rest1 will in the same time return to the same situation from which they departed2 and this may be demonstrated1 whatsoever be the points considered4 ,wherefore all the points shall in that time return to the same situation2 which was to be proved4 ,from hence it follows1 that if the perimeters ,b ,i and ,l ,k ,n be not commensurable1 then all the points will never return to have the same situation or configuration in respect of one another4 ,sidenote3 ,if a sphere have simple motion1 its motion will more dissipate heterogeneous bodies by how much it is more remote from the poles4 #g4 ,in simple motion1 if the body moved be of a spherical figure1 it hath less force towards its poles than towards its middle to dissipate heterogeneous1 or to congregate homogeneous bodies4 ,let there be a sphere 7as in the third figure7 whose centre is ,a and diameter ,b ,c2 and let it be conceived to be moved with simple circular motion2 of which motion let the axis be the strait line ,d ,e1 cutting the diameter ,b ,c at right angles in ,a4 ,let now the circle1 which is described by any point ,b of the sphere1 have ,b ,f for its diameter2 and taking ,f ,g equal to ,b ,c1 and dividing it in the middle in ,h1 the centre of the sphere ,a will1 when half a revolution is finished1 lie in ,h4 ,and seeing ,h ,f and ,a ,b are equal1 a circle described upon the centre ,h with the radius ,h ,f or ,h ,g1 will be equal to the circle whose centre is ,a and radius ,a ,b4 ,and if the same motion be continued1 the point ,b will at the end of another half revolution return to the place from whence it began to be moved2 and therefore at the end of half a revolution1 the point ,b will be carried to ,f1 and the whole hemisphere ,d ,b ,e into that hemisphere in which are the points ,l1 ,k and ,f4 ,wherefore that part of the fluid medium1 which is contiguous to the point ,f1 will in the same time go back the length of the strait line ,b ,f2 and in the return of the point ,f to ,b1 that is1 of ,g to ,c1 the fluid medium will go back as much in a strait line from the point ,c4 ,and this is the effect of simple motion in the middle of the sphere1 where the distance from the poles is greatest4 ,let now the point ,i be taken in the same sphere nearer to the pole ,e1 and through it let the strait line ,i ,k be drawn parallel to the strait line ,b ,f1 cutting the arch ,f ,l in ,k1 and the axis ,h ,l in ,m2 then connecting ,h ,k1 upon ,h ,f let the perpendicular ,k ,n be drawn4 ,in the same time therefore that ,b comes to ,f the point ,i will come to ,k1 ,b ,f and ,i ,k being equal and described with the same velocity4 ,now the motion in ,i ,k to the fluid medium upon which it works1 namely1 to that part of the medium which is contiguous to the point ,k1 is oblique1 whereas if it proceeded in the strait line ,h ,k it would be perpendicular2 and therefore the motion which proceeds in ,i ,k has less power than that which proceeds in ,h ,k with the same velocity4 ,but the motions in ,h ,k and ,h ,f do equally thrust back the medium2 and therefore the part of the sphere at ,k moves the medium less than the part at ,f1 namely1 so much less as ,k ,n is less than ,h ,f4 ,wherefore also the same motion hath less power to disperse heterogeneous1 and to congregate homogeneous bodies1 when it is nearer1 than when it is more remote from the poles2 which was to be proved4 ,coroll4 ,it is also necessary1 that in planes which are perpendicular to the axis1 and more remote than the pole itself from the middle of the sphere1 this simple motion have no effect4 ,for the axis ,d ,e with simple motion describes the superficies of a cylinder2 and towards the bases of the cylinder there is in this motion no endeavour at all4 ,sidenote3 ,if a simple circular motion of a fluid body be hindered by a body which is not fluid1 the fluid body will spread itself upon the superficies of that body4 #h4 ,if in a fluid medium moved about1 as hath been said1 with simple motion1 there be conceived to float some other spherical body which is not fluid1 the parts of the medium1 which are stopped by that body1 will endeavour to spread themselves every way upon the superficies of it4 ,and this is manifest enough by experience1 namely1 by the spreading of water poured out upon a pavement4 ,but the reason of it may be this4 ,seeing the sphere ,a 7in fig4 #c7 is moved towards ,b1 the medium also in which it is moved will have the same motion4 ,but because in this motion it falls upon a body not liquid1 as ,g1 so that it cannot go on2 and seeing the small parts of the medium cannot go forwards1 nor can they go directly backwards against the force of the movent2 it remains1 therefore1 that they diffuse themselves upon the superficies of that body1 as towards ,o and ,p2 which was to be proved4 ,sidenote3 ,circular motion about a fixed centre casteth off by the tangent such things as lie upon the circumference stick not to it4 #i4 ,compounded circular motion1 in which all the parts of the moved body do at once describe circumferences1 some greater1 others less1 according to the proportion of their several distances from the common centre1 carries about with it such bodies1 as being not fluid1 adhere to the body so moved2 and such as do not adhere1 it casteth forwards in a strait line which is a tangent to the point from which they are cast off4 ,for let there be a circle whose radius is ,a ,b 7in fig4 #d72 and let a body be placed in the circumference in ,b1 which if it be fixed there1 will necessarily be carried about with it1 as is manifest of itself4 ,but whilst the motion proceeds1 let us suppose that body to be unfixed in ,b4 ,i say1 the body will continue its motion in the tangent ,b ,c4 ,for let both the radius ,a ,b and the sphere ,b be conceived to consist of hard matter2 and let us suppose the radius ,a ,b to be stricken in the point ,b by some other body which falls upon it in the tangent ,d ,b4 ,now1 therefore1 there will be a motion made by the concourse of two things1 the one1 endeavour towards ,c in the strait line ,d ,b produced1 in which the body ,b would proceed1 if it were not retained by the radius ,a ,b2 the other1 the retention itself4 ,but the retention alone causeth no endeavour towards the centre2 and1 therefore1 the retention being taken away1 which is done by the unfixing of ,b1 there will remain but one endeavour in ,b1 namely1 that in the tangent ,b ,c4 ,wherefore the motion of the body ,b unfixed will proceed in the tangent ,b ,c2 which was to be proved4 ,by this demonstration it is manifest1 that circular motion about an unmoved axis shakes off and puts further from the centre of its motion such things as touch1 but do not stick fast to its superficies2 and the more1 by how much the distance is greater from the poles of the circular motion2 and so much the more also1 by how much the things1 that are shaken off1 are less driven towards the centre by the fluid ambient1 for other causes4 ,sidenote3 ,such things as are moved with simple circular motion1 beget simple circular motion4 #aj4 ,if in a fluid medium a spherical body be moved with simple circular motion1 and in the same medium there float another sphere whose matter is not fluid1 this sphere also shall be moved with simple circular motion4 ,let ,b ,c ,d 7in fig4 #e7 be a circle1 whose centre is ,a1 and in whose circumference there is a sphere so moved1 that it describes with simple motion the perimeter ,b ,c ,d4 ,let also ,e ,f ,g be another sphere of consistent matter1 whose semidiameter is ,e ,h1 and centre ,h2 and with the radius ,a ,h let the circle ,h ,i be described4 ,i say1 the sphere ,e ,f ,g will1 by the motion of the body in ,b ,c ,d1 be moved in the circumference ,h ,i with simple motion4 ,for seeing the motion in ,b ,c ,d 7by art4 #d of this chapter7 makes all the points of the fluid medium describe in the same time circular lines equal to one another1 the points ,e1 ,h and ,g of the strait line ,e ,h ,g will in the same time describe with equal radii equal circles4 ,let ,e ,b be drawn equal and parallel to the strait line ,a ,h2 and let ,a ,b be connected1 which will therefore be equal and parallel to ,e ,h2 and therefore also1 if upon the centre ,b and radius ,b ,e the arch ,e ,k be drawn equal to the arch ,h ,i1 and the strait lines ,a ,i1 ,b ,k and ,i ,k be drawn1 ,b ,k and ,a ,i will be equal2 and they will also be parallel1 because the two arches ,e ,k and ,h ,i1 that is1 the two angles ,k ,b ,e and ,i ,a ,h are equal2 and1 consequently1 the strait lines ,a ,b and ,k ,i1 which connect them1 will also be equal and parallel4 ,wherefore ,k ,i and ,e ,h are parallel4 ,seeing1 therefore1 ,e and ,h are carried in the same time to ,k and ,i1 the whole strait line ,i ,k will be parallel to ,e ,h1 from whence it departed4 ,and1 therefore1 seeing the sphere ,e ,f ,g is supposed to be of consistent matter1 so as all its points keep always the same situation1 it is necessary that every other strait line1 taken in the same sphere1 be carried always parallel to the places in which it formerly was4 ,wherefore the sphere ,e ,f ,g is moved with simple circular motion2 which was to be demonstrated4 ,sidenote3 ,if that which is so moved have one side hard and the other side fluid1 its motion will not be perfectly circular4 #aa4 ,if in a fluid medium1 whose parts are stirred by a body moved with simple motion1 there float another body1 which hath its superficies either wholly hard1 or wholly fluid1 the parts of this body shall approach the centre equally on all sides2 that is to say1 the motion of the body shall be circular1 and concentric with the motion of the movent4 ,but if it have one side hard1 and the other side fluid1 then both those motions shall not have the same centre1 nor shall the floating body be moved in the circumference of a perfect circle4 ,let a body be moved in the circumference of the circle ,k ,l ,m ,n 7in fig #b47 whose centre is ,a4 ,and let there be another body at ,i1 whose superficies is either all hard or all fluid4 ,also let the medium1 in which both these bodies are placed1 be fluid4 ,i say1 the body at ,i will be moved in the circle ,i ,b about the centre ,a4 ,for this has been demonstrated in the last article4 ,wherefore let the superficies of the body at ,i be fluid on one side1 and hard on the other4 ,and first1 let the fluid side be towards the centre4 ,seeing1 therefore1 the motion of the medium is such1 as that its parts do continually change their places1 7as hath been shown in art #e72 if this change of place be considered in those parts of the medium which are contiguous to the fluid superficies1 it must needs be that the small parts of that superficies enter into the places of the small parts of the medium which are contiguous to them2 and the like change of place will be made with the next contiguous parts towards ,a4 ,and if the fluid parts of the body at ,i have any degree at all of tenacity 7for there are degrees of tenacity1 as in the air and water7 the whole fluid side will be lifted up a little1 but so much the less1 as its parts have less tenacity2 whereas the hard part of the superficies1 which is contiguous to the fluid part1 has no cause at all of elevation1 that is to say1 no endeavour towards ,a4 ,secondly1 let the hard superficies of the body at ,i be towards ,a4 ,by reason1 therefore1 of the said change of place of the parts which are contiguous to it1 the hard superficies must1 of necessity1 seeing by supposition there is no empty space1 either come nearer to ,a1 or else its smallest parts must supply the contiguous places of the medium1 which otherwise would be empty4 ,but this cannot be1 by reason of the supposed hardness2 and1 therefore1 the other must needs be1 namely1 that the body come nearer to ,a4 ,wherefore the body at ,i has greater endeavour towards the centre ,a1 when its hard side is next it1 than when it is averted from it4 ,but the body in ,i1 while it is moving in the circumference of the circle ,i ,b1 has sometimes one side1 sometimes another1 turned towards the centre2 and1 therefore1 it is sometimes nearer1 sometimes further off from the centre ,a4 ,wherefore the body at ,i is not carried in the circumference of a perfect circle2 which was to be demonstrated4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxi4 ,fig4 #a-e ---------------------------------------- -------------------------------- ------- ,,chapter ,,xxii4 ,,of ,,other ,,variety ,,of ,,motion4 #a4 ,endeavour and pressure how they differ4--#b4 ,two kinds of mediums in which bodies are moved4--#c4 ,propagation of motion1 what it is4--#d4 ,what motion bodies have1 when they press one another4--#e4 ,fluid bodies1 when they are pressed together1 penetrate one another4--#f4 ,when one body presseth another and doth not penetrate it1 the action of the pressing body is perpendicular to the superficies of the body pressed4--#g4 ,when a hard body1 pressing another body1 penetrates the same1 it doth not penetrate it perpendicularly1 unless it fall perpendicularly upon it4--#h4 ,motion sometimes opposite to that of the movent4--#i4 ,in a full medium1 motion is propagated to any distance4--#aj4 ,dilatation and contraction what they are4--#aa4 ,dilatation and contraction suppose mutation of the smallest parts in respect of their situation4--#ab4 ,all traction is pulsion4--#ac4 ,such things as being pressed or bent restore themselves1 have motion in their internal parts4--#ad4 ,though that which carrieth another be stopped1 the body carried will proceed4--#ae1 #af4 ,the effects of percussion not to be compared with those of weight4--#ag1 #ah4 ,motion cannot begin first in the internal parts of a body4--#ai4 ,action and reaction proceed in the same line4--#bj4 ,habit1 what it is4 ,sidenote3 ,endeavour and pressure how they differ4 #a4 ,i have already 7chapter ,,xv4 art4 #b7 defined endeavour to be motion through some length1 though not considered as length1 but as a point4 ,whether1 therefore1 there be resistance or no resistance1 the endeavour will still be the same4 ,for simply to endeavour is to go4 ,but when two bodies1 having opposite endeavours1 press one another1 then the endeavour of either of them is that which we call pressure1 and is mutual when their pressures are opposite4 ,sidenote3 ,two kinds of mediums in which bodies are moved4 #b4 ,bodies moved1 and also the mediums in which they are moved1 are of two kinds4 ,for either they have their parts coherent in such manner1 as no part of the moved body will easily yield to the movent1 except the whole body yield also1 and such are the things we call hard3 or else their parts1 while the whole remains unmoved1 will easily yield to the movent1 and these we call fluid or soft bodies4 ,for the words fluid1 soft1 tough1 and hard1 in the same manner as great and little1 are used only comparatively2 and are not different kinds1 but different degrees of quality4 ,sidenote3 ,propagation of motion1 what it is4 #c4 ,to do1 and to suffer1 is to move and to be moved2 and nothing is moved but by that which toucheth it and is also moved1 as has been formerly shown4 ,and how great soever the distance be1 we say the first movent moveth the last moved body1 but mediately2 namely so1 as that the first moveth the second1 the second the third1 and so on1 till the last of all be touched4 ,when therefore one body1 having opposite endeavour to another body1 moveth the same1 and that moveth a third1 and so on1 ,i call that action propagation of motion4 ,sidenote3 ,what motion bodies have when they press one another4 #d4 ,when two fluid bodies1 which are in a free and open space1 press one another1 their parts will endeavour1 or be moved1 towards the sides2 not only those parts which are there where the mutual contact is1 but all the other parts4 ,for in the first contact1 the parts1 which are pressed by both the endeavouring bodies1 have no place either forwards or backwards in which they can be moved2 and therefore they are pressed out towards the sides4 ,and this expressure1 when the forces are equal1 is in a line perpendicular to the bodies pressing4 ,but whensoever the foremost parts of both the bodies are pressed1 the hindermost also must be pressed at the same time2 for the motion of the hindermost parts cannot in an instant be stopped by the resistance of the foremost parts1 but proceeds for some time2 and therefore1 seeing they must have some place in which they may be moved1 and that there is no place at all for them forwards1 it is necessary that they be moved into the places which are towards the sides every way4 ,and this effect follows of necessity1 not only in fluid1 but in consistent and hard bodies1 though it be not always manifest to sense4 ,for though from the compression of two stones we cannot with our eyes discern any swelling outwards towards the sides1 as we perceive in two bodies of wax2 yet we know well enough by reason1 that some tumour must needs be there1 though it be but little4 ,sidenote3 ,fluid bodies1 when they are pressed together1 penetrate one another4 #e4 ,but when the space is enclosed1 and both the bodies be fluid1 they will1 if they be pressed together1 penetrate one another1 though differently1 according to their different endeavours4 ,for suppose a hollow cylinder of hard matter1 well stopped at both ends1 but filled first1 below with some heavy fluid body1 as quicksilver1 and above with water or air4 ,if now the bottom of the cylinder be turned upwards1 the heaviest fluid body1 which is now at the top1 having the greatest endeavour downwards1 and being by the hard sides of the vessel hindered from extending itself sideways1 must of necessity either be received by the lighter body1 that it may sink through it1 or else it must open a passage through itself1 by which the lighter body may ascend4 ,for of the two bodies1 that1 whose parts are most easily separated1 will be the first divided2 which being done1 it is not necessary that the parts of the other suffer any separation at all4 ,and therefore when two liquors1 which are enclosed in the same vessel1 change their places1 there is no need that their smallest parts should be mingled with one another2 for a way being opened through one of them1 the parts of the other need not be separated4 ,now if a fluid body1 which is not enclosed1 press a hard body1 its endeavour will indeed be towards the internal parts of that hard body2 but being excluded by the resistance of it1 the parts of the fluid body will be moved every way according to the superficies of the hard body1 and that equally1 if the pressure be perpendicular2 for when all the parts of the cause are equal1 the effects will be equal also4 ,but if the pressure be not perpendicular1 then the angles of the incidence being unequal1 the expansion also will be unequal1 namely1 greater on that side where the angle is greater1 because that motion is most direct which proceeds by the directest line4 ,sidenote3 ,when one body presseth another and doth not penetrate it1 the action of the pressing body is perpendicular to the superficies of the body pressed4 #f4 ,if a body1 pressing another body1 do not penetrate it1 it will nevertheless give to the part it presseth an endeavour to yield1 and recede in a strait line perpendicular to its superficies in that point in which it is pressed4 ,let ,a ,b ,c ,d 7in fig4 #a7 be a hard body1 and let another body1 falling upon it in the strait line ,e ,a1 with any inclination or without inclination1 press it in the point ,a4 ,i say the body so pressing1 and not penetrating it1 will give to the part ,a an endeavour to yield or recede in a strait line perpendicular to the line ,a ,d4 ,for let ,a ,b be perpendicular to ,a ,d1 and let ,b ,a be produced to ,f4 ,if therefore ,a ,f be coincident with ,a ,e1 it is of itself manifest that the motion in ,e ,a will make ,a to endeavour in the line ,a ,b4 ,let now ,e ,a be oblique to ,a ,d1 and from the point ,e let the strait line ,e ,c be drawn1 cutting ,a ,d at right angles in ,d1 and let the rectangles ,a ,b ,c ,d and ,a ,d ,e ,f be completed4 ,i have shown 7in the #hth article of chapter ,,xvi7 that the body will be carried from ,e to ,a by the concourse of two uniform motions1 the one in ,e ,f and its parallels1 the other in ,e ,d and its parallels4 ,but the motion in ,e ,f and its parallels1 whereof ,d ,a is one1 contributes nothing to the body in ,a to make it endeavour or press towards ,b2 and therefore the whole endeavour1 which the body hath in the inclined line ,e ,a to pass or press the strait line ,a ,d1 it hath it all from the perpendicular motion or endeavour in ,f ,a4 ,wherefore the body ,e1 after it is in ,a1 will have only that perpendicular endeavour which proceeds from the motion in ,f ,a1 that is1 in ,a ,b2 which was to be proved4 ,sidenote3 ,when a hard body1 pressing another body1 penetrates the same1 it doth not penetrate it perpendicularly1 unless it fall perpendicularly upon it4 #g4 ,if a hard body falling upon or pressing another body penetrate the same1 its endeavour after its first penetration will be neither in the inclined line produced1 nor in the perpendicular1 but sometimes betwixt both1 sometimes without them4 ,let ,e ,a ,g 7in the same fig4 #a7 be the inclined line produced2 and first1 let the passage through the medium1 in which ,e ,a is1 be easier than the passage through the medium in which ,a ,g is4 ,as soon therefore as the body is within the medium in which is ,a ,g1 it will find greater resistance to its motion in ,d ,a and its parallels1 than it did whilst it was above ,a ,d2 and therefore below ,a ,d it will proceed with slower motion in the parallels of ,d ,a1 than above it4 ,wherefore the motion which is compounded of the two motions in ,e ,f and ,e ,d will be slower below ,a ,d than above it2 and therefore also1 the body will not proceed from ,a in ,e ,a produced1 but below it4 ,seeing1 therefore1 the endeavour in ,a ,b is generated by the endeavour in ,f ,a2 if to the endeavour in ,f ,a there be added the endeavour in ,d ,a1 which is not all taken away by the immersion of the point ,a into the lower medium1 the body will not proceed from ,a in the perpendicular ,a ,b1 but beyond it2 namely1 in some strait line between ,a ,b and ,a ,g1 as in the line ,a ,h4 ,secondly1 let the passage through the medium ,e ,a be less easy than that through ,a ,g4 ,the motion1 therefore1 which is made by the concourse of the motions in ,e ,f and ,f ,b1 is slower above ,a ,d than below it2 and consequently1 the endeavour will not proceed from ,a in ,e ,a produced1 but beyond it1 as in ,a ,i4 ,wherefore1 if a hard body falling1 c42 which was to be proved4 ,this divergency of the strait line ,a ,h from the strait line ,a ,g is that which1 the writers of optics commonly called refraction1 which1 when the passage is easier in the first than in the second medium1 is made by diverging from the line of inclination towards the perpendicular2 and contrarily1 when the passage is not so easy in the first medium1 by departing further from the perpendicular4 ,sidenote3 ,motion sometimes opposite to that of the movent4 #h4 ,by the #fth theorem it is manifest1 that the force of the movent may be so placed1 as that the body moved by it may proceed in a way almost directly contrary to that of the movent1 as we see in the motion of ships4 ,for let ,a ,b 7in fig4 #b7 represent a ship1 whose length from the prow to the poop is ,a ,b1 and let the wind lie upon it in the strait parallel lines ,c ,b1 ,d ,e and ,f ,g2 and let ,d ,e and ,f ,g be cut in ,e and ,g by a strait line drawn from ,b perpendicular to ,a ,b2 also let ,b ,e and ,e ,g be equal1 and the angle ,a ,b ,c any angle how small soever4 ,then between ,b ,c and ,b ,a let the strait line ,b ,i be drawn2 and let the sail be conceived to be spread in the same line ,b ,i1 and the wind to fall upon it in the points ,l1 ,m and ,b2 from which points1 perpendicular to ,b ,i1 let ,b ,k1 ,m ,q and ,l ,p be drawn4 ,lastly1 let ,e ,n and ,g ,o be drawn perpendicular to ,b ,g1 and cutting ,b ,k in ,h and ,k2 and let ,h ,n and ,k ,o be made equal to one another1 and severally equal to ,b ,a4 ,i say1 the ship ,b ,a1 by the wind falling upon it in ,c ,b1 ,d ,e1 ,f ,g1 and other lines parallel to them1 will be carried forwards almost opposite to the wind1 that is to say1 in a way almost contrary to the way of the movent4 ,for the wind that blows in the line ,c ,b will 7as hath been shown in art4 #f7 give to the point ,b an endeavour to proceed in a strait line perpendicular to the strait line ,b ,i1 that is1 in the strait line ,b ,k2 and to the points ,m and ,l an endeavour to proceed in the strait lines ,m ,q and ,l ,p1 which are parallel to ,b ,k4 ,let now the measure of the time be ,b ,g1 which is divided in the middle in ,e2 and let the point ,b be carried to ,h in the time ,b ,e4 ,in the same time1 therefore1 by the wind blowing in ,d ,m and ,f ,l1 and as many other lines as may be drawn parallel to them1 the whole ship will be applied to the strait line ,h ,n4 ,also at the end of the second time ,e ,g1 it will be applied to the strait line ,k ,o4 ,wherefore the ship will always go forward2 and the angle it makes with the wind will be equal to the angle ,a ,b ,c1 how small soever that angle be2 and the way it makes will in every time be equal to the strait line ,e ,h4 ,i say1 thus it would be1 if the ship might be moved with as great celerity sideways from ,b ,a towards ,k ,o1 as it may be moved forwards in the line ,b ,a4 ,but this is impossible1 by reason of the resistance made by the great quantity of water which presseth the side1 much exceeding the resistance made by the much smaller quantity which presseth the prow of the ship2 so that the way the ship makes sideways is scarce sensible2 and1 therefore1 the point ,b will proceed almost in the very line ,b ,a1 making with the wind the angle ,a ,b ,c1 how acute soever2 that is to say1 it will proceed almost in the strait line ,b ,c1 that is1 in a way almost contrary to the way of the movent2 which was to be demonstrated4 ,but the sail in ,b ,i must be so stretched as that there be left in it no bosom at all2 for otherwise the strait lines ,l ,p1 ,m ,q and ,b ,k will not be perpendicular to the plane of the sail1 but falling below ,p1 ,q and ,k1 will drive the ship backwards4 ,but by making use of a small board for a sail1 a little waggon with wheels for the ship1 and of a smooth pavement for the sea1 ,i have by experience found this to be so true1 that ,i could scarce oppose the board to the wind in any obliquity1 though never so small1 but the waggon was carried forwards by it4 ,by the same #fth theorem it may be found1 how much a stroke1 which falls obliquely1 is weaker than a stroke falling perpendicularly1 they being like and equal in all other respects4 ,let a stroke fall upon the wall ,a ,b obliquely1 as for example1 in the strait line ,c ,a 7in fig4 #c47 ,let ,c ,e be drawn parallel to ,a ,b1 and ,d ,a perpendicular to the same ,a ,b and equal to ,c ,a2 and let both the velocity and time of the motion in ,c ,a be equal to the velocity and time of the motion in ,d ,a4 ,i say1 the stroke in ,c ,a will be weaker than that in ,d ,a1 in the proportion of ,e ,a to ,d ,a4 ,for producing ,d ,a howsoever to ,f1 the endeavour of both the strokes will 7by art4 #f7 proceed from ,a in the perpendicular ,a ,f4 ,but the stroke in ,c ,a is made by the concourse of two motions in ,c ,e and ,e ,a1 of which that in ,c ,e contributes nothing to the stroke in ,a1 because ,c ,e and ,b ,a are parallels2 and1 therefore1 the stroke in ,c ,a is made by the motion which is in ,e ,a only4 ,but the velocity or force of the perpendicular stroke in ,e ,a1 to the velocity or force of the stroke in ,d ,a1 is as ,e ,a to ,d ,a4 ,wherefore the oblique stroke in ,c ,a is weaker than the perpendicular stroke in ,d ,a1 in the proportion of ,e ,a to ,d ,a or ,c ,a2 which was to be proved4 ,sidenote3 ,in a full medium1 motion is propagated to any distance4 #i4 ,in a full medium1 all endeavour proceeds as far as the medium itself reacheth2 that is to say1 if the medium be infinite1 the endeavour will proceed infinitely4 ,for whatsoever endeavoureth is moved1 and therefore whatsoever standeth in its way it maketh it yield1 at least a little1 namely1 so far as the movent itself is moved forwards4 ,but that which yieldeth is also moved1 and consequently maketh that to yield which is in its way1 and so on successively as long as the medium is full2 that is to say1 infinitely1 if the full medium be infinite2 which was to be proved4 ,now although endeavour thus perpetually propagated do not always appear to the senses as motion1 yet it appears as action1 or as the efficient cause of some mutation4 ,for if there be placed before our eyes some very little object1 as for example1 a small grain of sand1 which at a certain distance is visible2 it is manifest that it may be removed to such a distance as not to be any longer seen1 though by its action it still work upon the organs of sight1 as is manifest from that which was last proved1 that all endeavour proceeds infinitely4 ,let it be conceived therefore to be removed from our eyes to any distance how great soever1 and a sufficient number of other grains of sand of the same bigness added to it2 it is evident that the aggregate of all those sands will be visible2 and though none of them can be seen when it is single and severed from the rest1 yet the whole heap or hill which they make will manifestly appear to the sight2 which would be impossible1 if some action did not proceed from each several part of the whole heap4 ,sidenote3 ,dilatation contraction what they are4 #aj4 ,between the degrees of hard and soft are those things which we call tough1 tough being that which may be bent without being altered from what it was2 and the bending of a line is either the adduction or diduction of the extreme parts1 that is1 a motion from straitness to crookedness1 or contrarily1 whilst the line remains still the same it was2 for by drawing out the extreme points of a line to their greatest distance1 the line is made strait1 which otherwise is crooked4 ,so also the bending of a superficies is the diduction or adduction of its extreme lines1 that is1 their dilatation and contraction4 ,sidenote3 ,dilatation contraction suppose mutation of the smallest parts in respect of their situation4 #aa4 ,dilatation and contraction1 as also all flexion1 supposes necessarily that the internal parts of the body bowed do either come nearer to the external parts1 or go further from them4 ,for though flexion be considered only in the length of a body1 yet when that body is bowed1 the line which is made on one side will be convex1 and the line on the other side will be concave2 of which the concave1 being the interior line1 will1 unless something be taken from it and added to the convex line1 be the more crooked1 that is1 the greater of the two4 ,but they are equal2 and1 therefore1 in flexion there is an accession made from the interior to the exterior parts2 and1 on the contrary1 in tension1 from the exterior to the interior parts4 ,and as for those things which do not easily suffer such transposition of their parts1 they are called brittle2 and the great force they require to make them yield1 makes them also with sudden motion to leap asunder1 and break in pieces4 ,sidenote3 ,all traction is pulsion4 #ab4 ,also motion is distinguished into pulsion and traction4 ,and pulsion1 as ,i have already defined it1 is when that which is moved goes before that which moveth it4 ,but contrarily1 in traction the movent goes before that which is moved4 ,nevertheless1 considering it with greater attention1 it seemeth to be the same with pulsion4 ,for of two parts of a hard body1 when that which is foremost drives before it the medium in which the motion is made1 at the same time that which is thrust forwards thrusteth the next1 and this again the next1 and so on successively4 ,in which action1 if we suppose that there is no place void1 it must needs be1 that by continual pulsion1 namely1 when that action has gone round1 the movent will be behind that part1 which at the first seemed not to be thrust forwards1 but to be drawn2 so that now the body1 which was drawn1 goes before the body which gives it motion2 and its motion is no longer traction1 but pulsion4 ,sidenote3 ,such things as being pressed or bent restore themselves1 have motion in their internal parts4 #ac4 ,such things as are removed from their places by forcible compression or extension1 and1 as soon as the force is taken away1 do presently return and restore themselves to their former situation1 have the beginning of their restitution within themselves1 namely1 a certain motion in their internal parts1 which was there1 when1 before the taking away of the force1 they were compressed1 or extended4 ,for that restitution is motion1 and that which is at rest cannot be moved1 but by a moved and a contiguous movent4 ,nor doth the cause of their restitution proceed from the taking away of the force by which they were compressed or extended2 for the removing of impediments hath not the efficacy of a cause1 as has been shown at the end of the #crd article of chap4 ,,xv4 ,the cause therefore of their restitution is some motion either of the parts of the ambient1 or of the parts of the body compressed or extended4 ,but the parts of the ambient have no endeavour which contributes to their compression or extension1 nor to the setting of them at liberty1 or restitution4 ,it remains therefore that from the time of their compression or extension there be left some endeavour or motion1 by which1 the impediment being removed1 every part resumes its former place2 that is to say1 the whole restores itself4 ,sidenote3 ,though that which carrieth another be stopped1 the body carried will proceed4 #ad4 ,in the carriage of bodies1 if that body1 which carries another1 hit upon any obstacle1 or be by any means suddenly stopped1 and that which is carried be not stopped1 it will go on1 till its motion be by some external impediment taken away4 ,for ,i have demonstrated 7chap4 ,,viii1 art4 #ai7 that motion1 unless it be hindered by some external resistance1 will be continued eternally with the same celerity2 and in the #gth article of chap4 ,,ix1 that the action of an external agent is of no effect without contact4 ,when therefore that1 which carrieth another thing1 is stopped1 that stop doth not presently take away the motion of that which is carried4 ,it will therefore proceed1 till its motion be by little and little extinguished by some external resistance3 which was to be proved2 though experience alone had been sufficient to prove this4 ,in like manner1 if that body which carrieth another be put from rest into sudden motion1 that which is carried will not be moved forwards together with it1 but will be left behind4 ,for the contiguous part of the body carried hath almost the same motion with the body which carries it2 and the remote parts will receive different velocities according to their different distances from the body that carries them2 namely1 the more remote the parts are1 the less will be their degrees of velocity4 ,it is necessary1 therefore1 that the body1 which is carried1 be left accordingly more or less behind4 ,and this also is manifest by experience1 when at the starting forward of the horse the rider falleth backwards4 ,sidenote3 ,the effects of percussion not to be compared with those of weight4 #ae4 ,in percussion1 therefore1 when one hard body is in some small part of it stricken by another with great force1 it is not necessary that the whole body should yield to the stroke with the same celerity with which the stricken part yields4 ,for the rest of the parts receive their motion from the motion of the part stricken and yielding1 which motion is less propagated every way towards the sides1 than it is directly forwards4 ,and hence it is1 that sometimes very hard bodies1 which being erected can hardly be made to stand1 are more easily broken than thrown down by a violent stroke2 when1 nevertheless1 if all their parts together were by any weak motion thrust forwards1 they would easily be cast down4 #af4 ,though the difference between trusion and percussion consist only in this1 that in trusion the motion both of the movent and moved body begin both together in their very contact2 and in percussion the striking body is first moved1 and afterwards the body stricken2 yet their effects are so different1 that it seems scarce possible to compare their forces with one another4 ,i say1 any effect of percussion being propounded1 as for example1 the stroke of a beetle of any weight assigned1 by which a pile of any given length is to be driven into earth of any tenacity given1 it seems to me very hard1 if not impossible1 to define with what weight1 or with what stroke1 and in what time1 the same pile may be driven to a depth assigned into the same earth4 ,the cause of which difficulty is this1 that the velocity of the percutient is to be compared with the magnitude of the ponderant4 ,now velocity1 seeing it is computed by the length of space transmitted1 is to be accounted but as one dimension2 but weight is as a solid thing1 being measured by the dimension of the whole body4 ,and there is no comparison to be made of a solid body with a length1 that is1 with a line4 ,sidenote3 ,motion cannot begin first in the internal parts of a body4 #ag4 ,if the internal parts of a body be at rest1 or retain the same situation with one another for any time how little soever1 there cannot in those parts be generated any new motion or endeavour1 whereof the efficient cause is not without the body of which they are parts4 ,for if any small part1 which is comprehended within the superficies of the whole body1 be supposed to be now at rest1 and by and by to be moved1 that part must of necessity receive its motion from some moved and contiguous body4 ,but by supposition1 there is no such moved and contiguous part within the body4 ,wherefore1 if there be any endeavour or motion or change of situation in the internal parts of that body1 it must needs arise from some efficient cause that is without the body which contains them2 which was to be proved4 #ah4 ,in hard bodies1 therefore1 which are compressed or extended1 if1 that which compresseth or extendeth them being taken away1 they restore themselves to their former place or situation1 it must needs be that that endeavour or motion of their internal parts1 by which they were able to recover their former places or situations1 was not extinguished when the force by which they were compressed or extended was taken away4 ,therefore1 when the lath of a cross-bow bent doth1 as soon as it is at liberty1 restore itself1 though to him1 that judges by sense1 both it and all its parts seem to be at rest2 yet he1 that judging by reason doth not account the taking away of impediment for an efficient cause1 nor conceives that without an efficient cause any thing can pass from rest to motion1 will conclude that the parts were already in motion before they began to restore themselves4 ,sidenote3 ,action and reaction proceed in the same line4 #ai4 ,action and reaction proceed in the same line1 but from opposite terms4 ,for seeing reaction is nothing but endeavour in the patient to restore itself to that situation from which it was forced by the agent2 the endeavour or motion both of the agent and patient or reagent will be propagated between the same terms2 yet so1 as that in action the term1 from which1 is in reaction the term to which4 ,and seeing all action proceeds in this manner1 not only between the opposite terms of the whole line in which it is propagated1 but also in all the parts of that line1 the terms from which and to which1 both of the action and reaction1 will be in the same line4 ,wherefore action and reaction proceed in the same line1 c4 ,sidenote3 ,habit1 what it is4 #bj4 ,to what has been said of motion1 ,i will add what ,i have to say concerning habit4 ,habit1 therefore1 is a generation of motion1 not of motion simply1 but an easy conducting of the moved body in a certain and designed way4 ,and seeing it is attained by the weakening of such endeavours as divert its motion1 therefore such endeavours are to be weakened by little and little4 ,but this cannot be done but by the long continuance of action1 or by actions often repeated2 and therefore custom begets that facility1 which is commonly and rightly called habit2 and it may be defined thus3 ,,habit is motion made more easy and ready by custom2 that is to say1 by perpetual endeavour1 or by iterated endeavours in a way differing from that in which the motion proceeded from the beginning1 and opposing such endeavours as resist4 ,and to make this more perspicuous by example1 we may observe1 that when one that has no skill in music first puts his hand to an instrument1 he cannot after the first stroke carry his hand to the place where he would make the second stroke1 without taking it back by a new endeavour1 and1 as it were beginning again1 pass from the first to the second4 ,nor will he be able to go on to the third place without another new endeavour2 but he will be forced to draw back his hand again1 and so successively1 by renewing his endeavour at every stroke2 till at the last1 by doing this often1 and by compounding many interrupted motions or endeavours into one equal endeavour1 he be able to make his hand go readily on from stroke to stroke in that order and way which was at the first designed4 ,nor are habits to be observed in living creatures only1 but also in bodies inanimate4 ,for we find that when the lath of a cross-bow is strongly bent1 and would if the impediment were removed return again with great force2 if it remain a long time bent1 it will get such a habit1 that when it is loosed and left to its own freedom1 it will not only not restore itself1 but will require as much force for the bringing of it back to its first posture1 as it did for the bending of it at the first4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxii4 ,fig4 #a-c ---------------------------------------- -------------------------------- ------- ,,chap4 ,,xxiii4 ,,of ,,the ,,centre ,,of ,,equiponderation2 ,,of ,,bodies ,,pressing ,,downwards ,,in ,,strait ,,parallel ,,lines4 #a4 ,definitions and suppositions4--#b4 ,two planes of equiponderation are not parallel4--#c4 ,the centre of equiponderation is in every plane of equiponderation4--#d4 ,the moments of equal ponderants are to one another as their distances from the centre of the scale4--#e1 #f4 ,the moments of unequal ponderants have their proportion to one another compounded of the proportions of their weights and distances from the centre of the scale4--#g4 ,if two ponderants have their weights and distances from the centre of the scale in reciprocal proportion1 they are equally poised2 and contrarily4--#h4 ,if the parts of any ponderant press the beams of the scale every where equally1 all the parts cut off1 reckoned from the centre of the scale1 will have their moments in the same proportion with that of the parts of a triangle cut off from the vertex by strait lines parallel to the base4--#i4 ,the diameter of equiponderation of figures1 which are deficient according to commensurable proportions of their altitudes and bases1 divides the axis1 so that the part taken next the vertex is to the other part of the complete figure to the deficient figure4--#aj4 ,the diameter of equiponderation of the complement of the half of any of the said deficient figures1 divides that line which is drawn through the vertex parallel to the base1 so that the part next the vertex is to the other part as the complete figure to the complement4--#aa4 ,the centre of equiponderation of the half of any of the deficient figures in the first row of the table of art4 #c1 chap4 ,,xvii1 may be found out by the numbers of the second row4--#ab4 ,the centre of equiponderation of the half of any of the figures of the second row of the same table1 may be found out by the numbers of the fourth row4--#ac4 ,the centre of equiponderation of the half of any of the figures in the same table being known1 the centre of the excess of the same figure above a triangle of the same altitude and base is also known4--#ad4 ,the centre of equiponderation of a solid sector is in the axis so divided1 that the part next the vertex be to the whole axis1 wanting half the axis of the portion of the sphere1 as #c to #d4 ,,definitions4 ,sidenote3 ,definitions4 ,i4 ,a scale is a strait line1 whose middle point is immovable1 all the rest of its points being at liberty2 and that part of the scale1 which reaches from the centre to either of the weights1 is called the beam4 ,,ii4 ,equiponderation is when the endeavour of one body1 which presses one of the beams1 resists the endeavour of another body pressing the other beam1 so that neither of them is moved2 and the bodies1 when neither of them is moved1 are said to be equally poised4 ,,iii4 ,weight is the aggregate of all the endeavours1 by which all the points of that body1 which presses the beam1 tend downwards in lines parallel to one another2 and the body which presses is called the ponderant4 ,,iv4 ,moment is the power which the ponderant has to move the beam1 by reason of a determined situation4 ,v4 ,the plane of equiponderation is that by which the ponderant is so divided1 that the moments on both sides remain equal4 ,,vi4 ,the diameter of equiponderation is the common section of the two planes of equiponderation1 and is in the strait line by which the weight is hanged4 ,,vii4 ,the centre of equiponderation is the common point of the two diameters of equiponderation4 ,,suppositions4 ,sidenote3 ,suppositions4 ,i4 ,when two bodies are equally poised1 if weight be added to one of them and not to the other1 their equiponderation ceases4 ,,ii4 ,when two ponderants of equal magnitude1 and of the same species or matter1 press the beam on both sides at equal distances from the centre of the scale1 their moments are equal4 ,also when two bodies endeavour at equal distances from the centre of the scale1 if they be of equal magnitude and of the same species1 their moments are equal4 ,sidenote3 ,two planes of equiponderation are not parallel4 #b4 ,no two planes of equiponderation are parallel4 ,let ,a ,b ,c ,d 7in fig4 #a7 be any ponderant whatsoever2 and in it let ,e ,f be a plane of equiponderation2 parallel to which1 let any other plane be drawn1 as ,g ,h4 ,i say1 ,g ,h is not a plane of equiponderation4 ,for seeing the parts ,a ,e ,f ,d and ,e ,b ,c ,f of the ponderant ,a ,b ,c ,d are equally poised2 and the weight ,e ,g ,h ,f is added to the part ,a ,e ,f ,d1 and nothing is added to the part ,e ,b ,c ,f1 but the weight ,e ,g ,h ,f is taken from it2 therefore1 by the first supposition1 the parts ,a ,g ,h ,d and ,g ,b ,c ,h will not be equally poised2 and consequently ,g ,h is not a plane of equiponderation4 ,wherefore1 no two planes of equiponderation are parallel2 which was to be proved4 ,sidenote3 ,the centre of equiponderation is in every plane of equiponderation4 #c4 ,the centre of equiponderation is in every plane of equiponderation4 ,for if another plane of equiponderation be taken1 it will not1 by the last article1 be parallel to the former plane2 and therefore both those planes will cut one another4 ,now that section 7by the #fth definition7 is the diameter of equiponderation4 ,again1 if another diameter of equiponderation be taken1 it will cut that former diameter2 and in that section 7by the #gth definition7 is the centre of equiponderation4 ,wherefore the centre of equiponderation is in that diameter which lies in the said plane of equiponderation4 ,sidenote3 ,the moments of equal ponderants are to one another as their distances from the centre of the scale4 #d4 ,the moment of any ponderant applied to one point of the beam1 to the moment of the same or an equal ponderant applied to any other point of the beam1 is as the distance of the former point from the centre of the scale1 to the distance of the latter point from the same centre4 ,or thus1 those moments are to one another1 as the arches of circles which are made upon the centre of the scale through those points1 in the same time4 ,or lastly thus1 they are as the parallel bases of two triangles1 which have a common angle at the centre of the scale4 ,let ,a 7in fig4 #b7 be the centre of the scale2 and let the equal ponderants ,d and ,e press the beam ,a ,b in the points ,b and ,c2 also let the strait lines ,b ,d and ,c ,e be diameters of equiponderation2 and the points ,d and ,e in the ponderants ,d and ,e be their centres of equiponderation4 ,let ,a ,g ,f be drawn howsoever1 cutting ,d ,b produced in ,f1 and ,e ,c in ,g2 and lastly1 upon the common centre ,a1 let the two arches ,b ,h and ,c ,i be described1 cutting ,a ,g ,f in ,h and ,i4 ,i say1 the moment of the ponderant ,d to the moment of the ponderant ,e is as ,a ,b to ,a ,c1 or as ,b ,h to ,c ,i1 or as ,b ,f to ,c ,g4 ,for the effect of the ponderant ,d1 in the point ,b1 is circular motion in the arch ,b ,h2 and the effect of the ponderant ,e1 in the point ,c1 circular motion in the arch ,c ,i2 and by reason of the equality of the ponderants ,d and ,e1 these motions are to one another as the quicknesses or velocities with which the points ,b and ,c describe the arches ,b ,h and ,c ,i1 that is1 as the arches themselves ,b ,h and ,c ,i1 or as the strait parallels ,b ,f and ,c ,g1 or as the parts of the beam ,a ,b and ,a ,c2 for ,a ,b4 ,a ,c33 ,b ,f4 ,c ,g33 ,b ,h4 ,c ,i4 are proportionals2 and therefore the effects1 that is1 by the #dth definition1 the moments of the equal ponderants applied to several points of the beam1 are to one another as ,a ,b and ,a ,c2 or as the distances of those points from the centre of the scale2 or as the parallel bases of the triangles which have a common angle at ,a2 or as the concentric arches ,b ,h and ,c ,i2 which was to be demonstrated4 ,sidenote3 ,the moments of unequal ponderants have their proportion to one another compounded of the proportions of their weights and distances from the centre of the scale4 #e4 ,unequal ponderants1 when they are applied to several points of the beam1 and hang at liberty1 that is1 so as the line by which they hang be the diameter of equiponderation1 whatsoever be the figure of the ponderant1 have their moments to one another in proportion compounded of the proportions of their distances from the centre of the scale1 and of their weights4 ,let ,a 7in fig4 #c7 be the centre of the scale1 and ,a ,b the beam2 to which let the two ponderants ,c and ,d be applied at the points ,b and ,e4 ,i say1 the proportion of the moment of the ponderant ,c to the moment of the ponderant ,d1 is compounded of the proportions of ,a ,b to ,a ,e1 and of the weight ,c to the weight ,d2 or1 if ,c and ,d be of the same species1 of the magnitude ,c to the magnitude ,d4 ,let either of them1 as ,c1 be supposed to be bigger than the other1 ,d4 ,if1 therefore1 by the addition of ,f1 ,f and ,d together be as one body equal to ,c1 the moment of ,c to the moment of ,f ,d will be 7by the last article7 as ,b ,g is to ,e ,h4 ,now as ,f ,d is to ,d1 so let ,e ,h be to another ,e ,i2 and the moment of ,f ,d1 that is of ,c1 to the moment of ,d1 will be as ,b ,g to ,e ,i4 ,but the proportion of ,b ,g to ,e ,i is compounded of the proportions of ,b ,g to ,e ,h1 that is1 of ,a ,b to ,a ,e1 and of ,e ,h to ,e ,i1 that is1 of the weight ,c to the weight ,d4 ,wherefore unequal ponderants1 when they are applied1 c4 ,which was to be proved4 #f4 ,the same figure remaining1 if ,i ,k be drawn parallel to the beam ,a ,b1 and cutting ,a ,g in ,k2 and ,k ,l be drawn parallel to ,b ,g1 cutting ,a ,b in ,l1 the distances ,a ,b and ,a ,l from the centre will be proportional to the moments of ,c and ,d4 ,for the moment of ,c is ,b ,g1 and the moment of ,d is ,e ,i1 to which ,k ,l is equal4 ,but as the distance ,a ,b from the centre is to the distance ,a ,l from the centre1 so is ,b ,g1 the moment of the ponderant ,c1 to ,l ,k1 or ,e ,i the moment of the ponderant ,d4 ,sidenote3 ,if two ponderants have their weights and distances from the centre of the scale in reciprocal proportion1 they are equally poised2 and contrarily4 #g4 ,if two ponderants have their weights and distances from the centre in reciprocal proportion1 and the centre of the scale be between the points to which the ponderants are applied1 they will be equally poised4 ,and contrarily1 if they be equally poised1 their weights and distances from the centre of the scale will be in reciprocal proportion4 ,let the centre of the scale 7in the same third figure7 be ,a1 the beam ,a ,b2 and let any ponderant ,c1 having ,b ,g for its moment1 be applied to the point ,b2 also let any other ponderant ,d1 whose moment is ,e ,i1 be applied to the point ,e4 ,through the point ,i let ,i ,k be drawn parallel to the beam ,a ,b1 cutting ,a ,g in ,k2 also let ,k ,l be drawn parallel to ,b ,g1 ,k ,l will then be the moment of the ponderant ,d2 and by the last article1 it will be as ,b ,g1 the moment of the ponderant ,c in the point ,b1 to ,l ,k the moment of the ponderant ,d in the point ,e1 so ,a ,b to ,a ,l4 ,on the other side of the centre of the scale1 let ,a ,n be taken equal to ,a ,l2 and to the point ,n let there be applied the ponderant ,o1 having to the ponderant ,c the proportion of ,a ,b to ,a ,n4 ,i say1 the ponderants in ,b and ,n will be equally poised4 ,for the proportion of the moment of the ponderant ,o1 in the point ,n1 to the moment of the ponderant ,c in the point ,b1 is by the #eth article1 compounded of the proportions of the weight ,o to the weight ,c1 and of the distance from the centre of the scale ,a ,n or ,a ,l to the distance from the centre of the scale ,a ,b4 ,but seeing we have supposed1 that the distance ,a ,b to the distance ,a ,n is in reciprocal proportion of the weight ,o to the weight ,c1 the proportion of the moment of the ponderant ,o1 in the point ,n1 to the moment of the ponderant ,c1 in the point ,b1 will be compounded of the proportions of ,a ,b to ,a ,n1 and of ,a ,n to ,a ,b4 ,wherefore1 setting in order ,a ,b1 ,a ,n1 ,a ,b1 the moment of ,o to the moment of ,c will be as the first to the last1 that is1 as ,a ,b to ,a ,b4 ,their moments therefore are equal2 and consequently the plane which passes through ,a will 7by the fifth definition7 be a plane of equiponderation4 ,wherefore they will be equally poised2 as was to be proved4 ,now the converse of this is manifest4 ,for if there be equiponderation and the proportion of the weights and distances be not reciprocal1 then both the weights will always have the same moments1 although one of them have more weight added to it or its distance changed4 ,coroll4 ,when ponderants are of the same species1 and their moments be equal2 their magnitudes and distances from the centre of the scale will be reciprocally proportional4 ,for in homogeneous bodies1 it is as weight to weight1 so magnitude to magnitude4 ,sidenote3 ,if the parts of any ponderant press the beams of the scale every where equally1 all the parts cut off1 reckoned from centre of the scale1 will have their moments in the same proportion with that of the parts of a triangle cut off from the vertex by strait lines parallel to the base4 #h4 ,if to the whole length of the beam there be applied a parallelogram1 or a parallelopipedum1 or a prisma1 or a cylinder1 or the superficies of a cylinder1 or of a sphere1 or of any portion of a sphere or prisma2 the parts of any of them cut off with planes parallel to the base will have their moments in the same proportion with the parts of a triangle1 which has its vertex in the centre of the scale1 and for one of its sides the beam itself1 which parts are cut off by planes parallel to the base4 ,first1 let the rectangled parallelogram ,a ,b ,c ,d 7in figure #d7 be applied to the whole length of the beam ,a ,b2 and producing ,c ,b howsoever to ,e1 let the triangle ,a ,b ,e be described4 ,let now any part of the parallelogram1 as ,a ,f1 be cut off by the plane ,f ,g1 parallel to the base ,c ,b2 and let ,f ,g be produced to ,a ,e in the point ,h4 ,i say1 the moment of the whole ,a ,b ,c ,d to the moment of its part ,a ,f1 is as the triangle ,a ,b ,e to the triangle ,a ,g ,h1 that is1 in proportion duplicate to that of the distances from the centre of the scale4 ,for1 the parallelogram ,a ,b ,c ,d being divided into equal parts1 infinite in number1 by strait lines drawn parallel to the base2 and supposing the moment of the strait line ,c ,b to be ,b ,e1 the moment of the strait line ,f ,g will 7by the #gth article7 be ,g ,h2 and the moments of all the strait lines of that parallelogram will be so many strait lines in the triangle ,a ,b ,e drawn parallel to the base ,b ,e2 all which parallels together taken are the moment of the whole parallelogram ,a ,b ,c ,d2 and the same parallels do also constitute the superficies of the triangle ,a ,b ,e4 ,wherefore the moment of the parallelogram ,a ,b ,c ,d is the triangle ,a ,b ,e2 and for the same reason1 the moment of the parallelogram ,a ,f is the triangle ,a ,g ,h2 and therefore the moment of the whole parallelogram to the moment of a parallelogram which is part of the same1 is as the triangle ,a ,b ,e to the triangle ,a ,g ,h1 or in proportion duplicate to that of the beams to which they are applied4 ,and what is here demonstrated in the case of a parallelogram may be understood to serve for that of a cylinder1 and of a prisma1 and their superficies2 as also for the superficies of a sphere1 of an hemisphere1 or any portion of a sphere4 ,for the parts of the superficies of a sphere have the same proportion with that of the parts of the axis cut off by the same parallels1 by which the parts of the superficies are cut off1 as ,archimedes has demonstrated2 and therefore when the parts of any of these figures are equal and at equal distances from the centre of the scale1 their moments also are equal1 in the same manner as they are in parallelograms4 ,secondly1 let the parallelogram ,a ,k ,i ,b not be rectangled2 the strait line ,i ,b will nevertheless press the point ,b perpendicularly in the strait line ,b ,e2 and the strait line ,l ,g will press the point ,g perpendicularly in the strait line ,g ,h2 and all the rest of the strait lines which are parallel to ,i ,b will do the like4 ,whatsoever therefore the moment be which is assigned to the strait line ,i ,b1 as here1 for example1 it is supposed to be ,b ,e1 if ,a ,e be drawn1 the moment of the whole parallelogram ,a ,i will be the triangle ,a ,b ,e2 and the moment of the part ,a ,l will be the triangle ,a ,g ,h4 ,wherefore the moment of any ponderant1 which has its sides equally applied to the beam1 whether they be applied perpendicularly or obliquely1 will be always to the moment of a part of the same in such proportion as the whole triangle has to a part of the same cut off by a plane which is parallel to the base4 ,sidenote3 ,the diameter of equiponderation of figures which are deficient according to commensurable proportions of their altitudes and bases1 divides the axis1 so that the part taken next the vertex is to the other part as the complete figure to the deficient figure4 #i4 ,the centre of equiponderation of any figure1 which is deficient according to commensurable proportions of the altitude and base diminished1 and whose complete figure is either a parallelogram or a cylinder1 or a parallelopipedum1 divides the axis1 so1 that the part next the vertex1 to the other part1 is as the complete figure to the deficient figure4 ,for let ,c ,i ,a ,p ,e 7in fig4 #e7 be a deficient figure1 whose axis is ,a ,b1 and whose complete figure is ,c ,d ,f ,e2 and let the axis ,a ,b be so divided in ,z1 that ,a ,z be to ,z ,b as ,c ,d ,f ,e is to ,c ,i ,a ,p ,e4 ,i say1 the centre of equiponderation of the figure ,c ,i ,a ,p ,e will be in the point ,z4 ,first1 that the centre of equiponderation of the figure ,c ,i ,a ,p ,e is somewhere in the axis ,a ,b is manifest of itself2 and therefore ,a ,b is a diameter of equiponderation4 ,let ,a ,e be drawn1 and let ,b ,e be put for the moment of the strait line ,c ,e2 the triangle ,a ,b ,e will therefore 7by the third article7 be the moment of the complete figure ,c ,d ,f ,e4 ,let the axis ,a ,b be equally divided in ,l1 and let ,g ,l ,h be drawn parallel and equal to the strait line ,c ,e1 cutting the crooked line ,c ,i ,a ,p ,e in ,i and ,p1 and the strait lines ,a ,c and ,a ,e in ,k and ,m4 ,moreover1 let ,z ,o be drawn parallel to the same ,c ,e2 and let it be1 as ,l ,g to ,l ,i1 so ,l ,m to another1 ,l ,n2 and let the same be done in all the rest of the strait lines possible1 parallel to the base2 and through all the points ,n1 let the line ,a ,n ,e be drawn2 the three-sided figure ,a ,n ,e ,b will therefore be the moment of the figure ,c ,i ,a ,p ,e4 ,now the triangle ,a ,b ,e is 7by the #ith article of chapter ,,xvii7 to the three-sided figure ,a ,n ,e ,b1 as ,a ,b ,c ,d ,a ,i ,c ,b is to ,a ,i ,c ,b twice taken1 that is1 as ,c ,d ,f ,e ,c ,i ,a ,p ,e is to ,c ,i ,a ,p ,e twice taken4 ,but as ,c ,i ,a ,p ,e is to ,c ,d ,f ,e1 that is1 as the weight of the deficient figure is to the weight of the complete figure1 so is ,c ,i ,a ,p ,e twice taken to ,c ,d ,f ,e twice taken4 ,wherefore1 setting in order ,c ,d ,f ,e ,c ,i ,a ,p ,e4 #b ,c ,i ,a ,p ,e4 #b ,c ,d ,f ,e2 the proportion of ,c ,d ,f ,e ,c ,i ,a ,p ,e to ,c ,d ,f ,e twice taken will be compounded of the proportion of ,c ,d ,f ,e ,c ,i ,a ,p ,e to ,c ,i ,a ,p ,e twice taken1 that is1 of the proportion of the triangle ,a ,b ,e to the three-sided figure ,a ,n ,e ,b1 that is1 of the moment of the complete figure to the moment of the deficient figure1 and of the proportion of ,c ,i ,a ,p ,e twice taken to ,c ,d ,f ,e twice taken1 that is1 to the proportion reciprocally taken of the weight of the deficient figure to the weight of the complete figure4 ,again1 seeing by supposition ,a ,z4 ,z ,b33 ,c ,d ,f ,e4 ,c ,i ,a ,p ,e are proportionals2 ,a ,b4 ,a ,z33 ,c ,d ,f ,e ,c ,i ,a ,p ,e4 ,c ,d ,f ,e will also1 by compounding1 be proportionals4 ,and seeing ,a ,l is the half of ,a ,b1 ,a ,l4 ,a ,z33 ,c ,d ,f ,e ,c ,i ,a ,p ,e4 #b ,c ,d ,f ,e will also be proportionals4 ,but the proportion of ,c ,d ,f ,e ,c ,i ,a ,p ,e to #b ,c ,d ,f ,e is compounded1 as was but now shown1 of the proportions of moment to moment1 c41 and therefore the proportion of ,a ,l to ,a ,z is compounded of the proportion of the moment of the complete figure ,c ,d ,f ,e to the moment of the deficient figure ,c ,i ,a ,p ,e1 and of the proportion of the weight of the deficient figure ,c ,i ,a ,p ,e to the weight of the complete figure ,c ,d ,f ,e2 but the proportion of ,a ,l to ,a ,z is compounded of the proportions of ,a ,l to ,b ,z and of ,b ,z to ,a ,z4 ,now the proportion of ,b ,z to ,a ,z is the proportion of the weights reciprocally taken1 that is to say1 of the weight ,c ,i ,a ,p ,e to the weight ,c ,d ,f ,e4 ,therefore the remaining proportion of ,a ,l to ,b ,z1 that is1 of ,l ,b to ,b ,z1 is the proportion of the moment of the weight ,c ,d ,f ,e to the moment of the weight ,c ,i ,a ,p ,e4 ,but the proportion of ,a ,l to ,b ,z is compounded of the proportions of ,a ,l to ,a ,z and of ,a ,z to ,z ,b2 of which proportions that of ,a ,z to ,z ,b is the proportion of the weight ,c ,d ,f ,e to the weight ,c ,i ,a ,p ,e4 ,wherefore 7by art4 #e of this chapter7 the remaining proportion of ,a ,l to ,a ,z is the proportion of the distances of the points ,z and ,l from the centre of the scale1 which is ,a4 ,and1 therefore1 7by art4 #f7 the weight ,c ,i ,a ,p ,e shall hang from ,o in the strait line ,o ,z4 ,so that ,o ,z is one diameter of equiponderation of the weight ,c ,i ,a ,p ,e4 ,but the strait line ,a ,b is the other diameter of equiponderation of the same weight ,c ,i ,a ,p ,e4 ,wherefore 7by the #gth definition7 the point ,z is the centre of the same equiponderation2 which point1 by construction1 divides the axis so1 that the part ,a ,z1 which is the part next the vertex1 is to the other part ,z ,b1 as the complete figure ,c ,d ,f ,e is to the deficient figure ,c ,i ,a ,p ,e2 which is that which was to be demonstrated4 ,coroll4 ,i4 ,the centre of equiponderation of any of those plane three-sided figures1 which are compared with their complete figures in the table of art4 #c1 chap4 ,,xvii1 is to be found in the same table1 by taking the denominator of the fraction for the part of the axis cut off next the vertex1 and the numerator for the other part next the base4 ,for example1 if it be required to find the centre of equiponderation of the second three-sided figure of four means1 there is in the concourse of the second column with the row of three-sided figures of four means this fraction #eg1 which signifies that that figure is to its parallelogram or complete figure as #eg to unity1 that is1 as #eg to #gg1 or as #e to #g2 and1 therefore the centre of equiponderation of that figure divides the axis1 so that the part next the vertex is to the other part as #g to #e4 ,coroll4 ,,ii4 ,the centre of equiponderation of any of the solids of those figures1 which are contained in the table of art4 #g of the same chap4 ,,xvii1 is exhibited in the same table4 ,for example1 if the centre of equiponderation of a cone be sought for1 the cone will be found to be #ac of its cylinder2 and1 therefore1 the centre of its equiponderation will so divide the axis1 that the part next the vertex to the other part will be as #c to #a4 ,also the solid of a three-sided figure of one mean1 that is1 a parabolical solid1 seeing it is #bd1 that is of its cylinder1 will have its centre of equiponderation in that point1 which divides the axis1 so that the part towards the vertex be double to the part towards the base4 ,sidenote3 ,the diameter of equiponderation of the complement of the half of any of the said deficient figures1 divides that line which is drawn through the vertex parallel to the base1 so that the part next the vertex is to the other part as the complete figure to the complement4 #aj4 ,the diameter of equiponderation of the complement of the half of any of those figures which are contained in the table of art4 #c1 chap4 ,,xvii1 divides that line which is drawn through the vertex parallel and equal to the base1 so that the part next the vertex will be to the other part1 as the complete figure to the complement4 ,for let ,a ,i ,c ,b 7in the same fig4 #e7 be the half of a parabola1 or of any other of those three-sided figures which are in the table of art4 #c1 chap4 ,,xvii1 whose axis is ,a ,b1 and base ,b ,c1 having ,a ,d drawn from the vertex1 equal and parallel to the base ,b ,c1 and whose complete figure is the parallelogram ,a ,b ,c ,d4 ,let ,i ,q be drawn at any distance from the side ,c ,d1 but parallel to it2 and let ,a ,d be the altitude of the complement ,a ,i ,c ,d1 and ,q ,i a line ordinately applied in it4 ,wherefore the altitude ,a ,l in the deficient figure ,a ,i ,c ,b is equal to ,q ,i the line ordinately applied in its complement2 and contrarily1 ,l ,i the line ordinately applied in the figure ,a ,i ,c ,b is equal to the altitude ,a ,q in its complement2 and so in all the rest of the ordinate lines and altitudes the mutation is such1 that that line1 which is ordinately applied in the figure1 is the altitude of its complement4 ,and1 therefore1 the proportion of the altitudes decreasing to that of the ordinate lines decreasing1 being multiplicate according to any number in the deficient figure1 is submultiplicate according to the same number in its complement4 ,for example1 if ,a ,i ,c ,b be a parabola1 seeing the proportion of ,a ,b to ,a ,l is duplicate to that of ,b ,c to ,l ,i1 the proportion of ,,ad to ,a ,q in the complement ,a ,i ,c ,d1 which is the same with that of ,b ,c to ,l ,i1 will be subduplicate to that of ,c ,d to ,q ,i1 which is the same with that of ,a ,b to ,a ,l2 and consequently1 in a parabola1 the complement will be to the parallelogram as #a to #c2 in a three-sided figure of two means1 as #a to #d2 in a three-sided figure of three means1 as #a to #e1 c4 ,but all the ordinate lines together in ,a ,i ,c ,d are its moment2 and all the ordinate lines in ,a ,i ,c ,b are its moment4 ,wherefore the moments of the complements of the halves of deficient figures in the table of art4 #c of chap4 ,,xvii1 being compared1 are as the deficient figures themselves2 and1 therefore1 the diameter of equiponderation will divide the strait line ,a ,d in such proportion1 that the part next the vertex be to the other part1 as the complete figure ,a ,b ,c ,d is to the complement ,a ,i ,c ,d4 ,coroll4 ,the diameter of equiponderation of these halves may be found by the table of art4 #c of chap4 ,,xvii1 in this manner4 ,let there be propounded any deficient figure1 namely1 the second three-sided figure of two means4 ,this figure is to the complete figure as #ce to #a1 that is #c to #e4 ,wherefore the complement to the same complete figure is as #b to #e2 and1 therefore1 the diameter of equiponderation of this complement will cut the strait line drawn from the vertex parallel to the base1 so that the part next the vertex will be to the other part as #e to #b4 ,and1 in like manner1 any other of the said three-sided figures being propounded1 if the numerator of its fraction found out in the table be taken from the denominator1 the strait line drawn from the vertex is to be divided1 so that the part next the vertex be to the other part1 as the denominator is to the remainder which that subtraction leaves4 ,sidenote3 ,the centre of equiponderation of the half of any of the deficient figures in the first row of the table of art4 #c1 chapter xvii1 may be found out by the numbers of the second row4 #aa4 ,the centre of equiponderation of the half of any of those crooked-lined figures1 which are in the first row of the table of art4 #c of chap4 ,,xvii1 is in that strait line which1 being parallel to the axis1 divides the base according to the numbers of the fraction next below it in the second row1 so that the numerator be answerable to that part which is towards the axis4 ,for example1 let the first figure of three means be taken1 whose half is ,a ,b ,c ,d 7in fig4 #f71 and let the rectangle ,a ,b ,e ,d be completed4 ,the complement therefore will be ,b ,c ,d ,e4 ,and seeing ,a ,b ,e ,d is to the figure ,a ,b ,c ,d 7by the table7 as #e to #d1 the same ,a ,b ,e ,d will be to the complement ,b ,c ,d ,e as #e to #a4 ,wherefore1 if ,f ,g be drawn parallel to the base ,d ,a1 cutting the axis so that ,a ,g be to ,g ,b as #d to #e1 the centre of equiponderation of the figure ,a ,b ,c ,d will1 by the precedent article1 be somewhere in the same ,f ,g4 ,again1 seeing1 by the same article1 the complete figure ,a ,b ,e ,d1 is to the complement ,b ,c ,d ,e as #e to #a1 therefore if ,b ,e and ,a ,d be divided in ,i and ,h as #e to #a the centre of equiponderation of the complement ,b ,c ,d ,e will be somewhere in the strait line which connects ,h and ,i4 ,let now the strait line ,l ,k be drawn through ,m the centre of the complete figure1 parallel to the base2 and the strait line ,n ,o through the same centre ,m1 perpendicular to it2 and let the strait lines ,l ,k and ,f ,g cut the strait line ,h ,i in ,p and ,q4 ,let ,p ,r be taken quadruple to ,p ,q2 and let ,r ,m be drawn and produced to ,f ,g in ,s4 ,r ,m therefore will be to ,m ,s as #d to #a1 that is1 as the figure ,a ,b ,c ,d to its complement ,b ,c ,d ,e4 ,wherefore1 seeing ,m is the centre of the complete figure ,a ,b ,e ,d1 and the distances of ,r and ,s from the centre ,m be in proportion reciprocal to that of the weight of the complement ,b ,c ,d ,e to the weight of the figure ,a ,b ,c ,d1 ,r and ,s will either be the centres of equiponderation of their own figures1 or those centres will be in some other points of the diameters of equiponderation ,h ,i and ,f ,g4 ,but this last is impossible4 ,for no other strait line can be drawn through the point ,m terminating in the strait lines ,h ,i and ,f ,g1 and retaining the proportion of ,m ,r to ,m ,s1 that is1 of the figure ,a ,b ,c ,d to its complement ,b ,c ,d ,e4 ,the centre1 therefore1 of equiponderation of the figure ,a ,b ,c ,d is in the point ,s4 ,now1 seeing ,p ,m hath the same proportion to ,q ,s which ,r ,p hath to ,r ,q1 ,q ,s will be #e of those parts of which ,p ,m is four1 that is1 of which ,i ,n is four4 ,but ,i ,n or ,p ,m is #b of those parts of which ,e ,b or ,f ,g is #f2 and1 therefore1 if it be as #d to #e1 so #b to a fourth1 that fourth will be #b4 ,wherefore ,q ,s is #b of those parts of which ,f ,g is #f4 ,but ,f ,q is #a2 and1 therefore1 ,f ,s is #c4 ,wherefore the remaining part ,g ,s is #b4 ,so that ,f ,g is so divided in ,s1 that the part towards the axis is in proportion to the other part1 as #b to #c1 that is as #e to #g2 which answereth to the fraction #eg in the second row1 next under the fraction #de in the first row4 ,wherefore drawing ,s ,t parallel to the axis1 the base will be divided in like manner4 ,by this method it is manifest1 that the base of a semiparabola will be divided into #c and #e2 and the base of the first three-sided figure of two means1 into #d and #f2 and of the first three-sided figure of four means1 into #f and #h4 ,the fractions1 therefore1 of the second row denote the proportions1 into which the bases of the figures of the first row are divided by the diameters of equiponderation4 ,but the first row begins one place higher than the second row4 ,sidenote3 ,the centre of equiponderation of the half of any of the figures of the second row of the same table may be found out by the numbers of the fourth row4 #ab4 ,the centre of equiponderation of the half of any of the figures in the second row of the same table of art4 #c1 chap4 ,,xvii1 is in a strait line parallel to the axis1 and dividing the base according to the numbers of the fraction in the fourth row1 two places lower1 so as that the numerator be answerable to that part which is next the axis4 ,let the half of the second three-sided figure of two means be taken2 and let it be ,a ,b ,c ,d 7in fig4 #g72 whose complement is ,b ,c ,d ,e1 and the rectangle completed ,a ,b ,e ,d4 ,let this rectangle be divided by the two strait lines ,l ,k and ,n ,o1 cutting one another in the centre ,m at right angles2 and because ,a ,b ,e ,d is to ,a ,b ,c ,d as #e to #c1 let ,a ,b be divided in ,g1 so that ,a ,g to ,b ,g be as #c to #e2 and let ,f ,g be drawn parallel to the base4 ,also because ,a ,b ,e ,d is 7by art4 #i7 to ,b ,c ,d ,e as #e to #b1 let ,b ,e be divided in the point ,i1 so that ,b ,i be to ,i ,e as #e to #b2 and let ,i ,h be drawn parallel to the axis1 cutting ,l ,k and ,f ,g in ,p and ,q4 ,let now ,p ,r be so taken1 that it be to ,p ,q as #c to #b1 and let ,r ,m be drawn and produced to ,f ,g in ,s4 ,seeing1 therefore1 ,r ,p is to ,p ,q1 that is1 ,r ,m to ,m ,s1 as ,a ,b ,c ,d is to its complement ,b ,c ,d ,e1 and the centres of equiponderation of ,a ,b ,c ,d and ,b ,c ,d ,e are in the strait lines ,f ,g and ,h ,i1 and the centre of equiponderation of them both together in the point ,m2 ,r will be the centre of the complement ,b ,c ,d ,e1 and ,s the centre of the figure ,a ,b ,c ,d4 ,and seeing ,p ,m1 that is ,i ,n1 is to ,q ,s1 as ,r ,p is to ,r ,q2 and ,i ,n or ,p ,m is #c of those parts1 of which ,b ,e1 that is ,f ,g1 is #ad2 therefore ,q ,s is #e of the same parts2 and ,e ,i1 that is ,f ,g1 #d2 and ,f ,s1 #i2 and ,g ,s1 #e4 ,wherefore the strait line ,s ,t being drawn parallel to the axis1 will divide the base ,a ,d into #e and #i4 ,but the fraction #ei is found in the fourth row of the table1 two places below the fraction in the second row4 ,by the same method1 if in the same second row there be taken the second three-sided figure of three means1 the centre of equiponderation of the half of it will be found to be in a strait line parallel to the axis1 dividing the base according to the numbers of the fraction #faj1 two places below in the fourth row4 ,and the same way serves for all the rest of the figures in the second row4 ,in like manner1 the centre of equiponderation of the third three-sided figure of three means will be found to be in a strait line parallel to the axis1 dividing the base1 so that the part next the axis be to the other part as #g to #ac1 c4 ,coroll4 ,the centres of equiponderation of the halves of the said figures are known1 seeing they are in the intersection of the strait lines ,s ,t and ,f ,g1 which are both known4 ,sidenote3 ,the centre of equiponderation of the half of any of the figures in the same table being known1 the centre of the excess of the same figure above a triangle of the same altitude and base is also known4 #ac4 ,the centre of equiponderation of the half of any of the figures1 which 7in the table of art4 #c1 chap4 ,,xvii7 are compared with their parallelograms1 being known2 the centre of equiponderation of the excess of the same figure above its triangle is also known4 ,for example1 let the semiparabola ,a ,b ,c ,d 7in fig4 #h7 be taken1 whose axis is ,a ,b2 whose complete figure is ,a ,b ,e ,d2 and whose excess above its triangle is ,b ,c ,d ,b4 ,its centre of equiponderation may be found out in this manner4 ,let ,f ,g be drawn parallel to the base1 so that ,a ,f be a third part of the axis2 and let ,h ,i be drawn parallel to the axis1 so that ,a ,h be a third part of the base4 ,this being done1 the centre of equiponderation of the triangle ,a ,b ,d will be ,i4 ,again1 let ,k ,l be drawn parallel to the base1 so that ,a ,k be to ,a ,b as #b to #e2 and ,m ,n parallel to the axis1 so that ,a ,m be to ,a ,d as #c to #h2 and let ,m ,n terminate in the strait line ,k ,l4 ,the centre1 therefore1 of equiponderation of the parabola ,a ,b ,c ,d is ,n2 and therefore we have the centres of equiponderation of the semiparabola ,a ,b ,c ,d1 and of its part the triangle ,a ,b ,d4 ,that we may now find the centre of equiponderation of the remaining part ,b ,c ,d ,b1 let ,i ,n be drawn and produced to ,o1 so that ,n ,o be triple to ,i ,n2 and ,o will be the centre sought for4 ,for seeing the weight of ,a ,b ,d to the weight of ,b ,c ,d ,b is in proportion reciprocal to that of the strait line ,n ,o to the strait line ,i ,n2 and ,n is the centre of the whole1 and ,i the centre of the triangle ,a ,b ,d2 ,o will be the centre of the remaining part1 namely1 of the figure ,b ,d ,c ,b2 which was to be found4 ,coroll4 ,the centre of equiponderation of the figure ,b ,d ,c ,b is in the concourse of two strait lines1 whereof one is parallel to the base1 and divides the axis1 so that the part next the base be or #iae of the whole axis2 the other is parallel to the axis1 and so divides the base1 that the part towards the axis be 1 or #abbd of the whole base4 ,for drawing ,o ,p parallel to the base1 it will be as ,i ,n to ,n ,o1 so ,f ,k to ,k ,p1 that is1 so #a to #c1 or #e to #ae4 ,but ,a ,f is #eae1 or of the whole ,a ,b2 and ,a ,k is #fae1 or 2 and ,f ,k #aae2 and ,,kp #cae2 and therefore ,a ,p is #iae of the axis ,a ,b4 ,also ,a ,h is 1 or #hbd2 and ,a ,m 1 or #ibd of the whole base2 and therefore ,o ,q being drawn parallel to the axis1 ,m ,q1 which is triple to ,h ,m1 will be #cbd4 ,wherefore ,a ,q is #abbd1 or of the base ,a ,d4 ,the excesses of the rest of the three-sided figures in the first row of the table of art4 #c1 chap4 ,,xvii1 have their centres of equiponderation in two strait lines1 which divide the axis and base according to those fractions1 which add #d to the numerators of the fractions of a parabola #iae and #abbd2 and #f to the denominators1 in this manner3-- ,in a parabola1 the axis #iae1 the base #abbd4 ,in the first three-sided figure1 the axis #acba1 the base #afcj4 ,in the second three-sided figure1 the axis #agbg1 the base #bjcf1 c4 ,and by the same method1 any man1 if it be worth the pains1 may find out the centres of equiponderation of the excesses above their triangles of the rest of the figures in the second and third row1 c4 ,sidenote3 ,the centre of equiponderation of a solid sector is in the axis so divided1 that the part next the vertex be to the whole axis1 wanting half the axis of the portion of the sphere1 as #c to #d4 #ad4 ,the centre of equiponderation of the sector of a sphere1 that is1 of a figure compounded of a right cone1 whose vertex is the centre of the sphere1 and the portion of the sphere whose base is the same with that of the cone1 divides the strait line which is made of the axis of the cone and half the axis of the portion together taken1 so that the part next the vertex be triple to the other part1 or to the whole strait line as #c to #d4 ,for let ,a ,b ,c 7in fig4 #i7 be the sector of a sphere1 whose vertex is the centre of the sphere ,a2 whose axis is ,a ,d2 and the circle upon ,b ,c is the common base of the portion of the sphere and of the cone whose vertex is ,a2 the axis of which portion is ,e ,d1 and the half thereof ,f ,d2 and the axis of the cone1 ,a ,e4 ,lastly1 let ,a ,g be #cd of the strait line ,a ,f4 ,i say1 ,g is the centre of equiponderation of the sector ,a ,b ,c4 ,let the strait line ,f ,h be drawn of any length1 making right angles with ,a ,f at ,f2 and drawing the strait line ,a ,h1 let the triangle ,a ,f ,h be made4 ,then upon the same centre ,a let any arch ,i ,k be drawn1 cutting ,a ,d in ,l2 and its chord1 cutting ,a ,d in ,m2 and dividing ,m ,l equally in ,n1 let ,n ,o be drawn parallel to the strait line ,f ,h1 and meeting with the strait line ,a ,h in ,o4 ,seeing now ,b ,d ,c is the spherical superficies of the portion cut off with a plane passing through ,b ,c1 and cutting the axis at right angles2 and seeing ,f ,h divides ,e ,d1 the axis of the portion1 into two equal parts in ,f2 the centre of equiponderation of the superficies ,b ,d ,c will be in ,f 7by art4 #h72 and for the same reason the centre of equiponderation of the superficies ,i ,l ,k1 ,k being in the strait line ,a ,c1 will be in ,n4 ,and in like manner1 if there were drawn1 between the centre of the sphere ,a and the outermost spherical superficies of the sector1 arches infinite in number1 the centres of equiponderation of the spherical superficies1 in which those arches are1 would be found to be in that part of the axis1 which is intercepted between the superficies itself and a plane passing along by the chord of the arch1 and cutting the axis in the middle at right angles4 ,let it now be supposed that the moment of the outermost spherical superficies ,b ,d ,c is ,f ,h4 ,seeing therefore the superficies ,b ,d ,c is to the superficies ,i ,l ,k in proportion duplicate to that of the arch ,b ,d ,c to the arch ,i ,l ,k1 that is1 of ,b ,e to ,i ,m1 that is1 of ,f ,h to ,n ,o2 let it be as ,f ,h to ,n ,o1 so ,n ,o to another ,n ,p2 and again1 as ,n ,o to ,n ,p1 so ,n ,p to another ,n ,q2 and let this be done in all the strait lines parallel to the base ,f ,h that can possibly be drawn between the base and the vertex of the triangle ,a ,f ,h4 ,if then through all the points ,q there be drawn the crooked line ,a ,q ,h1 the figure ,a ,f ,h ,q ,a will be the complement of the first three-sided figure of two means2 and the same will also be the moment of all the spherical superficies1 of which the solid sector ,a ,b ,c ,d is compounded2 and by consequent1 the moment of the sector itself4 ,let now ,f ,h be understood to be the semidiameter of the base of a right cone1 whose side is ,a ,h1 and axis ,a ,f ,wherefore1 seeing the bases of the cones1 which pass through ,f and ,n and the rest of the points of the axis1 are in proportion duplicate to that of the strait lines ,f ,h and ,n ,o1 c41 the moment of all the bases together1 that is1 of the whole cone1 will be the figure itself ,a ,f ,h ,q ,a2 and therefore the centre of equiponderation of the cone ,a ,f ,h is the same with that of the solid sector4 ,wherefore1 seeing ,a ,g is of the axis ,a ,f1 the centre of equiponderation of the cone ,a ,f ,h is in ,g2 and therefore the centre of the solid sector is in ,g also1 and divides the part ,a ,f of the axis so that ,a ,g is triple to ,g ,f2 that is1 ,a ,g is to ,a ,f as #c to #d2 which was to be demonstrated4 ,note1 that when the sector is a hemisphere1 the axis of the cone vanisheth into that point which is the centre of the sphere2 and therefore it addeth nothing to half the axis of the portion4 ,wherefore1 if in the axis of the hemisphere there be taken from the centre of half the axis1 that is1 of the semidiameter of the sphere1 there will be the centre of equiponderation of the hemisphere4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4,,xxiii4 ,fig4 #a-i ---------------------------------------- -------------------------------- ,,chapter ,,xxiv4 ,,of ,,refraction ,,and ,,reflection4 #a4 ,definitions4--#b4 ,in perpendicular motion there is no refraction4--#c4 ,things thrown out of a thinner into a thicker medium are so refracted that the angle refracted is greater than the angle of inclination4--#d4 ,endeavour1 which from one point tendeth every way1 will be so refracted1 as that the sine of the angle refracted will be to the sine of the angle of inclination1 as the density of the first medium is to the density of the second medium1 reciprocally taken4--#e4 ,the sine of the refracted angle in one inclination is to the sine of the refracted angle in another inclination1 as the sine of the angle of that inclination is to the sine of the angle of this inclination4--#f4 ,if two lines of incidence1 having equal inclination1 be the one in a thinner1 the other in a thicker medium1 the sine of the angle of inclination will be a mean proportional between the two sines of the refracted angles4--#g4 ,if the angle of inclination be semirect1 and the line of inclination be in the thicker medium1 and the proportion of their densities be the same with that of the diagonal to the side of a square1 and the separating superficies be plane1 the refracted line will be in the separating superficies4--#h4 ,if a body be carried in a strait line upon another body1 and do not penetrate the same1 but be reflected from it1 the angle of reflection will be equal to the angle of incidence4--#i4 ,the same happens in the generation of motion in the line of incidence4 ,,definitions4 ,sidenote3 ,definitions4 ,i4 ,refraction is the breaking of that strait line1 in which a body is moved or its action would proceed in one and the same medium1 into two strait lines1 by reason of the different natures of the two mediums4 ,,ii4 ,the former of these is called the line of incidence2 the latter the refracted line4 ,,iii4 ,the point of refraction is the common point of the line of incidence1 and of the refracted line4 ,,iv4 ,the refracting superficies1 which also is the separating superficies of the two mediums1 is that in which is the point of refraction4 ,v4 ,the angle refracted is that1 which the refracted line makes in the point of refraction with that line1 which from the same point is drawn perpendicular to the separating superficies in a different medium4 ,,vi4 ,the angle of refraction is that which the refracted line makes with the line of incidence produced4 ,,vii4 ,the angle of inclination is that which the line of incidence makes with that line1 which from the point of refraction is drawn perpendicular to the separating superficies4 ,,viii4 ,the angle of incidence is the complement to a right angle of the angle of inclination4 ,and so1 7in fig4 #a7 the refraction is made in ,a ,b ,f4 ,the refracted line is ,b ,f4 ,the line of incidence is ,a ,b4 ,the point of incidence and of refraction is ,b4 ,the refracting or separating superficies is ,d ,b ,e4 ,the line of incidence produced directly is ,a ,b ,c4 ,the perpendicular to the separating superficies is ,b ,h4 ,the angle of refraction is ,c ,b ,f4 ,the angle refracted is ,h ,b ,f4 ,the angle of inclination is ,a ,b ,g or ,h ,b ,c4 ,the angle of incidence is ,a ,b ,d4 ,,ix4 ,moreover the thinner medium is understood to be that in which there is less resistance to motion1 or to the generation of motion2 and the thicker that wherein there is greater resistance4 ,x4 ,and that medium in which there is equal resistance everywhere1 is a homogeneous medium4 ,all other mediums are heterogeneous4 ,sidenote3 ,in perpendicular motion there is no refraction4 ,if a body pass1 or there be generation of motion from one medium to another of different density1 in a line perpendicular to the separating superficies1 there will be no refraction4 ,for seeing on every side of the perpendicular all things in the mediums are supposed to be like and equal1 if the motion itself be supposed to be perpendicular1 the inclinations also will be equal1 or rather none at all2 and therefore there can be no cause from which refraction may be inferred to be on one side of the perpendicular1 which will not conclude the same refraction to be on the other side4 ,which being so1 refraction on one side will destroy refraction on the other side2 and consequently either the refracted line will be everywhere1 which is absurd1 or there will be no refracted line at all2 which was to be demonstrated4 ,coroll4 ,it is manifest from hence1 that the cause of refraction consisteth only in the obliquity of the line of incidence1 whether the incident body penetrate both the mediums1 or without penetrating1 propagate motion by pressure only4 ,sidenote3 ,things thrown out of a thinner into a thicker medium are so refracted that the angle refracted is greater than the angle of inclination4 #c4 ,if a body1 without any change of situation of its internal parts1 as a stone1 be moved obliquely out of the thinner medium1 and proceed penetrating the thicker medium1 and the thicker medium be such1 as that its internal parts being moved restore themselves to their former situation2 the angle refracted will be greater than the angle of inclination4 ,for let ,d ,b ,e 7in the same first figure7 be the separating superficies of two mediums2 and let a body1 as a stone thrown1 be understood to be moved as is supposed in the strait line ,a ,b ,c2 and let ,a ,b be in the thinner medium1 as in the air2 and ,b ,c in the thicker1 as in the water4 ,i say the stone1 which being thrown1 is moved in the line ,a ,b1 will not proceed in the line ,b ,c1 but in some other line1 namely1 that1 with which the perpendicular ,b ,h makes the refracted angle ,h ,b ,f greater than the angle of inclination ,h ,b ,c4 ,for seeing the stone coming from ,a1 and falling upon ,b1 makes that which is at ,b proceed towards ,h1 and that the like is done in all the strait lines which are parallel to ,b ,h2 and seeing the parts moved restore themselves by contrary motion in the same line2 there will be contrary motion generated in ,h ,b1 and in all the strait lines which are parallel to it4 ,wherefore1 the motion of the stone will be made by the concourse of the motions in ,a ,g1 that is1 in ,d ,b1 and in ,g ,b1 that is1 in ,b ,h1 and lastly1 in ,h ,b1 that is1 by the concourse of three motions4 ,but by the concourse of the motions in ,a ,g and ,b ,h1 the stone will be carried to ,c2 and therefore by adding the motion in ,h ,b1 it will be carried higher in some other line1 as in ,b ,f1 and make the angle ,h ,b ,f greater than the angle ,h ,b ,c4 ,and from hence may be derived the cause1 why bodies which are thrown in a very oblique line1 if either they be any thing flat1 or be thrown with great force1 will1 when they fall upon the water1 be cast up again from the water into the air4 ,for let ,a ,b 7in fig4 #b7 be the superficies of the water2 into which1 from the point ,c1 let a stone be thrown in the strait line ,c ,a1 making with the line ,b ,a produced a very little angle ,c ,a ,d2 and producing ,b ,a indefinitely to ,d1 let ,c ,d be drawn perpendicular to it1 and ,a ,e parallel to ,c ,d4 ,the stone therefore will be moved in ,c ,a by the concourse of two motions in ,c ,d and ,d ,a1 whose velocities are as the lines themselves ,c ,d and ,d ,a4 ,and from the motion in ,c ,d and all its parallels downwards1 as soon as the stone falls upon ,a1 there will be reaction upwards1 because the water restores itself to its former situation4 ,if now the stone be thrown with sufficient obliquity1 that is1 if the strait line ,c ,d be short enough1 that is1 if the endeavour of the stone downwards be less than the reaction of the water upwards1 that is1 less than the endeavour it hath from its own gravity 7for that may be71 the stone will by reason of the excess of the endeavour which the water hath to restore itself1 above that which the stone hath downwards1 be raised again above the superficies ,a ,b1 and be carried higher1 being reflected in a line which goes higher1 as the line ,a ,g4 ,sidenote3 ,endeavour1 which from one point tendeth every way1 will be so refracted1 as that the sine of the angle refracted will be to the sine of the angle of inclination1 as the density of the first medium is to the density of the second medium1 reciprocally taken4 #d4 ,if from a point1 whatsoever the medium be1 endeavour be propagated every way into all the parts of that medium2 and to the same endeavour there be obliquely opposed another medium of a different nature1 that is1 either thinner or thicker2 that endeavour will be so refracted1 that the sine of the angle refracted1 to the sine of the angle of inclination1 will be as the density of the first medium to the density of the second medium1 reciprocally taken4 ,first1 let a body be in the thinner medium in ,a 7fig4 #c71 and let it be understood to have endeavour every way1 and consequently1 that its endeavour proceed in the lines ,a ,b and ,a b2 to which let ,b b the superficies of the thicker medium be obliquely opposed in ,b and b1 so that ,a ,b and ,a b be equal2 and let the strait line ,b b be produced both ways4 ,from the points ,b and b1 let the perpendiculars ,b ,c and b c be drawn2 and upon the centres ,b and b1 and at the equal distances ,b ,a and b ,a1 let the circles ,a ,c and ,a c be described1 cutting ,b ,c and b c in ,c and c1 and the same ,c ,b and c b produced in ,d and d1 as also ,a ,b and ,a b produced in ,e and e4 ,then from the point ,a to the strait lines ,b ,c and b c let the perpendiculars ,a ,f and ,a f be drawn4 ,a ,f therefore will be the sine of the angle of inclination of the strait line ,a ,b1 and ,a f the sine of the angle of inclination of the strait line ,a h1 which two inclinations are by construction made equal4 ,i say1 as the density of the medium in which are ,b ,c and b c is to the density of the medium in which are ,b ,d and b d1 so is the sine of the angle refracted1 to the sine of the angle of inclination4 ,let the strait line ,f ,g be drawn parallel to the strait line ,a ,b1 meeting with the strait line b ,b produced in ,g4 ,seeing therefore ,a ,f and ,b ,g are also parallels1 they will be equal2 and consequently1 the endeavour in ,a ,f is propagated in the same time1 in which the endeavour in ,b ,g would be propagated if the medium were of the same density4 ,but because ,b ,g is in a thicker medium1 that is1 in a medium which resists the endeavour more than the medium in which ,a ,f is1 the endeavour will be propagated less in ,b ,g than in ,a ,f1 according to the proportion which the density of the medium1 in which ,a ,f is1 hath to the density of the medium in which ,b ,g is4 ,let therefore the density of the medium1 in which ,b ,g is1 be to the density of the medium1 in which ,a ,f is1 as ,b ,g is to ,b ,h2 and let the measure of the time be the radius of the circle4 ,let ,h ,i be drawn parallel to ,b ,d1 meeting with the circumference in ,i2 and from the point ,i let ,i ,k be drawn perpendicular to ,b ,d2 which being done1 ,b ,h and ,i ,k will be equal2 and ,i ,k will be to ,a ,f1 as the density of the medium in which is ,a ,f is to the density of the medium in which is ,i ,k4 ,seeing therefore in the time ,a ,b1 which is the radius of the circle1 the endeavour is propagated in ,a ,f in the thinner medium1 it will be propagated in the same time1 that is1 in the time ,b ,i in the thicker medium from ,k to ,i4 ,therefore1 ,b ,i is the refracted line of the line of incidence ,a ,b2 and ,i ,k is the sine of the angle refracted2 and ,a ,f the sine of the angle of inclination4 ,wherefore1 seeing ,i ,k is to ,a ,f1 as the density of the medium in which is ,a ,f to the density of the medium in which is ,i ,k2 it will be as the density of the medium in which is ,a ,f or ,b ,c to the density of the medium in which is ,i ,k or ,b ,d1 so the sine of the angle refracted to the sine of the angle of inclination4 ,and by the same reason it may be shown1 that as the density of the thinner medium is to the density of the thicker medium1 so will ,k ,i the sine of the angle refracted be to ,a ,f the sine of the angle of inclination4 ,secondly1 let the body1 which endeavoureth every way1 be in the thicker medium at ,i4 ,if1 therefore1 both the mediums were of the same density1 the endeavour of the body in ,i ,b would tend directly to ,l2 and the sine of the angle of inclination ,l ,m would be equal to ,i ,k or ,b ,h4 ,but because the density of the medium1 in which is ,i ,k1 to the density of the medium1 in which is ,l ,m1 is as ,b ,h to ,b ,g1 that is1 to ,a ,f1 the endeavour will be propagated further in the medium in which ,l ,m is1 than in the medium in which ,i ,k is1 in the proportion of density to density1 that is1 of ,m ,l to ,a ,f4 ,wherefore1 ,b ,a being drawn1 the angle refracted will be ,c ,b ,a1 and its sine ,a ,f4 ,but ,l ,m is the sine of the angle of inclination2 and therefore again1 as the density of one medium is to the density of the different medium1 so reciprocally is the sine of the angle refracted to the sine of the angle of inclination2 which was to be demonstrated4 ,in this demonstration1 ,i have made the separating superficies ,b b plane by construction4 ,but though it were concave or convex1 the theorem would nevertheless be true4 ,for the refraction being made in the point ,b of the plane separating superficies1 if a crooked line1 as ,p ,q1 be drawn1 touching the separating line in the point ,b2 neither the refracted line ,b ,i1 nor the perpendicular ,b ,d1 will be altered2 and the refracted angle ,k ,b ,i1 as also its sine ,k ,i1 will be still the same they were4 ,sidenote3 ,the sine of the refracted angle in one inclination is to the sine of the refracted angle in another inclination1 as the sine of the angle of that inclination is to the sine of the angle of this inclination4 #e4 ,the sine of the angle refracted in one inclination is to the sine of the angle refracted in another inclination1 as the sine of the angle of that inclination to the sine of the angle of this inclination4 ,for seeing the sine of the refracted angle is to the sine of the angle of inclination1 whatsoever that inclination be1 as the density of one medium to the density of the other medium2 the proportion of the sine of the refracted angle1 to the sine of the angle of inclination1 will be compounded of the proportions of density to density1 and of the sine of the angle of one inclination to the sine of the angle of the other inclination4 ,but the proportions of the densities in the same homogeneous body are supposed to be the same4 ,wherefore refracted angles in different inclinations are as the sines of the angles of those inclinations2 which was to be demonstrated4 ,sidenote3 ,if two lines of incidence1 having equal inclination1 be one in a thinner the other in a thicker medium1 the sine of the angle of inclination will be a mean proportional between the two sines of the refracted angles4 #f4 ,if two lines of incidence1 having equal inclination1 be the one in a thinner1 the other in a thicker medium1 the sine of the angle of their inclination will be a mean proportional between the two sines of their angles refracted4 ,for let the strait line ,a ,b 7in fig4 #c7 have its inclination in the thinner medium1 and be refracted in the thicker medium in ,b ,i2 and let ,e ,b have as much inclination in the thicker medium1 and be refracted in the thinner medium in ,b ,s2 and let ,r ,s1 the sine of the angle refracted1 be drawn4 ,i say1 the strait lines ,r ,s1 ,a ,f1 and ,i ,k are in continual proportion4 ,for it is1 as the density of the thicker medium to the density of the thinner medium1 so ,r ,s to ,a ,f4 ,but it is also as the density of the same thicker medium to that of the same thinner medium1 so ,a ,f to ,i ,k4 ,wherefore ,r ,s4 ,a ,f 33 ,a ,f4 ,i ,k are proportionals2 that is1 ,r ,s1 ,a ,f1 and ,i ,k are in continual proportion1 and ,a ,f is the mean proportional2 which was to be proved4 ,sidenote3 ,if the angle of inclination be semirect1 and the line of inclination be in the thicker medium1 and the proportion of their densities be the same with that of the diagonal to the side of a square1 and the separating superficies be plain1 the refracted line will be in the separating superficies4 #g4 ,if the angle of inclination be semirect1 and the line of inclination be in the thicker medium1 and the proportion of the densities be as that of a diagonal to the side of its square1 and the separating superficies be plain1 the refracted line will be in that separating superficies4 ,for in the circle ,a ,c 7fig4 #d7 let the angle of inclination ,a ,b ,c be an angle of #de degrees4 ,let ,c ,b be produced to the circumference in ,d2 and let ,c ,e1 the sine of the angle ,e ,b ,c1 be drawn1 to which let ,b ,f be taken equal in the separating line ,b ,g4 ,b ,c ,e ,f will therefore be a parallelogram1 and ,f ,e and ,b ,c1 that is ,f ,e and ,b ,g equal4 ,let ,a ,g be drawn1 namely the diagonal of the square whose side is ,b ,g1 and it will be1 as ,a ,g to ,e ,f so ,b ,g to ,b ,f2 and so1 by supposition1 the density of the medium1 in which ,c is1 to the density of the medium in which ,d is2 and so also the sine of the angle refracted to the sine of the angle of inclination4 ,drawing therefore ,f ,d1 and from ,d the line ,d ,h perpendicular to ,a ,b produced1 ,d ,h will be the sine of the angle of inclination4 ,and seeing the sine of the angle refracted is to the sine of the angle of inclination1 as the density of the medium1 in which is ,c1 is to the density of the medium in which is ,d1 that is1 by supposition1 as ,a ,g is to ,f ,e1 that is as ,b ,g is to ,d ,h2 and seeing ,d ,h is the sine of the angle of inclination1 ,b ,g will therefore be the sine of the angle refracted4 ,wherefore ,b ,g will be the refracted line1 and lye in the plain separating superficies2 which was to be demonstrated4 ,coroll4 ,it is therefore manifest1 that when the inclination is greater than #de degrees1 as also when it is less1 provided the density be greater1 it may happen that the refraction will not enter the thinner medium at all4 ,sidenote3 ,if a body be carried in a strait line upon another body1 and do not penetrate it1 but be reflected from it1 the angle of reflection will be equal to the angle of incidence4 #h4 ,if a body fall in a strait line upon another body1 and do not penetrate it1 but be reflected from it1 the angle of reflection will be equal to the angle of incidence4 ,let there be a body at ,a 7in fig4 #e71 which falling with strait motion in the line ,a ,c upon another body at ,c1 passeth no further1 but is reflected2 and let the angle of incidence be any angle1 as ,a ,c ,d4 ,let the strait line ,c ,e be drawn1 making with ,d ,c produced the angle ,e ,c ,f equal to the angle ,a ,c ,d2 and let ,a ,d be drawn perpendicular to the strait line ,d ,f4 ,also in the same strait line ,d ,f let ,c ,g be taken equal to ,c ,d2 and let the perpendicular ,g ,e be raised1 cutting ,c ,e in ,e4 ,this being done1 the triangles ,a ,c ,d and ,e ,c ,g will be equal and like4 ,let ,c ,h be drawn equal and parallel to the strait line ,a ,d2 and let ,h ,c be produced indefinitely to ,i4 ,lastly let ,e ,a be drawn1 which will pass through ,h1 and be parallel and equal to ,g ,d4 ,i say the motion from ,a to ,c1 in the strait line of incidence ,a ,c1 will be reflected in the strait line ,c ,e4 ,for the motion from ,a to ,c is made by two coefficient or concurrent motions1 the one in ,a ,h parallel to ,d ,g1 the other in ,a ,d perpendicular to the same ,d ,g2 of which two motions that in ,a ,h works nothing upon the body ,a after it has been moved as far as ,c1 because1 by supposition1 it doth not pass the strait line ,d ,g2 whereas the endeavour in ,a ,d1 that is in ,h ,c1 worketh further towards ,i4 ,but seeing it doth only press and not penetrate1 there will be reaction in ,h1 which causeth motion from ,c towards ,h2 and in the meantime the motion in ,h ,e remains the same it was in ,a ,h2 and therefore the body will now be moved by the concourse of two motions in ,c ,h and ,h ,e1 which are equal to the two motions it had formerly in ,a ,h and ,h ,c4 ,wherefore it will be carried on in ,c ,e4 ,the angle therefore of reflection will be ,e ,c ,g1 equal1 by construction1 to the angle ,a ,c ,d2 which was to be demonstrated4 ,now when the body is considered but as a point1 it is all one whether the superficies or line in which the reflection is made be strait or crooked2 for the point of incidence and reflection ,c is as well in the crooked line which toucheth ,d ,g in ,c1 as in ,d ,g itself4 ,sidenote3 ,the same happens in the generation of motion in the line of incidence4 #i4 ,but if we suppose that not a body be moved1 but some endeavour only be propagated from ,a to ,c1 the demonstration will nevertheless be the same4 ,for all endeavour is motion2 and when it hath reached the solid body in ,c1 it presseth it1 and endeavoureth further in ,c ,i4 ,wherefore the reaction will proceed in ,c ,h2 and the endeavour in ,c ,h concurring with the endeavour in ,h ,e1 will generate the endeavour in ,c ,e1 in the same manner as in the repercussion of bodies moved4 ,if therefore endeavour be propagated from any point to the concave superficies of a spherical body1 the reflected line with the circumference of a great circle in the same sphere will make an angle equal to the angle of incidence4 ,for if endeavour be propagated from ,a 7in fig4 #f7 to the circumference in ,b1 and the centre of the sphere be ,c1 and the line ,c ,b be drawn1 as also the tangent ,d ,b ,e2 and lastly if the angle ,f ,b ,d be made equal to the angle ,a ,b ,e1 the reflection will be made in the line ,b ,f1 as hath been newly shown4 ,wherefore the angles1 which the strait lines ,a ,b and ,f ,b make with the circumference1 will also be equal4 ,but it is here to be noted1 that if ,c ,b be produced howsoever to ,g1 the endeavour in the line ,g ,b ,c will proceed only from the perpendicular reaction in ,g ,b2 and that therefore there will be no other endeavour in the point ,b towards the parts which are within the sphere1 besides that which tends towards the centre4 ,and here ,i put an end to the third part of this discourse2 in which ,i have considered motion and magnitude by themselves in the abstract4 ,the fourth and last part1 concerning the phenomena of nature1 that is to say1 concerning the motions and magnitudes of the bodies which are parts of the world1 real and existent1 is that which follows4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxiv4 ,fig4 #a-f ---------------------------------------- -------------------------------- ,,part ,,iv4 ,,physics1 ,,or ,,the ,,phenomena ,,of ,,nature4 ------- ,,chapter ,,xxv4 ,,of ,,sense ,,and ,,animal ,,motion4 #a4 ,the connexion of what hath been said with that which followeth4--#b4 ,the investigation of the nature of sense1 and the definition of sense4--#c4 ,the subject and object of sense4--#d4 ,the organs of sense4--#e4 ,all bodies are not indued with sense4--#f4 ,but one phantasm at one and the same time4--#g4 ,imagination the remains of past sense1 which also is memory4 ,of sleep4--#h4 ,how phantasms succeed one another4--#i4 ,dreams1 whence they proceed4--#aj4 ,of the senses1 their kinds1 their organs1 and phantasms proper and common4--#aa4 ,the magnitude of images1 how and by what it is determined4--#ab4 ,pleasure1 pain1 appetite and aversion1 what they are4--#ac4 ,deliberation and will1 what4 ,sidenote3 ,the connexion of what hath been said with that which followeth4 #a4 ,i have1 in the first chapter1 defined philosophy to be knowledge of effects acquired by true ratiocination1 from knowledge first had of their causes and generation2 and of such causes or generations as may be1 from former knowledge of their effects or appearances4 ,there are1 therefore1 two methods of philosophy2 one1 from the generation of things to their possible effects2 and the other1 from their effects or appearances to some possible generation of the same4 ,in the former of these the truth of the first principles of our ratiocination1 namely definitions1 is made and constituted by ourselves1 whilst we consent and agree about the appellations of things4 ,and this part ,i have finished in the foregoing chapters2 in which1 if ,i am not deceived1 ,i have affirmed nothing1 saving the definitions themselves1 which hath not good coherence with the definitions ,i have given2 that is to say1 which is not sufficiently demonstrated to all those1 that agree with me in the use of words and appellations2 for whose sake only ,i have written the same4 ,i now enter upon the other part2 which is the finding out by the appearances or effects of nature1 which we know by sense1 some ways and means by which they may be1 ,i do not say they are1 generated4 ,the principles1 therefore1 upon which the following discourse depends1 are not such as we ourselves make and pronounce in general terms1 as definitions2 but such1 as being placed in the things themselves by the ,author of ,nature1 are by us observed in them2 and we make use of them in single and particular1 not universal propositions4 ,nor do they impose upon us any necessity of constituting theorems2 their use being only1 though not without such general propositions as have been already demonstrated1 to show us the possibility of some production or generation4 ,seeing1 therefore1 the science1 which is here taught1 hath its principles in the appearances of nature1 and endeth in the attaining of some knowledge of natural causes1 ,i have given to this part the title of ,,physics1 or the ,phenomena of ,nature4 ,now such things as appear1 or are shown to us by nature1 we call phenomena or appearances4 ,of all the phenomena or appearances which are near us1 the most admirable is apparition itself1 2 namely1 that some natural bodies have in themselves the patterns almost of all things1 and others of none at all4 ,so that if the appearances be the principles by which we know all other things1 we must needs acknowledge sense to be the principle by which we know those principles1 and that all the knowledge we have is derived from it4 ,and as for the causes of sense1 we cannot begin our search of them from any other phenomenon than that of sense itself4 ,but you will say1 by what sense shall we take notice of sense8 ,i answer1 by sense itself1 namely1 by the memory which for some time remains in us of things sensible1 though they themselves pass away4 ,for he that perceives that he hath perceived1 remembers4 ,in the first place1 therefore1 the causes of our perception1 that is1 the causes of those ideas and phantasms which are perpetually generated within us whilst we make use of our senses1 are to be enquired into2 and in what manner their generation proceeds4 ,to help which inquisition1 we may observe first of all1 that our phantasms or ideas are not always the same2 but that new ones appear to us1 and old ones vanish1 according as we apply our organs of sense1 now to one object1 now to another4 ,wherefore they are generated1 and perish4 ,and from hence it is manifest1 that they are some change or mutation in the sentient4 ,sidenote3 ,the investigation of the nature of sense1 and the definition of sense4 #b4 ,now that all mutation or alteration is motion or endeavour 7and endeavour also is motion7 in the internal parts of the thing that is altered1 hath been proved 7in art4 #i1 chap4 ,,viii7 from this1 that whilst even the least parts of any body remain in the same situation in respect of one another1 it cannot be said that any alteration1 unless perhaps that the whole body together hath been moved1 hath happened to it2 but that it both appeareth and is the same it appeared and was before4 ,sense1 therefore1 in the sentient1 can be nothing else but motion in some of the internal parts of the sentient2 and the parts so moved are parts of the organs of sense4 ,for the parts of our body1 by which we perceive any thing1 are those we commonly call the organs of sense4 ,and so we find what is the subject of our sense1 namely1 that in which are the phantasms2 and partly also we have discovered the nature of sense1 namely1 that it is some internal motion in the sentient4 ,i have shown besides 7in chap4 ,,ix1 art4 #g7 that no motion is generated but by a body contiguous and moved3 from whence it is manifest1 that the immediate cause of sense or perception consists in this1 that the first organ of sense is touched and pressed4 ,for when the uttermost part of the organ is pressed1 it no sooner yields1 but the part next within it is pressed also2 and1 in this manner1 the pressure or motion is propagated through all the parts of the organ to the innermost4 ,and thus also the pressure of the uttermost part proceeds from the pressure of some more remote body1 and so continually1 till we come to that from which1 as from its fountain1 we derive the phantasm or idea that is made in us by our sense4 ,and this1 whatsoever it be1 is that we commonly call the object1 ,sense1 therefore1 is some internal motion in the sentient1 generated by some internal motion of the parts of the object1 and propagated through all the media to the innermost part of the organ4 ,by which words ,i have almost defined what sense is4 ,moreover1 ,i have shown 7art4 #b1 chap4 ,,xv7 that all resistance is endeavour opposite to another endeavour1 that is to say1 reaction4 ,seeing1 therefore1 there is in the whole organ1 by reason of its own internal natural motion1 some resistance or reaction against the motion which is propagated from the object to the innermost part of the organ1 there is also in the same organ an endeavour opposite to the endeavour which proceeds from the object2 so that when that endeavour inwards is the last action in the act of sense1 then from the reaction1 how little soever the duration of it be1 a phantasm or idea hath its being2 which1 by reason that the endeavour is now outwards1 doth always appear as something situate without the organ4 ,so that now ,i shall give you the whole definition of sense1 as it is drawn from the explication of the causes thereof and the order of its generation1 thus3 ,,sense is a phantasm1 made by the reaction and endeavour outwards in the organ of sense1 caused by an endeavour inwards from the object1 remaining for some time more or less4 ,sidenote3 ,the subject and object of sense4 #c4 ,the subject of sense is the sentient itself1 namely1 some living creature2 and we speak more correctly1 when we say a living creature seeth1 than when we say the eye seeth4 ,the object is the thing received2 and it is more accurately said1 that we see the sun1 than that we see the light4 ,for light and colour1 and heat and sound1 and other qualities which are commonly called sensible1 are not objects1 but phantasms in the sentients4 ,for a phantasm is the act of sense1 and differs no otherwise from sense than fieri1 that is1 being a doing1 differs from factum esse1 that is1 being done2 which difference1 in things that are done in an instant1 is none at all2 and a phantasm is made in an instant4 ,for in all motion which proceeds by perpetual propagation1 the first part being moved moves the second1 the second the third1 and so on to the last1 and that to any distance1 how great soever4 ,and in what point of time the first or foremost part proceeded to the place of the second1 which is thrust on1 in the same point of time the last save one proceeded into the place of the last yielding part2 which by reaction1 in the same instant1 if the reaction be strong enough1 makes a phantasm2 and a phantasm being made1 perception is made together with it4 ,sidenote3 ,the organs of sense4 #d4 ,the organs of sense1 which are in the sentient1 are such parts thereof1 that if they be hurt1 the very generation of phantasms is thereby destroyed1 though all the rest of the parts remain entire4 ,now these parts in the most of living creatures are found to be certain spirits and membranes1 which1 proceeding from the pia mater1 involve the brain and all the nerves2 also the brain itself1 and the arteries which are in the brain2 and such other parts1 as being stirred1 the heart also1 which is the fountain of all sense1 is stirred together with them4 ,for whensoever the action of the object reacheth the body of the sentient1 that action is by some nerve propagated to the brain2 and if the nerve leading thither be so hurt or obstructed1 that the motion can be propagated no further1 no sense follows4 ,also if the motion be intercepted between the brain and the heart by the defect of the organ by which the action is propagated1 there will be no perception of the object4 ,sidenote3 ,all bodies are not endued with sense4 #e4 ,but though all sense1 as ,i have said1 be made by reaction1 nevertheless it is not necessary that every thing that reacteth should have sense4 ,i know there have been philosophers1 and those learned men1 who have maintained that all bodies are endued with sense4 ,nor do ,i see how they can be refuted1 if the nature of sense be placed in reaction only4 ,and1 though by the reaction of bodies inanimate a phantasm might be made1 it would nevertheless cease1 as soon as ever the object were removed4 ,for unless those bodies had organs1 as living creatures have1 fit for the retaining of such motion as is made in them1 their sense would be such1 as that they should never remember the same4 ,and therefore this hath nothing to do with that sense which is the subject of my discourse4 ,for by sense1 we commonly understand the judgment we make of objects by their phantasms2 namely1 by comparing and distinguishing those phantasms2 which we could never do1 if that motion in the organ1 by which the phantasm is made1 did not remain there for some time1 and make the same phantasm return4 ,wherefore sense1 as ,i here understand it1 and which is commonly so called1 hath necessarily some memory adhering to it1 by which former and later phantasms may be compared together1 and distinguished from one another4 ,sense1 therefore1 properly so called1 must necessarily have in it a perpetual variety of phantasms1 that they may be discerned one from another4 ,for if we should suppose a man to be made with clear eyes1 and all the rest of his organs of sight well disposed1 but endued with no other sense2 and that he should look only upon one thing1 which is always of the same colour and figure1 without the least appearance of variety1 he would seem to me1 whatsoever others may say1 to see1 no more than ,i seem to myself to feel the bones of my own limbs by my organs of feeling2 and yet those bones are always and on all sides touched by a most sensible membrane4 ,i might perhaps say he were astonished1 and looked upon it2 but ,i should not say he saw it2 it being almost all one for a man to be always sensible of one and the same thing1 and not to be sensible at all of any thing4 ,sidenote3 ,but one phantasm at one and the same time4 #f4 ,and yet such is the nature of sense1 that it does not permit a man to discern many things at once4 ,for seeing the nature of sense consists in motion2 as long as the organs are employed about one object1 they cannot be so moved by another at the same time1 as to make by both their motions one sincere phantasm of each of them at once4 ,and therefore two several phantasms will not be made by two objects working together1 but only one phantasm compounded from the action of both4 ,besides1 as when we divide a body1 we divide its place2 and when we reckon many bodies1 we must necessarily reckon as many places2 and contrarily1 as ,i have shown in the seventh chapter2 so what number soever we say there be of times1 we must understand the same number of motions also2 and as oft as we count many motions1 so oft we reckon many times4 ,for though the object we look upon be of divers colours1 yet with those divers colours it is but one varied object1 and not variety of objects4 ,moreover1 whilst those organs which are common to all the senses1 such as are those parts of every organ which proceed in men from the root of the nerves to the heart1 are vehemently stirred by a strong action from some one object1 they are1 by reason of the contumacy which the motion1 they have already1 gives them against the reception of all other motion1 made the less fit to receive any other impression from whatsoever other objects1 to what sense soever those objects belong4 ,and hence it is1 that an earnest studying of one object1 takes away the sense of all other objects for the present4 ,for study is nothing else but a possession of the mind1 that is to say1 a vehement motion made by some one object in the organs of sense1 which are stupid to all other motions as long as this lasteth2 according to what was said by ,terence1 8,populus studio stupidus in funambulo animum occuparat40 ,for what is stupor but that which the ,greeks call 1 that is1 a cessation from the sense of other things8 ,wherefore at one and the same time1 we cannot by sense perceive more than one single object2 as in reading1 we see the letters successively one by one1 and not all together1 though the whole page be presented to our eye2 and though every several letter be distinctly written there1 yet when we look upon the whole page at once1 we read nothing4 ,from hence it is manifest1 that every endeavour of the organ outwards1 is not to be called sense1 but that only1 which at several times is by vehemence made stronger and more predominant than the rest2 which deprives us of the sense of other phantasms1 no otherwise than the sun deprives the rest of the stars of light1 not by hindering their action1 but by obscuring and hiding them with his excess of brightness4 ,sidenote3 ,imagination1 the remains of past sense2 which also is memory4 ,of sleep4 #g4 ,but the motion of the organ1 by which a phantasm is made1 is not commonly called sense1 except the object be present4 ,and the phantasm remaining after the object is removed or past by1 is called fancy1 and in ,latin imaginatio2 which word1 because all phantasms are not images1 doth not fully answer the signification of the word fancy in its general acceptation4 ,nevertheless ,i may use it safely enough1 by understanding it for the ,greek 4 ,,imagination therefore is nothing else but sense decaying1 or weakened1 by the absence of the object4 ,but what may be the cause of this decay or weakening8 ,is the motion the weaker1 because the object is taken away8 ,if it were1 then phantasms would always and necessarily be less clear in the imagination1 than they are in sense2 which is not true4 ,for in dreams1 which are the imaginations of those that sleep1 they are no less clear than in sense itself4 ,but the reason why in men waking the phantasms of things past are more obscure than those of things present1 is this1 that their organs being at the same time moved by other present objects1 those phantasms are the less predominant4 ,whereas in sleep1 the passages being shut up1 external action doth not at all disturb or hinder internal motion4 ,if this be true1 the next thing to be considered1 will be1 whether any cause may be found out1 from the supposition whereof it will follow1 that the passage is shut up from the external objects of sense to the internal organ4 ,i suppose1 therefore1 that by the continual action of objects1 to which a reaction of the organ1 and more especially of the spirits1 is necessarily consequent1 the organ is wearied1 that is1 its parts are no longer moved by the spirits without some pain2 and consequently the nerves being abandoned and grown slack1 they retire to their fountain1 which is the cavity either of the brain or of the heart2 by which means the action which proceeded by the nerves is necessarily intercepted4 ,for action upon a patient1 that retires from it1 makes but little impression at the first2 and at last1 when the nerves are by little and little slackened1 none at all4 ,and therefore there is no more reaction1 that is1 no more sense1 till the organ being refreshed by rest1 and by a supply of new spirits recovering strength and motion1 the sentient awaketh4 ,and thus it seems to be always1 unless some other preternatural cause intervene2 as heat in the internal parts from lassitude1 or from some disease stirring the spirits and other parts of the organ in some extraordinary manner4 ,sidenote3 ,how phantasms succeed one another4 #h4 ,now it is not without cause1 nor so casual a thing as many perhaps think it1 that phantasms in this their great variety proceed from one another2 and that the same phantasms sometimes bring into the mind other phantasms like themselves1 and at other times extremely unlike4 ,for in the motion of any continued body1 one part follows another by cohesion2 and therefore1 whilst we turn our eyes and other organs successively to many objects1 the motion which was made by every one of them remaining1 the phantasms are renewed as often as any one of those motions comes to be predominant above the rest2 and they become predominant in the same order in which at any time formerly they were generated by sense4 ,so that when by length of time very many phantasms have been generated within us by sense1 then almost any thought may arise from any other thought2 insomuch that it may seem to be a thing indifferent and casual1 which thought shall follow which4 ,but for the most part this is not so uncertain a thing to waking as to sleeping men4 ,for the thought or phantasm of the desired end brings in all the phantasms1 that are means conducing to that end1 and that in order backwards from the last to the first1 and again forwards from the beginning to the end4 ,but this supposes both appetite1 and judgment to discern what means conduce to the end1 which is gotten by experience2 and experience is store of phantasms1 arising from the sense of very many things4 ,for and meminisse1 fancy and memory1 differ only in this1 that memory supposeth the time past1 which fancy doth not4 ,in memory1 the phantasms we consider are as if they were worn out with time2 but in our fancy we consider them as they are2 which distinction is not of the things themselves1 but of the considerations of the sentient4 ,for there is in memory something like that which happens in looking upon things at a great distance2 in which as the small parts of the object are not discerned1 by reason of their remoteness2 so in memory1 many accidents and places and parts of things1 which were formerly perceived by sense1 are by length of time decayed and lost4 ,the perpetual arising of phantasms1 both in sense and imagination1 is that which we commonly call discourse of the mind1 and is common to men with other living creatures4 ,for he that thinketh1 compareth the phantasms that pass1 that is1 taketh notice of their likeness or unlikeness to one another4 ,and as he that observes readily the likenesses of things of different natures1 or that are very remote from one another1 is said to have a good fancy2 so he is said to have a good judgment1 that finds out the unlikenesses or differences of things that are like one another4 ,now this observation of differences is not perception made by a common organ of sense1 distinct from sense or perception properly so called1 but is memory of the differences of particular phantasms remaining for some time2 as the distinction between hot and lucid1 is nothing else but the memory both of a heating1 and of an enlightening object4 ,sidenote3 ,dreams1 whence they proceed4 #i4 ,the phantasms of men that sleep1 are dreams4 ,concerning which we are taught by experience these five things4 ,first1 that for the most part there is neither order nor coherence in them4 ,secondly1 that we dream of nothing but what is compounded and made up of the phantasms of sense past4 ,thirdly1 that sometimes they proceed1 as in those that are drowsy1 from the interruption of their phantasms by little and little1 broken and altered through sleepiness2 and sometimes also they begin in the midst of sleep4 ,fourthly1 that they are clearer than the imaginations of waking men1 except such as are made by sense itself1 to which they are equal in clearness4 ,fifthly1 that when we dream1 we admire neither the places nor the looks of the things that appear to us4 ,now from what hath been said1 it is not hard to show what may be the causes of these phenomena4 ,for as for the first1 seeing all order and coherence proceeds from frequent looking back to the end1 that is1 from consultation2 it must needs be1 that seeing in sleep we lose all thought of the end1 our phantasms succeed one another1 not in that order which tends to any end1 but as it happeneth1 and in such manner1 as objects present themselves to our eyes when we look indifferently upon all things before us1 and see them1 not because we would see them1 but because we do not shut our eyes2 for then they appear to us without any order at all4 ,the second proceeds from this1 that in the silence of sense there is no new motion from the objects1 and therefore no new phantasm1 unless we call that new1 which is compounded of old ones1 as a chimera1 a golden mountain1 and the like4 ,as for the third1 why a dream is sometimes as it were the continuation of sense1 made up of broken phantasms1 as in men distempered with sickness1 the reason is manifestly this1 that in some of the organs sense remains1 and in others it faileth4 ,but how some phantasms may be revived1 when all the exterior organs are benumbed with sleep1 is not so easily shown4 ,nevertheless that1 which hath already been said1 contains the reason of this also4 ,for whatsoever strikes the pia mater1 reviveth some of those phantasms that are still in motion in the brain2 and when any internal motion of the heart reacheth that membrane1 then the predominant motion in the brain makes the phantasm4 ,now the motions of the heart are appetites and aversions1 of which ,i shall presently speak further4 ,and as appetites and aversions are generated by phantasms1 so reciprocally phantasms are generated by appetites and aversions4 ,for example1 heat in the heart proceeds from anger and fighting2 and again1 from heat in the heart1 whatsoever be the cause of it1 is generated anger and the image of an enemy1 in sleep4 ,and as love and beauty stir up heat in certain organs2 so heat in the same organs1 from whatsoever it proceeds1 often causeth desire and the image of an unresisting beauty4 ,lastly1 cold doth in the same manner generate fear in those that sleep1 and causeth them to dream of ghosts1 and to have phantasms of horror and danger2 as fear also causeth cold in those that wake4 ,so reciprocal are the motions of the heart and brain4 ,the fourth1 namely1 that the things we seem to see and feel in sleep1 are as clear as in sense itself1 proceeds from two causes2 one1 that having then no sense of things without us1 that internal motion which makes the phantasm1 in the absence of all other impressions1 is predominant2 and the other1 that the parts of our phantasms which are decayed and worn out by time1 are made up with other fictitious parts4 ,to conclude1 when we dream1 we do not wonder at strange places and the appearances of things unknown to us1 because admiration requires that the things appearing be new and unusual1 which can happen to none but those that remember former appearances2 whereas in sleep1 all things appear as present4 ,but it is here to be observed1 that certain dreams1 especially such as some men have when they are between sleeping and waking1 and such as happen to those that have no knowledge of the nature of dreams and are withal superstitious1 were not heretofore nor are now accounted dreams4 ,for the apparitions men thought they saw1 and the voices they thought they heard in sleep1 were not believed to be phantasms1 but things subsisting of themselves1 and objects without those that dreamed4 ,for to some men1 as well sleeping as waking1 but especially to guilty men1 and in the night1 and in hallowed places1 fear alone1 helped a little with the stories of such apparitions1 hath raised in their minds terrible phantasms1 which have been and are still deceitfully received for things really true1 under the names of ghosts and incorporeal substances4 ,sidenote3 ,of the senses1 their kinds1 their organs and phantasms1 proper and common4 #aj4 ,in most living creatures there are observed five kinds of senses1 which are distinguished by their organs1 and by their different kinds of phantasms2 namely1 sight1 hearing1 smell1 taste1 and touch2 and these have their organs partly peculiar to each of them severally1 and partly common to them all4 ,the organ of sight is partly animate1 and partly inanimate4 ,the inanimate parts are the three humours2 namely1 the watery humour1 which by the interposition of the membrane called uvea1 the perforation whereof is called the apple of the eye1 is contained on one side by the first concave superficies of the eye1 and on the other side by the ciliary processes1 and the coat of the crystalline humour2 the crystalline1 which1 hanging in the midst between the ciliary processes1 and being almost of spherical figure1 and of a thick consistence1 is enclosed on all sides with its own transparent coat2 and the vitreous or glassy humour1 which filleth all the rest of the cavity of the eye1 and is somewhat thicker then the watery humour1 but thinner than the crystalline4 ,the animate part of the organ is1 first1 the membrane choroeides1 which is a part of the pia mater1 saving that it is covered with a coat derived from the marrow of the optic nerve1 which is called the retina2 and this choroeides1 seeing it is part of the pia mater1 is continued to the beginning of the medulla spinalis within the scull1 in which all the nerves which are within the head have their roots4 ,wherefore all the animal spirits that the nerves receive1 enter into them there2 for it is not imaginable that they can enter into them anywhere else4 ,seeing therefore sense is nothing else but the action of objects propagated to the furthest part of the organ2 and seeing also that animal spirits are nothing but vital spirits purified by the heart1 and carried from it by the arteries2 it follows necessarily1 that the action is derived from the heart by some of the arteries to the roots of the nerves which are in the head1 whether those arteries be the plexus retiformis1 or whether they be other arteries which are inserted into the substance of the brain4 ,and1 therefore1 those arteries are the complement or the remaining part of the whole organ of sight4 ,and this last part is a common organ to all the senses2 whereas1 that which reacheth from the eye to the roots of the nerves is proper only to sight4 ,the proper organ of hearing is the tympanum of the ear and its own nerve2 from which to the heart the organ is common4 ,so the proper organs of smell and taste are nervous membranes1 in the palate and tongue for the taste1 and in the nostrils for the smell2 and from the roots of those nerves to the heart all is common4 ,lastly1 the proper organ of touch are nerves and membranes dispersed through the whole body2 which membranes are derived from the root of the nerves4 ,and all things else belonging alike to all the senses seem to be administered by the arteries1 and not by the nerves4 ,the proper phantasm of sight is light2 and under this name of light1 colour also1 which is nothing but perturbed light1 is comprehended4 ,wherefore the phantasm of a lucid body is light2 and of a coloured body1 colour4 ,but the object of sight1 properly so called1 is neither light nor colour1 but the body itself which is lucid1 or enlightened1 or coloured4 ,for light and colour1 being phantasms of the sentient1 cannot be accidents of the object4 ,which is manifest enough from this1 that visible things appear oftentimes in places in which we know assuredly they are not1 and that in different places they are of different colours1 and may at one and the same time appear in divers places4 ,motion1 rest1 magnitude1 and figure1 are common both to the sight and touch2 and the whole appearance together of figure1 and light or colour1 is by the ,greeks commonly called 1 and 1 and 2 and by the ,latins1 species and imago2 all which names signify no more but appearance4 ,the phantasm1 which is made by hearing1 is sound2 by smell1 odour2 by taste1 savour2 and by touch1 hardness and softness1 heat and cold1 wetness1 oiliness1 and many more1 which are easier to be distinguished by sense than words4 ,smoothness1 roughness1 rarity1 and density1 refer to figure1 and are therefore common both to touch and sight4 ,and as for the objects of hearing1 smell1 taste1 and touch1 they are not sound1 odour1 savour1 hardness1 c41 but the bodies themselves from which sound1 odour1 savour1 hardness1 c4 proceed2 of the causes of which1 and of the manner how they are produced1 ,i shall speak hereafter4 ,but these phantasms1 though they be effects in the sentient1 as subject1 produced by objects working upon the organs2 yet there are also other effects besides these1 produced by the same objects in the same organs2 namely certain motions proceeding from sense1 which are called animal motions4 ,for seeing in all sense of external things there is mutual action and reaction1 that is1 two endeavours opposing one another1 it is manifest that the motion of both of them together will be continued every way1 especially to the confines of both the bodies4 ,and when this happens in the internal organ1 the endeavour outwards will proceed in a solid angle1 which will be greater1 and consequently the idea greater1 than it would have been if the impression had been weaker4 ,sidenote3 ,the magnitude of images1 how and by what it is determined4 #aa4 ,from hence the natural cause is manifest1 first1 why those things seem to be greater1 which1 cteris paribus1 are seen in a greater angle3 secondly1 why in a serene cold night1 when the moon doth not shine1 more of the fixed stars appear than at another time4 ,for their action is less hindered by the serenity of the air1 and not obscured by the greater light of the moon1 which is then absent2 and the cold1 making the air more pressing1 helpeth or strengtheneth the action of the stars upon our eyes2 in so much as stars may then be seen which are seen at no other time4 ,and this may suffice to be said in general concerning sense made by the reaction of the organ4 ,for1 as for the place of the image1 the deceptions of sight1 and other things of which we have experience in ourselves by sense1 seeing they depend for the most part upon the fabric itself of the eye of man1 ,i shall speak of them then when ,i come to speak of man4 ,sidenote3 ,pleasure1 pain1 appetite1 and aversion1 what they are4 #ab4 ,but there is another kind of sense1 of which ,i will say something in this place1 namely1 the sense of pleasure and pain1 proceeding not from the reaction of the heart outwards1 but from continual action from the outermost part of the organ towards the heart4 ,for the original of life being in the heart1 that motion in the sentient1 which is propagated to the heart1 must necessarily make some alteration or diversion of vital motion1 namely1 by quickening or slackening1 helping or hindering the same4 ,now when it helpeth1 it is pleasure2 and when it hindereth1 it is pain1 trouble1 grief1 c4 ,and as phantasms seem to be without1 by reason of the endeavour outwards1 so pleasure and pain1 by reason of the endeavour of the organ inwards1 seem to be within2 namely1 there where the first cause of the pleasure or pain is2 as when the pain proceeds from a wound1 we think the pain and the wound are both in the same place4 ,sidenote3 ,pleasure1 pain1 appetite1 and aversion1 what they are4 ,now vital motion is the motion of the blood1 perpetually circulating 7as hath been shown from many infallible signs and marks by ,doctor ,harvey1 the first observer of it7 in the veins and arteries4 ,which motion1 when it is hindered by some other motion made by the action of sensible objects1 may be restored again either by bending or setting strait the parts of the body2 which is done when the spirits are carried now into these1 now into other nerves1 till the pain1 as far as is possible1 be quite taken away4 ,but if vital motion be helped by motion made by sense1 then the parts of the organ will be disposed to guide the spirits in such manner as conduceth most to the preservation and augmentation of that motion1 by the help of the nerves4 ,and in animal motion this is the very first endeavour1 and found even in the embryo2 which while it is in the womb1 moveth its limbs with voluntary motion1 for the avoiding of whatsoever troubleth it1 or for the pursuing of what pleaseth it4 ,and this first endeavour1 when it tends towards such things as are known by experience to be pleasant1 is called appetite1 that is1 an approaching2 and when it shuns what is troublesome1 aversion1 or flying from it4 ,and little infants1 at the beginning and as soon as they are born1 have appetite to very few things1 as also they avoid very few1 by reason of their want of experience and memory2 and therefore they have not so great a variety of animal motion as we see in those that are more grown4 ,for it is not possible1 without such knowledge as is derived from sense1 that is1 without experience and memory1 to know what will prove pleasant or hurtful2 only there is some place for conjecture from the looks or aspects of things4 ,and hence it is1 that though they do not know what may do them good or harm1 yet sometimes they approach and sometimes retire from the same thing1 as their doubt prompts them4 ,but afterwards1 by accustoming themselves by little and little1 they come to know readily what is to be pursued and what to be avoided2 and also to have a ready use of their nerves and other organs1 in the pursuing and avoiding of good and bad4 ,wherefore appetite and aversion are the first endeavours of animal motion4 ,consequent to this first endeavour1 is the impulsion into the nerves and retraction again of animal spirits1 of which it is necessary there be some receptacle or place near the original of the nerves2 and this motion or endeavour is followed by a swelling and relaxation of the muscles2 and lastly1 these are followed by contraction and extension of the limbs1 which is animal motion4 ,sidenote3 ,deliberation and will1 what4 #ac4 ,the considerations of appetites and aversions are divers4 ,for seeing living creatures have sometimes appetite and sometimes aversion to the same thing1 as they think it will either be for their good or their hurt2 while that vicissitude of appetites and aversions remains in them1 they have that series of thoughts which is called deliberation2 which lasteth as long as they have it in their power to obtain that which pleaseth1 or to avoid that which displeaseth them4 ,appetite1 therefore1 and aversion are simply so called as long as they follow not deliberation4 ,but if deliberation have gone before1 then the last act of it1 if it be appetite1 is called will2 if aversion1 unwillingness4 ,so that the same thing is called both will and appetite2 but the consideration of them1 namely1 before and after deliberation1 is divers4 ,nor is that which is done within a man whilst he willeth any thing1 different from that which is done in other living creatures1 whilst1 deliberation having preceded1 they have appetite4 ,neither is the freedom of willing or not willing1 greater in man1 than in other living creatures4 ,for where there is appetite1 the entire cause of appetite hath preceded2 and1 consequently1 the act of appetite could not choose but follow1 that is1 hath of necessity followed 7as is shown in chapter ,,ix1 article #e74 ,and therefore such a liberty as is free from necessity1 is not to be found in the will either of men or beasts4 ,but if by liberty we understand the faculty or power1 not of willing1 but of doing what they will1 then certainly that liberty is to be allowed to both1 and both may equally have it1 whensoever it is to be had4 ,again1 when appetite and aversion do with celerity succeed one another1 the whole series made by them hath its name sometimes from one1 sometimes from the other4 ,for the same deliberation1 whilst it inclines sometimes to one1 sometimes to the other1 is from appetite called hope1 and from aversion1 fear4 ,for where there is no hope1 it is not to be called fear1 but hate2 and where no fear1 not hope1 but desire4 ,to conclude1 all the passions1 called passions of the mind1 consist of appetite and aversion1 except pure pleasure and pain1 which are a certain fruition of good or evil2 as anger is aversion from some imminent evil1 but such as is joined with appetite of avoiding that evil by force4 ,but because the passions and perturbations of the mind are innumerable1 and many of them not to be discerned in any creatures besides men2 ,i will speak of them more at large in that section which is concerning man4 ,as for those objects1 if there be any such1 which do not at all stir the mind1 we are said to contemn them4 ,and thus much of sense in general4 ,in the next place ,i shall speak of sensible objects4 ------- ,,chapter ,,xxvi4 ,,of ,,the ,,world ,,and ,,of ,,the ,,stars4 #a4 ,the magnitude and duration of the world1 inscrutable4--#b4 ,no place in the world empty4--#c4 ,the arguments of ,lucretius for vacuum1 invalid4--#d4 ,other arguments for the establishing of vacuum1 invalid4--#e4 ,six suppositions for the salving of the phenomena of nature4--#f4 ,possible causes of the motions annual and diurnal2 and of the apparent direction1 station1 and retrogradation of the planets4--#g4 ,the supposition of simple motion1 why likely4--#h4 ,the cause of the eccentricity of the annual motion of the earth4--#i4 ,the cause why the moon hath always one and the same face turned towards the earth4--#aj4 ,the cause of the tides of the ocean4--#aa4 ,the cause of the precession of the equinoxes4 ,sidenote3 ,the magnitude and duration of the world1 inscrutable4 #a4 ,consequent to the contemplation of sense is the contemplation of bodies1 which are the efficient causes or objects of sense4 ,now every object is either a part of the whole world1 or an aggregate of parts4 ,the greatest of all bodies1 or sensible objects1 is the world itself2 which we behold when we look round about us from this point of the same which we call the earth4 ,concerning the world1 as it is one aggregate of many parts1 the things that fall under inquiry are but few2 and those we can determine1 none4 ,of the whole world we may inquire what is its magnitude1 what its duration1 and how many there be1 but nothing else4 ,for as for place and time1 that is to say1 magnitude and duration1 they are only our own fancy of a body simply so called1 that is to say1 of a body indefinitely taken1 as ,i have shown before in chapter ,,vii4 ,all other phantasms are of bodies or objects1 as they are distinguished from one another2 as colour1 the phantasm of coloured bodies2 sound1 of bodies that move the sense of hearing1 c4 ,the questions concerning the magnitude of the world are whether it be finite or infinite1 full or not full2 concerning its duration1 whether it had a beginning1 or be eternal2 and concerning the number1 whether there be one or many2 though as concerning the number1 if it were of infinite magnitude1 there could be no controversy at all4 ,also if it had a beginning1 then by what cause and of what matter it was made2 and again1 from whence that cause and that matter had their being1 will be new questions2 till at last we come to one or many eternal cause or causes4 ,and the determination of all these things belongeth to him that professeth the universal doctrine of philosophy1 in case as much could be known as can be sought4 ,but the knowledge of what is infinite can never be attained by a finite inquirer4 ,whatsoever we know that are men1 we learn it from our phantasms2 and of infinite1 whether magnitude or time1 there is no phantasm at all2 so that it is impossible either for a man or any other creature to have any conception of infinite4 ,and though a man may from some effect proceed to the immediate cause thereof1 and from that to a more remote cause1 and so ascend continually by right ratiocination from cause to cause2 yet he will not be able to proceed eternally1 but wearied will at last give over1 without knowing whether it were possible for him to proceed to an end or not4 ,but whether we suppose the world to be finite or infinite1 no absurdity will follow4 ,for the same things which now appear1 might appear1 whether the ,creator had pleased it should be finite or infinite4 ,besides1 though from this1 that nothing can move itself1 it may rightly be inferred that there was some first eternal movent2 yet it can never be inferred1 though some used to make such inference1 that that movent was eternally immoveable1 but rather eternally moved4 ,for as it is true1 that nothing is moved by itself2 so it is true also that nothing is moved but by that which is already moved4 ,the questions therefore about the magnitude and beginning of the world1 are not to be determined by philosophers1 but by those that are lawfully authorized to order the worship of ,god4 ,for as ,almighty ,god1 when he had brought his people into ,juda1 allowed the priests the first fruits reserved to himself2 so when he had delivered up the world to the disputations of men1 it was his pleasure that all opinions concerning the nature of infinite and eternal1 known only to himself1 should1 as the first fruits of wisdom1 be judged by those whose ministry he meant to use in the ordering of religion4 ,i cannot therefore commend those that boast they have demonstrated1 by reasons drawn from natural things1 that the world had a beginning4 ,they are contemned by idiots1 because they understand them not2 and by the learned1 because they understand them2 by both deservedly4 ,for who can commend him that demonstrates thus8 ,if the world be eternal1 then an infinite number of days1 or other measures of time1 preceded the birth of ,abraham4 ,but the birth of ,abraham preceded the birth of ,isaac2 and therefore one infinite is greater than another infinite1 or one eternal than another eternal2 which1 he says1 is absurd4 ,this demonstration is like his1 who from this1 that the number of even numbers is infinite1 would conclude that there are as many even numbers as there are numbers simply1 that is to say1 the even numbers are as many as all the even and odd together4 ,they1 which in this manner take away eternity from the world1 do they not by the same means take away eternity from the ,creator of the world8 ,from this absurdity therefore they run into another1 being forced to call eternity nunc stans1 a standing still of the present time1 or an abiding now2 and1 which is much more absurd1 to give to the infinite number of numbers the name of unity4 ,but why should eternity be called an abiding now1 rather than an abiding then8 ,wherefore there must either be many eternities1 or now and then must signify the same4 ,with such demonstrators as these1 that speak in another language1 it is impossible to enter into disputation4 ,and the men1 that reason thus absurdly1 are not idiots1 but1 which makes the absurdity unpardonable1 geometricians1 and such as take upon them to be judges1 impertinent1 but severe judges of other men's demonstrations4 ,the reason is this1 that as soon as they are entangled in the words infinite and eternal1 of which we have in our mind no idea1 but that of our own insufficiency to comprehend them1 they are forced either to speak something absurd1 or1 which they love worse1 to hold their peace4 ,for geometry hath in it somewhat like wine1 which1 when new1 is windy2 but afterwards though less pleasant1 yet more wholesome4 ,whatsoever therefore is true1 young geometricians think demonstrable2 but elder not4 ,wherefore ,i purposely pass over the questions of infinite and eternal2 contenting myself with that doctrine concerning the beginning and magnitude of the world1 which ,i have been persuaded to by the holy ,scriptures and fame of the miracles which confirm them2 and by the custom of my country1 and reverence due to the laws4 ,and so ,i pass on to such things as it is not unlawful to dispute of4 ,sidenote3 ,no place in the world empty4 #b4 ,concerning the world it is further questioned1 whether the parts thereof be contiguous to one another1 in such manner as not to admit of the least empty space between2 and the disputation both for and against it is carried on with probability enough4 ,for the taking away of vacuum1 ,i will instance in only one experiment1 a common one1 but ,i think unanswerable4 ,let ,a ,b 7in fig4 #a7 represent a vessel1 such as gardeners use to water their gardens withal2 whose bottom ,b is full of little holes2 and whose mouth ,a may be stopped with one's finger1 when there shall be need4 ,if now this vessel be filled with water1 the hole at the top ,a being stopped1 the water will not flow out at any of the holes in the bottom ,b4 ,but if the finger be removed to let in the air above1 it will run out at them all2 and as2 soon as the finger is applied to it again1 the water will suddenly and totally be stayed again from running out4 ,the cause whereof seems to be no other but this1 that the water cannot by its natural endeavour to descend drive down the air below it1 because there is no place for it to go into1 unless either by thrusting away the next contiguous air1 it proceed by continual endeavour to the hole ,a1 where it may enter and succeed into the place of the water that floweth out1 or else1 by resisting the endeavour of the water downwards1 penetrate the same and pass up through it4 ,by the first of these ways1 while the hole at ,a remains stopped1 there is no possible passage2 nor by the second1 unless the holes be so great that the water1 flowing out at them1 can by its own weight force the air at the same time to ascend into the vessel by the same holes3 as we see it does in a vessel whose mouth is wide enough1 when we turn suddenly the bottom upwards to pour out the water2 for then the air being forced by the weight of the water1 enters1 as is evident by the sobbing and resistance of the water1 at the sides or circumference of the orifice4 ,and this ,i take for a sign that all space is full2 for without this1 the natural motion of the water1 which is a heavy body1 downwards1 would not be hindered4 ,sidenote3 ,the arguments of ,lucretius for vacuum invalid4 #c4 ,on the contrary1 for the establishing of vacuum1 many and specious arguments and experiments have been brought4 ,nevertheless there seems to be something wanting in all of them to conclude it firmly4 ,these arguments for vacuum are partly made by the followers of the doctrine of ,epicurus2 who taught that the world consists of very small spaces not filled by any body1 and of very small bodies that have within them no empty space1 which by reason of their hardness he calls atoms2 and that these small bodies and spaces are every where intermingled4 ,their arguments are thus delivered by ,lucretius4 ,and first he says1 that unless it were so1 there could be no motion4 ,for the office and property of bodies is to withstand and hinder motion4 ,if1 therefore1 the universe were filled with body1 motion would everywhere be hindered1 so as to have no beginning anywhere2 and consequently there would be no motion at all4 ,it is true that in whatsoever is full and at rest in all its parts1 it is not possible motion should have beginning4 ,but nothing is drawn from hence for the proving of vacuum4 ,for though it should be granted that there is vacuum1 yet if the bodies which are intermingled with it1 should all at once and together be at rest1 they would never be moved again4 ,for it has been demonstrated above1 in chap4 ,,ix1 art4 #g1 that nothing can be moved but by that which is contiguous and already moved4 ,but supposing that all things are at rest together1 there can be nothing contiguous and moved1 and therefore no beginning of motion4 ,now the denying of the beginning of motion1 doth not take away present motion1 unless beginning be taken away from body also4 ,for motion may be either co-eternal1 or concreated with body4 ,nor doth it seem more necessary that bodies were first at rest1 and afterwards moved1 than that they were first moved1 and rested1 if ever they rested at all1 afterwards4 ,neither doth there appear any cause1 why the matter of the world should1 for the admission of motion1 be intermingled with empty spaces rather than full2 ,i say full1 but withal fluid4 ,nor1 lastly1 is there any reason why those hard atoms may not also1 by the motion of intermingled fluid matter1 be congregated and brought together into compounded bodies of such bigness as we see4 ,wherefore nothing can by this argument be concluded1 but that motion was either coeternal1 or of the same duration with that which is moved2 neither of which conclusions consisteth with the doctrine of ,epicurus1 who allows neither to the world nor to motion any beginning at all4 ,the necessity1 therefore1 of vacuum is not hitherto demonstrated4 ,and the cause1 as far as ,i understand from them that have discoursed with me of vacuum1 is this1 that whilst they contemplate the nature of fluid1 they conceive it to consist1 as it were1 of small grains of hard matter1 in such manner as meal is fluid1 made so by grinding of the corn2 when nevertheless it is possible to conceive fluid to be of its own nature as homogeneous as either an atom1 or as vacuum itself4 ,the second of their arguments is taken from weight1 and is contained in these verses of ,lucretius3 ,corporis officium est quoniam premere omnia deorsum2 ,contra autem natura manet sine pondere inanis2 ,ergo1 quod magnum est que1 leviusque videtur1 ,nimirum plus esse sibi declarat inanis4--,i4 #cfc-ff4 ,that is to say1 seeing the office and property of body is to press all things downwards2 and on the contrary1 seeing the nature of vacuum is to have no weight at all2 therefore when of two bodies of equal magnitude1 one is lighter than the other1 it is manifest that the lighter body hath in it more vacuum than the other4 ,to say nothing of the assumption concerning the endeavour of bodies downwards1 which is not rightly assumed1 because the world hath nothing to do with downwards1 which is a mere fiction of ours2 nor of this1 that if all things tended to the same lowest part of the world1 either there would be no coalescence at all of bodies1 or they would all be gathered together into the same place3 this only is sufficient to take away the force of the argument1 that air1 intermingled with those his atoms1 had served as well for his purpose as his intermingled vacuum4 ,the third argument is drawn from this1 that lightning1 sound1 heat and cold1 do penetrate all bodies1 except atoms1 how solid soever they be4 ,but this reason1 except it be first demonstrated that the same things cannot happen without vacuum by perpetual generation of motion1 is altogether invalid4 ,but that all the same things may so happen1 shall in due place be demonstrated4 ,lastly1 the fourth argument is set down by the same ,lucretius in these verses3 ,duo de concursu corpora lata ,si cita dissiliant1 nempe aer omne necesse est1 ,inter corpora quod fuerat1 possidat inane4 ,is porro quamvis circum celerantibus auris ,confluat1 haud poterit tamen uno tempore totum ,compleri spatium2 nam primum quemque necesse est ,occupet ille locum1 deinde omnia possideantur4--,i4 #che-ia4 ,that is1 if two flat bodies be suddenly pulled asunder1 of necessity the air must come between them to fill up the space they left empty4 ,but with what celerity soever the air flow in1 yet it cannot in one instant of time fill the whole space1 but first one part of it1 then successively all4 ,which nevertheless is more repugnant to the opinion of ,epicurus1 than of those that deny vacuum4 ,for though it be true1 that if two bodies were of infinite hardness1 and were joined together by their superficies which were most exactly plane1 it would be impossible to pull them asunder1 in regard it could not be done but by motion in an instant2 yet1 if as the greatest of all magnitudes cannot be given1 nor the swiftest of all motions1 so neither the hardest of all bodies2 it might be1 that by the application of very great force1 there might be place made for a successive flowing in of the air1 namely1 by separating the parts of the joined bodies by succession1 beginning at the outermost and ending at the innermost part4 ,he ought1 therefore1 first to have proved1 that there are some bodies extremely hard1 not relatively as compared with softer bodies1 but absolutely1 that is to say1 infinitely hard2 which is not true4 ,but if we suppose1 as ,epicurus doth1 that atoms are indivisible1 and yet have small superficies of their own2 then if two bodies should be joined together by many1 or but one only small superficies of either of them1 then ,i say this argument of ,lucretius would be a firm demonstration1 that no two bodies made up of atoms1 as he supposes1 could ever possibly be pulled asunder by any force whatsoever4 ,but this is repugnant to daily experience4 ,sidenote3 ,other arguments for the establishing of vacuum1 invalid4 #d4 ,and thus much of the arguments of ,lucretius4 ,let us now consider the arguments which are drawn from the experiments of later writers4 ,i4 ,the first experiment is this3 that if a hollow vessel be thrust into water with the bottom upwards1 the water will ascend into it2 which they say it could not do1 unless the air within were thrust together into a narrower place2 and that this were also impossible1 except there were little empty places in the air4 ,also1 that when the air is compressed to a certain degree1 it can receive no further compression1 its small particles not suffering themselves to be pent into less room4 ,this reason1 if the air could not pass through the water as it ascends within the vessel1 might seem valid4 ,but it is sufficiently known1 that air will penetrate water by the application of a force equal to the gravity of the water4 ,if therefore the force1 by which the vessel is thrust down1 be greater or equal to the endeavour by which the water naturally tendeth downwards1 the air will go out that way where the resistance is made1 namely1 towards the edges of the vessel4 ,for1 by how much the deeper is the water which is to be penetrated1 so much greater must be the depressing force4 ,but after the vessel is quite under water1 the force by which it is depressed1 that is to say1 the force by which the water riseth up1 is no longer increased4 ,there is therefore such an equilibration between them1 as that the natural endeavour of the water downwards is equal to the endeavour by which the same water is to be penetrated to the increased depth4 ,,ii4 ,the second experiment is1 that if a concave cylinder of sufficient length1 made of glass1 that the experiment may be the better seen1 having one end open and the other close shut1 be filled with quicksilver1 and the open end being stopped with one's finger1 be together with the finger dipped into a dish or other vessel1 in which also there is quicksilver1 and the cylinder be set upright1 we shall1 the finger being taken away to make way for the descent of the quicksilver1 see it descend into the vessel under it1 till there be only so much remaining within the cylinder as may fill about twenty-six inches of the same2 and thus it will always happen whatsoever be the cylinder1 provided that the length be not less than twenty-six inches4 ,from whence they conclude that the cavity of the cylinder above the quicksilver remains empty of all body4 ,but in this experiment ,i find no necessity at all of vacuum4 ,for when the quicksilver which is in the cylinder descends1 the vessel under it must needs be filled to a greater height1 and consequently so much of the contiguous air must be thrust away as may make place for the quicksilver which is descended4 ,now if it be asked whither that air goes1 what can be answered but this1 that it thrusteth away the next air1 and that the next1 and so successively1 till there be a return to the place where the propulsion first began4 ,and there1 the last air thus thrust on will press the quicksilver in the vessel with the same force with which the first air was thrust away2 and if the force with which the quicksilver descends be great enough1 which is greater or less as it descends from a place of greater or less height1 it will make the air penetrate the quicksilver in the vessel1 and go up into the cylinder to fill the place which they thought was left empty4 ,but because the quicksilver hath not in every degree of height force enough to cause such penetration1 therefore in descending it must of necessity stay somewhere1 namely1 there1 where its endeavour downwards1 and the resistance of the same to the penetration of the air1 come to an equilibrium4 ,and by this experiment it is manifest1 that this equilibrium will be at the height of twenty-six inches1 or thereabouts4 ,,iii4 ,the third experiment is1 that when a vessel hath as much air in it as it can naturally contain1 there may nevertheless be forced into it as much water as will fill three quarters of the same vessel4 ,and the experiment is made in this manner4 ,into the glass bottle1 represented 7in figure #b7 by the sphere ,f ,g1 whose centre is ,a1 let the pipe ,b ,a ,c be so fitted1 that it may precisely fill the mouth of the bottle2 and let the end ,b be so near the bottom1 that there may be only space enough left for the free passage of the water which is thrust in above4 ,let the upper end of this pipe have a cover at ,d1 with a spout at ,e1 by which the water1 when it ascends in the pipe1 may run out4 ,also let ,h ,c be a cock1 for the opening or shutting of the passage of the water between ,b and ,d1 as there shall be occasion4 ,let the cover ,d ,e be taken off1 and the cock ,h ,c being opened1 let a syringe full of water be forced in2 and before the syringe be taken away1 let the cock be turned to hinder the going out of the air4 ,and in this manner let the injection of water be repeated as often as it shall be requisite1 till the water rise within the bottle2 for example1 to ,g ,f4 ,lastly1 the cover being fastened on again1 and the cock ,h ,c opened1 the water will run swiftly out at ,e1 and sink by little and little from ,g ,f to the bottom of the pipe ,b4 ,from this phenomenon1 they argue for the necessity of vacuum in this manner4 ,the bottle1 from the beginning1 was full of air2 which air could neither go out by penetrating so great a length of water as was injected by the pipe1 nor by any other way4 ,of necessity1 therefore1 all the water as high as ,f ,g1 as also all the air that was in the bottle before the water was forced in1 must now be in the same place1 which at first was filled by the air alone2 which were impossible1 if all the space within the bottle were formerly filled with air precisely1 that is1 without any vacuum4 ,besides1 though some man perhaps may think the air1 being a thin body1 may pass through the body of the water contained in the pipe1 yet from that other phenomenon1 namely1 that all the water which is in the space ,b ,f ,g is cast out again by the spout at ,e1 for which it seems impossible that any other reason can be given besides the force by which the air frees itself from compression1 it follows1 that either there was in the bottle some space empty1 or that many bodies may be together in the same place4 ,but this last is absurd2 and therefore the former is true1 namely1 that there was vacuum4 ,this argument is infirm in two places4 ,for first1 that is assumed which is not to be granted2 and in the second place1 an experiment is brought1 which ,i think is repugnant to vacuum4 ,that which is assumed is1 that the air can have no passage out through the pipe4 ,nevertheless1 we see daily that air easily ascends from the bottom to the superficies of a river1 as is manifest by the bubbles that rise2 nor doth it need any other cause to give it this motion1 than the natural endeavour downwards of the water4 ,why1 therefore1 may not the endeavour upwards of the same water1 acquired by the injection1 which endeavour upwards is greater than the natural endeavour of the water downwards1 cause the air in the bottle to penetrate in like manner the water that presseth it downwards2 especially1 seeing the water1 as it riseth in the bottle1 doth so press the air that is above it1 as that it generateth in every part thereof an endeavour towards the external superficies of the pipe1 and consequently maketh all the parts of the enclosed air to tend directly towards the passage at ,b8 ,i say1 this is no less manifest1 than that the air which riseth up from the bottom of a river should penetrate the water1 how deep soever it be4 ,wherefore ,i do not yet see any cause why the force1 by which the water is injected1 should not at the same time eject the air4 ,and as for their arguing the necessity of vacuum from the rejection of the water2 in the first place1 supposing there is vacuum1 ,i demand by what principle of motion that ejection is made4 ,certainly1 seeing this motion is from within outwards1 it must needs be caused by some agent within the bottle2 that is to say1 by the air itself4 ,now the motion of that air1 being caused by the rising of the water1 begins at the bottom1 and tends upwards2 whereas the motion by which it ejecteth the water ought to begin above1 and tend downwards4 ,from whence therefore hath the enclosed air this endeavour towards the bottom8 ,to this question ,i know not what answer can be given1 unless it be said1 that the air descends of its own accord to expel the water4 ,which1 because it is absurd1 and that the air1 after the water is forced in1 hath as much room as its magnitude requires1 there will remain no cause at all why the water should be forced out4 ,wherefore the assertion of vacuum is repugnant to the very experiment which is here brought to establish it4 ,many other phenomena are usually brought for vacuum1 as those of weather-glasses1 olipyles1 wind-guns1 c4 which would all be very hard to be salved1 unless water be penetrable by air1 without the intermixture of empty space4 ,but now1 seeing air may with no great endeavour pass through not only water1 but any other fluid body though never so stubborn1 as quicksilver1 these phenomena prove nothing4 ,nevertheless1 it might in reason be expected1 that he that would take away vacuum1 should without vacuum show us such causes of these phenomena1 as should be at least of equal1 if not greater probability4 ,this therefore shall be done in the following discourse1 when ,i come to speak of these phenomena in their proper places4 ,but first1 the most general hypotheses of natural philosophy are to be premised4 ,and seeing that suppositions are put for the true causes of apparent effects1 every supposition1 except such as be absurd1 must of necessity consist of some supposed possible motion2 for rest can never be the efficient cause of anything2 and motion supposeth bodies moveable2 of which there are three kinds1 fluid1 consistent1 and mixed of both4 ,fluid are those1 whose parts may by very weak endeavour be separated from one another2 and consistent those for the separation of whose parts greater force is to be applied4 ,there are therefore degrees of consistency2 which degrees1 by comparison with more or less consistent1 have the names of hardness or softness4 ,wherefore a fluid body is always divisible into bodies equally fluid1 as quantity into quantities2 and soft bodies1 of whatsoever degree of softness1 into soft bodies of the same degree4 ,and though many men seem to conceive no other difference of fluidity1 but such as ariseth from the different magnitudes of the parts1 in which sense dust1 though of diamonds1 may be called fluid2 yet ,i understand by fluidity1 that which is made such by nature equally in every part of the fluid body2 not as dust is fluid1 for so a house which is falling in pieces may be called fluid2 but in such manner as water seems fluid1 and to divide itself into parts perpetually fluid4 ,and this being well understood1 ,i come to my suppositions4 ,sidenote3 ,six suppositions for the salving of the phenomena of nature4 #e4 ,first1 therefore1 ,i suppose that the immense space1 which we call the world1 is the aggregate of all bodies which are either consistent and visible1 as the earth and the stars2 or invisible1 as the small atoms which are disseminated through the whole space between the earth and the stars2 and lastly1 that most fluid ether1 which so fills all the rest of the universe1 as that it leaves in it no empty place at all4 ,secondly1 ,i suppose with ,copernicus1 that the greater bodies of the world1 which are both consistent and permanent1 have such order amongst themselves1 as that the sun hath the first place1 ,mercury the second1 ,venus the third1 the ,earth with the moon going about it the fourth1 ,mars the fifth1 ,jupiter with his attendants the sixth1 ,saturn the seventh2 and after these1 the fixed stars have their several distances from the sun4 ,thirdly1 ,i suppose that in the sun and the rest of the planets there is and always has been a simple circular motion4 ,fourthly1 ,i suppose that in the body of the air there are certain other bodies intermingled1 which are not fluid2 but withal that they are so small1 that they are not perceptible by sense2 and that these also have their proper simple motion1 and are some of them more1 some less hard or consistent4 ,fifthly1 ,i suppose with ,kepler that as the distance between the sun and the earth is to the distance between the moon and the earth1 so the distance between the moon and the earth is to the semidiameter of the earth4 ,as for the magnitude of the circles1 and the times in which they are described by the bodies which are in them1 ,i will suppose them to be such as shall seem most agreeable to the phenomena in question4 ,sidenote3 ,possible causes of the motions annual and diurnal2 and of the apparent direction1 station1 and retrogradation of the planets4 #f4 ,the causes of the different seasons of the year1 and of the several variations of days and nights in all the parts of the superficies of the earth1 have been demonstrated1 first by ,copernicus1 and since by ,kepler1 ,galileus1 and others1 from the supposition of the earth's diurnal revolution about its own axis1 together with its annual motion about the sun in the ecliptic according to the order of the signs2 and thirdly1 by the annual revolution of the same earth about its own centre1 contrary to the order of the signs4 ,i suppose with ,copernicus1 that the diurnal revolution is from the motion of the earth1 by which the equinoctial circle is described about it4 ,and as for the other two annual motions1 they are the efficient cause of the earth's being carried about in the ecliptic in such manner1 as that its axis is always kept parallel to itself4 ,which parallelism was for this reason introduced1 lest by the earth's annual revolution its poles should seem to be necessarily carried about the sun1 contrary to experience4 ,i have1 in art4 #aj1 chap4 ,,xxi1 demonstrated1 from the supposition of simple circular motion in the sun1 that the earth is so carried about the sun1 as that its axis is thereby kept always parallel to itself4 ,wherefore1 from these two supposed motions in the sun1 the one simple circular motion1 the other circular motion about its own centre1 it may be demonstrated that the year hath both the same variations of days and nights1 as have been demonstrated by ,copernicus4 ,for if the circle a b c d 7in fig4 #c7 be the ecliptic1 whose centre is e1 and diameter a e c2 and the earth be placed in a1 and the sun be moved in the little circle f g h i1 namely1 according to the order f1 g1 h1 and i1 it hath been demonstrated1 that a body placed in a will be moved in the same order through the points of the ecliptic a1 b1 c1 and d1 and will always keep its axis parallel to itself4 ,but if1 as ,i have supposed1 the earth also be moved with simple circular motion in a plane that passeth through a1 cutting the plane of the ecliptic so as that the common section of both the planes be in a c1 thus also the axis of the earth will be kept always parallel to itself4 ,for let the centre of the earth be moved about in the circumference of the epicycle1 whose diameter is l a k1 which is a part of the strait line l a c2 therefore l a k1 the diameter of the epicycle1 passing through the centre of the earth1 will be in the plane of the ecliptic4 ,wherefore seeing that by reason of the earth's simple motion both in the ecliptic and in its epicycle1 the strait line l a k is kept always parallel to itself1 every other strait line also taken in the body of the earth1 and consequently its axis1 will in like manner be kept always parallel to itself2 so that in what part soever of the ecliptic the centre of the epicycle be found1 and in what part soever of the epicycle the centre of the earth be found at the same time1 the axis of the earth will be parallel to the place where the same axis would have been1 if the centre of the earth had never gone out of the ecliptic4 ,now as ,i have demonstrated the simple annual motion of the earth from the supposition of simple motion in the sun2 so from the supposition of simple motion in the earth may be demonstrated the monthly simple motion of the moon4 ,for if the names be but changed1 the demonstration will be the same1 and therefore need not be repeated4 ,sidenote3 ,the supposition of simple motion1 why likely4 #g4 ,that which makes this supposition of the sun's simple motion in the epicycle f g h i probable1 is first1 that the periods of all the planets are not only described about the sun1 but so described1 as that they are all contained within the zodiac1 that is to say1 within the latitude of about sixteen degrees2 for the cause of this seems to depend upon some power in the sun1 especially in that part of the sun which respects the zodiac4 ,secondly1 that in the whole compass of the heavens there appears no other body from which the cause of this phenomenon can in probability be derived4 ,besides1 ,i could not imagine that so many and such various motions of the planets should have no dependance at all upon one another4 ,but1 by supposing motive power in the sun1 we suppose motion also2 for power to move without motion is no power at all4 ,i have therefore supposed that there is in the sun for the governing of the primary planets1 and in the earth for the governing of the moon1 such motion1 as being received by the primary planets and by the moon1 makes them necessarily appear to us in such manner as we see them4 ,whereas1 that circular motion1 which is commonly attributed to them1 about a fixed axis1 which is called conversion1 being a motion of their parts only1 and not of their whole bodies1 is insufficient to salve their appearances4 ,for seeing whatsoever is so moved1 hath no endeavour at all towards those parts which are without the circle1 they have no power to propagate any endeavour to such bodies as are placed without it4 ,and as for them that suppose this may be done by magnetical virtue1 or by incorporeal and immaterial species1 they suppose no natural cause2 nay1 no cause at all4 ,for there is no such thing as an incorporeal movent1 and magnetical virtue is a thing altogether unknown2 and whensoever it shall be known1 it will be found to be a motion of body4 ,it remains1 therefore1 that if the primary planets be carried about by the sun1 and the moon by the earth1 they have the simple circular motions of the sun and the earth for the causes of their circulations4 ,otherwise1 if they be not carried about by the sun and the earth1 but that every planet hath been moved1 as it is now moved1 ever since it was made1 there will be of their motions no cause natural4 ,for either these motions were concreated with their bodies1 and their cause is supernatural2 or they are coeternal with them1 and so they have no cause at all4 ,for whatsoever is eternal was never generated4 ,i may add besides1 to confirm the probability of this simple motion1 that as almost all learned men are now of the same opinion with ,copernicus concerning the parallelism of the axis of the earth1 it seemed to me to be more agreeable to truth1 or at least more handsome1 that it should be caused by simple circular motion alone1 than by two motions1 one in the ecliptic1 and the other about the earth's own axis the contrary way1 neither of them simple1 nor either of them such as might be produced by any motion of the sun4 ,i thought best therefore to retain this hypothesis of simple motion1 and from it to derive the causes of as many of the phenomena as ,i could1 and to let such alone as ,i could not deduce from thence4 ,it will perhaps be objected1 that although by this supposition the reason may be given of the parallelism of the axis of the earth1 and of many other appearances1 nevertheless1 seeing it is done by placing the body of the sun in the centre of that orb which the earth describes with its annual motion1 the supposition itself is false2 because this annual orb is eccentric to the sun4 ,in the first place1 therefore1 let us examine what that eccentricity is1 and whence it proceeds4 ,sidenote3 ,the cause of the eccentricity of the annual motion of the earth4 #h4 ,let the annual circle of the earth a b c d 7in fig4 #c7 be divided into four equal parts by the strait lines a c and b d1 cutting one another in the centre e2 and let a be the beginning of ,libra1 b of ,capricorn1 c of ,aries and d of ,cancer2 and let the whole orb a b c d be understood1 according to ,copernicus1 to have every way so great distance from the zodiac of the fixed stars1 that it be in comparison with it but as a point4 ,let the earth be now supposed to be in the beginning of ,libra at a4 ,the sun1 therefore1 will appear in the beginning of ,aries at c4 ,wherefore1 if the earth be moved from a to b1 the apparent motion of the sun will be from c to the beginning of ,cancer in d2 and the earth being moved forwards from b to c1 the sun also will appear to be moved forwards to the beginning of ,libra in a2 wherefore c d a will be the summer arch1 and the winter arch will be a b c4 ,now1 in the time1 of the sun's apparent motion in the summer arch1 there are numbered #ahf days2 and1 consequently1 the earth makes in the same time the same number of diurnal conversions in the arch a b c2 and1 therefore1 the earth in its motion through the arch c d a will make only #agh diurnal conversions4 ,wherefore the arch a b c ought to be greater than the arch c d a by #h days1 that is to say1 by almost so many degrees4 ,let the arch a r1 as also c s1 be each of them an arch of two degrees and #aaf4 ,wherefore the arch r b s will be greater than the semicircle a b c by #d degrees1 and greater than the arch s d r by #h degrees4 ,the equinoxes1 therefore1 will be in the points r and s2 and therefore also1 when the earth is in r1 the sun will appear in s4 ,wherefore the true place of the sun will be in t1 that is to say1 without the centre of the earth's annual motion by the quantity of the sine of the arch a r1 or the sine of two degrees and #af minutes4 ,now this sine1 putting #ajj1jjj for the radius1 will be near #cehj parts thereof4 ,and so much is the eccentricity of the earth's annual motion1 provided that that motion be in a perfect circle2 and s and r are the equinoctial parts4 ,and the strait lines s r and c a1 produced both ways till they reach the zodiac of the fixed stars1 will fall still upon the same fixed stars2 because the whole orb a b c d is supposed to have no magnitude at all in respect of the great distance of the fixed stars4 ,supposing now the sun to be in c1 it remains that ,i show the cause why the earth is nearer to the sun1 when in its annual motion it is found to be in d1 than when it is in b4 ,and ,i take the cause to be this4 ,when the earth is in the beginning of ,capricorn at b1 the sun appears in the beginning of ,cancer at d2 and then is the midst of summer4 ,but in the midst of summer1 the northern parts of the earth are towards the sun1 which is almost all dry land1 containing all ,europe and much the greatest part of ,asia and ,america4 ,but when the earth is in the beginning of ,cancer at d1 it is the midst of winter1 and that part of the earth is towards the sun1 which contains those great seas called the ,south ,sea and the ,indian ,sea1 which are of far greater extent than all the dry land in that hemisphere4 ,wherefore by the last article of chapter ,,xxi1 when the earth is in d1 it will come nearer to its first movent1 that is1 to the sun which is in t2 that is to say1 the earth is nearer to the sun in the midst of winter when it is in d1 than in the midst of summer when it is in b2 and1 therefore1 during the winter the sun is in its ,perigum1 and in its ,apogum during the summer4 ,and thus ,i have shown a possible cause of the eccentricity of the earth2 which was to be done4 ,i am1 therefore1 of ,kepler's opinion in this1 that he attributes the eccentricity of the earth to the difference of the parts thereof1 and supposes one part to be affected1 and another disaffected to the sun4 ,and ,i dissent from him in this1 that he thinks it to be by magnetic virtue1 and that this magnetic virtue or attraction and thrusting back of the earth is wrought by immateriate species2 which cannot be1 because nothing can give motion but a body moved and contiguous4 ,for if those bodies be not moved which are contiguous to a body unmoved1 how this body should begin to be moved is not imaginable2 as has been demonstrated in art4 #g1 chap4 ,,ix1 and often inculcated in other places1 to the end that philosophers might at last abstain from the use of such unconceivable connexions of words4 ,i dissent also from him in this1 that he says the similitude of bodies is the cause of their mutual attraction4 ,for if it were so1 ,i see no reason why one egg should not be attracted by another4 ,if1 therefore1 one part of the earth be more affected by the sun than another part1 it proceeds from this1 that one part hath more water1 the other more dry land4 ,and from hence it is1 as ,i showed above1 that the earth comes nearer to the sun when it shines upon that part where there is more water1 than when it shines upon that where there is more dry land4 ,sidenote3 ,the cause why the moon hath always one and the same face turned towards the earth4 #i4 ,this eccentricity of the earth is the cause why the way of its annual motion is not a perfect circle1 but either an elliptical1 or almost an elliptical line2 as also why the axis of the earth is not kept exactly parallel to itself in all places1 but only in the equinoctial points4 ,now seeing ,i have said that the moon is carried about by the earth1 in the same manner that the earth is by the sun2 and that the earth goeth about the sun in such manner as that it shows sometimes one hemisphere1 sometimes the other to the sun2 it remains to be enquired1 why the moon has always one and the same face turned towards the earth4 ,suppose1 therefore1 the sun to be moved with simple motion in the little circle f g h i1 7in fig4 #d7 whose centre is t2 and let be the annual circle of the earth2 and a the beginning of ,libra4 ,about the point a let the little circle l k be described2 and in it let the centre of the earth be understood to be moved with simple motion2 and both the sun and the earth to be moved according to the order of the signs4 ,upon the centre a let the way of the moon m n o p be described2 and let q r be the diameter of a circle cutting the globe of the moon into two hemispheres1 whereof one is seen by us when the moon is at the full1 and the other is turned from us4 ,the diameter therefore of the moon q o r will be perpendicular to the strait line t a4 ,wherefore the moon is carried1 by reason of the motion of the earth1 from o towards p4 ,but by reason of the motion of the sun1 if it were in p it would at the same time be carried from p towards o2 and by these two contrary movents the strait line q r will be turned about2 and1 in a quadrant of the circle m n o p1 it will be turned so much as makes the fourth part of its whole conversion4 ,wherefore when the moon is in p1 q r will be parallel to the strait line m o4 ,secondly1 when the moon is in m1 the strait line q r will1 by reason of the motion of the earth1 be in m o4 ,but by the working of the sun's motion upon it in the quadrant p m1 the same q r will be turned so much as makes another quarter of its whole conversion4 ,when1 therefore1 the moon is in m1 q r will be perpendicular to the strait line o m4 ,by the same reason1 when the moon is in n1 q r will be parallel to the strait line m o2 and1 the moon returning to o1 the same q r will return to its first place2 and the body of the moon will in one entire period make also one entire conversion upon her own axis4 ,in the making of which1 it is manifest1 that one and the same face of the moon is always turned towards the earth4 ,and if any diameter were taken in that little circle1 in which the moon were supposed to be carried about with simple motion1 the same effect would follow2 for if there were no action from the sun1 every diameter of the moon would be carried about always parallel to itself4 ,wherefore ,i have given a possible cause why one and the same face of the moon is always turned towards the earth4 ,but it is to be noted1 that when the moon is without the ecliptic1 we do not always see the same face precisely4 ,for we see only that part which is illuminated4 ,but when the moon is without the ecliptic1 that part which is towards us is not exactly the same with that which is illuminated4 ,sidenote3 ,the cause of the tides of the ocean4 #aj4 ,to these three simple motions1 one of the sun1 another of the moon1 and the third of the earth1 in their own little circles f g h i1 l k1 and q r1 together with the diurnal conversion of the earth1 by which conversion all things that adhere to its superficies are necessarily carried about with it1 may be referred the three phenomena concerning the tides of the ocean4 ,whereof the first is the alternate elevation and depression of the water at the shores1 twice in the space of twenty-four hours and near upon fifty-two minutes2 for so it has constantly continued in all ages4 ,the second1 that at the new and full moons1 the elevations of the water are greater than at other times between4 ,and the third1 that when the sun is in the equinoctial1 they are yet greater than at any other time4 ,for the salving of which phenomena1 we have already the four above-mentioned motions2 to which ,i assume also this1 that the part of the earth which is called ,america1 being higher than the water1 and extended almost the space of a whole semicircle from north to south1 gives a stop to the motion of the water4 ,this being granted1 in the same #dth figure1 where l b k c is supposed to be in the plane of the moon's monthly motion1 let the little circle l d k e be described about the same centre a in the plane of the equinoctial4 ,this circle therefore will decline from the circle l b k c in an angle of almost #bh degrees2 for the greatest declination of the ecliptic is #bc1 to which adding #e for the greatest declination of the moon from the ecliptic1 the sum will be #bh degrees4 ,seeing now the waters1 which are under the circle of the moon's course1 are by reason of the earth's simple motion in the plane of the same circle moved together with the earth1 that is to say1 together with their own bottoms1 neither outgoing nor outgone2 if we add the diurnal motion1 by which the other waters which are under the equinoctial are moved in the same order1 and consider withal that the circles of the moon and of the equinoctial intersect one another2 it will be manifest1 that both those waters1 which are under the circle of the moon1 and under the equinoctial1 will run together under the equinoctial2 and consequently1 that their motion will not only be swifter than the ground that carries them2 but also that the waters themselves will have greater elevation whensoever the earth is in the equinoctial4 ,wherefore1 whatsoever the cause of the tides may be1 this may be the cause of their augmentation at that time4 ,again1 seeing ,i have supposed the moon to be carried about by the simple motion of the earth in the little circle l b k c2 and demonstrated1 at the #dth article of chapter ,,xxi1 that whatsoever is moved by a movent that hath simple motion1 will be moved always with the same velocity2 it follows1 that the centre of the earth will be carried in the circumference l b k c with the same velocity with which the moon is carried in the circumference m n o p4 ,wherefore the time1 in which the moon is carried about in m n o p1 is to the time1 in which the earth is carried about in l b k c1 as one circumference to the other1 that is1 as a o to a k4 ,but a o is observed to be to the semidiameter of the earth as #ei to #a2 and therefore the earth1 if a k be put for its semidiameter1 will make fifty-nine revolutions in l b k c in the time that the moon makes one monthly circuit in m n o p4 ,but the moon makes her monthly circuit in little more than twenty-nine days4 ,wherefore the earth shall make its circuit in the circumference l b k c in twelve hours and a little more1 namely1 about twenty-six minutes more2 that is to say1 it shall make two circuits in twenty-four hours and almost fifty-two minutes2 which is observed to be the time between the high-water of one day and the high-water of the day following4 ,now the course of the waters being hindered by the southern part of ,america1 their motion will be interrupted there2 and consequently1 they will be elevated in those places1 and sink down again by their own weight1 twice in the space of twenty-four hours and fifty-two minutes4 ,and thus ,i have given a possible cause of the diurnal reciprocation of the ocean4 ,now from this swelling of the ocean in those parts of the earth1 proceed the flowings and ebbings in the ,atlantic1 ,spanish1 ,british1 and ,german seas2 which though they have their set times1 yet upon several shores they happen at several hours of the day4 ,and they receive some augmentation from the north1 by reason that the shores of ,china and ,tartary1 hindering the general course of the waters1 make them swell there1 and discharge themselves in part through the strait of ,anian into the ,northern ,ocean1 and so into the ,german ,sea4 ,as for the spring tides which happen at the time of the new and full moons1 they are caused by that simple motion1 which at the beginning ,i supposed to be always in the moon4 ,for as1 when ,i showed the cause of the eccentricity of the earth1 ,i derived the elevation of the waters from the simple motion of the sun2 so the same may here be derived from the simple motion of the moon4 ,for though from the generation of clouds1 there appear in the sun a more manifest power of elevating the waters than in the moon2 yet the power of increasing moisture in vegetables and living creatures appears more manifestly in the moon than in the sun2 which may perhaps proceed from this1 that the sun raiseth up greater1 and the moon lesser drops of water4 ,nevertheless1 it is more likely1 and more agreeable to common observation1 that rain is raised not only by the sun1 but also by the moon2 for almost all men expect change of weather at the time of the conjunctions of the sun and moon with one another and with the earth1 more than in the time of their quarters4 ,in the last place1 the cause why the spring tides are greater at the time of the equinoxes hath been already sufficiently declared in this article1 where ,i have demonstrated1 that the two motions of the earth1 namely1 its simple motion in the little circle l b k c1 and its diurnal motion in l d k e1 cause necessarily a greater elevation of waters when the sun is about the equinoxes1 than when he is in other places4 ,i have therefore given possible causes of the phenomenon of the flowing and ebbing of the ocean4 ,sidenote3 ,cause of the precession of the equinoxes4 #aa4 ,as for the explication of the yearly precession of the equinoctial points1 we must remember that1 as ,i have already shown1 the annual motion of the earth is not in the circumference of a circle1 but of an ellipsis1 or a line not considerably different from that of an ellipsis4 ,in the first place1 therefore1 this elliptical line is to be described4 ,let the ecliptic 7in fig4 #e7 be divided into four equal parts by the two strait lines a b and 1 cutting one another at right angles in the centre c4 ,and taking the arch b d of two degrees and sixteen minutes1 let the strait line d e be drawn parallel to a b1 and cutting in f2 which being done1 the eccentricity of the earth will be c f4 ,seeing therefore the annual motion of the earth is in the circumference of an ellipsis1 of which is the greater axis1 a b cannot be the lesser axis2 for a b and are equal4 ,wherefore the earth passing through a and b1 will either pass above 1 as through g1 or passing through 1 will fall between c and a2 it is no matter which4 ,let it pass therefore through g2 and let g l be taken equal to the strait line 2 and dividing g l equally in i1 g i will be equal to f1 and i l equal to f 2 and consequently the point i will cut the eccentricity c f into two equal parts2 and taking i h equal to i f1 h i will be the whole eccentricity4 ,if now a strait line1 namely1 the line i 1 be drawn through i parallel to the strait lines a b and e d1 the way of the sun in summer1 namely1 the arch g 1 will be greater than his way in winter1 by #h degrees4 ,wherefore the true equinoxes will be in the strait line i 2 and therefore the ellipsis of the earth's annual motion will not pass through a1 g1 b1 and l2 but through 1 g1 and l4 ,wherefore the annual motion of the earth is in the ellipsis g l2 and cannot be1 the eccentricity being salved1 in any other line4 ,and this perhaps is the reason1 why ,kepler1 against the opinion of all the astronomers of former time1 thought fit to bisect the eccentricity of the earth1 or1 according to the ancients1 of the sun1 not by diminishing the quantity of the same eccentricity1 7because the true measure of that quantity is the difference by which the summer arch exceeds the winter arch71 but by taking for the centre of the ecliptic of the great orb the point c nearer to f1 and so placing the whole great orb as much nearer to the ecliptic of the fixed stars towards 1 as is the distance between c and i4 ,for seeing the whole great orb is but as a point in respect of the immense distance of the fixed stars1 the two strait lines and a b1 being produced both ways to the beginnings of ,aries and ,libra1 will fall upon the same points of the sphere of the fixed stars4 ,let therefore the diameter of the earth m n be in the plane of the earth's annual motion4 ,if now the earth be moved by the sun's simple motion in the circumference of the ecliptic about the centre i1 this diameter will be kept always parallel to itself and to the strait line g l4 ,but seeing the earth is moved in the circumference of an ellipsis without the ecliptic1 the point n1 whilst it passeth through 1 will go in a lesser circumference than the point m2 and consequently1 as soon as ever it begins to be moved1 it will lose its parallelism with the strait line 2 so that m n produced will at last cut the strait line g l produced4 ,and contrarily1 as soon as m n is past 1 the earth making its way in the internal elliptical line l 1 the same m n produced towards m1 will cut l g produced4 ,and when the earth hath almost finished its whole circumference1 the same m n shall again make a right angle with a line drawn from the centre i1 a little short of the point from which the earth began its motion4 ,and there the next year shall be one of the equinoctial points1 namely1 near the end of 2 the other shall be opposite to it near the end of 4 ,and thus the points in which the days and nights are made equal do every year fall back2 but with so slow a motion1 that1 in a whole year1 it makes but #ea first minutes4 ,and this relapse being contrary to the order of the signs1 is commonly called the precession of the equinoxes4 ,of which ,i have from my former suppositions deduced a possible cause2 which was to be done4 ,according to what ,i have said concerning the cause of the eccentricity of the earth2 and according to ,kepler1 who for the cause thereof supposeth one part of the earth to be affected to the sun1 the other part to be disaffected2 the apogum and perigum of the sun should be moved every year in the same order1 and with the same velocity1 with which the equinoctial points are moved2 and their distance from them should always be the quadrant of a circle2 which seems to be otherwise4 ,for astronomers say1 that the equinoxes are now1 the one about #bh degrees gone back from the first star of ,aries1 the other as much from the beginning of ,libra2 so that the apogum of the sun or the aphelium of the earth ought to be about the #bhth degree of ,cancer4 ,but it is reckoned to be in the #gth degree4 ,seeing1 therefore1 we have not sufficient evidence of the 7that so it is17 it is in vain to seek for the 7why it is so47 ,wherefore1 as long as the motion of the apogum is not observable by reason of the slowness thereof1 and as long as it remains doubtful whether their distance from the equinoctial points be more or less than a quadrant precisely2 so long it may be lawful for me to think they proceed both of them with equal velocity4 ,also1 ,i do not at all meddle with the causes of the eccentricities of ,saturn1 ,jupiter1 ,mars1 and ,mercury4 ,nevertheless1 seeing the eccentricity of the earth may1 as ,i have shewn1 be caused by the unlike constitution of the several parts of the earth which are alternately turned towards the sun1 it is credible also1 that like effects may be produced in these other planets from their having their superficies of unlike parts4 ,and this is all ,i shall say concerning ,sidereal ,philosophy4 ,and1 though the causes ,i have here supposed be not the true causes of these phenomena1 yet ,i have demonstrated that they are sufficient to produce them1 according to what ,i at first propounded4 ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxvi4 ,fig4 #a-e ------- ,,chapter ,,xxvii4 ,,of ,,light1 ,,heat1 ,,and ,,of ,,colours4 #a4 ,of the immense magnitude of some bodies1 and the unspeakable littleness of others4--#b4 ,of the cause of the light of the sun4--#c4 ,how light heateth4--#d4 ,the generation of fire from the sun4--#e4 ,the generation of fire from collision4--#f4 ,the cause of light in glow-worms1 rotten wood1 and the ,bolognan stone4--#g4 ,the cause of light in the concussion of sea water4--#h4 ,the cause of flame1 sparks1 and colliquation4--#i4 ,the cause why wet hay sometimes burns of its own accord2 also the cause of lightning4--#aj4 ,the cause of the force of gunpowder2 and what is to be ascribed to the coals1 what to the brimstone1 and what to the nitre4--#aa4 ,how heat is caused by attrition4--#ab4 ,the distinction of light into first1 second1 c4--#ac4 ,the causes of the colours we see in looking through a prisma of glass1 namely1 of red1 yellow1 blue1 and violet colour4--#ad4 ,why the moon and the stars appear redder in the horizon than in the midst of the heaven4--#ae4 ,the cause of whiteness4--#af4 ,the cause of blackness4 ,sidenote3 ,of the immense magnitude of some bodies1 and the unspeakable littleness of others4 #a4 ,besides the stars1 of which ,i have spoken in the last chapter1 whatsoever other bodies there be in the world1 they may be all comprehended under the name of intersidereal bodies4 ,and these ,i have already supposed to be either the most fluid ther1 or such bodies whose parts have some degree of cohesion4 ,now1 these differ from one another in their several consistencies1 magnitudes1 motions1 and figures4 ,in consistency1 ,i suppose some bodies to be harder1 others softer through all the several degrees of tenacity4 ,in magnitude1 some to be greater1 others less1 and many unspeakably little4 ,for we must remember that1 by the understanding1 quantity is divisible into divisibles perpetually4 ,and1 therefore1 if a man could do as much with his hands as he can with his understanding1 he would be able to take from any given magnitude a part which should be less than any other magnitude given4 ,but the ,omnipotent ,creator of the world can actually from a part of any thing take another part1 as far as we by our understanding can conceive the same to be divisible4 ,wherefore there is no impossible smallness of bodies4 ,and what hinders but that we may think this likely8 ,for we know there are some living creatures so small that we can scarce see their whole bodies4 ,yet even these have their young ones2 their little veins and other vessels1 and their eyes so small as that no microscope can make them visible4 ,so that we cannot suppose any magnitude so little1 but that our very supposition is actually exceeded by nature4 ,besides1 there are now such microscopes commonly made1 that the things we see with them appear a hundred thousand times bigger than they would do if we looked upon them with our bare eyes4 ,nor is there any doubt but that by augmenting the power of these microscopes 7for it may be augmented as long as neither matter nor the hands of workmen are wanting7 every one of those hundred thousandth parts might yet appear a hundred thousand times greater than they did before4 ,neither is the smallness of some bodies to be more admired than the vast greatness of others4 ,for it belongs to the same ,infinite ,power1 as well to augment infinitely as infinitely to diminish4 ,to make the great orb1 namely1 that whose radius reacheth from the earth to the sun1 but as a point in respect of the distance between the sun and the fixed stars2 and1 on the contrary1 to make a body so little1 as to be in the same proportion less than any other visible body1 proceeds equally from one and the same ,author of ,nature4 ,but this of the immense distance of the fixed stars1 which for a long time was accounted an incredible thing1 is now believed by almost all the learned4 ,why then should not that other1 of the smallness of some bodies1 become credible at some time or other8 ,for the ,majesty of ,god appears no less in small things than in great2 and as it exceedeth human sense in the immense greatness of the universe1 so also it doth in the smallness of the parts thereof4 ,nor are the first elements of compositions1 nor the first beginnings of actions1 nor the first moments of times more credible1 than that which is now believed of the vast distance of the fixed stars4 ,some things are acknowledged by mortal men to be very great1 though finite1 as seeing them to be such4 ,they acknowledge also that some things1 which they do not see1 may be of infinite magnitude4 ,but they are not presently nor without great study persuaded1 that there is any mean between infinite and the greatest of those things which either they see or imagine4 ,nevertheless1 when after meditation and contemplation many things which we wondered at before are now grown more familiar to us1 we then believe them1 and transfer our admiration from the creatures to the ,creator4 ,but how little soever some bodies may be1 yet ,i will not suppose their quantity to be less than is requisite for the salving of the phenomena4 ,and in like manner ,i shall suppose their motion1 namely1 their velocity and slowness1 and the variety of their figures1 to be only such as the explication of their natural causes requires4 ,and lastly1 ,i suppose1 that the parts of the pure ther1 as if it were the first matter1 have no motion at all but what they receive from bodies which float in them1 and are not themselves fluid4 ,sidenote3 ,of the cause of the light of the sun4 #b4 ,having laid these grounds1 let us come to speak of causes2 and in the first place let us inquire what may be the cause of the light of the sun4 ,seeing1 therefore1 the body of the sun doth by its simple circular motion thrust away the ambient ethereal substance sometimes one way sometimes another1 so that those parts1 which are next the sun1 being moved by it1 do propagate that motion to the next remote parts1 and these to the next1 and so on continually2 it must needs be that1 notwithstanding any distance1 the foremost part of the eye will at last be pressed2 and by the pressure of that part1 the motion will be propagated to the innermost part of the organ of sight1 namely1 to the heart2 and from the reaction of the heart1 there will proceed an endeavour back by the same way1 ending in the endeavour outwards of the coat of the eye1 called the retina4 ,but this endeavour outwards1 as has been defined in chapter ,,xxv1 is the thing which is called light1 or the phantasm of a lucid body4 ,for it is by reason of this phantasm that an object is called lucid4 ,wherefore we have a possible cause of the light of the sun2 which ,i undertook to find4 ,sidenote3 ,how light heateth4 #c4 ,the generation of the light of the sun is accompanied with the generation of heat4 ,now every man knows what heat is in himself1 by feeling it when he grows hot2 but what it is in other things1 he knows only by ratiocination4 ,for it is one thing to grow hot1 and another thing to heat or make hot4 ,and therefore though we perceive that the fire or the sun heateth1 yet we do not perceive that it is itself hot4 ,that other living creatures1 whilst they make other things hot1 are hot themselves1 we infer by reasoning from the like sense in ourselves4 ,but this is not a necessary inference4 ,for though it may truly be said of living creatures1 that they heat1 therefore they are themselves hot2 yet it cannot from hence be truly inferred that fire heateth1 therefore it is itself hot2 no more than this1 fire causeth pain1 therefore it is itself in pain4 ,wherefore1 that is only and properly called hot1 which when we feel we are necessarily hot4 ,now when we grow hot1 we find that our spirits and blood1 and whatsoever is fluid within us1 is called out from the internal to the external parts of our bodies1 more or less1 according to the degree of the heat2 and that our skin swelleth4 ,he1 therefore1 that can give a possible cause of this evocation and swelling1 and such as agrees with the rest of the phenomena of heat1 may be thought to have given the cause of the heat of the sun4 ,it hath been shown1 in the #eth article of chapter ,,xxi1 that the fluid medium1 which we call the air1 is so moved by the simple circular motion of the sun1 as that all its parts1 even the least1 do perpetually change places with one another2 which change of places is that which there ,i called fermentation4 ,from this fermentation of the air1 ,i have1 in the #hth article of the last chapter1 demonstrated that the water may be drawn up into the clouds4 ,and ,i shall now show that the fluid parts may1 in like manner1 by the same fermentation1 be drawn out from the internal to the external parts of our bodies4 ,for seeing that wheresoever the fluid medium is contiguous to the body of any living creature1 there the parts of that medium are1 by perpetual change of place1 separated from one another2 the contiguous parts of the living creature must1 of necessity1 endeavour to enter into the spaces of the separated parts4 ,for otherwise those parts1 supposing there is no vacuum1 would have no place to go into4 ,and therefore that1 which is most fluid and separable in the parts of the living creature which are contiguous to the medium1 will go first out2 and into the place thereof will succeed such other parts as can most easily transpire through the pores of the skin4 ,and from hence it is necessary that the rest of the parts1 which are not separated1 must altogether be moved outwards1 for the keeping of all places full4 ,but this motion outwards of all parts together must1 of necessity1 press those parts of the ambient air which are ready to leave their places2 and therefore all the parts of the body1 endeavouring at once that way1 make the body swell4 ,wherefore a possible cause is given of heat from the sun2 which was to be done4 ,sidenote3 ,the generation of fire from the sun4 #d4 ,we have now seen how light and heat are generated2 heat by the simple motion of the medium1 making the parts perpetually change places with one another2 and light by this1 that by the same simple motion action is propagated in a strait line4 ,but when a body hath its parts so moved1 that it sensibly both heats and shines at the same time1 then it is that we say fire is generated4 ,now by fire ,i do not understand a body distinct from matter combustible or glowing1 as wood or iron1 but the matter itself1 not simply and always1 but then only when it shineth and heateth4 ,he1 therefore1 that renders a cause possible and agreeable to the rest of the phenomena1 namely1 whence1 and from what action1 both the shining and heating proceed1 may be thought to have given a possible cause of the generation of fire4 ,let1 therefore1 ,a ,b ,c 7in the first figure7 be a sphere1 or the portion of a sphere1 whose centre is ,d2 and let it be transparent and homogeneous1 as crystal1 glass1 or water1 and objected to the sun4 ,wherefore1 the foremost part ,a ,b ,c will1 by the simple motion of the sun1 by which it thrusts forwards the medium1 be wrought upon by the sunbeams in the strait lines ,e ,a1 ,f ,b1 and ,g ,c2 which strait lines may1 in respect of the great distance of the sun1 be taken for parallels4 ,and seeing the medium within the sphere is thicker than the medium without it1 those beams will be refracted towards the perpendiculars4 ,let the strait lines ,e ,a and ,g ,c be produced till they cut the sphere in ,h and ,i2 and drawing the perpendiculars ,a ,d and ,c ,d1 the refracted beams ,e ,a and ,g ,c will of necessity fall1 the one between ,a ,h and ,a ,d1 the other between ,c ,i and ,c ,d4 ,let those refracted beams be ,a ,k and ,c ,l4 ,and again1 let the lines ,d ,k ,m and ,d ,l ,n be drawn perpendicular to the sphere2 and let ,a ,k and ,c ,l be produced till they meet with the strait line ,b ,d produced in ,o4 ,seeing1 therefore1 the medium within the sphere is thicker than that without it1 the refracted line ,a ,k will recede further from the perpendicular ,k ,m than ,k ,o will recede from the same4 ,wherefore ,k ,o will fall between the refracted line and the perpendicular4 ,let1 therefore1 the refracted line be ,k ,p1 cutting ,f ,o in ,p2 and for the same reason the strait line ,l ,p will be the refracted line of the strait line ,c ,l4 ,wherefore1 seeing the beams are nothing else but the ways in which the motion is propagated1 the motion about ,p will be so much more vehement than the motion about ,a ,b ,c1 by how much the base of the portion ,a ,b ,c is greater than the base of a like portion in the sphere1 whose centre is ,p1 and whose magnitude is equal to that of the little circle about ,p1 which comprehendeth all the beams that are propagated from ,a ,b ,c2 and this sphere being much less than the sphere ,a ,b ,c1 the parts of the medium1 that is1 of the air about ,p1 will change places with one another with much greater celerity than those about ,a ,b ,c4 ,if1 therefore1 any matter combustible1 that is to say1 such as may be easily dissipated1 be placed in ,p1 the parts of that matter1 if the proportion be great enough between ,a ,c and a like portion of the little circle about ,p1 will be freed from their mutual cohesion1 and being separated will acquire simple motion4 ,but vehement simple motion generates in the beholder a phantasm of lucid and hot1 as ,i have before demonstrated of the simple motion of the sun2 and therefore the combustible matter which is placed in ,p will be made lucid and hot1 that is to say1 will be fire4 ,wherefore ,i have rendered a possible cause of fire2 which was to be done4 ,sidenote3 ,the generation of fire from collision4 #e4 ,from the manner by which the sun generateth fire1 it is easy to explain the manner by which fire may be generated by the collision of two flints4 ,for by that collision some of those particles of which the stone is compacted1 are violently separated and thrown off2 and being withal swiftly turned round1 the eye is moved by them1 as it is in the generation of light by the sun4 ,wherefore they shine2 and falling upon matter which is already half dissipated1 such as is tinder1 they thoroughly dissipate the parts thereof1 and make them turn round4 ,from whence1 as ,i have newly shown1 light and heat1 that is to say fire1 is generated4 ,sidenote3 ,the cause of light in glow-worms1 rotten wood1 and the ,bolognan stone4 #f4 ,the shining of glow-worms1 some kinds of rotten wood1 and of a kind of stone made at ,bologna1 may have one common cause1 namely1 the exposing of them to the hot sun4 ,we find by experience that the ,bologna stone shines not1 unless it be so exposed2 and after it has been exposed it shines but for a little time1 namely1 as long as it retains a certain degree of heat4 ,and the cause may be that the parts1 of which it is made1 may together with heat have simple motion imprinted in them by the sun4 ,which if it be so1 it is necessary that it shine in the dark1 as long as there is sufficient heat in it2 but this ceasing1 it will shine no longer4 ,also we find by experience that in the glow-worm there is a certain thick humour1 like the crystalline humour of the eye2 which if it be taken out and held long enough in one's fingers1 and then be carried into the dark1 it will shine by reason of the warmth it received from the fingers2 but as soon as it is cold it will cease shining4 ,from whence1 therefore1 can these creatures have their light1 but from lying all day in the sunshine in the hottest time of summer8 ,in the same manner1 rotten wood1 except it grow rotten in the sunshine1 or be afterwards long enough exposed to the sun1 will not shine4 ,that this doth not happen in every worm1 nor in all kinds of rotten wood1 nor in all calcined stones1 the cause may be that the parts1 of which the bodies are made1 are different both in motion and figure from the parts of bodies of other kinds4 ,sidenote3 ,the cause of light in the concussion of sea water4 #g4 ,also the sea water shineth when it is either dashed with the strokes of oars1 or when a ship in its course breaks strongly through it2 but more or less1 according as the wind blows from different points4 ,the cause whereof may be this1 that the particles of salt1 though they never shine in the salt-pits1 where they are but slowly drawn up by the sun1 being here beaten up into the air in greater quantities and with more force1 are withal made to turn round1 and consequently to shine1 though weakly4 ,i have1 therefore1 given a possible cause of this phenomenon4 ,sidenote3 ,the cause of flame1 sparks1 colliquation4 #h4 ,if such matter as is compounded of hard little bodies be set on fire1 it must needs be1 that1 as they fly out in greater or less quantities1 the flame which is made by them will be greater or less4 ,and if the ethereal or fluid part of that matter fly out together with them1 their motion will be the swifter1 as it is in wood and other things which flame with a manifest mixture of wind4 ,when1 therefore1 these hard particles by their flying out move the eye strongly1 they shine bright2 and a great quantity of them flying out together1 they make a great shining body4 ,for flame being nothing but an aggregate of shining particles1 the greater the aggregate is1 the greater and more manifest will be the flame4 ,i have1 therefore1 shown a possible cause of flame4 ,and from hence the cause appears evidently1 why glass is so easily and quickly melted by the small flame of a candle blown1 which will not be melted without blowing but by a very strong fire4 ,now1 if from the same matter there be a part broken off1 namely1 such a part as consisteth of many of the small particles1 of this is made a spark4 ,for from the breaking off it hath a violent turning round1 and from hence it shines4 ,but though from this matter there fly neither flame nor sparks1 yet some of the smallest parts of it may be carried out as far as to the superficies1 and remain there as ashes2 the parts whereof are so extremely small1 that it cannot any longer be doubted how far nature may proceed in dividing4 ,lastly1 though by the application of fire to this matter there fly little or nothing from it1 yet there will be in the parts an endeavour to simple motion2 by which the whole body will either be melted1 or1 which is a degree of melting1 softened4 ,for all motion has some effect upon all matter whatsoever1 as has been shown at art4 #c1 chap4 ,,xv4 ,now if it be softened to such a degree1 as that the stubbornness of the parts be exceeded by their gravity1 then we say it is melted2 otherwise1 softened and made pliant and ductile4 ,again1 the matter having in it some particles hard1 others ethereal or watery2 if1 by the application of fire1 these latter be called out1 the former will thereby come to a more full contact with one another2 and1 consequently1 will not be so easily separated2 that is to say1 the whole body will be made harder4 ,and this may be the cause why the same fire makes some things soft1 others hard4 ,sidenote3 ,the cause why wet hay sometimes burns of its own accord2 also the cause of lightning4 #i4 ,it is known by experience that if hay be laid wet together in a heap1 it will after a time begin to smoke1 and then burn as it were of itself4 ,the cause whereof seems to be this1 that in the air1 which is enclosed within the hay1 there are those little bodies1 which1 as ,i have supposed1 are moved freely with simple motion4 ,but this motion being by degrees hindered more and more by the descending moisture1 which at the last fills and stops all the passages1 the thinner parts of the air ascend by penetrating the water2 and those hard little bodies1 being so thrust together that they touch and press one another1 acquire stronger motion2 till at last by the increased strength of this motion the watery parts are first driven outwards1 from whence appears vapour2 and by the continued increase of this motion1 the smallest particles of the dried hay are forced out1 and recovering their natural simple motion1 they grow hot and shine1 that is to say1 they are set on fire4 ,the same also may be the cause of lightning1 which happens in the hottest time of the year1 when the water is raised up in greatest quantity and carried highest4 ,for after the first clouds are raised1 others after others follow them2 and being congealed above1 they happen1 whilst some of them ascend and others descend1 to fall one upon another in such manner1 as that in some places all their parts are joined together1 in others they leave hollow spaces between them2 and into these spaces1 the ethereal parts being forced out by the compressure of the clouds1 many of the harder little bodies are so pent together1 as they have not the liberty of such motion as is natural to the air4 ,wherefore their endeavour grows more vehement1 till at last they force their way through the clouds1 sometimes in one place1 sometimes in another2 and1 breaking through with great noise1 they move the air violently1 and striking our eyes1 generate light1 that is to say1 they shine4 ,and this shining is that we call lightning4 ,sidenote3 ,the cause of the force of gunpowder2 and what is to be ascribed to the coals1 what to the brimstone1 and what to the nitre4 #aj4 ,the most common phenomenon proceeding from fire1 and yet the most admirable of all others1 is the force of gunpowder fired2 which being compounded of nitre1 brimstone and coals1 beaten small1 hath from the coals its first taking fire2 from the brimstone its nourishment and flame1 that is to say1 light and motion1 and from the nitre the vehemence of both4 ,now if a piece of nitre1 before it is beaten1 be laid upon a burning coal1 first it melts1 and1 like water1 quencheth that part of the coal it toucheth4 ,then vapour or air1 flying out where the coal and nitre join1 bloweth the coal with great swiftness and vehemence on all sides4 ,and from hence it comes to pass1 that by two contrary motions1 the one1 of the particles which go out of the burning coal1 the other1 of those of the ethereal and watery substance of the nitre1 is generated that vehement motion and inflammation4 ,and1 lastly1 when there is no more action from the nitre1 that is to say1 when the volatile parts of the nitre are flown out1 there is found about the sides a certain white substance1 which being thrown again into the fire1 will grow red-hot again1 but will not be dissipated1 at least unless the fire be augmented4 ,if now a possible cause of this be found out1 the same will also be a possible cause why a grain of gunpowder set on fire doth expand itself with such vehement motion1 and shine4 ,and it may be caused in this manner4 ,let the particles1 of which nitre consisteth1 be supposed to be some of them hard1 others watery1 and the rest ethereal4 ,also let the hard particles be supposed to be spherically hollow1 like small bubbles1 so that many of them growing together may constitute a body1 whose little caverns are filled with a substance which is either watery1 or ethereal1 or both4 ,as soon1 therefore1 as the hard particles are dissipated1 the watery and ethereal particles will necessarily fly out2 and as they fly1 of necessity blow strongly the burning coals and brimstone which are mingled together2 whereupon there will follow a great expansion of light1 with vehement flame1 and a violent dissipation of the particles of the nitre1 the brimstone and the coals4 ,wherefore ,i have given a possible cause of the force of fired gunpowder4 ,it is manifest from hence1 that for the rendering of the cause why a bullet of lead or iron1 shot from a piece of ordnance1 flies with so great velocity1 there is no necessity to introduce such rarefaction1 as1 by the common definition of it1 makes the same matter to have sometimes more1 sometimes less quantity2 which is inconceivable4 ,for every thing is said to be greater or less1 as it hath more or less quantity4 ,the violence with which a bullet is thrust out of a gun1 proceeds from the swiftness of the small particles of the fired powder2 at least it may proceed from that cause without the supposition of any empty space4 ,sidenote3 ,how heat is caused by attrition4 #aa4 ,besides1 by the attrition or rubbing of one body against another1 as of wood against wood1 we find that not only a certain degree of heat1 but fire itself is sometimes generated4 ,for such motion is the reciprocation of pressure1 sometimes one way1 sometimes the other2 and by this reciprocation whatsoever is fluid in both the pieces of wood is forced hither and thither2 and consequently1 to an endeavour of getting out2 and at last by breaking out makes fire4 ,sidenote3 ,the distinction of light into first1 second1 c4 #ab4 ,now light is distinguished into1 first1 second1 third1 and so on infinitely4 ,and we call that first light1 which is in the first lucid body2 as the sun1 fire1 c43 second1 that which is in such bodies1 as being not transparent are illuminated by the sun2 as the moon1 a wall1 c43 and third1 that which is in bodies not transparent1 but illuminated by second light1 c4 ,sidenote3 ,the causes of the colours we see in looking through a prisma of glass1 namely1 of red1 yellow1 blue1 violet colour4 #ac4 ,colour is light1 but troubled light1 namely1 such as is generated by perturbed motion2 as shall be made manifest by the red1 yellow1 blue and purple1 which are generated by the interposition of a diaphanous prisma1 whose opposite bases are triangular1 between the light and that which is enlightened4 ,sidenote3 ,discussion of ,figure #bg4b ,for let there be a prisma of glass1 or of any other transparent matter which is of greater density than air2 and let the triangle ,a ,b ,c be the base of this prisma4 ,also let the strait line ,d ,e be the diameter of the sun's body1 having oblique position to the strait line ,a ,b2 and let the sunbeams pass in the lines ,d ,a and ,e ,b ,c4 ,and lastly1 let the strait lines ,d ,a and ,e ,c be produced indefinitely to ,f and ,g4 ,seeing therefore the strait line ,d ,a1 by reason of the density of the glass1 is refracted towards the perpendicular2 let the line refracted at the point ,a be the strait line ,a ,h4 ,and again1 seeing the medium below ,a ,c is thinner than that above it1 the other refraction1 which will be made there1 will diverge from the perpendicular4 ,let therefore this second refracted line be ,a ,i4 ,also let the same be done at the point ,c1 by making the first refracted line to be ,c ,k1 and the second ,c ,l4 ,seeing therefore the cause of the refraction in the point ,a of the strait line of ,a ,b is the excess of the resistance of the medium in ,a ,b above the resistance of the air1 there must of necessity be reaction from the point ,a towards the point ,b2 and consequently the medium at ,a within the triangle ,a ,b ,c will have its motion troubled1 that is to say1 the strait motion in ,a ,f and ,a ,h will be mixed with the transverse motion between the same ,a ,f and ,a ,h1 represented by the short transverse lines in the triangle ,a ,f ,h4 ,again1 seeing at the point ,a of the strait line ,a ,c there is a second refraction from ,a ,h in ,a ,i1 the motion of the medium will again be perturbed by reason of the transverse reaction from ,a towards ,c1 represented likewise by the short transverse lines in the triangle ,a ,h ,i4 ,and in the same manner there is a double perturbation represented by the transverse lines in the triangles ,c ,g ,k and ,c ,k ,l4 ,but as for the light between ,a ,i and ,c ,g1 it will not be perturbed2 because1 if there were in all the points of the strait lines ,a ,b and ,a ,c the same action which is in the points ,a and ,c1 then the plane of the triangle ,c ,g ,k would be everywhere coincident with the plane of the triangle ,a ,f ,h2 by which means all would appear alike between ,a and ,c4 ,besides1 it is to be observed1 that all the reaction at ,a tends towards the illuminated parts which are between ,a and ,c1 and consequently perturbeth the first light4 ,and on the contrary1 that all the reaction at ,c tends towards the parts without the triangle or without the prisma ,a ,b ,c1 where there is none but second light2 and that the triangle ,a ,f ,h shows that perturbation of light which is made in the glass itself2 as the triangle ,a ,h ,i shows that perturbation of light which is made below the glass4 ,in like manner1 that ,c ,g ,k shows the perturbation of light within the glass2 and ,c ,k ,l that which is below the glass4 ,from whence there are four divers motions1 or four different illuminations or colours1 whose differences appear most manifestly to the sense in a prisma1 whose base is an equilateral triangle1 when the sunbeams that pass through it are received upon a white paper4 ,for the triangle ,a ,f ,h appears red to the sense2 the triangle ,a ,h ,i yellow2 the triangle ,c ,g ,k green1 and approaching to blue2 and lastly1 the triangle ,c ,k ,l appears purple4 ,it is therefore evident that when weak but first light passeth through a more resisting diaphanous body1 as glass1 the beams1 which fall upon it transversely1 make redness2 and when the same first light is stronger1 as it is in the thinner medium below the strait line ,a ,c1 the transverse beams make yellowness4 ,also when second light is strong1 as it is in the triangle ,c ,g ,k1 which is nearest to the first light1 the transverse beams make greenness2 and when the same second light is weaker1 as in the triangle ,c ,k ,l1 they make a purple colour4 ,sidenote3 ,why the moon and the stars appear redder in the horizon than in the midst of the heaven4 #ad4 ,from hence may be deduced a cause1 why the moon and stars appear bigger and redder near the horizon than in the mid-heaven4 ,for between the eye and the apparent horizon there is more impure air1 such as is mingled with watery and earthy little bodies1 than is between the same eye and the more elevated part of heaven4 ,but vision is made by beams which constitute a cone1 whose base1 if we look upon the moon1 is the moon's face1 and whose vertex is in the eye2 and therefore1 many beams from the moon must needs fall upon little bodies that are without the visual cone1 and be by them reflected to the eye4 ,but these reflected beams tend all in lines which are transverse to the visual cone1 and make at the eye an angle which is greater than the angle of the cone4 ,wherefore1 the moon appears greater in the horizon1 than when she is more elevated4 ,and because those reflected beams go transversely1 there will be generated1 by the last article1 redness4 ,a possible cause therefore is shown1 why the moon as also the stars appear greater and redder in the horizon1 than in the midst of heaven4 ,the same also may be the cause1 why the sun appears in the horizon greater and of a colour more degenerating to yellow1 than when he is higher elevated4 ,for the reflection from the little bodies between1 and the transverse motion of the medium1 are still the same4 ,but the light of the sun is much stronger than that of the moon2 and therefore1 by the last article1 his splendour must needs by this perturbation degenerate into yellowness4 ,but for the generation of these four colours1 it is not necessary that the figure of the glass be a prisma2 for if it were spherical it would do the same4 ,for in a sphere the sunbeams are twice refracted and twice reflected4 ,and this being observed by ,des ,cartes1 and withal that a rainbow never appears but when it rains2 as also1 that the drops of rain have their figures almost spherical2 he hath shown from thence the cause of the colours in the rainbow2 which therefore need not be repeated4 ,sidenote3 ,the cause of whiteness4 #ae4 ,whiteness is light1 but light perturbed by the reflections of many beams of light coming to the eye together within a little space4 ,for if glass or any other diaphanous body be reduced to very small parts by contusion or concussion1 every one of those parts1 if the beams of a lucid body be from any one point of the same reflected to the eye1 will represent to the beholder an idea or image of the whole lucid body1 that is to say1 a phantasm of white4 ,for the strongest light is the most white2 and therefore many such parts will make many such images4 ,wherefore1 if those parts lie thick and close together1 those many images will appear confusedly1 and will by reason of the confused light represent a white colour4 ,so that from hence may be deduced a possible cause1 why glass beaten1 that is1 reduced to powder1 looks white4 ,also why water and snow are white2 they being nothing but a heap of very small diaphanous bodies1 namely1 of little bubbles1 from whose several convex superficies there are by reflection made several confused phantasms of the whole lucid body1 that is to say1 whiteness4 ,for the same reason1 salt and nitre are white1 as consisting of small bubbles which contain within them water and air2 as is manifest in nitre1 from this1 that being thrown into the fire it violently blows the same2 which salt also doth1 but with less violence4 ,but if a white body be exposed1 not to the light of the day1 but to that of the fire or of a candle1 it will not at the first sight be easily judged whether it be white or yellow2 the cause whereof may be this1 that the light of those things1 which burn and flame1 is almost of a middle colour between whiteness and yellowness4 ,sidenote3 ,the cause of blackness4 #af4 ,as whiteness is light1 so blackness is the privation of light1 or darkness4 ,and1 from hence it is1 first1 that all holes1 from which no light can be reflected to the eye1 appear black4 ,secondly1 that when a body hath little eminent particles erected straight up from the superficies1 so that the beams of light which fall upon them are reflected not to the eye but to the body itself1 that superficies appears black2 in the same manner as the sea appears black when ruffled by the wind4 ,thirdly1 that any combustible matter is by the fire made to look black before it shines4 ,for the endeavour of the fire being to dissipate the smallest parts of such bodies as are thrown into it1 it must first raise and erect those parts before it can work their dissipation4 ,if1 therefore1 the fire be put out before the parts are totally dissipated1 the coal will appear black2 for the parts being only erected1 the beams of light falling upon them will not be reflected to the eye1 but to the coal itself4 ,fourthly1 that burning glasses do more easily burn black things than white4 ,for in a white superficies the eminent parts are convex1 like little bubbles2 and therefore the beams of light1 which fall upon them1 are reflected every way from the reflecting body4 ,but in a black superficies1 where the eminent particles are more erected1 the beams of light falling upon them are all necessarily reflected towards the body itself2 and1 therefore1 bodies that are black are more easily set on fire by the sun beams1 than those that are white4 ,fifthly1 that all colours that are made of the mixture of white and black proceed from the different position of the particles that rise above the superficies1 and their different forms of asperity4 ,for1 according to these differences1 more or fewer beams of light are reflected from several bodies to the eye4 ,but in regard those differences are innumerable1 and the bodies themselves so small that we cannot perceive them2 the explication and precise determination of the causes of all colours is a thing of so great difficulty1 that ,i dare not undertake it4 ---------------------------------------- -------------------------------- ,illustration3 ,vol4 ,i4 ,lat4 ,eng4 ,c4 ,,xxvii4 ,fig4 #a-b ,fig #a ,fig #b ---------------------------------------- -------------------------------- ------- ,,chapter ,,xxviii4 ,,of ,,cold1 ,,wind1 ,,hard1 ,,ice1 ,,restitution ,,of ,,bodies ,,bent1 ,,diaphanous1 ,,lightning ,,and ,,thunder2 ,,and ,,of ,,the ,,heads ,,of ,,rivers4 #a4 ,why breath from the same mouth sometimes heats and sometimes cools4--#b4 ,wind1 and the inconstancy of winds1 whence4--#c ,why there is a constant1 though not a great wind1 from east to west1 near the equator4--#d4 ,what is the effect of air pent in between the clouds4--#e4 ,no change from soft to hard1 but by motion4--#f4 ,what is the cause of cold near the poles4--#g44 ,the cause of ice2 and why the cold is more remiss in rainy than in clear weather4 ,why water doth not freeze in deep wells as it doth near the superficies of the earth4 ,why ice is not so heavy as water2 and why wine is not so easily frozen as water4--#h4 ,another cause of hardness from the fuller contact of atoms2 also1 how hard things are broken4--#i4 ,a third cause of hardness from heat4--#aj4 ,a fourth cause of hardness from the motion of atoms enclosed in a narrow space4--#aa4 ,how hard things are softened4--#ab4 ,whence proceed the spontaneous restitution of things bent4--#ac4 ,diaphanous and opacous1 what they are1 and whence4--#ad4 ,the cause of lightning and thunder4--#ae4 ,whence it proceeds that clouds can fall again after they are once elevated and frozen4--#af4 ,how it could be that the moon was eclipsed1 when she was not diametrically opposite to the sun4--#ag4 ,by what means many suns may appear at once4--#ah4 ,of the heads of rivers4 ,sidenote3 ,why breath from the same mouth sometimes heats and sometimes cools4 #a4 ,as1 when the motion of the ambient ethereal substance makes the spirits and fluid parts of our bodies tend outwards1 we acknowledge heat2 so1 by the endeavour inwards of the same spirits and humours1 we feel cold4 ,so that to cool is to make the exterior parts of the body endeavour inwards1 by a motion contrary to that of calefaction1 by which the internal parts are called outwards4 ,he1 therefore1 that would know the cause of cold1 must find by what motion or motions the exterior parts of any body endeavour to retire inwards4 ,to begin with those phenomena which are the most familiar4 ,there is almost no man but knows1 that breath blown strongly1 and which comes from the mouth with violence1 that is to say1 the passage being strait1 will cool the hand2 and that the same breath blown gently1 that is to say1 through a greater aperture1 will warm the same4 ,the cause of which phenomenon may be this1 the breath going out hath two motions2 the one1 of the whole and direct1 by which the foremost parts of the hand are driven inwards2 the other1 simple motion of the small particles of the same breath1 which1 7as ,i have shown in the #crd article of the last chapter1 causeth heat4 ,according1 therefore1 as either of these motions is predominant1 so there is the sense sometimes of cold1 sometimes of heat4 ,wherefore1 when the breath is softly breathed out at a large passage1 that simple motion which causeth heat prevaileth1 and consequently heat is felt2 and when1 by compressing the lips1 the breath is more strongly blown out1 then is the direct motion prevalent1 which makes us feel cold4 ,for1 the direct motion of the breath or air is wind2 and all wind cools or diminisheth former heat4 ,sidenote3 ,wind1 and the inconstancy of winds1 whence4 #b4 ,and seeing not only great wind1 but almost any ventilation and stirring of the air1 doth refrigerate2 the reason of many experiments concerning cold cannot well be given without finding first what are the causes of wind4 ,now1 wind is nothing else but the direct motion of the air thrust forwards2 which1 nevertheless1 when many winds concur1 may be circular or otherwise indirect1 as it is in whirlwinds4 ,wherefore1 in the first place we are to enquire into the causes of winds4 ,wind is air moved in a considerable quantity1 and that either in the manner of waves1 which is both forwards and also up and down1 or else forwards only4 ,supposing1 therefore1 the air both clear and calm for any time how little soever1 yet1 the greater bodies of the world being so disposed and ordered as has been said1 it will be necessary that a wind presently arise somewhere4 ,for1 seeing that motion of the parts of the air1 which is made by the simple motion of the sun in his own epicycle1 causeth an exhalation of the particles of water from the seas and all other moist bodies1 and those particles make clouds2 it must needs follow1 that1 whilst the particles of water pass upwards1 the particles of air1 for the keeping of all spaces full1 be jostled out on every side1 and urge the next particles1 and these the next2 till having made their circuit1 there comes continually so much air to the hinder parts of the earth as there went water from before it4 ,wherefore1 the ascending vapours move the air towards the sides every way2 and all direct motion of the air being wind1 they make a wind4 ,and if this wind meet often with other vapours which arise in other places1 it is manifest that the force thereof will be augmented1 and the way or course of it changed4 ,besides1 according as the earth1 by its diurnal motion1 turns sometimes the drier1 sometimes the moister part towards the sun1 so sometimes a greater1 sometimes a less1 quantity of vapours will be raised2 that is to say1 sometimes there will be a less1 sometimes a greater wind4 ,wherefore1 ,i have rendered a possible cause of such winds as are generated by vapours2 and also of their inconstancy4 ,from hence it follows that these winds cannot be made in any place1 which is higher than that to which vapours may ascend4 ,nor is that incredible which is reported of the highest mountains1 as the ,peak of ,teneriffe and the ,andes of ,peru1 namely1 that they are not at all troubled with these inconstant winds4 ,and if it were certain that neither rain nor snow were ever seen in the highest tops of those mountains1 it could not be doubted but that they are higher than any place to which vapours use to ascend4 ,sidenote3 ,why there is a constant1 though not a great wind1 from east to west1 near the equator4 #c4 ,nevertheless1 there may be wind there1 though not that which is made by the ascent of vapours1 yet a less and more constant wind1 like the continued blast of a pair of bellows1 blowing from the east4 ,and this may have a double cause2 the one1 the diurnal motion of the earth2 the other1 its simple motion in its own epicycle4 ,for these mountains being1 by reason of their height1 more eminent than all the rest of the parts of the earth1 do by both these motions drive the air from the west eastwards4 ,to which1 though the diurnal motion contribute but little1 yet seeing ,i have supposed that the simple motion of the earth1 in its own epicycle1 makes two revolutions in the same time in which the diurnal motion makes but one1 and that the semidiameter of the epicycle is double to the semidiameter of the diurnal conversion1 the motion of every point of the earth in its own epicycle will have its velocity quadruple to that of the diurnal motion2 so that by both these motions together1 the tops of those hills will sensibly be moved against the air2 and consequently a wind will be felt4 ,for whether the air strike the sentient1 or the sentient the air1 the perception of motion will be the same4 ,but this wind1 seeing it is not caused by the ascent of vapours1 must necessarily be very constant4 ,sidenote3 ,what is the effect of air pent in between the clouds4 #d4 ,when one cloud is already ascended into the air1 if another cloud ascend towards it1 that part of the air1 which is intercepted between them both1 must of necessity be pressed out every way4 ,also when both of them1 whilst the one ascends and the other either stays or descends1 come to be joined in such manner as that the ethereal substance be shut within them on every side1 it will by this compression also go out by penetrating the water4 ,but in the meantime1 the hard particles1 which are mingled with the air and are agitated1 as ,i have supposed1 with simple motion1 will not pass through the water of the clouds1 but be more straitly compressed within their cavities4 ,and this ,i have demonstrated at the #dth and #eth articles of chapter ,,xxii4 ,besides1 seeing the globe of the earth floateth in the air which is agitated by the sun's motion1 the parts of the air resisted by the earth will spread themselves every way upon the earth's superficies2 as ,i have shown at the #hth article of chapter ,,xxi4 ,sidenote3 ,no change from soft to hard1 but by motion4 #e4 ,we perceive a body to be hard1 from this1 that1 when touching it1 we would thrust forwards that part of the same which we touch1 we cannot do it otherwise than by thrusting forwards the whole body4 ,we may indeed easily and sensibly thrust forwards any particle of the air or water which we touch1 whilst yet the rest of its parts remain to sense unmoved4 ,but we cannot do so to any part of a stone4 ,wherefore ,i define a hard body to be that whereof no part can be sensibly moved1 unless the whole be moved4 ,whatsoever therefore is soft or fluid1 the same can never be made hard but by such motion as makes many of the parts together stop the motion of some one part1 by resisting the same4 ,sidenote3 ,what it the cause of cold near the poles4 #f4 ,those things premised1 ,i shall show a possible cause why there is greater cold near the poles of the earth1 than further from them4 ,the motion of the sun between the tropics1 driving the air towards that part of the earth's superficies which is perpendicularly under it1 makes it spread itself every way2 and the velocity of this expansion of the air grows greater and greater1 as the superficies of the earth comes to be more and more straitened1 that is to say1 as the circles which are parallel to the equator come to be less and less4 ,wherefore this expansive motion of the air drives before it the parts of the air1 which are in its way1 continually towards the poles more and more strongly1 as its force comes to be more and more united1 that is to say1 as the circles which are parallel to the equator are less and less2 that is1 so much the more1 by how much they are nearer to the poles of the earth4 ,in those places1 therefore1 which are nearer to the poles1 there is greater cold than in those which are more remote from them4 ,now this expansion of the air upon the superficies of the earth1 from east to west1 doth1 by reason of the sun's perpetual accession to the places which are successively under it1 make it cold at the time of the sun's rising and setting2 but as the sun comes to be continually more and more perpendicular to those cooled places1 so by the heat1 which is generated by the supervening simple motion of the sun1 that cold is again remitted2 and can never be great1 because the action by which it was generated is not permanent4 ,wherefore ,i have rendered a possible cause of cold in those places that are near the poles1 or where the obliquity of the sun is great4 ,sidenote3 ,the cause of ice2 and why the cold is more remiss in rainy than in clear weather4 ,why water doth not freeze in deep wells1 as it doth near the superficies of the earth4 ,why ice is not so heavy as water2 and why wine is not so easily frozen as water4 #g4 ,how water may be congealed by cold1 may be explained in this manner4 ,let ,a 7in figure #a7 represent the sun1 and ,b the earth4 ,a will therefore be much greater than ,b4 ,let ,e ,f be in the plane of the equinoctial2 to which let ,g ,h1 ,i ,k1 and ,l ,c be parallel4 ,lastly1 let ,c and ,d be the poles of the earth4 ,the air1 therefore1 by its action in those parallels1 will rake the superficies of the earth2 and that with motion so much the stronger1 by how much the parallel circles towards the poles grow less and less4 ,from whence must arise a wind1 which will force together the uppermost parts of the water1 and withal raise them a little1 weakening their endeavour towards the centre of the earth4 ,and from their endeavour towards the centre of the earth1 joined with the endeavour of the said wind1 the uppermost parts of the water will be pressed together and coagulated1 that is to say1 the top of the water will be skinned over and hardened4 ,and so again1 the water next the top will be hardened in the same manner1 till at length the ice be thick4 ,and this ice1 being now compacted of little hard bodies1 must also contain many particles of air received into it4 ,as rivers and seas1 so also in the same manner may the clouds be frozen4 ,for when1 by the ascending and descending of several clouds at the same time1 the air intercepted between them is by compression forced out1 it rakes1 and by little and little hardens them4 ,and though those small drops1 which usually make clouds1 be not yet united into greater bodies1 yet the same wind will be made2 and by it1 as water is congealed into ice1 so will vapours in the same manner be congealed into snow4 ,from the same cause it is that ice may be made by art1 and that not far from the fire4 ,for it is done by the mingling of snow and salt together1 and by burying in it a small vessel full of water4 ,now while the snow and salt1 which have in them a great deal of air1 are melting1 the air1 which is pressed out every way in wind1 rakes the sides of the vessel2 and as the wind by its motion rakes the vessel1 so the vessel by the same motion and action congeals the water within it4 ,we find by experience1 that cold is always more remiss in places where it rains1 or where the weather is cloudy1 things being alike in all other respects1 than where the air is clear4 ,and this agreeth very well with what ,i have said before4 ,for in clear weather1 the course of the wind which1 as ,i said even now1 rakes the superficies of the earth1 as it is free from all interruption1 so also it is very strong4 ,but when small drops of water are either rising or falling1 that wind is repelled1 broken1 and dissipated by them2 and the less the wind is1 the less is the cold4 ,we find also by experience1 that in deep wells the water freezeth not so much as it doth upon the superficies of the earth4 ,for the wind1 by which ice is made1 entering into the earth by reason of the laxity of its parts1 more or less1 loseth some of its force1 though not much4 ,so that if the well be not deep1 it will freeze2 whereas if it be so deep1 as that the wind which causeth cold cannot reach it1 it will not freeze4 ,we find moreover by experience1 that ice is lighter than water4 ,the cause whereof is manifest from that which ,i have already shown1 namely1 that air is received in and mingled with the particles of the water whilst it is congealing4 ,lastly1 wine is not so easily congealed as water1 because in wine there are particles1 which1 being not fluid1 are moved very swiftly1 and by their motion congelation is retarded4 ,but if the cold prevail against this motion1 then the outermost parts of the wine will be first frozen1 and afterwards the inner parts2 whereof this is a sign1 that the wine which remains unfrozen in the midst will be very strong4 ,sidenote3 ,another cause of hardness from the fuller contact of atoms4 ,also how hard things are broken4 #h4 ,we have seen one way of making things hard1 namely1 by congelation4 ,another way is thus4 ,having already supposed that innumerable atoms1 some harder than others and that have several simple motions of their own1 are intermingled with the ethereal substance2 it follows necessarily from hence1 that by reason of the fermentation of the whole air1 of which ,i have spoken in chapter ,,xxi1 some of those atoms meeting with others will cleave together1 by applying themselves to one another in such manner as is agreeable to their motions and mutual contacts2 and1 seeing there is no vacuum1 cannot be pulled asunder1 but by so much force as is sufficient to overcome their hardness4 ,now there are innumerable degrees of hardness4 ,as for example1 there is a degree of it in water1 as is manifest from this1 that upon a plane it may be drawn any way at pleasure by one's finger4 ,there is a greater degree of it in clammy liquors1 which1 when they are poured out1 do in falling downwards dispose themselves into one continued thread2 which thread1 before it be broken1 will by little and little diminish its thickness1 till at last it be so small1 as that it seems to break only in a point2 and in their separation the external parts break first from one another1 and then the more internal parts successively one after another4 ,in wax there is yet a greater degree of hardness4 ,for when we would pull one part of it from another1 we first make the whole mass slenderer1 before we can pull it asunder4 ,and how much the harder anything is which we would break1 so much the more force we must apply to it4 ,wherefore1 if we go on to harder things1 as ropes1 wood1 metals1 stones1 c41 reason prompteth us to believe that the same1 though not always sensibly1 will necessarily happen2 and that even the hardest things are broken asunder in the same manner1 namely1 by solution of their continuity begun in the outermost superficies1 and proceeding successively to the innermost parts4 ,in like manner1 when the parts of bodies are to be separated1 not by pulling them asunder1 but by breaking them1 the first separation will necessarily be in the convex superficies of the bowed part of the body1 and afterwards in the concave superficies4 ,for in all bowing there is in the convex superficies an endeavour in the parts to go one from another1 and in the concave superficies to penetrate one another4 ,this being well understood1 a reason may be given how two bodies1 which are contiguous in one common superficies1 may by force be separated without the introduction of vacuum2 though ,lucretius thought otherwise1 believing that such separation was a strong establishment of vacuum4 ,for a marble pillar being made to hang by one of its bases1 if it be long enough1 it will by its own weight be broken asunder2 and yet it will not necessarily follow that there should be vacuum1 seeing the solution of its continuity may begin in the circumference1 and proceed successively to the midst thereof4 ,sidenote3 ,a third cause of hardness1 from heat4 #i4 ,another cause of hardness in some things may be in this manner4 ,if a soft body consist of many hard particles1 which by the intermixture of many other fluid particles cohere but loosely together1 those fluid parts1 as hath been shown in the last article of chapter ,,xxi1 will be exhaled2 by which means each hard particle will apply itself to the next to it according to a greater superficies1 and consequently they will cohere more closely to one another1 that is to say1 the whole mass will be made harder4 ,sidenote3 ,a fourth cause of hardness1 from the motion of atoms enclosed in a narrow space4 #aj4 ,again1 in some things hardness may be made to a certain degree in this manner4 ,when any fluid substance hath in it certain very small bodies intermingled1 which1 being moved with simple motion of their own1 contribute like motion to the parts of the fluid substance1 and this be done in a small enclosed space1 as in the hollow of a little sphere1 or a very slender pipe1 if the motion be vehement and there be a great number of these small enclosed bodies1 two things will happen2 the one1 that the fluid substance will have an endeavour of dilating itself at once every way2 the other1 that if those small bodies can nowhere get out1 then from their reflection it will follow1 that the motion of the parts of the enclosed fluid substance1 which was vehement before1 will now be much more vehement4 ,wherefore1 if any one particle of that fluid substance should be touched and pressed by some external movent1 it could not yield but by the application of very sensible force4 ,wherefore the fluid substance1 which is enclosed and so moved1 hath some degree of hardness4 ,now1 greater and less degree of hardness depends upon the quantity and velocity of those small bodies1 and upon the narrowness of the place both together4 ,sidenote3 ,how hard things are softened4 #aa4 ,such things as are made hard by sudden heat1 namely such as are hardened by fire1 are commonly reduced to their former soft form by maceration4 ,for fire hardens by evaporation1 and therefore if the evaporated moisture be restored again1 the former nature and form is restored together with it4 ,and such things as are frozen with cold1 if the wind by which they were frozen change into the opposite quarter1 they will be unfrozen again1 unless they have gotten a habit of new motion or endeavour by long continuance in that hardness4 ,nor is it enough to cause thawing1 that there be a cessation of the freezing wind2 for the taking away of the cause doth not destroy a produced effect2 but the thawing also must have its proper cause1 namely1 a contrary wind1 or at least a wind opposite in some degree4 ,and this we find to be true by experience4 ,for1 if ice be laid in a place so well enclosed that the motion of the air cannot get to it1 that ice will remain unchanged1 though the place be not sensibly cold4 ,sidenote3 ,whence proceeds the spontaneous restitution of things bent4 #ab4 ,of hard bodies1 some may manifestly be bowed2 others not1 but are broken in the very first moment of their bending4 ,and of such bodies as may manifestly be bended1 some being bent1 do1 as soon as ever they are set at liberty1 restore themselves to their former posture2 others remain still bent4 ,now if the cause of this restitution be asked1 ,i say1 it may be in this manner1 namely1 that the particles of the bended body1 whilst it is held bent1 do nevertheless retain their motion2 and by this motion they restore it as soon as the force is removed by which it was bent4 ,for when any thing is bent1 as a plate of steel1 and1 as soon as the force is removed1 restores itself again1 it is evident that the cause of its restitution cannot be referred to the ambient air2 nor can it be referred to the removal of the force by which it was bent2 for in things that are at rest the taking away of impediments is not a sufficient cause of their future motion2 there being no other cause of motion1 but motion4 ,the cause therefore of such restitution is in the parts of the steel itself4 ,wherefore1 whilst it remains bent1 there is in the parts1 of which it consisteth1 some motion though invisible2 that is to say1 some endeavour at least that way by which the restitution is to be made2 and therefore this endeavour of all the parts together is the first beginning of restitution2 so that the impediment being removed1 that is to say1 the force by which it was held bent1 it will be restored again4 ,now the motion of the parts1 by which this done1 is that which ,i called simple motion1 or motion returning into itself4 ,when therefore in the bending of a plate the ends are drawn together1 there is on one side a mutual compression of the parts2 which compression is one endeavour opposite to another endeavour3 and on the other side a divulsion of the parts4 ,the endeavour therefore of the parts on one side tends to the restitution of the plate from the middle towards the ends2 and on the other side1 from the ends towards the middle4 ,wherefore the impediment being taken away1 this endeavour1 which is the beginning of restitution1 will restore the plate to its former posture4 ,and thus ,i have given a possible cause why some bodies1 when they are bent1 restore themselves again2 which was to be done4 ,as for stones1 seeing they are made by the accretion of many very hard particles within the earth2 which particles have no great coherence1 that is to say1 touch one another in small latitude1 and consequently admit many particles of air2 it must needs be that1 in bending of them1 their internal parts will not easily be compressed1 by reason of their hardness4 ,and because their coherence is not firm1 as soon as the external hard particles are disjoined1 the ethereal parts will necessarily break out1 and so the body will suddenly be broken4 ,sidenote3 ,diaphanous and opacous1 what they are1 and whence4 #ac4 ,those bodies are called diaphanous1 upon which1 whilst the beams of a lucid body do work1 the action of every one of those beams is propagated in them in such manner1 as that they still retain the same order amongst themselves1 or the inversion of that order2 and therefore bodies1 which are perfectly diaphanous1 are also perfectly homogeneous4 ,on the contrary1 an opacous body is that1 which1 by reason of its heterogeneous nature1 doth by innumerable reflections and refractions in particles of different figures and unequal hardness1 weaken the beams that fall upon it before they reach the eye4 ,and of diaphanous bodies1 some are made such by nature from the beginning2 as the substance of the air1 and of the water1 and perhaps also some parts of stones1 unless these also be water that has been long congealed4 ,others are made so by the power of heat1 which congregates homogeneous bodies4 ,but such1 as are made diaphanous in this manner1 consist of parts which were formerly diaphanous4 ,sidenote3 ,the cause of lightning and thunder4 #ad4 ,in what manner clouds are made by the motion of the sun1 elevating the particles of water from the sea and other moist places1 hath been explained in chapter ,,xxvi4 ,also how clouds come to be frozen1 hath been shown above at the #gth article4 ,now from this1 that air may be enclosed as it were in caverns1 and pent together more and more by the meeting of ascending and descending clouds1 may be deduced a possible cause of thunder and lightning4 ,for seeing the air consists of two parts1 the one ethereal1 which has no proper motion of its own1 as being a thing divisible into the least parts2 the other hard1 namely1 consisting of many hard atoms1 which have every one of them a very swift simple motion of its own3 whilst the clouds by their meeting do more and more straiten such cavities as they intercept1 the ethereal parts will penetrate and pass through their watery substance2 but the hard parts will in the meantime be the more thrust together1 and press one another2 and consequently1 by reason of their vehement motions1 they will have an endeavour to rebound from each other4 ,whensoever1 therefore1 the compression is great enough1 and the concave parts of the clouds are1 for the cause ,i have already given1 congealed into ice1 the cloud will necessarily be broken2 and this breaking of the cloud produceth the first clap of thunder4 ,afterwards the air1 which was pent in1 having now broken through1 makes a concussion of the air without1 and from hence proceeds the roaring and murmur which follows2 and both the first clap and the murmur that follows it make that noise which is called thunder4 ,also1 from the same air breaking through the clouds and with concussion falling upon the eye1 proceeds that action upon our eye1 which causeth in us a perception of that light1 which we call lightning4 ,wherefore ,i have given a possible cause of thunder and lightning4 ,sidenote3 ,whence it proceeds that clouds can fall again1 after they are once elevated and frozen4 #ae4 ,but if the vapours1 which are raised into clouds1 do run together again into water or be congealed into ice1 from whence is it1 seeing both ice and water are heavy1 that they are sustained in the air8 ,or rather1 what may the cause be1 that being once elevated1 they fall down again8 ,for there is no doubt but the same force which could carry up that water1 could also sustain it there4 ,why therefore being once carried up1 doth it fall again8 ,i say it proceeds from the same simple motion of the sun1 both that vapours are forced to ascend1 and that water gathered into clouds is forced to descend4 ,for in chapter ,,xxi1 article #aa1 ,i have shown how vapours are elevated2 and in the same chapter1 article #e1 ,i have also shown how by the same motion homogeneous bodies are congregated1 and heterogeneous dissipated2 that is to say1 how such things1 as have a like nature to that of the earth1 are driven towards the earth2 that is to say1 what is the cause of the descent of heavy bodies4 ,now if the action of the sun be hindered in the raising of vapours1 and be not at all hindered in the casting of them down1 the water will descend4 ,but a cloud cannot hinder the action of the sun in making things of an earthly nature descend to the earth1 though it may hinder it in making vapours ascend4 ,for the lower part of a thick cloud is so covered by its upper part1 as that it cannot receive that action of the sun by which vapours are carried up2 because vapours are raised by the perpetual fermentation of the air1 or by the separating of its smallest parts from one another1 which is much weaker when a thick cloud is interposed1 than when the sky is clear4 ,and therefore1 whensoever a cloud is made thick enough1 the water1 which would not descend before1 will then descend1 unless it be kept up by the agitation of the wind4 ,wherefore ,i have rendered a possible cause1 both why the clouds may be sustained in the air1 and also why they may fall down again to the earth2 which was propounded to be done4 ,sidenote3 ,how it could be that the moon was eclipsed1 when she was not diametrically opposite to the sun4 #af4 ,granting that the clouds may be frozen1 it is no wonder if the moon have been seen eclipsed at such time as she hath been almost two degrees above the horizon1 the sun at the same time appearing in the horizon2 for such an eclipse was observed by ,mstlin1 at ,tubingen1 in the year #aeij4 ,for it might happen that a frozen cloud was then interposed between the sun and the eye of the observer4 ,and if it were so1 the sun1 which was then almost two degrees below the horizon1 might appear to be in it1 by reason of the passing of his beams through the ice4 ,and it is to be noted that those1 that attribute such refractions to the atmosphere1 cannot attribute to it so great a refraction as this4 ,wherefore not the atmosphere1 but either water in a continued body1 or else ice1 must be the cause of that refraction4 ,sidenote3 ,by what means many suns may appear at once4 #ag4 ,again1 granting that there may be ice in the clouds1 it will be no longer a wonder that many suns have sometimes appeared at once4 ,for looking-glasses may be so placed1 as by reflections to show the same object in many places4 ,and may not so many frozen clouds serve for so many looking-glasses8 ,and may they not be fitly disposed for that purpose8 ,besides1 the number of appearances may be increased by refractions also2 and therefore it would be a greater wonder to me1 if such phenomena as these should never happen4 ,and were it not for that one phenomenon of the new star which was seen in ,cassiopea1 ,i should think comets were made in the same manner1 namely1 by vapours drawn not only from the earth but from the rest of the planets also1 and congealed into one continued body4 ,for ,i could very well from hence give a reason both of their hair1 and of their motions4 ,but seeing that star remained sixteen whole months in the same place amongst the fixed stars1 ,i cannot believe the matter of it was ice4 ,wherefore ,i leave to others the disquisition of the cause of comets2 concerning which nothing that hath hitherto been published1 besides the bare histories of them1 is worth considering4 ,sidenote3 ,of the heads of rivers4 #ah4 ,the heads of rivers may be deduced from rain-water1 or from melted snows1 very easily2 but from other causes1 very hardly1 or not at all4 ,for both rain-water and melted snows run down the descents of mountains2 and if they descend only by the outward superficies1 the showers or snows themselves may be accounted the springs or fountains2 but if they enter the earth and descend within it1 then1 wheresoever they break out1 there are their springs4 ,and as these springs make small streams1 so1 many small streams running together make rivers4 ,now1 there was never any spring found1 but where the water which flowed to it1 was either further1 or at least as far from the centre of the earth1 as the spring itself4 ,and whereas it has been objected by a great philosopher1 that in the top of ,mount ,cenis1 which parts ,savoy from ,piedmont1 there springs a river which runs down by ,susa2 it is not true4 ,for there are above that river1 for two miles length1 very high hills on both sides1 which are almost perpetually covered with snow2 from which innumerable little streams running down do manifestly supply that river with water sufficient for its magnitude4 ------- ,,chap4 ,,xxix4 ,,of ,,sound1 ,,odour1 ,,savour1 ,,and ,,touch4 #a44 ,the definition of sound1 and the distinctions of sounds4--#b4 ,the cause of the degrees of sounds4--#c4 ,the difference between sounds acute and grave4--#d4 ,the difference between clear and hoarse sounds1 whence4--#e4 ,the sound of thunder and of a gun1 whence it proceeds4--#f4 ,whence it is that pipes1 by blowing into them1 have a clear sound4--#g4 ,of reflected sound4--#h4 ,from whence it is that sound is uniform and lasting4--#i4 ,how sound may be helped and hindered by the wind4--#aj4 ,not only air1 but other bodies how hard soever they be1 convey sound4--#aa4 ,the causes of grave and acute sounds1 and of concent4--#ab4 ,phenomena for smelling4--#ac4 ,the first organ and the generation of smelling4--#ad4 ,how it is helped by heat and by wind4--#ae4 ,why such bodies are least smelt1 which have least intermixture of air in them4--#af4 ,why odorous things become more odorous by being bruised4--#ag4 ,the first organ of tasting2 and why some savours cause nauseousness4--#ah4 ,the first organ of feeling2 and how we come to the knowledge of such objects as are common to the touch and other senses4 ,sidenote3 ,the definition of sound1 and the distinction of sounds4 #a4 ,,sound is sense generated by the action of the medium1 when its motion reacheth the ear and the rest of the organs of sense4 ,now1 the motion of the medium is not the sound itself1 but the cause of it4 ,for the phantasm which is made in us1 that is to say1 the reaction of the organ1 is properly that which we call sound4 ,the principal distinctions of sounds are these2 first1 that one sound is stronger1 another weaker4 ,secondly1 that one is more grave1 another more acute4 ,thirdly1 that one is clear1 another hoarse4 ,fourthly1 that one is primary1 another derivative4 ,fifthly1 that one is uniform1 another not4 ,sixthly1 that one is more durable1 another less durable4 ,of all which distinctions the members may be subdistinguished into parts distinguishable almost infinitely4 ,for the variety of sounds seems to be not much less than that of colours4 ,as vision1 so hearing is generated by the motion of the medium1 but not in the same manner4 ,for sight is from pressure1 that is1 from an endeavour2 in which there is no perceptible progression of any of the parts of the medium2 but one part urging or thrusting on another propagateth that action successively to any distance whatsoever2 whereas the motion of the medium1 by which sound is made1 is a stroke4 ,for when we hear1 the drum of the ear1 which is the first organ of hearing1 is stricken2 and the drum being stricken1 the pia mater is also shaken1 and with it the arteries which are inserted into it2 by which the action being propagated to the heart itself1 by the reaction of the heart a phantasm is made which we call sound2 and because the reaction tendeth outwards1 we think it is without4 ,sidenote3 ,the cause of the degrees of sounds4 #b4 ,and seeing the effects produced by motion are greater or less1 not only when the velocity is greater or less1 but also when the body hath greater or less magnitude though the velocity be the same2 a sound may be greater or less both these ways4 ,and because neither the greatest nor the least magnitude or velocity can be given1 it may happen that either the motion may be of so small velocity1 or the body itself of so small magnitude1 as to produce no sound at all2 or either of them may be so great1 as to take away the faculty of sense by hurting the organ4 ,from hence may be deduced possible causes of the strength and weakness of sounds in the following phenomena4 ,the first whereof is this1 that if a man speak through a trunk which hath one end applied to the mouth of the speaker1 and the other to the ear of the hearer1 the sound will come stronger than it would do through the open air4 ,and the cause1 not only the possible1 but the certain and manifest cause is this1 that the air which is moved by the first breath and carried forwards in the trunk1 is not diffused as it would be in the open air1 and is consequently brought to the ear almost with the same velocity with which it was first breathed out4 ,whereas1 in the open air1 the first motion diffuseth itself every way into circles1 such as are made by the throwing of a stone into a standing water1 where the velocity grows less and less as the undulation proceeds further and further from the beginning of its motion4 ,the second is this1 that if the trunk be short1 and the end which is applied to the mouth be wider than that which is applied to the ear1 thus also the sound will be stronger than if it were made in the open air4 ,and the cause is the same1 namely1 that by how much the wider end of the trunk is less distant from the beginning of the sound1 by so much the less is the diffusion4 ,the third1 that it is easier for one1 that is within a chamber1 to hear what is spoken without1 than for him1 that stands without1 to hear what is spoken within4 ,for the windows and other inlets of the moved air are as the wide end of the trunk4 ,and for this reason some creatures seem to hear the better1 because nature has bestowed upon them wide and capacious ears4 ,the fourth is this1 that though he1 which standeth upon the sea-shore1 cannot hear the collision of the two nearest waves1 yet nevertheless he hears the roaring of the whole sea4 ,and the cause seems to be this1 that though the several collisions move the organ1 yet they are not severally great enough to cause sense2 whereas nothing hinders but that all of them together may make sound4 ,sidenote3 ,the difference between sounds acute and grave4 #c4 ,that bodies when they are stricken do yield some a more grave1 others a more acute sound1 the cause may consist in the difference of the times in which the parts stricken and forced out of their places return to the same places again4 ,for in some bodies1 the restitution of the moved parts is quick1 in others slow4 ,and this also may be the cause1 why the parts of the organ1 which are moved by the medium1 return to their rest again1 sometimes sooner1 sometimes later4 ,now1 by how much the vibrations or the reciprocal motions of the parts are more frequent1 by so much doth the whole sound made at the same time by one stroke consist of more1 and consequently of smaller parts4 ,for what is acute in sound1 the same is subtle in matter2 and both of them1 namely acute sound and subtle matter1 consist of very small parts1 that of time1 and this of the matter itself4 ,the third distinction of sounds cannot be conceived clearly enough by the names ,i have used of clear and hoarse1 nor by any other that ,i know2 and therefore it is needful to explain them by examples4 ,when ,i say hoarse1 ,i understand whispering and hissing1 and whatsoever is like to these1 by what appellation soever it be expressed4 ,and sounds of this kind seem to be made by the force of some strong wind1 raking rather than striking such hard bodies as it falls upon4 ,on the contrary1 when ,i use the word clear1 ,i do not understand such a sound as may be easily and distinctly heard2 for so whispers would be clear2 but such as is made by somewhat that is broken1 and such as is clamour1 tinkling1 the sound of a trumpet1 c4 and to express it significantly in one word1 noise4 ,and seeing no sound is made but by the concourse of two bodies at the least1 by which concourse it is necessary that there be as well reaction as action1 that is to say1 one motion opposite to another2 it follows that according as the proportion between those two opposite motions is diversified1 so the sounds which are made will be different from one another4 ,and whensoever the proportion between them is so great1 as that the motion of one of the bodies be insensible if compared with the motion of the other1 then the sound will not be of the same kind2 as when the wind falls very obliquely upon a hard body1 or when a hard body is carried swiftly through the air2 for then there is made that sound which ,i call a hoarse sound1 in ,greek 4 ,therefore the breath blown with violence from the mouth makes a hissing1 because in going out it rakes the superficies of the lips1 whose reaction against the force of the breath is not sensible4 ,and this is the cause why the winds have that hoarse sound4 ,also if two bodies1 how hard soever1 be rubbed together with no great pressure1 they make a hoarse sound4 ,and this hoarse sound1 when it is made1 as ,i have said1 by the air raking the superficies of a hard body1 seemeth to be nothing but the dividing of the air into innumerable and very small files4 ,for the asperity of the superficies doth1 by the eminences of its innumerable parts1 divide or cut in pieces the air that slides upon it4 ,sidenote3 ,the difference between clear hoarse sounds1 whence4 #d4 ,noise1 or that which ,i call clear sound1 is made two ways2 one1 by two hoarse sounds made by opposite motions2 the other1 by collision1 or by the sudden pulling asunder of two bodies1 whereby their small particles are put into commotion1 or being already in commotion suddenly restore themselves again2 which motion1 making impression upon the medium1 is propagated to the organ of hearing4 ,and seeing there is in this collision or divulsion an endeavour in the particles of one body1 opposite to the endeavour of the particles of the other body1 there will also be made in the organ of hearing a like opposition of endeavours1 that is to say1 of motions2 and consequently the sound arising from thence will be made by two opposite motions1 that is to say1 by two opposite hoarse sounds in one and the same part of the organ4 ,for1 as ,i have already said1 a hoarse sound supposeth the sensible motion of but one of the bodies4 ,and this opposition of motions in the organ is the cause why two bodies make a noise1 when they are either suddenly stricken against one another1 or suddenly broken asunder4 ,sidenote3 ,the sound of thunder and of a gun1 whence it proceeds4 #e4 ,this being granted1 and seeing withal that thunder is made by the vehement eruption of the air out of the cavities of congealed clouds1 the cause of the great noise or clap may be the sudden breaking asunder of the ice4 ,for in this action it is necessary that there be not only a great concussion of the small particles of the broken parts1 but also that this concussion1 by being communicated to the air1 be carried to the organ of hearing1 and make impression upon it4 ,and then1 from the first reaction of the organ proceeds that first and greatest sound1 which is made by the collision of the parts whilst they restore themselves4 ,and seeing there is in all concussion a reciprocation of motion forwards and backwards in the parts stricken2 for opposite motions cannot extinguish one another in an instant1 as ,i have shown in the #aath article of chapter ,,viii2 it follows necessarily that the sound will both continue1 and grow weaker and weaker1 till at last the action of the reciprocating air grow so weak1 as to be imperceptible4 ,wherefore a possible cause is given both of the first fierce noise of the thunder1 and also of the murmur that follows it4 ,the cause of the great sound from a discharged piece of ordnance is like that of a clap of thunder4 ,for the gunpowder being fired doth1 in its endeavour to go out1 attempt every way the sides of the metal in such manner1 as that it enlargeth the circumference all along1 and withal shorteneth the axis2 so that whilst the piece of ordnance is in discharging1 it is made both wider and shorter than it was before2 and therefore also presently after it is discharged its wideness will be diminished1 and its length increased again by the restitution of all the particles of the matter1 of which it consisteth1 to their former position4 ,and this is done with such motions of the parts1 as are not only very vehement1 but also opposite to one another2 which motions1 being communicated to the air1 make impression upon the organ1 and by the reaction of the organ create a sound1 which lasteth for some time2 as ,i have already shown in this article4 ,i note by the way1 as not belonging to this place1 that the possible cause why a gun recoils when it is shot off1 may be this2 that being first swollen by the force of the fire1 and afterwards restoring itself1 from this restitution there proceeds an endeavour from all the sides towards the cavity2 and consequently this endeavour is in those parts which are next the breech2 which being not hollow1 but solid1 the effect of the restitution is by it hindered and diverted into the length2 and by this means both the breech and the whole gun is thrust backwards2 and the more forcibly by how much the force is greater1 by which the part next the breech is restored to its former posture1 that is to say1 by how much the thinner is that part4 ,the cause1 therefore1 why guns recoil1 some more some less1 is the difference of their thickness towards the breech2 and the greater that thickness is1 the less they recoil2 and contrarily4 ,sidenote3 ,whence it is that pipes1 by blowing into them1 have a clear sound4 #f4 ,also the cause why the sound of a pipe1 which is made by blowing into it1 is nevertheless clear1 is the same with that of the sound which is made by collision4 ,for if the breath1 when it is blown into a pipe1 do only rake its concave superficies1 or fall upon it with a very sharp angle of incidence1 the sound will not be clear1 but hoarse4 ,but if the angle be great enough1 the percussion1 which is made against one of the hollow sides1 will be reverberated to the opposite side2 and so successive repercussions will be made from side to side1 till at last the whole concave superficies of the pipe be put into motion2 which motion will be reciprocated1 as it is in collision2 and this reciprocation being propagated to the organ1 from the reaction of the organ will arise a clear sound1 such as is made by collision1 or by breaking asunder of hard bodies4 ,in the same manner it is with the sound of a man's voice4 ,for when the breath passeth out without interruption1 and doth but lightly touch the cavities through which it is sent1 the sound it maketh is a hoarse sound4 ,but if in going out it strike strongly upon the larynx1 then a clear sound is made1 as in a pipe4 ,and the same breath1 as it comes in divers manners to the palate1 the tongue1 the lips1 the teeth1 and other organs of speech1 so the sounds into which it is articulated become different from one another4 ,sidenote3 ,of reflected sound4 #g4 ,i call that primary sound1 which is generated by motion from the sounding body to the organ in a strait line without reflection2 and ,i call that reflected sound1 which is generated by one or more reflections1 being the same with that we call echo1 and is iterated as often as there are reflections made from the object to the ear4 ,and these reflections are made by hills1 walls1 and other resisting bodies1 so placed as that they make more or fewer reflections of the motion1 according as they are themselves more or fewer in number2 and they make them more or less frequently1 according as they are more or less distant from one another4 ,now the cause of both these things is to be sought for in the situation of the reflecting bodies1 as is usually done in sight4 ,for the laws of reflection are the same in both1 namely1 that the angles of incidence and reflection be equal to one another4 ,if1 therefore1 in a hollow elliptic body1 whose inside is well polished1 or in two right parabolical solids1 which are joined together by one common base1 there be placed a sounding body in one of the burning points1 and the ear in the other1 there will be heard a sound by many degrees greater than in the open air2 and both this1 and the burning of such combustible things1 as being put in the same places are set on fire by the sun-beams1 are effects of one and the same cause4 ,but1 as when the visible object is placed in one of the burning points1 it is not distinctly seen in the other1 because every part of the object being seen in every line1 which is reflected from the concave superficies to the eye1 makes a confusion in the sight2 so neither is sound heard articulately and distinctly when it comes to the ear in all those reflected lines4 ,and this may be the reason why in churches which have arched roofs1 though they be neither elliptical nor parabolical1 yet because their figure is not much different from these1 the voice from the pulpit will not be heard so articulately as it would be1 if there were no vaulting at all4 ,sidenote3 ,from whence it is that sound is uniform and lasting4 #h4 ,concerning the uniformity and duration of sounds1 both which have one common cause1 we may observe1 that such bodies as being stricken yield an unequal or harsh sound1 are very heterogeneous1 that is to say1 they consist of parts which are very unlike both in figure and hardness1 such as are wood1 stones1 and others not a few4 ,when these are stricken1 there follows a concussion of their internal particles1 and a restitution of them again4 ,but they are neither moved alike1 nor have they the same action upon one another2 some of them recoiling from the stroke1 whilst others which have already finished their recoilings are now returning2 by which means they hinder and stop one another4 ,and from hence it is that their motions are not only unequal and harsh1 but also that their reciprocations come to be quickly extinguished4 ,whensoever1 therefore1 this motion is propagated to the ear1 the sound it makes is unequal and of small duration4 ,on the contrary1 if a body that is stricken be not only sufficiently hard1 but have also the particles of which it consisteth like to one another both in hardness and figure1 such as are the particles of glass and metals1 which being first melted do afterwards settle and harden2 the sound it yieldeth will1 because the motions of its parts and their reciprocations are like and uniform1 be uniform and pleasant1 and be more or less lasting1 according as the body stricken hath greater or less magnitude4 ,the possible cause1 therefore1 of sounds uniform and harsh1 and of their longer or shorter duration1 may be one and the same likeness and unlikeness of the internal parts of the sounding body1 in respect both of their figure and hardness4 ,besides1 if two plane bodies of the same matter and of equal thickness1 do both yield an uniform sound1 the sound of that body1 which hath the greatest extent of length1 will be the longest heard4 ,for the motion1 which in both of them hath its beginning from the point of percussion1 is to be propagated in the greater body through a greater space1 and consequently that propagation will require more time2 and therefore also the parts which are moved1 will require more time for their return4 ,wherefore all the reciprocations cannot be finished but in longer time2 and being carried to the ear1 will make the sound last the longer4 ,and from hence it is manifest1 that of hard bodies which yield an uniform sound1 the sound lasteth longer which comes from those that are round and hollow1 than from those that are plane1 if they be like in all other respects4 ,for in circular lines the action1 which begins at any point1 hath not from the figure any end of its propagation1 because the line in which it is propagated returns again to its beginning2 so that the figure hinders not but that the motion may have infinite progression4 ,whereas in a plane1 every line hath its magnitude finite1 beyond which the action cannot proceed4 ,if1 therefore1 the matter be the same1 the motion of the parts of that body whose figure is round and hollow1 will last longer than of that which is plane4 ,also1 if a string which is stretched be fastened at both ends to a hollow body1 and be stricken1 the sound will last longer than if it were not so fastened2 because the trembling or reciprocation which it receives from the stroke1 is by reason of the connection communicated to the hollow body2 and this trembling1 if the hollow body be great1 will last the longer by reason of that greatness4 ,wherefore also1 for the reason above mentioned1 the sound will last the longer4 ,sidenote3 ,how sound may be helped and hindered by the wind4 #i4 ,in hearing it happens1 otherwise than in seeing1 that the action of the medium is made stronger by the wind when it blows the same way1 and weaker when it blows the contrary way4 ,the cause whereof cannot proceed from anything but the different generation of sound and light4 ,for in the generation of light1 none of the parts of the medium between the object and the eye are moved from their own places to other places sensibly distant2 but the action is propagated in spaces imperceptible2 so that no contrary wind can diminish1 nor favourable wind encrease the light1 unless it be so strong as to remove the object further off or bring it nearer to the eye4 ,for the wind1 that is to say the air moved1 doth not by its interposition between the object and the eye work otherwise than it would do1 if it were still and calm4 ,for1 where the pressure is perpetual1 one part of the air is no sooner carried away1 but another1 by succeeding it1 receives the same impression1 which the part carried away had received before4 ,but in the generation of sound1 the first collision or breaking asunder beateth away and driveth out of its place the nearest part of the air1 and that to a considerable distance1 and with considerable velocity2 and as the circles grow by their remoteness wider and wider1 so the air being more and more dissipated1 hath also its motion more and more weakened4 ,whensoever therefore the air is so stricken as to cause sound1 if the wind fall upon it1 it will move it all nearer to the ear1 if it blow that way1 and further from it if it blow the contrary way2 so that according as it blows from or towards the object1 so the sound which is heard will seem to come from a nearer or remoter place2 and the reaction1 by reason of the unequal distances1 be strengthened or debilitated4 ,from hence may be understood the reason why the voice of such as are said to speak in their bellies1 though it be uttered near hand1 is nevertheless heard1 by those that suspect nothing1 as if it were a great way off4 ,for having no former thought of any determined place from which the voice should proceed1 and judging according to the greatness1 if it be weak they think it a great way off1 if strong near4 ,these ventriloqui1 therefore1 by forming their voice1 not as others by the emission of their breath1 but by drawing it inwards1 do make the same appear small and weak2 which weakness of the voice deceives those1 that neither suspect the artifice nor observe the endeavour which they use in speaking2 and so1 instead of thinking it weak1 they think it far off4 ,sidenote3 ,not only air1 but other bodies1 how hard soever they be1 convey sound4 #aj4 ,as for the medium1 which conveys sound1 it is not air only4 ,for water1 or any other body how hard soever1 may be that medium4 ,for the motion may be propagated perpetually in any hard continuous body2 but by reason of the difficulty1 with which the parts of hard bodies are moved1 the motion in going out of hard matter makes but a weak impression upon the air4 ,nevertheless1 if one end of a very long and hard beam be stricken1 and the ear be applied at the same time to the other end1 so that1 when the action goeth out of the beam1 the air1 which it striketh1 may immediately be received by the ear1 and be carried to the tympanum1 the sound will be considerably strong4 ,in like manner1 if in the night1 when all other noise which may hinder sound ceaseth1 a man lay his ear to the ground1 he will hear the sound of the steps of passengers1 though at a great distance2 because the motion1 which by their treading they communicate to the earth1 is propagated to the ear by the uppermost parts of the earth which receiveth it from their feet4 ,sidenote3 ,the causes of grave and acute sounds1 and of concent4 #aa4 ,i have shown above1 that the difference between grave and acute sounds consisteth in this1 that by how much the shorter the time is1 in which the reciprocations of the parts of a body stricken are made1 by so much the more acute will be the sound4 ,now by how much a body of the same bigness is either more heavy or less stretched1 by so much the longer will the reciprocations last2 and therefore heavier and less stretched bodies1 if they be like in all other respects1 will yield a graver sound than such as be lighter and more stretched4 ,as for the concent of sounds1 it is to be considered that the reciprocation or vibration of the air1 by which sound is made1 after it hath reached the drum of the ear1 imprinteth a like vibration upon the air that is inclosed within it2 by which means the sides of the drum within are stricken alternately4 ,now the concent of two sounds consists in this1 that the tympanum receives its sounding stroke from both the sounding bodies in equal and equally frequent spaces of time2 so that when two strings make their vibrations in the same times1 the concent they produce is the most exquisite of all other4 ,for the sides of the tympanum1 that is to say of the organ of hearing1 will be stricken by both those vibrations together at once1 on one side or other4 ,for example1 if the two equal strings ,a ,b and ,c ,d be stricken together1 and the latitudes of their vibrations ,e ,f and ,g ,h be also equal1 and the points ,e1 ,g1 ,f and ,h be in the concave superficies of the tympanum1 so that it receive strokes from both the strings together in ,e and ,g1 and again together in ,f and ,h1 the sound1 which is made by the vibrations of each string1 will be so like1 that it may be taken for the same sound1 and is called unison2 which is the greatest concord4 ,again1 the string ,a ,b retaining still its former vibration ,e ,f1 let the string ,c ,d be stretched till its vibration have double the swiftness it had before1 and let ,e ,f be divided equally in ,i4 ,in what time therefore the string ,c ,d makes one part of its vibration from ,g to ,h1 in the same time the string ,a ,b will make one part of its vibration from ,e to ,i2 and in what time the string ,c ,d hath made the other part of its vibration back from ,h to ,g1 in the same time another part of the vibration of the string ,a ,b will be made from ,i to ,f4 ,but the points ,f and ,g are both in the sides of the organ1 and therefore they will strike the organ both together1 not at every stroke1 but at every other stroke4 ,and this is the nearest concord to unison1 and makes that sound which is called an eighth4 ,again1 the vibration of the string ,a ,b remaining still the same it was1 let ,c ,d be stretched till its vibration be swifter than that of the string ,a ,b in the proportion of #c to #b1 and let ,e ,f be divided into three equal parts in ,k and ,l4 ,in what time therefore the string ,c ,d makes one third part of its vibration1 which third part is from ,g to ,h1 the string ,a ,b will make one third part of its vibration1 that is to say1 two-thirds of ,e ,f1 namely1 ,e ,l4 ,and in what time the string ,c ,d makes another third part of its vibration1 namely ,h ,g1 the string ,a ,b will make another third part of its vibration1 namely from ,l to ,f1 and back again from ,f to ,l4 ,lastly1 whilst the string ,c ,d makes the last third part of its vibration1 that is from ,g to ,h1 the string ,a ,b will make the last third part of its vibration from ,l to ,e4 ,but the points ,e and ,h are both in the sides of the organ4 ,wherefore1 at every third time1 the organ will be stricken by the vibration of both the strings together1 and make that concord which is called a fifth4 ,a-------,b ,c-------,d ,g ,e ,k ,i ,l ,h ,f ,sidenote3 ,phenomena of smelling4 #ab4 ,for the finding out the cause of smells1 ,i shall make use of the evidence of these following phenomena4 ,first1 that smelling is hindered by cold1 and helped by heat4 ,secondly1 that when the wind bloweth from the object1 the smell is the stronger2 and1 contrarily1 when it bloweth from the sentient towards the object1 the weaker2 both which phenomena are1 by experience1 manifestly found to be true in dogs1 which follow the track of beasts by the scent4 ,thirdly1 that such bodies1 as are less pervious to the fluid medium1 yield less smell than such as are more pervious2 as may be seen in stones and metals1 which1 compared with plants and living creatures1 and their parts1 fruits and excretions1 have very little or no smell at all4 ,fourthly1 that such bodies1 as are of their own nature odorous1 become yet more odorous when they are bruised4 ,fifthly1 that when the breath is stopped1 at least in men1 nothing can be smelt4 ,sixthly1 that the sense of smelling is also taken away by the stopping of the nostrils1 though the mouth be left open4 ,sidenote3 ,the first organ and the generation of smelling4 #ac4 ,by the fifth and sixth phenomenon it is manifest1 that the first and immediate organ of smelling is the innermost cuticle of the nostrils1 and that part of it1 which is below the passage common to the nostrils and the palate4 ,for when we draw breath by the nostrils we draw it into the lungs4 ,that breath1 therefore1 which conveys smells is in the way which passeth to the lungs1 that is to say1 in that part of the nostrils which is below the passage through which the breath goeth4 ,for1 nothing is smelt1 neither beyond the passage of the breath within1 nor at all without the nostrils4 ,and seeing that from different smells there must necessarily proceed some mutation in the organ1 and all mutation is motion2 it is therefore also necessary that1 in smelling1 the parts of the organ1 that is to say of that internal cuticle and the nerves that are inserted into it1 must be diversely moved by different smells4 ,and seeing also1 that it hath been demonstrated1 that nothing can be moved but by a body that is already moved and contiguous2 and that there is no other body contiguous to the internal membrane of the nostrils but breath1 that is to say attracted air1 and such little solid invisible bodies1 if there be any such1 as are intermingled with the air2 it follows necessarily1 that the cause of smelling is either the motion of that pure air or ethereal substance1 or the motion of those small bodies4 ,but this motion is an effect proceeding from the object of smell1 and1 therefore1 either the whole object itself or its several parts must necessarily be moved4 ,now1 we know that odorous bodies make odour1 though their whole bulk be not moved4 ,wherefore the cause of odour is the motion of the invisible parts of the odorous body4 ,and these invisible parts do either go out of the object1 or else1 retaining their former situation with the rest of the parts1 are moved together with them1 that is to say1 they have simple and invisible motion4 ,they that say1 there goes something out of the odorous body1 call it an effluvium2 which effluvium is either of the ethereal substance1 or of the small bodies that are intermingled with it4 ,but1 that all variety of odours should proceed from the effluvia of those small bodies that are intermingled with the ethereal substance1 is altogether incredible1 for these considerations2 first1 that certain unguents1 though very little in quantity1 do nevertheless send forth very strong odours1 not only to a great distance of place1 but also for a great continuance of time1 and are to be smelt in every point both of that place and time2 so that the parts issued out are sufficient to fill ten thousand times more space1 than the whole odorous body is able to fill2 which is impossible4 ,secondly1 that whether that issuing out be with strait or with crooked motion1 if the same quantity should flow from any other odorous body with the same motion1 it would follow that all odorous bodies would yield the same smell4 ,thirdly1 that seeing those effluvia have great velocity of motion 7as is manifest from this1 that noisome odours proceeding from caverns are presently smelt at a great distance7 it would follow1 that1 by reason there is nothing to hinder the passage of those effluvia to the organ1 such motion alone were sufficient to cause smelling2 which is not so2 for we cannot smell at all1 unless we draw in our breath through our nostrils4 ,smelling1 therefore1 is not caused by the effluvium of atoms2 nor1 for the same reason1 is it caused by the effluvium of ethereal substance2 for so also we should smell without the drawing in of our breath4 ,besides1 the ethereal substance being the same in all odorous bodies1 they would always affect the organ in the same manner2 and1 consequently1 the odours of all things would be alike4 ,it remains1 therefore1 that the cause of smelling must consist in the simple motion of the parts of odorous bodies without any efflux or diminution of their whole substance4 ,and by this motion there is propagated to the organ1 by the intermediate air1 the like motion1 but not strong enough to excite sense of itself without the attraction of air by respiration4 ,and this is a possible cause of smelling4 ,sidenote3 ,how smelling is helped by heat and by wind4 #ad4 ,the cause why smelling is hindered by cold and helped by heat may be this2 that heat1 as hath been shown in chapter ,,xxi1 generateth simple motion2 and therefore also1 wheresoever it is already1 there it will increase it2 and the cause of smelling being increased1 the smell itself will also be increased4 ,as for the cause why the wind blowing from the object makes the smell the stronger1 it is all one with that for which the attraction of air in respiration doth the same4 ,for1 he that draws in the air next to him1 draws with it by succession that air in which is the object4 ,now1 this motion of the air is wind1 and1 when another wind bloweth from the object1 will be increased by it4 ,sidenote3 ,why such bodies are least smelt1 which have the least intermixture of air in them4 #ae4 ,that bodies which contain the least quantity of air1 as stones and metals1 yield less smell than plants and living creatures2 the cause may be1 that the motion1 which causeth smelling1 is a motion of the fluid parts only2 which parts1 if they have any motion from the hard parts in which they are contained1 they communicate the same to the open air1 by which it is propagated to the organ4 ,where1 therefore1 there are no fluid parts as in metals1 or where the fluid parts receive no motion from the hard parts1 as in stones1 which are made hard by accretion1 there can be no smell4 ,and therefore also the water1 whose parts have little or no motion1 yieldeth no smell4 ,but1 if the same water1 by seeds and the heat of the sun1 be together with particles of earth raised into a plant1 and be afterwards pressed out again1 it will be odorous1 as wine from the vine4 ,and as water passing through plants is by the motion of the parts of those plants made an odorous liquor2 so also of air1 passing through the same plants whilst they are growing1 are made odorous airs4 ,and thus also it is with the juices and spirits1 which are bred in living creatures4 ,sidenote3 ,why odorous things become more odorous1 when bruised4 #af4 ,that odorous bodies may be made more odorous by contrition proceeds from this1 that being broken into many parts1 which are all odorous1 the air1 which by respiration is drawn from the object towards the organ1 doth in its passage touch upon all those parts1 and receive their motion4 ,now1 the air toucheth the superficies only2 and a body having less superficies whilst it is whole1 than all its parts together have after it is reduced to powder1 it follows that the same odorous body yieldeth less smell whilst it is whole1 than it will do after it is broken into smaller parts4 ,and thus much of smells4 ,sidenote3 ,the first organ of tasting2 and why some savours cause nauseousness4 #ag4 ,the taste follows2 whose generation hath this difference from that of the sight1 hearing1 and smelling1 that by these we have sense of remote objects2 whereas1 we taste nothing but what is contiguous1 and doth immediately touch either the tongue or palate1 or both4 ,from whence it is evident1 that the cuticles of the tongue and palate1 and the nerves inserted into them are the first organ of taste2 and 7because from the concussion of the parts of these1 there followeth necessarily a concussion of the pia mater7 that the action communicated to these is propagated to the brain1 and from thence to the farthest organ1 namely1 the heart1 in whose reaction consisteth the nature of sense4 ,now1 that savours1 as well as odours1 do not only move the brain but the stomach also1 as is manifest by the loathings that are caused by them both2 they1 that consider the organ of both these senses1 will not wonder at all2 seeing the tongue1 the palate and the nostrils1 have one and the same continued cuticle1 derived from the dura mater4 ,and that effluvia have nothing to do in the sense of tasting1 is manifest from this1 that there is no taste where the organ and the object are not contiguous4 ,by what variety of motions the different kinds of tastes1 which are innumerable1 may be distinguished1 ,i know not4 ,i might with others derive them from the divers figures of those atoms1 of which whatsoever may be tasted consisteth2 or from the diverse motions which ,i might1 by way of supposition1 attribute to those atoms2 conjecturing1 not without some likelihood of truth1 that such things as taste sweet have their particles moved with slow circular motion1 and their figures spherical2 which makes them smooth and pleasing to the organ2 that bitter things have circular motion1 but vehement1 and their figures full of angles1 by which they trouble the organ2 and that sour things have strait and reciprocal motion1 and their figures long and small1 so that they cut and wound the organ4 ,and in like manner ,i might assign for the causes of other tastes such several motions and figures of atoms1 as might in probability seem to be the true causes4 ,but this would be to revolt from philosophy to divination4 ,sidenote3 ,the first organ of feeling2 and how we came to the knowledge of such objects as are common to the touch and to other senses4 #ah4 ,by the touch1 we feel what bodies are cold or hot1 though they be distant from us4 ,others1 as hard1 soft1 rough1 and smooth1 we cannot feel unless they be contiguous4 ,the organ of touch is every one of those membranes1 which being continued from the pia mater are so diffused throughout the whole body1 as that no part of it can be pressed1 but the pia mater is pressed together with it4 ,whatsoever therefore presseth it1 is felt as hard or soft1 that is to say1 as more or less hard4 ,and as for the sense of rough1 it is nothing else but innumerable perceptions of hard and hard succeeding one another by short intervals both of time and place4 ,for we take notice of rough and smooth1 as also of magnitude and figure1 not only by the touch1 but also by memory4 ,for though some things are touched in one point1 yet rough and smooth1 like quantity and figure1 are not perceived but by the flux of a point1 that is to say1 we have no sense of them without time2 and we can have no sense of time without memory4 ------- ,,chapter ,,xxx4 ,,of ,,gravity4 #a4 ,a thick body doth not contain more matter1 unless also more place1 than a thin4--#b4 ,that the descent of heavy bodies proceeds not from their own appetite1 but from some power of the earth4--#c4 ,the difference of gravities proceedeth from the difference of the impetus with which the elements1 whereof heavy bodies are made1 do fall upon the earth4--#d4 ,the cause of the descent of heavy bodies4--#e4 ,in what proportion the descent of heavy bodies is accelerated4--#f4 ,why those that dive do not1 when they are under water1 feel the weight of the water above them4--#g4 ,the weight of a body that floateth1 is equal to the weight of so much water as would fill the space1 which the immersed part of the body takes up within the water4--#h4 ,if a body be lighter than water1 then how big soever that body be1 it may float upon any quantity of water1 how little soever4--#i4 ,how water may be lifted up and forced out of a vessel by air4--#aj4 ,why a bladder is heavier when blown full of air1 than when it is empty4--#aa4 ,the cause of the ejection upwards of heavy bodies from a wind-gun4--#ab4 ,the cause of the ascent of water in a weather-glass4--#ac4 ,the cause of motion upwards in living creatures4--#ad4 ,that there is in nature a kind of body heavier than air1 which nevertheless is not by sense distinguishable from it4--#ae4 ,of the cause of magnetical virtue4 ,sidenote3 ,a thick body doth not contain more matter1 unless also more place1 than a thin4 #a4 ,in chapter ,,xxi ,i have defined thick and thin1 as that place required1 so1 as that by thick was signified a more resisting body1 and by thin1 a body less resisting2 following the custom of those that have before me discoursed of refraction4 ,now if we consider the true and vulgar signification of those words1 we shall find them to be names collective1 that is to say1 names of multitude2 as thick to be that1 which takes up more parts of a space given1 and thin that1 which contains fewer parts of the same magnitude in the same space1 or in a space equal to it4 ,thick therefore is the same with frequent1 as a thick troop2 and thin the same with unfrequent1 as a thin rank1 thin of houses2 not that there is more matter in one place than in another equal place1 but a greater quantity of some named body4 ,for there is not less matter or body1 indefinitely taken1 in a desert1 than there is in a city2 but fewer houses1 or fewer men4 ,nor is there in a thick rank a greater quantity of body1 but a greater number of soldiers1 than in a thin4 ,wherefore the multitude and paucity of the parts contained within the same space do constitute density and rarity1 whether those parts be separated by vacuum or by air4 ,but the consideration of this is not of any great moment in philosophy2 and therefore ,i let it alone1 and pass on to the search of the causes of gravity4 ,sidenote3 ,that the descent of heavy bodies proceeds not from their own appetite1 but from some power of the earth4 #b4 ,now we call those bodies heavy1 which1 unless they be hindered by some force1 are carried towards the centre of the earth1 and that by their own accord1 for aught we can by sense perceive to the contrary4 ,some philosophers therefore have been of opinion1 that the descent of heavy bodies proceeded from some internal appetite1 by which1 when they were cast upwards1 they descended again1 as moved by themselves1 to such place as was agreeable to their nature4 ,others thought they were attracted by the earth4 ,to the former ,i cannot assent1 because ,i think ,i have already clearly enough demonstrated that there can be no beginning of motion1 but from an external and moved body2 and consequently1 that whatsoever hath motion or endeavour towards any place1 will always move or endeavour towards that same place1 unless it be hindered by the reaction of some external body4 ,heavy bodies1 therefore1 being once cast upwards1 cannot be cast down again but by external motion4 ,besides1 seeing inanimate bodies have no appetite at all1 it is ridiculous to think that by their own innate appetite they should1 to preserve themselves1 not understanding what preserves them1 forsake the place they are in1 and transfer themselves to another place2 whereas man1 who hath both appetite and understanding1 cannot1 for the preservation of his own life1 raise himself by leaping above three or four feet from the ground4 ,lastly1 to attribute to created bodies the power to move themselves1 what is it else than to say that there be creatures which have no dependance upon the ,creator8 ,to the latter1 who attribute the descent of heavy bodies to the attraction of the earth1 ,i assent4 ,but by what motion this is done1 hath not as yet been explained by any man4 ,i shall therefore in this place say somewhat of the manner and of the way by which the earth by its action attracteth heavy bodies4 ,sidenote3 ,the difference of gravities proceedeth from the difference of the impetus with which the elements1 whereof heavy bodies are made1 do fall upon the earth4 #c4 ,that by the supposition of simple motion in the sun1 homogeneous bodies are congregated and heterogeneous dissipated1 has already been demonstrated in the #eth article of chapter ,,xxi4 ,i have also supposed1 that there are intermingled with the pure air certain little bodies1 or1 as others call them1 atoms2 which by reason of their extreme smallness are invisible1 and differing from one another in consistence1 figure1 motion1 and magnitude2 from whence it comes to pass that some of them are congregated to the earth1 others to other planets1 and others are carried up and down in the spaces between4 ,and seeing those1 which are carried to the earth1 differ from one another in figure1 motion1 and magnitude1 they will fall upon the earth1 some with greater1 others with less impetus4 ,and seeing also that we compute the several degrees of gravity no otherwise than by this their falling upon the earth with greater or less impetus2 it follows1 that we conclude those to be the more heavy that have the greater impetus1 and those to be less heavy that have the less impetus4 ,our inquiry therefore must be1 by what means it may come to pass1 that of bodies1 which descend from above to the earth1 some are carried with greater1 others with less impetus2 that is to say1 some are more heavy than others4 ,we must also inquire1 by what means such bodies1 as settle upon the earth1 may by the earth itself be forced to ascend4 ,sidenote3 ,the cause of the descent of heavy bodies4 #d4 ,let the circle made upon the centre ,c 7in fig4 #b7 be a great circle in the superficies of the earth1 passing through the points ,a and ,b4 ,also let any heavy body1 as the stone ,a ,d1 be placed anywhere in the plane of the equator2 and let it be conceived to be cast up from ,a ,d perpendicularly1 or to be carried in any other line to ,e1 and supposed to rest there4 ,therefore1 how much space soever the stone took up in ,a ,d1 so much space it takes up now in ,e4 ,and because all place is supposed to be full1 the space ,a ,d will be filled by the air which flows into it first from the nearest places of the earth1 and afterwards successively from more remote places4 ,upon the centre ,c let a circle be understood to be drawn through ,e2 and let the plane space1 which is between the superficies of the earth and that circle1 be divided into plane orbs equal and concentric2 of which let that be the first1 which is contained by the two perimeters that pass through ,a and ,d4 ,whilst therefore the air1 which is in the first orb1 filleth the place ,a ,d1 the orb itself is made so much less1 and consequently its latitude is less than the strait line ,a ,d4 ,wherefore there will necessarily descend so much air from the orb next above4 ,in like manner1 for the same cause1 there will also be a descent of air from the orb next above that2 and so by succession from the orb in which the stone is at rest in ,e4 ,either therefore the stone itself1 or so much air1 will descend4 ,and seeing air is by the diurnal revolution of the earth more easily thrust away than the stone1 the air1 which is in the orb that contains the stone1 will be forced further upwards than the stone4 ,but this1 without the admission of vacuum1 cannot be1 unless so much air descend to ,e from the place next above2 which being done1 the stone will be thrust downwards4 ,by this means therefore the stone now receives the beginning of its descent1 that is to say1 of its gravity4 ,furthermore1 whatsoever is once moved1 will be moved continually 7as hath been shown in the #aith article of chapter ,,viii7 in the same way1 and with the same celerity1 except it be retarded or accelerated by some external movent4 ,now the air1 which is the only body that is interposed between the earth ,a and the stone above it ,e1 will have the same action in every point of the strait line ,e ,a1 which it hath in ,e4 ,but it depressed the stone in ,e2 and therefore also it will depress it equally in every point of the strait line ,e ,a4 ,wherefore the stone will descend from ,e to ,a with accelerated motion4 ,the possible cause therefore of the descent of heavy bodies under the equator1 is the diurnal motion of the earth4 ,and the same demonstration will serve1 if the stone be placed in the plane of any other circle parallel to the equator4 ,but because this motion hath1 by reason of its greater slowness1 less force to thrust off the air in the parallel circles than in the equator1 and no force at all at the poles1 it may well be thought 7for it is a certain consequent7 that heavy bodies descend with less and less velocity1 as they are more and more remote from the equator2 and that at the poles themselves1 they will either not descend at all1 or not descend by the axis2 which whether it be true or false1 experience must determine4 ,but it is hard to make the experiment1 both because the times of their descents cannot be easily measured with sufficient exactness1 and also because the places near the poles are inaccessible4 ,nevertheless1 this we know1 that by how much the nearer we come to the poles1 by so much the greater are the flakes of the snow that falls2 and by how much the more swiftly such bodies descend as are fluid and dissipable1 by so much the smaller are the particles into which they are dissipated4 ,sidenote3 ,in what proportion the descent of heavy bodies is accelerated4 #e4 ,supposing1 therefore1 this to be the cause of the descent of heavy bodies1 it will follow that their motion will be accelerated in such manner1 as that the spaces1 which are transmitted by them in the several times1 will have to one another the same proportion which the odd numbers have in succession from unity4 ,for if the strait line ,e ,a be divided into any number of equal parts1 the heavy body descending will1 by reason of the perpetual action of the diurnal motion1 receive from the air in every one of those times1 in every several point of the strait line ,e ,a1 a several new and equal impulsion2 and therefore also in every one of those times1 it will acquire a several and equal degree of celerity4 ,and from hence it follows1 by that which ,galileus hath in his ,dialogues of ,motion demonstrated1 that heavy bodies descend in the several times with such differences of transmitted spaces1 as are equal to the differences of the square numbers that succeed one another from unity2 which square numbers being #a1 #d1 #i1 #af1 c4 their differences are #c1 #e1 #g1 c42 that is to say1 the odd numbers which succeed one another from unity4 ,against this cause of gravity which ,i have given1 it will perhaps be objected1 that if a heavy body be placed in the bottom of some hollow cylinder of iron or adamant1 and the bottom be turned upwards1 the body will descend1 though the air above cannot depress it1 much less accelerate its motion4 ,but it is to be considered that there can be no cylinder or cavern1 but such as is supported by the earth1 and being so supported is1 together with the earth1 carried about by its diurnal motion4 ,for by this means the bottom of the cylinder will be as the superficies of the earth2 and by thrusting off the next and lowest air1 will make the uppermost air depress the heavy body1 which is at the top of the cylinder1 in such manner as is above explicated4 ,sidenote3 ,why those that dive1 do not1 when they are under water1 feel the weight of the water above them4 #f4 ,the gravity of water being so great as by experience we find it is1 the reason is demanded by many1 why those that dive1 how deep soever they go under water1 do not at all feel the weight of the water which lies upon them4 ,and the cause seems to be this1 that all bodies by how much the heavier they are1 by so much the greater is the endeavour by which they tend downwards4 ,but the body of a man is heavier than so much water as is equal to it in magnitude1 and therefore the endeavour downwards of a man's body is greater than that of water4 ,and seeing all endeavour is motion1 the body also of a man will be carried towards the bottom with greater velocity than so much water4 ,wherefore there is greater reaction from the bottom2 and the endeavour upwards is equal to the endeavour downwards1 whether the water be pressed by water1 or by another body which is heavier than water4 ,and therefore by these two opposite equal endeavours1 the endeavour both ways in the water is taken away2 and consequently1 those that dive are not at all pressed by it4 ,coroll4 ,from hence also it is manifest1 that water in water hath no weight at all1 because all the parts of water1 both the parts above1 and the parts that are directly under1 tend towards the bottom with equal endeavour and in the same strait lines4 ,sidenote3 ,the weight of a body that floateth1 is equal to the weight of so much water as would fill the space which the immersed part of the body takes up within the water4 #g4 ,if a body float upon the water1 the weight of that body is equal to the weight of so much water as would fill the place which the immersed part of the body takes up within the water4 ,let ,e ,f 7in fig4 #c7 be a body floating in the water ,a ,b ,c ,d2 and let the part ,e be above1 and the other part ,f under the water4 ,i say1 the weight of the whole body ,e ,f is equal to the weight of so much water as the space ,f will receive4 ,for seeing the weight of the body ,e ,f forceth the water out of the space ,f1 and placeth it upon the superficies ,a ,b1 where it presseth downwards2 it follows1 that from the resistance of the bottom there will also be an endeavour upwards4 ,and seeing again1 that by this endeavour of the water upwards1 the body ,e ,f is lifted up1 it follows1 that if the endeavour of the body downwards be not equal to the endeavour of the water upwards1 either the whole body ,e ,f will1 by reason of that inequality of their endeavours or moments1 be raised out of the water1 or else it will descend to the bottom4 ,but it is supposed to stand so1 as neither to ascend nor descend4 ,wherefore there is an equilibrium between the two endeavours2 that is to say1 the weight of the body ,e ,f is equal to the weight of so much water as the space ,f will receive2 which was to be proved4 ,sidenote3 ,if a body be lighter than water1 then how big soever that body be1 it may float upon any quantity of water1 how little soever4 #h4 ,from hence it follows1 that any body1 of how great magnitude soever1 provided it consist of matter less heavy than water1 may nevertheless float upon any quantity of water1 how little soever4 ,let ,a ,b ,c ,d 7in fig4 #d7 be a vessel2 and in it let ,e ,f ,g ,h be a body consisting of matter which is less heavy than water2 and let the space ,a ,g ,c ,f be filled with water4 ,i say1 the body ,e ,f ,g ,h will not sink to the bottom ,d ,c4 ,for seeing the matter of the body ,e ,f ,g ,h is less heavy than water1 if the whole space without ,a ,b ,c ,d were fall of water1 yet some part of the body ,e ,f ,g ,h1 as ,e ,f ,i ,k1 would be above the water2 and the weight of so much water as would fill the space ,i ,g ,h ,k would be equal to the weight of the whole body ,e ,f ,g ,h2 and consequently ,g ,h would not touch the bottom ,d ,c4 ,as for the sides of the vessel1 it is no matter whether they be hard or fluid2 for they serve only to terminate the water2 which may be done as well by water as by any other matter how hard soever2 and the water without the vessel is terminated somewhere1 so as that it can spread no farther4 ,the part therefore ,e ,f ,i ,k will be extant above the water ,a ,g ,c ,f which is contained in the vessel4 ,wherefore the body ,e ,f ,g ,h will also float upon the water ,a ,g ,c ,f1 how little soever that water be2 which was to be demonstrated4 ,sidenote3 ,how water may be lifted up and forced out of a vessel by air4 #i4 ,in the #dth article of chapter ,,xxvi1 there is brought for the proving of vacuum the experiment of water enclosed in a vessel2 which water1 the orifice above being opened1 is ejected upwards by the impulsion of the air4 ,it is therefore demanded1 seeing water is heavier than air1 how that can be done4 ,let the second figure of the same1 chapter ,,xxvi be considered1 where the water is with great force injected by a syringe into the space ,f ,g ,b4 ,in that injection1 the air 7but pure air7 goeth with the same force out of the vessel through the injected water4 ,but as for those small bodies1 which formerly ,i supposed to be intermingled with air and to be moved with simple motion1 they cannot1 together with the pure air1 penetrate the water2 but remaining behind are necessarily thrust together into a narrower place1 namely into the space which is above the water ,f ,g4 ,the motions therefore of those small bodies will be less and less free1 by how much the quantity of the injected water is greater and greater2 so that by their motions falling upon one another1 the same small bodies will mutually compress each other1 and have a perpetual endeavour of regaining their liberty1 and of depressing the water that hinders them4 ,wherefore1 as soon as the orifice above is opened1 the water which is next it will have an endeavour to ascend1 and will therefore necessarily go out4 ,but it cannot go out1 unless at the same time there enter in as much air2 and therefore both the water will go out1 and the air enter in1 till those small bodies which were left within the vessel have recovered their former liberty of motion2 that is to say1 till the vessel be again filled with air1 and no water be left of sufficient height to stop the passage at ,b4 ,wherefore ,i have shown a possible cause of this phenomenon1 namely1 the same with that of thunder4 ,for as in the generation of thunder1 the small bodies enclosed within the clouds1 by being too closely pent together1 do by their motion break the clouds1 and restore themselves to their natural liberty2 so here also the small bodies enclosed within the space which is above the strait line ,f ,g1 do by their own motion expel the water as soon as the passage is opened above4 ,and if the passage be kept stopped1 and these small bodies be more vehemently compressed by the perpetual forcing in of more water1 they will at last break the vessel itself with great noise4 ,sidenote3 ,why a bladder is heavier when blown full of air1 than when it is empty4 #aj4 ,if air be blown into a hollow cylinder1 or into a bladder1 it will increase the weight of either of them a little1 as many have found by experience1 who with great accurateness have tried the same4 ,and it is no wonder1 seeing1 as ,i have supposed1 there are intermingled with the common air a great number of small hard bodies1 which are heavier than the pure air4 ,for1 the ethereal substance1 being on all sides equally agitated by the motion of the sun1 hath an equal endeavour towards all the parts of the universe2 and1 therefore1 it hath no gravity at all4 ,sidenote3 ,the cause of the ejection upwards of heavy bodies from a wind-gun4 #aa4 ,we find also by experience1 that1 by the force of air enclosed in a hollow cannon1 a bullet of lead may with considerable violence be shot out of a gun of late invention1 called the wind-gun4 ,in the end of this cannon there are two holes1 with their valves on the inside1 to shut them close2 one of them serving for the admission of air1 and the other for the letting of it out4 ,also1 to that end which serves for the receiving in of air1 there is joined another cannon of the same metal and bigness1 in which there is fitted a rammer which is perforated1 and hath also a valve opening towards the former cannon4 ,by the help of this valve the rammer is easily drawn back1 and letteth in air from without2 and being often drawn back and returned again with violent strokes1 it forceth some part of that air into the former cannon1 so long1 till at last the resistance of the enclosed air is greater than the force of the stroke4 ,and by this means men think there is now a greater quantity of air in the cannon than there was formerly1 though it were full before4 ,also1 the air thus forced in1 how much soever it be1 is hindered from getting out again by the aforesaid valves1 which the very endeavour of the air to get out doth necessarily shut4 ,lastly1 that valve being opened which was made for the letting out of the air1 it presently breaketh out with violence1 and driveth the bullet before it with great force and velocity4 ,as for the cause of this1 ,i could easily attribute it1 as most men do1 to condensation1 and think that the air1 which had at the first but its ordinary degree of rarity1 was afterwards1 by the forcing in of more air1 condensed1 and last of all1 rarified again by being let out and restored to its natural liberty4 ,but ,i cannot imagine how the same place can be always full1 and1 nevertheless1 contain some times a greater1 sometimes a less quantity of matter2 that is to say1 that it can be fuller than full4 ,nor can ,i conceive how fulness can of itself be an efficient cause of motion4 ,for both these are impossible4 ,wherefore we must seek out some other possible cause of this phenomenon4 ,whilst1 therefore1 the valve which serves for the letting in of air1 is opened by the first stroke of the rammer1 the air within doth with equal force resist the entering of the air from without2 so that the endeavours between the internal and external air are opposite1 that is1 there are two opposite motions whilst the one goeth in and the other cometh out2 but no augmentation at all of air within the cannon4 ,for there is driven out by the stroke as much pure air1 which passeth between the rammer and the sides of the cannon1 as there is forced in of air impure by the same stroke4 ,and thus1 by many forcible strokes1 the quantity of small hard bodies will be increased within the cannon1 and their motions also will grow stronger and stronger1 as long as the matter of the cannon is able to endure their force2 by which1 if it be not broken1 it will at least be urged every way by their endeavour to free themselves2 and as soon as the valve1 which serves to let them out1 is opened1 they will fly out with violent motion1 and carry with them the bullet which is in their way4 ,wherefore1 ,i have given a possible cause of this phenomenon4 ,sidenote3 ,the cause of the ascent of water in a weather-glass4 #ab4 ,water1 contrary to the custom of heavy bodies1 ascendeth in the weather-glass2 but it doth it when the air is cold3 for when it is warm it descendeth again4 ,and this organ is called a thermometer or thermoscope1 because the degrees of heat and cold are measured and marked by it4 ,it is made in this manner4 ,let ,a ,b ,c ,d 7in fig4 #e7 be a vessel full of water1 and ,e ,f ,g a hollow cylinder of glass1 closed at ,e and open at ,g4 ,let it be heated1 and set upright within the water to ,f2 and let the open end reach to ,g4 ,this being done1 as the air by little and little grows colder1 the water will ascend slowly within the cylinder from ,f towards ,e2 till at last the external and internal air coming to be both of the same temper1 it will neither ascend higher nor descend lower1 till the temper of the air be changed4 ,suppose it1 therefore1 to be settled anywhere1 as at ,h4 ,if now the heat of the air be augmented1 the water will descend below ,h2 and if the heat be diminished1 it will ascend above it4 ,which1 though it be certainly known to be true by experience1 the cause1 nevertheless1 hath not as yet been discovered4 ,in the sixth and seventh articles of chapter ,,xxviii1 where ,i consider the cause of cold1 ,i have shown1 that fluid bodies are made colder by the pressure of the air1 that is to say1 by a constant wind that presseth them4 ,for the same cause it is1 that the superficies of the water is pressed at ,f2 and having no place1 to which it may retire from this pressure1 besides the cavity of the cylinder between ,h and ,e1 it is therefore necessarily forced thither by the cold1 and consequently it ascendeth more or less1 according as the cold is more or less increased4 ,and again1 as the heat is more intense or the cold more remiss1 the same water will be depressed more or less by its own gravity1 that is to say1 by the cause of gravity above explicated4 ,sidenote3 ,cause of motion upwards in living creatures #ac4 ,also living creatures1 though they be heavy1 can by leaping1 swimming and flying1 raise themselves to a certain degree of height4 ,but they cannot do this except they be supported by some resisting body1 as the earth1 the water and the air4 ,for these motions have their beginning from the contraction1 by the help of the muscles1 of the body animate4 ,for to this contraction there succeedeth a distension of their whole bodies2 by which distension1 the earth1 the water1 or the air1 which supporteth them1 is pressed2 and from hence1 by the reaction of those pressed bodies1 living creatures acquire an endeavour upwards1 but such as by reason of the gravity of their bodies is presently lost again4 ,by this endeavour1 therefore1 it is1 that living creatures raise themselves up a little way by leaping1 but to no great purpose3 but by swimming and flying they raise themselves to a greater height2 because1 before the effect of their endeavour is quite extinguished by the gravity of their bodies1 they can renew the same endeavour again4 ,that by the power of the soul1 without any antecedent contraction of the muscles or the help of something to support him1 any man can be able to raise his body upwards1 is a childish conceit4 ,for if it were true1 a man might raise himself to what height he pleased4 ,sidenote3 ,that there is in nature a kind of body heavier than air1 which nevertheless is not by sense distinguishable from it4 #ad4 ,the diaphanous medium1 which surrounds the eye on all sides1 is invisible2 nor is air to be seen in air1 nor water in water1 nor anything but that which is more opacous4 ,but in the confines of two diaphanous bodies1 one of them may be distinguished from the other4 ,it is not therefore a thing so very ridiculous for ordinary people to think all that space empty1 in which we say is air2 it being the work of reason to make us conceive that the air is anything4 ,for by which of our senses is it1 that we take notice of the air1 seeing we neither see1 nor hear1 nor taste1 nor smell1 nor feel it to be anything8 ,when we feel heat1 we do not impute it to the air1 but to the fire3 nor do we say the air is cold1 but we ourselves are cold2 and when we feel the wind1 we rather think something is coming1 than that any thing is already come4 ,also1 we do not at all feel the weight of water in water1 much less of air in air4 ,that we come to know that to be a body1 which we call air1 it is by reasoning2 but it is from one reason only1 namely1 because it is impossible for remote bodies to work upon our organs of sense but by the help of bodies intermediate1 without which we could have no sense of them1 till they come to be contiguous4 ,wherefore1 from the senses alone1 without reasoning from effects1 we cannot have sufficient evidence of the nature of bodies4 ,for there is underground1 in some mines of coals1 a certain matter of a middle nature between water and air1 which nevertheless cannot by sense be distinguished from air2 for it is as diaphanous as the purest air2 and1 as far as sense can judge1 equally penetrable4 ,but if we look upon the effect1 it is like that of water4 ,for when that matter breaks out of the earth into one of those pits1 it fills the same either totally or to some degree2 and if a man or fire be then let down in it1 it extinguishes them in almost as little time as water would do4 ,but for the better understanding of this phenomenon1 ,i shall describe the #fth figure4 ,in which let ,a ,b represent the pit of the mine2 and let part thereof1 namely ,c ,b1 be supposed to be filled with that matter4 ,if now a lighted candle be let down into it below ,c1 it will as suddenly be extinguished as if it were thrust into water4 ,also1 if a grate filled with coals thoroughly kindled and burning never so brightly1 be let down1 as soon as ever it is below ,c1 the fire will begin to grow pale1 and shortly after1 losing its light1 be extinguished1 no otherwise than if it were quenched in water4 ,but if the grate be drawn up again presently1 whilst the coals are still very hot1 the fire will1 by little and little1 be kindled again1 and shine as before4 ,there is1 indeed1 between this matter and water this considerable difference1 that it neither wetteth1 nor sticketh to such things as are put down into it1 as water doth2 which1 by the moisture it leaveth1 hindereth the kindling again of the matter once extinguished4 ,in like manner1 if a man be let down below ,c1 he will presently fall into a great difficulty of breathing1 and immediately after into a swoon1 and die unless he be suddenly drawn up again4 ,they1 therefore1 that go down into these pits1 have this custom1 that as soon as ever they feel themselves sick1 they shake the rope by which they were let down1 to signify they are not well1 and to the end that they may speedily be pulled up again4 ,for if a man be drawn out too late1 void of sense and motion1 they dig up a turf1 and put his face and mouth into the fresh earth2 by which means1 unless he be quite dead1 he comes to himself again1 by little and little1 and recovers life by breathing out1 as it were1 of that suffocating matter1 which he had sucked in whilst he was in the pit2 almost in the same manner as they that are drowned come to themselves again by vomiting up the water4 ,but this doth not happen in all mines1 but in some only2 and in those not always1 but often4 ,in such pits as are subject to it1 they use this remedy4 ,they dig another pit1 as ,d ,e1 close by it1 of equal depth1 and joining them both together with one common channel1 ,e ,b1 they make a fire in the bottom ,e1 which carries out at ,d the air contained in the pit ,d ,e2 and this draws with it the air contained in the channel ,e ,b2 which1 in like manner1 is followed by the noxious matter contained in ,c ,b2 and1 by this means1 the pit is for that time made healthful4 ,out of this history1 which ,i write only to such as have had experience of the truth of it1 without any design to support my philosophy with stories of doubtful credit1 may be collected the following possible cause of this phenomenon2 namely1 that there is a certain matter fluid and most transparent1 and not much lighter than water1 which1 breaking out of the earth1 fills the pit to ,c2 and that in this matter1 as in water1 both fire and living creatures are extinguished4 ,sidenote3 ,of the cause of magnetical virtue4 #ae4 ,about the nature of heavy bodies1 the greatest difficulty ariseth from the contemplation of those things which make other heavy bodies ascend to them2 such as jet1 amber1 and the loadstone4 ,but that which troubles men most is the loadstone1 which is also called ,lapis ,herculeus2 a stone1 though otherwise despicable1 yet of so great power that it taketh up iron from the earth1 and holds it suspended in the air1 as ,hercules did ,antus4 ,nevertheless1 we wonder at it somewhat the less1 because we see jet draw up straws1 which are heavy bodies1 though not so heavy as iron4 ,but as for jet1 it must first be excited by rubbing1 that is to say1 by motion to and fro2 whereas the loadstone hath sufficient excitation from its own nature1 that is to say1 from some internal principle of motion peculiar to itself4 ,now1 whatsoever is moved1 is moved by some contiguous and moved body1 as hath been formerly demonstrated4 ,and from hence it follows evidently1 that the first endeavour1 which iron hath towards the loadstone1 is caused by the motion of that air which is contiguous to the iron3 also1 that this motion is generated by the motion of the next air1 and so on successively1 till by this succession we find that the motion of all the intermediate air taketh its beginning from some motion which is in the loadstone itself2 which motion1 because the loadstone seems to be at rest1 is invisible4 ,it is therefore certain1 that the attractive power of the loadstone is nothing else but some motion of the smallest particles thereof4 ,supposing1 therefore1 that those small bodies1 of which the loadstone is in the bowels of the earth composed1 have by nature such motion or endeavour as was above attributed to jet1 namely1 a reciprocal motion in a line too short to be seen1 both those stones will have one and the same cause of attraction4 ,now in what manner and in what order of working this cause produceth the effect of attraction1 is the thing to be enquired4 ,and first we know1 that when the string of a lute or viol is stricken1 the vibration1 that is1 the reciprocal motion of that string in the same strait line1 causeth like vibration in another string which hath like tension4 ,we know also1 that the dregs or small sands1 which sink to the bottom of a vessel1 will be raised up from the bottom by any strong and reciprocal agitation of the water1 stirred with the hand or with a staff4 ,why1 therefore1 should not reciprocal motion of the parts of the loadstone contribute as much towards the moving of iron8 ,for1 if in the loadstone there be supposed such reciprocal motion1 or motion of the parts forwards and backwards1 it will follow that the like motion will be propagated by the air to the iron1 and consequently that there will be in all the parts of the iron the same reciprocations or motions forwards and backwards4 ,and from hence also it will follow1 that the intermediate air between the stone and the iron will1 by little and little1 be thrust away2 and the air being thrust away1 the bodies of the loadstone and the iron will necessarily come together4 ,the possible cause therefore why the loadstone and jet draw to them1 the one iron1 the other straws1 may be this1 that those attracting bodies have reciprocal motion either in a strait line1 or in an elliptical line1 when there is nothing in the nature of the attracted bodies which is repugnant to such a motion4 ,but why the loadstone1 if with the help of cork it float at liberty upon the top of the water1 should from any position whatsoever so place itself in the plane of the meridian1 as that the same points1 which at one time of its being at rest respect the poles of the earth1 should at all other times respect the same poles1 the cause may be this2 that the reciprocal motion1 which ,i supposed to be in the parts of the stone1 is made in a line parallel to the axis of the earth1 and has been in those parts ever since the stone was generated4 ,seeing therefore1 the stone1 whilst it remains in the mine1 and is carried about together with the earth by its diurnal motion1 doth by length of time get a habit of being moved in a line which is perpendicular to the line of its reciprocal motion1 it will afterwards1 though its axis be removed from the parallel situation it had with the axis of the earth1 retain its endeavour of returning to that situation again2 and all endeavour being the beginning of motion1 and nothing intervening that may hinder the same1 the loadstone will therefore return to its former situation4 ,for1 any piece of iron that has for a long time rested in the plane of the meridian1 whensoever it is forced from that situation and afterwards left to its own liberty again1 will of itself return to lie in the meridian again2 which return is caused by the endeavour it acquired from the diurnal motion of the earth in the parallel circles which are perpendicular to the meridians4 ,if iron be rubbed by the loadstone drawn from one pole to the other1 two things will happen2 one1 that the iron will acquire the same direction with the loadstone1 that is to say1 that it will lie in the meridian1 and have its axis and poles in the same position with those of the stone2 the other1 that the like poles of the stone and of the iron will avoid one another1 and the unlike poles approach one another4 ,and the cause of the former may be this1 that iron being touched by motion which is not reciprocal1 but drawn the same way from pole to pole1 there will be imprinted in the iron also an endeavour from the same pole to the same pole4 ,for seeing the loadstone differs from iron no otherwise than as ore from metal1 there will be no repugnance at all in the iron to receive the same motion which is in the stone4 ,from whence it follows1 that seeing they are both affected alike by the diurnal motion of the earth1 they will both equally return to their situation in the meridian1 whensoever they are put from the same4 ,also1 of the latter this may be the cause1 that as the loadstone in touching the iron doth by its action imprint in the iron an endeavour towards one of the poles1 suppose towards the ,north ,pole2 so reciprocally1 the iron by its action upon the loadstone doth imprint in it an endeavour towards the other pole1 namely towards the ,south ,pole4 ,it happens therefore in these reciprocations or motions forwards and backwards of the particles of the stone and of the iron betwixt the north and the south1 that whilst in one of them the motion is from north to south1 and the return from south to north1 in the other the motion will be from south to north1 and the return from north to south2 which motions being opposite to one another1 and communicated to the air1 the north pole of the iron1 whilst the attraction is working1 will be depressed towards the south pole of the loadstone2 or contrarily1 the north pole of the loadstone will be depressed towards the south pole of the iron2 and the axis both of the loadstone and of the iron will be situate in the same strait line4 ,the truth whereof is taught us by experience4 ,as for the propagation of this magnetical virtue1 not only through the air1 but through any other bodies how hard soever1 it is not to be wondered at1 seeing no motion can be so weak1 but that it may be propagated infinitely through a space filled with body of any hardness whatsoever4 ,for in a full medium1 there can be no motion which doth not make the next part yield1 and that the next1 and so successively without end2 so that there is no effect whatsoever1 but to the production thereof something is necessarily contributed by the several motions of all the several things that are in the world4 ,sidenote3 ,conclusion4 ,and thus much concerning the nature of body in general2 with which ,i conclude this my first section of the ,elements of ,philosophy4 ,in the first1 second1 and third parts1 where the principles of ratiocination consist in our own understanding1 that is to say1 in the legitimate use of such words as we ourselves constitute1 all the theorems1 if ,i be not deceived1 are rightly demonstrated4 ,the fourth part depends upon hypotheses2 which unless we know them to be true1 it is impossible for us to demonstrate that those causes1 which ,i have there explicated1 are the true causes of the things whose productions ,i have derived from them4 ,nevertheless1 seeing ,i have assumed no hypothesis1 which is not both possible and easy to be comprehended2 and seeing also that ,i have reasoned aright from those assumptions1 ,i have withal sufficiently demonstrated that they may be the true causes2 which is the end of physical contemplation4 ,if any other man from other hypotheses shall demonstrate the same or greater things1 there will be greater praise and thanks due to him than ,i demand for myself1 provided his hypotheses be such as are conceivable4 ,for as for those that say anything may be moved or produced by itself1 by species1 by its own power1 by substantial forms1 by incorporeal substances1 by instinct1 by antiperistasis1 by antipathy1 sympathy1 occult quality1 and other empty words of schoolmen1 their saying so is to no purpose4 ,and now ,i proceed to the phenomena of man's body2 where ,i shall speak of the optics1 and of the dispositions1 affections1 and manners of men1 if it shall please ,god to give me life1 and show their causes4 ,illustration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxviii ,,xxx4 ,fig4 #a-f ,,end ,,of ,,vol4 ,i4 ,,london3 ,c4 ,,richards1 ,,st4 ,,martin',s ,,lane1 ,,charing ,,cross4 ---------------------------------------- -------------------------------- ,transcriber's ,note ,figures #ah4b and #bg4b are not mentioned by name in the text4 ,figure #ad4aj is mentioned only in ,chapter ,,xiii4 ,however1 ,figure #b of ,chapter ,,xviii would appear to be the subject of a discussion on p4 #bgj4 ,a sidenote has been added1 and includes a reference to the figure4 ,likewise ,figure #b of ,chapter ,,xxvii1 depicting a #8prisma'1 is discussed in detail on p4 #dei et seq44 ,again1 a sidenote has been interpolated at the opening of that description1 and includes a reference to the figure4 ,the diacritics used in the ,greek phrases is followed as printed1 though they sometimes seem non-standard 7i4e41 rather than 74 ,in each chapter's summary1 the article numbers are prefixed with a dash if they appeared midline4 ,dashes have been added to the text in the case where the article number happened to appear in the first column4 ,on rare occasions1 there are errors in the numbering of the articles in the chapter summary4 ,for example1 in ,chapter ,,xvii1 article #g was misnumbered as a second #8#f'4 ,in article #ah of ,chapter ,,xiii1 a reference to article #aj of that chapter used the ,roman #8,x' which is the only instance of that usage4 ,it is now referred to as article #aj4 ,other errors deemed most likely to be the printer's have been corrected1 and are noted here4 ,the references are to the page and line in the original4 #eb4ag every man is a living creature ,restored4 #gc4ad in the first place1 we we are to search ,removed4 #bee4aa seeing the proproportion ,removed4 #bfg4bb ,and therefore if it it be again as ,r to ,t ,removed4 #cjg4bd the spiral of ,archi_/emides ,replaced4 #cbi4bi with simple motion the the perimeter ,removed4 #cci4aa be cut in ,e and and ,g by a strait line ,removed4 #cgb4ab to the base ,f ,h that that can possibly be ,removed4 drawn #dfd4bc the sea appears black when ruffled ,inserted4 #dci4b will make fifty-nine revolutions ,inserted4 ,chapter ,,xiv ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f4 ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h4 ,illustration3 ,fig4 #i4 ,illustration3 ,fig4 #aj4 ,,chapter ,,xvi ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f4 ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h4 ,illustration3 ,fig4 #i4 ,illustration3 ,fig4 #aj4 ,illustration3 ,fig4 #aa4 ,chapter ,,xvii ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f4 ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h4 ,chapter ,,xviii ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,chapter ,,xix ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f4 ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h4 ,illustration3 ,fig4 #i4 ,illustration3 ,fig4 #aj4 ,chapter ,,xx ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f4 ,chapter ,,xxi ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d4 ,illustration3 ,fig4 #e4 ,chapter ,,xxii ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,illustration3 ,fig4 #c4 ,chapter ,,xxiii ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b ,illustration3 ,fig4 #c4 ,illustration3 ,fig4 #d ,illustration3 ,fig4 #e4 ,illustration3 ,fig4 #f ,illustration3 ,fig4 #g4 ,illustration3 ,fig4 #h ,illustration3 ,fig #i4 ,,chapter ,,xxiv4 ,illustration3 ,fig #a4 ,illustration3 ,fig #b4 ,illustration3 ,fig #c4 ,illustration3 ,fig #d4 ,illustration3 ,fig #e4 ,illustration3 ,fig #f4 ,,chapter ,,xxvi4 ,illustration3 ,fig #a4 ,illustration3 ,fig #b4 ,illustration3 ,fig #c4 ,illustration3 ,fig #d4 ,illustration3 ,fig #e4 ,chapter ,,xxvii4 ,illustration3 ,fig #a4 ,illustration3 ,fig #b4 ,chapter ,,xxx4 ,illustration3 ,fig #a4 ,illustration3 ,fig #b4 ,illustration3 ,fig #c4 ,illustration3 ,fig #d4 ,illustration3 ,fig #e4 ,illustration3 ,fig #f4