,,transcriber',s ,,notes3 --,obvious print and punctuation errors were corrected4 --,bold text has been rendered as bold text4 --,spaced out text 7gesperrt7 has been rendered as spaced text4 --,superscript letters have been rendered as ab and abc4 ,a ,,view ,,of ,sir ,,isaac ,,newton's ,,philosophy4 ,illustration ,,london3 ,printed by ,s4 ,,palmer1 #agbh4 ,illustration ,to the ,noble and ,right ,honourable ,,sir ,,robert ,,walpole4 ,,sir1 ,i ,take the liberty to send you this view of ,sir ,,isaac ,,newton',s philosophy1 which1 if it were performed suitable to the dignity of the subject1 might not be a present unworthy the acceptance of the greatest person4 ,for his philosophy operations of nature1 which for so many ages had imployed the curiosity of mankind2 though no one before him was furnished with the strength of mind necessary to go any depth in this difficult search4 ,however1 ,i am encouraged to hope1 that this attempt1 imperfect as it is1 to give our countrymen in general some conception of the labours of a person1 who shall always be the boast of this nation1 may be received with indulgence by one1 under whose influence these kingdoms enjoy so much happiness4 ,indeed my admiration at the surprizing inventions of this great man1 carries me to conceive of him as a person1 who not only must raise the glory of the country1 which gave him birth2 but that he has even done honour to human nature1 by having extended the greatest and most noble of our faculties1 reason1 to subjects1 which1 till he attempted them1 appeared to be wholly beyond the reach of our limited capacities4 ,and what can give us a more pleasing prospect of our own condition1 than to see so exalted a proof of the strength of that faculty1 whereon the conduct of our lives1 and our happiness depends2 our passions and all our motives to action being in such manner guided by our opinions1 that where these are just1 our whole behaviour will be praise-worthy8 ,but why do ,i presume to detain you1 ,,sir1 with such reflections as these1 who must have the fullest experience within your own mind1 of the effects of right reason8 ,for to what other source can be ascribed that amiable frankness and unreserved condescension among your friends1 or that masculine perspicuity and strength of argument1 whereby you draw the admiration of the publick1 while you are engaged in the most important of all causes1 the liberties of mankind8 99 99 99 99 99 ,i humbly crave leave to make the only acknowledgement within my power1 for the benefits1 which ,i receive in common with the rest of my countrymen from these high talents1 by subscribing my self ,,sir1 ,your most faithful1 and ,most humble ,servant1 ,,henry ,,pemberton4 ,,preface4 ,i ,drew up the following papers many years ago at the desire of some friends1 who1 upon my taking care of the late edition of ,sir ,,isaac ,,newton',s ,principia1 perswaded me to make them publick4 ,i laid hold of that opportunity1 when my thoughts were afresh employed on this subject1 to revise what ,i had formerly written4 ,and ,i now send it abroad not without some hopes of answering these two ends4 ,my first intention was to convey to such1 as are not used to mathematical reasoning1 some idea of the philosophy of a person1 who has acquired an universal reputation1 and rendered our nation famous for these speculations in the learned world4 ,to which purpose ,i have avoided using terms of art as much as possible1 and taken care to define such as ,i was obliged to use4 ,though this caution was the less necessary at present1 since many of them are become familiar words to our language1 from the great number of books wrote in it upon philosophical subjects1 and the courses of experiments1 that have of late years been given by several ingenious men4 ,the other view ,i had1 was to encourage such young gentlemen as have a turn for the mathematical sciences1 to pursue those studies the more chearfully1 in order to understand in our author himself the demonstrations of the things ,i here declare4 ,and to facilitate their progress herein1 ,i intend to proceed still farther in the explanation of ,sir ,,isaac ,,newton',s philosophy4 ,for as ,i have received very much pleasure from perusing his writings1 ,i hope it is no illaudable ambition to endeavour the rendering them more easily understood1 that greater numbers may enjoy the same satisfaction4 ,it will perhaps be expected1 that ,i should say something particular of a person1 to whom ,i must always acknowledge my self to be much obliged4 ,what ,i have to declare on this head will be but short2 for it was in the very last years of ,sir ,,isaac's life1 that ,i had the honour of his acquaintance4 ,this happened on the following occasion4 ,mr4 ,polenus1 a ,professor in the ,university of ,padua1 from a new experiment of his1 thought the common opinion about the force of moving bodies was overturned1 and the truth of ,mr4 ,libnitz's notion in that matter fully proved4 ,the contrary of what ,polenus had asserted ,i demonstrated in a paper1 which ,dr4 ,,mead1 who takes all opportunities of obliging his friends1 was pleased to shew ,sir ,,isaac ,,newton ,this was so well approved of by him1 that he did me the honour to become a fellow-writer with me1 by annexing to what ,i had written1 a demonstration of his own drawn from another consideration4 ,when ,i printed my discourse in the philosophical transactions1 ,i put what ,sir ,,isaac had written in a scholium by it self1 that ,i might not seem to usurp what did not belong to me4 ,but ,i concealed his name1 not being then sufficiently acquainted with him to ask whether he was willing ,i might make use of it or not4 ,in a little time after he engaged me to take care of the new edition he was about making if his ,principia4 ,this obliged me to be very frequently with him1 and as he lived at some distance from me1 a great number of letters passed between us on this account4 ,when ,i had the honour of his conversation1 ,i endeavoured to learn his thoughts upon mathematical subjects1 and something historical concerning his inventions1 that ,i had not been before acquainted with4 ,i found1 he had read fewer of the modern mathematicians1 than one could have expected2 but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook4 ,i have often heard him censure the handling geometrical subjects by algebraic calculations2 and his book of ,algebra he called by the name of ,universal ,arithmetic1 in opposition to the injudicious title of ,geometry1 which ,des ,cartes had given to the treatise1 wherein he shews1 how the geometer may assist his invention by such kind of computations4 ,he frequently praised ,slusius1 ,barrow and ,huygens for not being influenced by the false taste1 which then began to prevail4 ,he used to commend the laudable attempt of ,hugo de ,omerique to restore the ancient analysis1 and very much esteemed ,apollonius's book ,de sectione rationis for giving us a clearer notion of that analysis than we had before4 ,dr4 ,barrow may be esteemed as having shewn a compass of invention equal1 if not superior to any of the moderns1 our author only excepted2 but ,sir ,,isaac ,,newton has several times particularly recommended to me ,huygens's stile and manner4 ,he thought him the most elegant of any mathematical writer of modern times1 and the most just imitator of the antients4 ,of their taste1 and form of demonstration ,sir ,,isaac always professed himself a great admirer3 ,i have heard him even censure himself for not following them yet more closely than he did2 and speak with regret of his mistake at the beginning of his mathematical studies1 in applying himself to the works of ,des ,cartes and other algebraic writers1 before he had considered the elements of ,euclide with that attention1 which so excellent a writer deserves4 ,as to the history of his inventions1 what relates to his discoveries of the methods of series and fluxions1 and of his theory of light and colours1 the world has been sufficiently informed of already4 ,the first thoughts1 which gave rise to his ,principia1 he had1 when he retired from ,cambridge in #afff on account of the plague4 ,as he sat alone in a garden1 he fell into a speculation on the power of gravity3 that as this power is not found sensibly diminished at the remotest distance from the center of the earth1 to which we can rise1 neither at the tops of the loftiest buildings1 nor even on the summits of the highest mountains2 it appeared to him reasonable to conclude1 that this power must extend much farther than was usually thought2 why not as high as the moon1 said he to himself8 and if so1 her motion must be influenced by it2 perhaps she is retained in her orbit thereby4 ,however1 though the power of gravity is not sensibly weakened in the little change of distance1 at which we can place our selves from the center of the earth2 yet it is very possible1 that so high as the moon this power may differ much in strength from what it is here4 ,to make an estimate1 what might be the degree of this diminution1 he considered with himself1 that if the moon be retained in her orbit by the force of gravity1 no doubt the primary planets are carried round the sun by the like power4 ,and by comparing the periods of the several planets with their distances from the sun1 he found1 that if any power like gravity held them in their courses1 its strength must decrease in the duplicate proportion of the increase of distance4 ,this be concluded by supposing them to move in perfect circles concentrical to the sun1 from which the orbits of the greatest part of them do not much differ4 ,supposing therefore the power of gravity1 when extended to the moon1 to decrease in the same manner1 he computed whether that force would be sufficient to keep the moon in her orbit4 ,in this computation1 being absent from books1 he took the common estimate in use among geographers and our seamen1 before ,norwood had measured the earth1 that #fj ,english miles were contained in one degree of latitude on the surface of the earth4 ,but as this is a very faulty supposition1 each degree containing about #fi of our miles1 his computation did not answer expectation2 whence he concluded1 that some other cause must at least join with the action of the power of gravity on the moon4 ,on this account he laid aside for that time any farther thoughts upon this matter4 ,but some years after1 a letter which he received from ,dr4 ,hook1 put him on inquiring what was the real figure1 in which a body let fall from any high place descends1 taking the motion of the earth round its axis into consideration4 ,such a body1 having the same motion1 which by the revolution of the earth the place has whence it falls1 is to be considered as projected forward and at the same time drawn down to the center of the earth4 ,this gave occasion to his resuming his former thoughts concerning the moon2 and ,picart in ,france having lately measured the earth1 by using his measures the moon appeared to be kept in her orbit purely by the power of gravity2 and consequently1 that this power decreases as you recede from the center of the earth in the manner our author had formerly conjectured4 ,upon this principle he found the line described by a falling body to be an ellipsis1 the center of the earth being one focus4 ,and the primary planets moving in such orbits round the sun1 he had the satisfaction to see1 that this inquiry1 which he had undertaken merely out of curiosity1 could be applied to the greatest purposes4 ,hereupon he composed near a dozen propositions relating to the motion of the primary planets about the sun4 ,several years after this1 some discourse he had with ,dr4 ,halley1 who at ,cambridge made him a visit1 engaged ,sir ,,isaac ,,newton to resume again the consideration of this subject2 and gave occasion to his writing the treatise which he published under the title of mathematical principles of natural philosophy4 ,this treatise1 full of such a variety of profound inventions1 was composed by him from scarce any other materials than the few propositions before mentioned1 in the space of one year and an half4 ,though his memory was much decayed1 ,i found he perfectly understood his own writings1 contrary to what ,i had frequently heard in discourse from many persons4 ,this opinion of theirs might arise perhaps from his not being always ready at speaking on these subjects1 when it might be expected he should4 ,but as to this1 it may be observed1 that great genius's are frequently liable to be absent1 not only in relation to common life1 but with regard to some of the parts of science they are the best informed of4 ,inventors seem to treasure up in their minds1 what they have found out1 after another manner than those do the same things1 who have not this inventive faculty4 ,the former1 when they have occasion to produce their knowledge1 are in some measure obliged immediately to investigate part of what they want4 ,for this they are not equally fit at all times3 so it has often happened1 that such as retain things chiefly by means of a very strong memory1 have appeared off hand more expert than the discoverers themselves4 ,as to the moral endowments of his mind1 they were as much to be admired as his other talents4 ,but this is a field ,i leave others to exspatiate in4 ,i only touch upon what ,i experienced myself during the few years ,i was happy in his friendship4 ,but this ,i immediately discovered in him1 which at once both surprized and charmed me3 ,neither his extreme great age1 nor his universal reputation had rendred him stiff in opinion1 or in any degree elated4 ,of this ,i had occasion to have almost daily experience4 ,the ,remarks ,i continually sent him by letters on his ,principia were received with the utmost goodness4 ,these were so far from being any ways displeasing to him1 that on the contrary it occasioned him to speak many kind things of me to my friends1 and to honour me with a publick testimony of his good opinion4 ,he also approved of the following treatise1 a great part of which we read together4 ,as many alterations were made in the late edition of his ,principia1 so there would have been many more if there had been a sufficient time4 ,but whatever of this kind may be thought wanting1 ,i shall endeavour to supply in my comment on that book4 ,i had reason to believe he expected such a thing from me1 and ,i intended to have published it in his life time1 after ,i had printed the following discourse1 and a mathematical treatise ,sir ,,isaac ,,newton had written a long while ago1 containing the first principles of fluxions1 for ,i had prevailed on him to let that piece go abroad4 ,i had examined all the calculations1 and prepared part of the figures2 but as the latter part of the treatise had never been finished1 he was about letting me have other papers1 in order to supply what was wanting4 ,but his death put a stop to that design4 ,as to my comment on the ,principia1 ,i intend there to demonstrate whatever ,sir ,,isaac ,,newton has set down without express proof1 and to explain all such expressions in his book1 as ,i shall judge necessary4 ,this comment ,i shall forthwith put to the press1 joined to an english translation of his ,principia1 which ,i have had some time by me4 ,a more particular account of my whole design has already been published in the new memoirs of literature for the month of march #agbg4 ,i have presented my readers with a copy of verses on ,sir ,,isaac ,,newton1 which ,i have just received from a young ,gentleman1 whom ,i am proud to reckon among the number of my dearest friends4 ,if ,i had any apprehension that this piece of poetry stood in need of an apology1 ,i should be desirous the reader might know1 that the author is but sixteen years old1 and was obliged to finish his composition in a very short space of time4 ,but ,i shall only take the liberty to observe1 that the boldness of the digressions will be best judged of by those who are acquainted with ,,pindar4 ,a ,,poem ,,on ,sir ,,isaac ,,newton4 ,,to ,,newton's genius1 and immortal fame ,th' advent'rous muse with trembling pinion soars4 ,thou1 heav'nly truth1 from thy seraphick throne ,look favourable down1 do thou assist ,my lab'ring thought1 do thou inspire my song4 ,,newton1 who first th' almighty's works display'd1 ,and smooth'd that mirror1 in whose polish'd face ,the great creator now conspicuous shines2 ,who open'd nature's adamantine gates1 ,and to our minds her secret powers expos'd2 ,,newton demands the muse2 his sacred hand ,shall guide her infant steps2 his sacred hand ,shall raise her to the ,heliconian height1 ,where1 on its lofty top inthron'd1 her head ,shall mingle with the ,stars4 ,hail nature1 hail1 ,o ,goddess1 handmaid of th' ethereal power1 ,now lift thy head1 and to th' admiring world ,shew thy long hidden beauty4 ,thee the wise ,of ancient fame1 immortal ,,plato's self1 ,the ,stagyrite1 and ,syracusian sage1 ,from black obscurity's abyss to raise1 7,drooping and mourning o'er thy wondrous works7 ,with vain inquiry sought4 ,like meteors these ,in their dark age bright sons of wisdom shone3 ,but at thy ,,newton all their laurels fade1 ,they shrink from all the honours of their names4 ,so glimm'ring stars contract their feeble rays1 ,when the swift lustre of ,,aurora's face ,flows o'er the skies1 and wraps the heav'ns in light4 ,,the ,deity's omnipotence1 the cause1 ,th' original of things long lay unknown4 ,alone the beauties prominent to sight 7,of the celestial power the outward form7 ,drew praise and wonder from the gazing world4 ,as when the deluge overspread the earth1 ,whilst yet the mountains only rear'd their heads ,above the surface of the wild expanse1 ,whelm'd deep below the great foundations lay1 ,till some kind angel at heav'n's high command ,roul'd back the rising tides1 and haughty floods1 ,and to the ocean thunder'd out his voice3 ,quick all the swelling and imperious waves1 ,the foaming billows and obscuring surge1 ,back to their channels and their ancient seats ,recoil affrighted3 from the darksome main ,earth raises smiling1 as new-born1 her head1 ,and with fresh charms her lovely face arrays4 ,so his extensive thought accomplish'd first ,the mighty task to drive th' obstructing mists ,of ignorance away1 beneath whose gloom ,th' inshrouded majesty of ,nature lay4 ,he drew the veil and swell'd the spreading scene4 ,how had the moon around th' ethereal void ,rang'd1 and eluded lab'ring mortals care1 ,till his invention trac'd her secret steps1 ,while she inconstant with unsteady rein ,through endless mazes and meanders guides ,in its unequal course her changing carr3 ,whether behind the sun's superior light ,she hides the beauties of her radiant face1 ,or1 when conspicuous1 smiles upon mankind1 ,unveiling all her night-rejoicing charms4 ,when thus the silver-tressed moon dispels ,the frowning horrors from the brow of night1 ,and with her splendors chears the sullen gloom1 ,while sable-mantled darkness with his veil ,the visage of the fair horizon shades1 ,and over nature spreads his raven wings2 ,let me upon some unfrequented green ,while sleep sits heavy on the drowsy world1 ,seek out some solitary peaceful cell1 ,where darksome woods around their gloomy brows ,bow low1 and ev'ry hill's protended shade ,obscures the dusky vale1 there silent dwell1 ,where contemplation holds its still abode1 ,there trace the wide and pathless void of heav'n1 ,and count the stars that sparkle on its robe4 ,or else in fancy's wild'ring mazes lost ,upon the verdure see the fairy elves ,dance o'er their magick circles1 or behold1 ,in thought enraptur'd with the ancient bards1 ,medea's baleful incantations draw ,down from her orb the paly queen of night4 ,but chiefly ,,newton let me soar with thee1 ,and while surveying all yon starry vault ,with admiration ,i attentive gaze1 ,thou shalt descend from thy celestial seat1 ,and waft aloft my high-aspiring mind1 ,shalt shew me there how nature has ordain'd ,her fundamental laws1 shalt lead my thought ,through all the wand'rings of th' uncertain moon1 ,and teach me all her operating powers4 ,she and the sun with influence conjoint ,wield the huge axle of the whirling earth1 ,and from their just direction turn the poles1 ,slow urging on the progress of the years4 ,the constellations seem to leave their seats1 ,and o'er the skies with solemn pace to move4 ,you1 splendid rulers of the day and night1 ,the seas obey1 at your resistless sway ,now they contract their waters1 and expose ,the dreary desart of old ocean's reign4 ,the craggy rocks their horrid sides disclose2 ,trembling the sailor views the dreadful scene1 ,and cautiously the threat'ning ruin shuns4 ,but where the shallow waters hide the sands1 ,there ravenous destruction lurks conceal'd1 ,there the ill-guided vessel falls a prey1 ,and all her numbers gorge his greedy jaws4 ,but quick returning see th' impetuous tides ,back to th' abandon'd shores impell the main4 ,again the foaming seas extend their waves1 ,again the rouling floods embrace the shoars1 ,and veil the horrours of the empty deep4 ,thus the obsequious seas your power confess1 ,while from the surface healthful vapours rise ,plenteous throughout the atmosphere diffus'd1 ,or to supply the mountain's heads with springs1 ,or fill the hanging clouds with needful rains1 ,that friendly streams1 and kind refreshing show'rs ,may gently lave the sun-burnt thirsty plains1 ,or to replenish all the empty air ,with wholsome moisture to increase the fruits ,of earth1 and bless the labours of mankind4 ,o ,,newton1 whether flies thy mighty soul1 ,how shall the feeble muse pursue through all ,the vast extent of thy unbounded thought1 ,that even seeks th' unseen recesses dark ,to penetrate of providence immense4 ,and thou the great dispenser of the world ,propitious1 who with inspiration taught'st ,our greatest bard to send thy praises forth2 ,thou1 who gav'st ,,newton thought2 who smil'dst serene1 ,when to its bounds he stretch'd his swelling soul2 ,who still benignant ever blest his toil1 ,and deign'd to his enlight'ned mind t' appear ,confess'd around th' interminated world3 ,to me ,o thy divine infusion grant 7,o thou in all so infinitely good7 ,that ,i may sing thy everlasting works1 ,thy inexhausted store of providence1 ,in thought effulgent and resounding verse4 ,o could ,i spread the wond'rous theme around1 ,where the wind cools the oriental world1 ,to the calm breezes of the ,zephir's breath1 ,to where the frozen hyperborean blasts4 ,to where the boist'rous tempest-leading south ,from their deep hollow caves send forth their storms4 ,thou still indulgent parent of mankind1 ,left humid emanations should no more ,flow from the ocean1 but dissolve away ,through the long series of revolving time2 ,and left the vital principle decay1 ,by which the air supplies the springs of life2 ,thou hast the fiery visag'd comets form'd ,with vivifying spirits all replete1 ,which they abundant breathe about the void1 ,renewing the prolifick soul of things4 ,no longer now on thee amaz'd we call1 ,no longer tremble at imagin'd ills1 ,when comets blaze tremendous from on high1 ,or when extending wide their flaming trains ,with hideous grasp the skies engirdle round1 ,and spread the terrors of their burning locks4 ,for these through orbits in the length'ning space ,of many tedious rouling years compleat ,around the sun move regularly on2 ,and with the planets in harmonious orbs1 ,and mystick periods their obeysance pay ,to him majestick ruler of the skies ,upon his throne of circled glory fixt4 ,he or some god conspicuous to the view1 ,or else the substitute of nature seems1 ,guiding the courses of revolving worlds4 ,he taught great ,,newton the all-potent laws ,of gravitation1 by whose simple power ,the universe exists4 ,nor here the sage ,big with invention still renewing staid4 ,but ,o bright angel of the lamp of day1 ,how shall the muse display his greatest toil8 ,let her plunge deep in ,aganippe's waves1 ,or in ,castalia's ever-flowing stream1 ,that re-inspired she may sing to thee1 ,how ,,newton dar'd advent'rous to unbraid ,the yellow tresses of thy shining hair4 ,or didst thou gracious leave thy radiant sphere1 ,and to his hand thy lucid splendours give1 ,t' unweave the light-diffusing wreath1 and part ,the blended glories of thy golden plumes8 ,he with laborious1 and unerring care1 ,how different and imbodied colours form ,thy piercing light1 with just distinction found4 ,he with quick sight pursu'd thy darting rays1 ,when penetrating to th' obscure recess ,of solid matter1 there perspicuous saw1 ,how in the texture of each body lay ,the power that separates the different beams4 ,hence over nature's unadorned face ,thy bright diversifying rays dilate ,their various hues3 and hence when vernal rains ,descending swift have burst the low'ring clouds1 ,thy splendors through the dissipating mists ,in its fair vesture of unnumber'd hues ,array the show'ry bow4 ,at thy approach ,the morning risen from her pearly couch ,with rosy blushes decks her virgin cheek2 ,the ev'ning on the frontispiece of heav'n ,his mantle spreads with many colours gay2 ,the mid-day skies in radiant azure clad1 ,the shining clouds1 and silver vapours rob'd ,in white transparent intermixt with gold1 ,with bright variety of splendor cloath ,all the illuminated face above4 ,when hoary-headed winter back retires ,to the chill'd pole1 there solitary sits ,encompass'd round with winds and tempests bleak ,in caverns of impenetrable ice1 ,and from behind the dissipated gloom ,like a new ,venus from the parting surge ,the gay-apparell'd spring advances on2 ,when thou in thy meridian brightness sitt'st1 ,and from thy throne pure emanations flow ,of glory bursting o'er the radiant skies3 ,then let the muse ,olympus' top ascend1 ,and o'er ,thessalia's plain extend her view1 ,and count1 ,o ,tempe1 all thy beauties o'er4 ,mountains1 whose summits grasp the pendant clouds1 ,between their wood-invelop'd slopes embrace ,the green-attired vallies4 ,every flow'r ,here in the pride of bounteous nature clad ,smiles on the bosom of th' enamell'd meads4 ,over the smiling lawn the silver floods ,of fair ,peneus gently roul along1 ,while the reflected colours from the flow'rs1 ,and verdant borders pierce the lympid waves1 ,and paint with all their variegated hue ,the yellow sands beneath4 ,smooth gliding on ,the waters hasten to the neighbouring sea4 ,still the pleas'd eye the floating plain pursues2 ,at length1 in ,neptune's wide dominion lost1 ,surveys the shining billows1 that arise ,apparell'd each in ,phbus' bright attire3 ,or from a far some tall majestick ship1 ,or the long hostile lines of threat'ning fleets1 ,which o'er the bright uneven mirror sweep1 ,in dazling gold and waving purple deckt2 ,such as of old1 when haughty ,athens power ,their hideous front1 and terrible array ,against ,pallene's coast extended wide1 ,and with tremendous war and battel stern ,the trembling walls of ,potida shook4 ,crested with pendants curling with the breeze ,the upright masts high bristle in the air1 ,aloft exalting proud their gilded heads4 ,the silver waves against the painted prows ,raise their resplendent bosoms1 and impearl ,the fair vermillion with their glist'ring drops3 ,and from on board the iron-cloathed host ,around the main a gleaming horrour casts2 ,each flaming buckler like the mid-day sun1 ,each plumed helmet like the silver moon1 ,each moving gauntlet like the light'ning's blaze1 ,and like a star each brazen pointed spear4 ,but lo the sacred high-erected fanes1 ,fair citadels1 and marble-crowned towers1 ,and sumptuous palaces of stately towns ,magnificent arise1 upon their heads ,bearing on high a wreath of silver light4 ,but see my muse the high ,pierian hill1 ,behold its shaggy locks and airy top1 ,up to the skies th' imperious mountain heaves ,the shining verdure of the nodding woods4 ,see where the silver ,hippocrene flows1 ,behold each glitt'ring rivulet1 and rill ,through mazes wander down the green descent1 ,and sparkle through the interwoven trees4 ,here rest a while and humble homage pay1 ,here1 where the sacred genius1 that inspir'd ,sublime ,m,,onides and ,,pindar',s breast1 ,his habitation once was fam'd to hold4 ,here thou1 ,o ,,homer1 offer'dst up thy vows1 ,thee1 the kind muse ,,calliop,a heard1 ,and led thee to the empyrean feats1 ,there manifested to thy hallow'd eyes ,the deeds of gods2 thee wise ,,minerva taught ,the wondrous art of knowing human kind2 ,harmonious ,,ph,,bus tun'd thy heav'nly mind1 ,and swell'd to rapture each exalted sense2 ,even ,,mars the dreadful battle-ruling god1 ,,mars taught thee war1 and with his bloody hand ,instructed thine1 when in thy sounding lines ,we hear the rattling of ,bellona's carr1 ,the yell of discord1 and the din of arms4 ,,pindar1 when mounted on his fiery steed1 ,soars to the sun1 opposing eagle like ,his eyes undazled to the fiercest rays4 ,he firmly seated1 not like ,,glaucus' son1 ,strides his swift-winged and fire-breathing horse1 ,and born aloft strikes with his ringing hoofs ,the brazen vault of heav'n1 superior there ,looks down upon the stars1 whose radiant light ,illuminates innumerable worlds1 ,that through eternal orbits roul beneath4 ,but thou all hail immortalized son ,of harmony1 all hail thou ,thracian bard1 ,to whom ,,apollo gave his tuneful lyre4 ,o might'st thou1 ,,orpheus1 now again revive1 ,and ,,newton should inform thy list'ning ear ,how the soft notes1 and soul-inchanting strains ,of thy own lyre were on the wind convey'd4 ,he taught the muse1 how sound progressive floats ,upon the waving particles of air1 ,when harmony in ever-pleasing strains1 ,melodious melting at each lulling fall1 ,with soft alluring penetration steals ,through the enraptur'd ear to inmost thought1 ,and folds the senses in its silken bands4 ,so the sweet musick1 which from ,,orpheus' touch ,and fam'd ,,amphion',s1 on the sounding string ,arose harmonious1 gliding on the air1 ,pierc'd the tough-bark'd and knotty-ribbed woods1 ,into their saps soft inspiration breath'd ,and taught attention to the stubborn oak4 ,thus when great ,,henry1 and brave ,,marlb',,rough led ,th' imbattled numbers of ,,britannia',s sons1 ,the trump1 that swells th' expanded cheek of fame1 ,that adds new vigour to the gen'rous youth1 ,and rouzes sluggish cowardize it self1 ,the trumpet with its ,mars-inciting voice1 ,the winds broad breast impetuous sweeping o'er ,fill'd the big note of war4 ,th' inspired host ,with new-born ardor press the trembling ,,gaul2 ,nor greater throngs had reach'd eternal night1 ,not if the fields of ,agencourt had yawn'd ,exposing horrible the gulf of fate2 ,or roaring ,danube spread his arms abroad1 ,and overwhelm'd their legions with his floods4 ,but let the wand'ring muse at length return2 ,nor yet1 angelick genius of the sun1 ,in worthy lays her high-attempting song ,has blazon'd forth thy venerated name4 ,then let her sweep the loud-resounding lyre ,again1 again o'er each melodious string ,teach harmony to tremble with thy praise4 ,and still thine ear ,o favourable grant1 ,and she shall tell thee1 that whatever charms1 ,whatever beauties bloom on nature's face1 ,proceed from thy all-influencing light4 ,that when arising with tempestuous rage1 ,the ,north impetuous rides upon the clouds ,dispersing round the heav'ns obstructive gloom1 ,and with his dreaded prohibition stays ,the kind effusion of thy genial beams2 ,pale are the rubies on ,,aurora',s lips1 ,no more the roses blush upon her cheeks1 ,black are ,peneus' streams and golden sands ,in ,tempe's vale dull melancholy sits1 ,and every flower reclines its languid head4 ,by what high name shall ,i invoke thee1 say1 ,thou life-infusing deity1 on thee ,i call1 and look propitious from on high1 ,while now to thee ,i offer up my prayer4 ,o had great ,,newton1 as he found the cause1 ,by which sound rouls thro' th' undulating air1 ,o had he1 baffling times resistless power1 ,discover'd what that subtle spirit is1 ,or whatsoe'er diffusive else is spread ,over the wide-extended universe1 ,which causes bodies to reflect the light1 ,and from their straight direction to divert ,the rapid beams1 that through their surface pierce4 ,but since embrac'd by th' icy arms of age1 ,and his quick thought by times cold hand congeal'd1 ,ev'n ,,newton left unknown this hidden power2 ,thou from the race of human kind select ,some other worthy of an angel's care1 ,with inspiration animate his breast1 ,and him instruct in these thy secret laws4 ,o let not ,,newton1 to whose spacious view1 ,now unobstructed1 all th' extensive scenes ,of the ethereal ruler's works arise2 ,when he beholds this earth he late adorn'd1 ,let him not see philosophy in tears1 ,like a fond mother solitary sit1 ,lamenting him her dear1 and only child4 ,but as the wise ,,pythagoras1 and he1 ,whose birth with pride the fam'd ,abdera boasts1 ,with expectation having long survey'd ,this spot their ancient seat1 with joy beheld ,divine philosophy at length appear ,in all her charms majestically fair1 ,conducted by immortal ,,newton',s hand4 ,so may he see another sage arise1 ,that shall maintain her empire3 then no more ,imperious ignorance with haughty sway ,shall stalk rapacious o'er the ravag'd globe3 ,then thou1 ,o ,,newton1 shalt protect these lines4 ,the humble tribute of the grateful muse2 ,ne'er shall the sacrilegious hand despoil ,her laurel'd temples1 whom his name preserves3 ,and were she equal to the mighty theme1 ,futurity should wonder at her song2 ,time should receive her with extended arms1 ,seat her conspicuous in his rouling carr1 ,and bear her down to his extreamest bound4 ,,fables with wonder tell how ,terra's sons ,with iron force unloos'd the stubborn nerves ,of hills1 and on the cloud-inshrouded top ,of ,pelion ,ossa pil'd4 ,but if the vast ,gigantick deeds of savage strength demand ,astonishment from men1 what then shalt thou1 ,o what expressive rapture of the soul1 ,when thou before us1 ,,newton1 dost display ,the labours of thy great excelling mind2 ,when thou unveilest all the wondrous scene1 ,the vast idea of th' eternal king1 ,not dreadful bearing in his angry arm ,the thunder hanging o'er our trembling heads2 ,but with th' effulgency of love replete1 ,and clad with power1 which form'd th' extensive heavens4 ,o happy he1 whose enterprizing hand ,unbars the golden and relucid gates ,of th' empyrean dome1 where thou enthron'd ,philosophy art seated4 ,thou sustain'd ,by the firm hand of everlasting truth ,despisest all the injuries of time2 ,thou never know'st decay when all around1 ,antiquity obscures her head4 ,behold ,th' ,egyptian towers1 the ,babylonian walls1 ,and ,thebes with all her hundred gates of brass1 ,behold them scatter'd like the dust abroad4 ,whatever now is flourishing and proud1 ,whatever shall1 must know devouring age4 ,euphrates' stream1 and seven-mouthed ,nile1 ,and ,danube1 thou that from ,germania's soil ,to the black ,euxine's far remoted shore1 ,o'er the wide bounds of mighty nations sweep'st ,in thunder loud thy rapid floods along4 ,ev'n you shall feel inexorable time2 ,to you the fatal day shall come2 no more ,your torrents then shall shake the trembling ground1 ,no longer then to inundations swol'n ,th' imperious waves the fertile pastures drench1 ,but shrunk within a narrow channel glide2 ,or through the year's reiterated course ,when time himself grows old1 your wond'rous streams ,lost ev'n to memory shall lie unknown ,beneath obscurity1 and ,chaos whelm'd1 ,but still thou sun illuminatest all ,the azure regions round1 thou guidest still ,the orbits of the planetary spheres2 ,the moon still wanders o'er her changing course1 ,and still1 ,o ,,newton1 shall thy name survive3 ,as long as nature's hand directs the world1 ,when ev'ry dark obstruction shall retire1 ,and ev'ry secret yield its hidden store1 ,which thee dim-sighted age forbad to see ,age that alone could stay thy rising soul4 ,and could mankind among the fixed stars1 ,e'en to th' extremest bounds of knowledge reach1 ,to those unknown innumerable suns1 ,whose light but glimmers from those distant worlds1 ,ev'n to those utmost boundaries1 those bars ,that shut the entrance of th' illumin'd space ,where angels only tread the vast unknown1 ,thou ever should'st be seen immortal there3 ,in each new sphere1 each new-appearing sun1 ,in farthest regions at the very verge ,of the wide universe should'st thou be seen4 ,and lo1 th' all-potent goddess ,,nature takes ,with her own hand thy great1 thy just reward ,of immortality2 aloft in air ,see she displays1 and with eternal grasp ,uprears the trophies of great ,,newton's fame4 ,r4 ,,glover4 ,,the ,,contents4 ,,introduction concerning ,sir ,,isaac ,,newton's method of reasoning in philosophy pag4 #a ,,book ,i4 ,,chap4 #a4 ,of the laws of motion ,the first law of motion proved p4 #bi ,the second law of motion proved p4 #bi ,the third law of motion proved p4 #ca ,,chap4 #b4 ,further proofs of the laws of motion ,the effects of percussion p4 #di ,the perpendicular descent of bodies p4 #ee ,the oblique descent of bodies in a straight line p4 #eg ,the curvilinear descent of bodies p4 #eh ,the perpendicular ascent of bodies ibid4 ,the oblique ascent of bodies p4 #ei ,the power of gravity proportional to the quantity of matter in each body p4 #fj ,the centre of gravity of bodies p4 #fb ,the mechanical powers p4 #fi ,the lever p4 #ga ,the wheel and axis p4 #gg ,the pulley p4 #hj ,the wedge p4 #hc ,the screw ibid4 ,the inclined plain p4 #hd ,the pendulum p4 #hf ,vibrating in a circle ibid4 ,vibrating in a cycloid p4 #ia ,the line of swiftest descent p4 #ic ,the centre of oscillation p4 #id ,experiments upon the percussion of bodies made by pendulums p4 #ih ,the centre of percussion p4 #ajj ,the motion of projectiles p4 #ajb ,the description of the conic sections p4 #ajf ,the difference between absolute and relative motion1 as also between absolute and relative time p4 #aab ,,chap4 #c4 ,of centripetal forces p4 #aag ,,chap4 #d4 ,of the resistance of fluids p4 #adc ,bodies are resisted in the duplicate proportion of their velocities p4 #adg ,of elastic fluids and their resistance p4 #adi ,how fluids may be rendered elastic p4 #aej ,the degree of resistance in regard to the proportion between the density of the body and of the fluid ,in rare and uncompressed fluids p4 #aec ,in compressed fluids p4 #aee ,the degree of resistance as it depends upon the figure of bodies ,in rare and uncompressed fluids p4 #aee ,in compressed fluids p4 #aeh ,,book ,,ii4 ,,chap4 #a4 ,that the planets move in a space empty of sensible matter p4 #afa ,the system of the world described p4 #afb ,the planets suffer no sensible resistance in their motion p4 #aff ,they are not kept in motion by a fluid p4 #afh ,that all space is not full of matter without vacancies p4 #afi ,,chap4 #b4 ,concerning the cause that keeps in motion the primary planets p4 #aga ,they are influenced by a centripetal power directed to the sun p4 #aga ,the strength of this power is reciprocally in the duplicate proportion of the distance ibid4 ,the cause of the irregularities in the motions of the planets p4 #age ,a correction of their motions p4 #agh ,that the frame of the world is not eternal p4 #ahj ,,chap4 #c4 ,of the motion of the moon and the other secondary planets ,that they are influenced by a centripetal force directed toward their primary1 as the primary are influenced by the sun p4 #ahb ,that the power usually called gravity extends to the moon p4 #ahi ,that the sun acts on the secondary planets p4 #aij ,the variation of the moon p4 #aic ,that the circuit of the moons orbit is increased by the sun in the quarters1 and diminished in the conjunction and opposition p4 #aih ,the distance of the moon from the earth in the quarters and in the conjunction and opposition is altered by the sun p4 #bjj ,these irregularities in the moon's motion varied by the change of distance between the earth and sun p4 #bja ,the period of the moon round the earth and her distance varied by the same means ibid4 ,the motion of the nodes and the inclination of the moons orbit p4 #bjb ,the motion of the apogeon and change of the eccentricity p4 #bah ,the inequalities of the other secondary planets deducible from these of the moon p4 #bbi ,,chap4 #d4 ,of comets ,they are not meteors1 nor placed totally without the planetary system p4 #bcj ,the sun acts on them in the same manner as on the planets p4 #bca ,their orbits are near to parabola's p4 #bcc ,the comet that appeared at the end of the year #afhj1 probably performs its period in #ege years1 and another comet in #ge years p4 #bcd ,why the comets move in planes more different from one another than the planets p4 #bce ,the tails of comets p4 #bch ,the use of them p4 #bdc #bdd ,the possible use of the comet it self p4 #bde #bdf ,,chap4 #e4 ,of the bodies of the sun and planets ,that each of the heavenly bodies is endued with an attractive power1 and that the force of the same body on others is proportional to the quantity of matter in the body attracted p4 #bdg ,this proved in the earth p4 #bdh ,in the sun p4 #bej ,in the rest of the planets p4 #bea ,that the attractive power is of the same nature in the sun and in all the planets1 and therefore is the same with gravity p4 #beb ,that the attractive power in each of these bodies is proportional to the quantity of matter in the body attracting ibid4 ,that each particle of which the sun and planets are composed is endued with an attracting power1 the strength of which is reciprocally in the duplicate proportion of the distance p4 #beg ,the power of gravity universally belongs to all matter p4 #bei ,the different weight of the same body upon the surface of the sun1 the earth1 ,jupiter and ,saturn2 the respective densities of these bodies1 and the proportion between their diameters p4 #bfa ,,chap4 #f4 ,of the fluid parts of the planets ,the manner in which fluids press p4 #bfd ,the motion of waves on the surface of water p4 #bfi ,the motion of sound through the air p4 #bgj ,the velocity of sound p4 #bhb ,concerning the tides p4 #bhc ,the figure of the earth p4 #bif ,the effect of this figure upon the power of gravity p4 #cjj ,the effect it has upon pendulums p4 #cjb ,bodies descend perpendicularly to the surface of the earth p4 #cjd ,the axis of the earth changes its direction twice a year1 and twice a month p4 #cac ,the figure of the secondary planets ibid4 ,,book ,,iii4 ,,chap4 #a4 ,concerning the cause of colours inherent in the light ,the sun's light is composed of rays of different colours p4 #cah ,the refraction of light p4 #cai #cbj ,bodies appear of different colour by day-light1 because some reflect one kind of light more copiously than the rest1 and other bodies other kinds of light p4 #cbi ,the effect of mixing rays of different colours p4 #ccd ,,chap4 #b4 ,of the properties of bodies whereon their colours depend4 ,light is not reflected by impinging against the solid parts of bodies p4 #cci ,the particles which compose bodies are transparent p4 #cda ,cause of opacity p4 #cdb ,why bodies in the open day-light have different colours p4 #cdd ,the great porosity of bodies considered p4 #cee ,,chap4 #c4 ,of the refraction1 reflection1 and inflection of light4 ,rays of different colours are differently refracted p4 #ceg ,the sine of the angle of incidence in each kind of rays bears a given proportion to the sine of refraction p4 #cfa ,the proportion between the refractive powers in different bodies p4 #cff ,unctuous bodies refract most in proportion to their density p4 #cfh ,the action between light and bodies is mutual p4 #cfi ,light has alternate fits of easy transmission and reflection p4 #cga ,the fits found to return alternately many thousand times p4 #cge ,why bodies reflect part of the light incident upon them and transmit another part ibid4 ,sir ,,isaac ,,newton's conjecture concerning the cause of this alternate reflection and transmission of light p4 #cgf ,the inflection of light p4 #cgg ,,chap4 #d4 ,of optic glasses4 ,how the rays of light are refracted by a spherical surface of glass p4 #cgh ,how they are refracted by two such surfaces p4 #chj ,how the image of objects is formed by a convex glass p4 #cha ,why convex glasses help the sight in old age1 and concave glasses assist short-sighted people p4 #chc ,the manner in which vision is performed by the eye p4 #che ,of telescopes with two convex glasses p4 #chf ,of telescopes with four convex glasses p4 #chh ,of telescopes with one convex and one concave glass ibid4 ,of microscopes p4 #chi ,of the imperfection of telescopes arising from the different refrangibility of the light p4 #cij ,of the reflecting telescope p4 #cic ,,chap4 #e4 ,of the rainbow ,of the inner rainbow p4 #cid #cie #cih #cii ,of the outter bow p4 #cif #cig #djj ,of a particular appearance in the inner rainbow p4 #dja ,conclusion p4 #dje ,,errata4 ,,page #be4 line #d4 read ,in these ,precepts4 p4 #dj4 l4 #bd4 for ,i read ,k4 p4 #ec4 l4 penult4 f4 4 r4 ,f4 p4 #hb4 l4 ult4 f4 #dj4 r4 #da4 p4 #hc l4 ult4 f4 #dc4 r4 #de4 p4 #ia4 l4 #c4 f4 #dh4 r4 #ej4 ibid4 l4 #be4 for #di4 r4 #ea4 p4 #ib4 l4 #ah4 f4 ,a ,g ,f ,e4 r4 ,h ,g ,f ,c4 p4 #if4 l4 #bc4 dele the comma after 4 p4 #adj4 l4 #ab4 dele and4 p4 #add4 l4 #ae4 f4 threefold4 r4 two-fold4 p4 #afb4 l4 #be4 f4 4 r4 4 p4 #aic4 #a4 #b4 r4 always4 p4 #aii4 l4 penult4 and p4 #bjj4 l4 #c4 #e4 f4 ,f4 r4 ,c4 p4 #bja4 l4 #h4 f4 ascends4 r4 must ascend4 ibid4 l4 #aj4 f4 it descends4 r4 descend4 p4 #bjh4 l4 #ad4 f4 ,w ,t ,o4 r4 ,n ,t ,o4 ,in fig4 #aaj4 draw a line from ,i through ,t1 till it meets the circle ,a ,d ,c ,b1 where place ,w4 p4 #baf4 l4 penult4 f4 action4 r4 motion4 p4 #bba4 l4 #bc4 f4 ,a ,f4 r4 ,a ,h4 p4 #bcb4 l4 #bc4 after invention put a full point4 p4 #bec4 l4 penult4 delete the comma after remarkable4 p4 #bee4 l4 ult4 f4 ,d ,e4 r4 ,b ,e4 p4 #bgh4 l4 #ag4 f4 4 r4 4 p4 #bii4 l4 #ai r4 the4 p4 #cfa4 l4 #ab4 f4 ,i4 r4 t4 p4 #cfi4 l4 #b1 #c4 r4 ,pseudo-topaz4 p4 #cgh4 l4 #ab4 f4 that4 r4 than4 p4 #cgi4 l4 #ae4 f4 converge4 r4 diverge4 p4 #chd4 l4 #g4 f4 optic-glass4 r4 optic-nerve4 p4 #cia4 l4 #ah4 r4 as #ej to #gh4 p4 #cib4 l4 #ah4 after telescope add be about #ajj feet long and the4 in fig4 #afa4 f4 put 4 p4 #cii4 l4 #h4 r4 ,a n1 ,a x4 c4 p4 #djj4 #a4 #ai4 r4 ,a 1 ,a 4 ,a 1 ,a 4 ,a 4 p4 #dja4 l4 #ad4 r4 fig4 #afc4 ,the pages #cgd1 #cge1 #cgf are erroneously numbered #cge1 #cgf1 #cgg2 and the pages #chb1 #chc are numbered #cha1 #chb4 ,a ,,list of such of the ,,subscribers ,,names ,as are come to the ,,hand of the ,,author4 ,a ,m,onseigneur d',aguesseau1 ,chancelier de ,france ,reverend ,mr ,abbot1 of ,emanuel ,coll4 ,camb4 ,capt4 ,george ,abell ,the ,hon4 ,sir ,john ,anstruther1 ,bar4 ,thomas ,abney1 ,esq2 ,mr4 ,nathan ,abraham ,sir ,arthur ,acheson1 ,bart4 ,mr ,william ,adair ,rev4 ,mr ,john ,adams1 ,fellow of ,sidney ,coll4 ,cambridge ,mr ,william ,adams ,mr ,george ,adams ,mr ,william ,adamson1 ,scholar of ,caius ,coll4 ,camb4 ,mr ,samuel ,adee1 ,fell4 of ,corp4 ,chr4 ,coll4 ,oxon ,mr ,andrew ,adlam ,mr ,john ,adlam ,mr ,stephen ,ainsworth ,mrs ,aiscot ,mr ,robert ,akenhead1 ,bookseller at ,newcastle upon ,tyne ,s4 ,b4 ,albinus1 ,m4 ,d4 ,anatom4 and ,chirurg in ,acad4 ,l4 ,b4 ,prof4 ,george ,aldridge1 ,m4 ,d4 ,mr ,george ,algood ,mr ,aliffe ,robert ,allen1 ,esq2 ,mr ,zach4 ,allen ,rev4 ,mr ,allerton1 ,fellow of ,sidney ,coll4 ,cambridge ,mr ,st4 ,amand ,mr ,john ,anns ,thomas ,anson1 ,esq2 ,rev4 ,dr4 ,christopher ,anstey ,mr ,isaac ,antrabus ,mr ,joshua ,appleby ,john ,arbuthnot1 ,m4 ,d4 ,william ,archer1 ,esq2 ,mr ,john ,archer1 ,merchant of ,amsterdam ,thomas ,archer1 ,esq2 ,coll4 ,john ,armstrong1 ,surveyor-,general of ,his ,majesty's ,ordnance ,mr ,armytage ,mr ,street ,arnold1 ,surgeon ,mr ,richard ,arnold ,mr ,ascough ,mr ,charles ,asgill ,richard ,ash1 ,esq2 of ,antigua ,mr ,ash1 ,fellow-,commoner of ,jesus ,coll4 ,cambridge ,william ,ashurst1 ,esq2 of ,castle ,henningham1 ,essex ,mr ,thomas ,ashurst ,mr ,samuel ,ashurst ,mr ,john ,askew1 ,merchant ,mr ,edward ,athawes1 ,merchant ,mr ,abraham ,atkins ,mr ,edward ,kensey ,atkins ,mr ,ayerst ,mr ,jonathan ,ayleworth1 ,jun4 ,rowland ,aynsworth1 ,esq2 ,b ,his ,grace the ,duke of ,bedford ,right ,honourable the ,marquis of ,bowmont ,right ,hon4 the ,earl of ,burlington ,right ,honourable ,lord ,viscount ,bateman ,rt4 ,rev4 ,ld4 ,bp4 of ,bath and ,wells ,rt4 ,rev4 ,lord ,bishop of ,bristol ,right ,hon4 ,lord ,bathurst ,richard ,backwell1 ,esq2 ,mr ,william ,backshell1 ,merch4 ,edmund ,backwell1 ,gent4 ,sir ,edmund ,bacon ,richard ,bagshaw1 of ,oakes1 ,esq2 ,tho4 ,bagshaw1 of ,bakewell1 ,esq2 ,rev4 ,mr4 ,bagshaw ,sir ,robert ,baylis ,honourable ,george ,baillie1 ,esq2 ,giles ,bailly1 ,m4 ,d4 of ,bristol ,mr ,serjeant ,baines ,rev4 ,mr4 ,samuel ,baker1 ,residen4 of ,st4 ,paul's4 ,mr ,george ,baker ,mr ,francis ,baker ,mr ,robert ,baker ,mr ,john ,bakewell ,anthony ,balam1 ,esq2 ,charles ,bale1 ,m4 ,d4 ,mr ,atwell1 ,fellow of ,exeter ,coll4 ,oxon ,mr ,savage ,atwood ,mr ,john ,atwood ,mr ,james ,audley ,sir ,robert ,austen1 ,bart4 ,sir ,john ,austen ,benjamin ,avery1 ,l4 ,l4 ,d4 ,mr ,balgay ,rev4 ,mr ,tho4 ,ball1 ,prebendary of ,chichester ,mr ,pappillon ,ball1 ,merchant ,mr ,levy ,ball ,rev4 ,mr ,jacob ,ball1 of ,andover ,rev4 ,mr ,edward ,ballad1 of ,trin4 ,coll4 ,cambridge ,mr ,baller ,john ,bamber1 ,m4 ,d4 ,rev4 ,mr ,banyer1 ,fellow of ,emanuel ,coll4 ,cambridge ,mr ,henry ,banyer1 of ,wisbech1 ,surgeon ,mr ,john ,barber1 ,apothecary in ,coventry ,henry ,steuart ,barclay1 of ,colairny1 ,esq2 ,rev4 ,mr ,barclay1 ,canon of ,windsor ,mr ,david ,barclay ,mr ,benjamin ,barker1 ,bookseller in ,london ---- ,barker1 ,esq2 ,mr ,francis ,barkstead ,rev4 ,mr ,barnard ,thomas ,barrett1 ,esq2 ,mr ,barrett ,richard ,barret1 ,m4 ,d4 ,mr ,barrow1 ,apothecary ,william ,barrowby1 ,m4 ,d4 ,edward ,barry1 ,m4 ,d4 of ,corke ,mr ,humphrey ,bartholomew1 of ,university ,college1 ,oxon ,mr ,benjamin ,bartlett ,mr ,henry ,bartlett ,mr ,james ,bartlett ,mr ,newton ,barton1 of ,trinity ,college1 ,cambridge ,rev4 ,mr4 ,barton ,william ,barnsley1 ,esq2 ,mr ,samuel ,bateman ,mr ,thomas ,bates ,peter ,barhurst1 ,esq2 ,mark ,barr1 ,esq2 ,thomas ,bast1 ,esq2 ,mr ,batley1 ,bookseller in ,london ,mr ,christopher ,batt1 jun4 ,mr ,william ,batt1 ,apothecary ,rev4 ,mr ,battely1 ,m4 ,a4 ,student of ,christ ,church1 ,oxon ,mr ,edmund ,baugh ,rev4 ,mr4 ,thomas ,bayes ,edward ,bayley1 ,m4 ,d4 of ,havant ,john ,bayley1 ,m4 ,d4 of ,chichester ,mr4 ,alexander ,baynes1 ,professor of ,law in the ,university of ,edinburgh ,mr ,benjamin ,beach ,thomas ,beacon1 ,esq2 ,rev4 ,mr ,philip ,bearcroft ,mr ,thomas ,bearcroft ,mr ,william ,beachcroft ,richard ,beard1 ,m4 ,d4 of ,worcester ,mr ,joseph ,beasley ,rev4 ,mr ,beats1 ,m4 ,a4 ,fellow of ,magdalen ,college1 ,cambridge ,sir ,george ,beaumont ,john ,beaumont1 ,esq2 of ,clapham ,william ,beecher1 of ,howberry1 ,esq2 ,mr ,michael ,beecher ,mr ,finney ,beifield1 of the ,inner-,temple ,mr ,benjamin ,bell ,mr ,humphrey ,bell ,mr ,phineas ,bell ,leonard ,belt1 ,gent4 ,william ,benbow1 ,esq2 ,mr ,martin ,bendall ,mr ,george ,bennet1 of ,cork1 ,bookseller ,rev4 ,mr ,martin ,benson1 ,archdeacon of ,berks ,samuel ,benson1 ,esq2 ,william ,benson1 ,esq2 ,rev4 ,richard ,bently1 ,d4 ,d4 ,master of ,trinity ,coll4 ,cambridge ,thomas ,bere1 ,esq2 ,the ,hon4 ,john ,berkley1 ,esq2 ,mr ,maurice ,berkley1 sen4 ,surgeon ,john ,bernard1 ,esq2 ,mr ,charles ,bernard ,hugh ,bethell1 of ,rise in ,yorkshire1 ,esq2 ,hugh ,bethell1 of ,swindon in ,yorkshire1 ,esq2 ,mr ,silvanus ,bevan1 ,apothecary ,mr ,calverly ,bewick1 jun4 ,henry ,bigg1 ,b4 ,d4 ,warden of ,new ,college1 ,oxon ,sir ,william ,billers ---- ,billers1 ,esq2 ,mr ,john 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,gloucester ,mr ,joseph ,stephens ,sir ,james ,steuart of ,gutters1 ,bar4 ,mr ,robert ,steuart1 ,professor of ,natural ,philosophy1 in the ,university of ,edinburgh ,rev4 ,mr ,stevens1 ,fellow of ,corp4 ,chr4 ,coll4 ,cambridge ,mr ,john ,stevens1 of ,trinity ,college1 ,oxon ,rev4 ,mr ,bennet ,stevenson ,hon4 ,richard ,stewart1 ,esq2 ,major ,james ,stewart ,capt ,bartholomew ,stibbs ,mr ,denham ,stiles ,mr ,thomas ,stiles1 sen4 ,mr ,thomas ,stiles1 jun4 ,rev4 ,mr ,stillingfleet ,mr ,edward ,stillingfleet ,mr ,john ,stillingfleet ,mr ,william ,stith ,mr ,stock1 of ,rochdall in ,lancashire ,mr ,stocton1 ,watch-,maker ,mr ,robert ,stogdon ,rev4 ,mr ,richard ,stonehewer ,thomas ,stoner1 ,esq2 ,mr ,george ,story1 of ,trinity ,college1 ,cambridge ,mr ,thomas ,story ,william ,strahan1 ,l4 ,l4 ,d4 ,mr ,thomas ,stratfield ,rev4 ,dr4 ,stratford1 ,canon of ,christ ,church1 ,oxford ,capt4 ,william ,stratton ,rev4 ,mr ,streat ,samuel ,strode1 ,esq2 ,mr ,george ,strode ,rev4 ,mr ,john ,strong ,hon4 ,commodore ,stuart ,alexander ,stuart1 ,m4 ,d4 ,charles ,stuart1 ,m4 ,d4 ,lewis ,stucly ,mr ,john ,sturges1 of ,bloomsbury ,mr ,sturgeon1 ,surgeon in ,bury ,hon4 ,lady ,suasso ,mr ,gerrard ,suffield ,mr ,william ,sumner1 of ,windsor ,sir ,robert ,sutton1 ,kt4 of the ,bath ,rev4 ,mr ,john ,sutton ,mr ,gerrard ,swartz ,mr ,thomas ,swayne ,william ,swinburn1 ,esq2 ,rev4 ,mr4 ,john ,swinton1 ,m4 ,a4 ,mr ,joshua ,symmonds1 ,surgeon ,rev4 ,mr ,edward ,synge ,t4 ,his ,grace the ,archbishop of ,tuam ,right ,hon4 ,earl of ,tankerville ,rt4 ,hon4 ,ld4 ,viscount ,townshend1 ,one of ,his ,majesty's ,principal ,secretaries of ,state ,right ,honourable ,lady ,viscountess ,townshend ,right ,hon ,ld ,viscount ,tyrconnel ,the ,honourable ,lord ,trevor ,charles ,talbot1 ,esq2 ,solicitor-,general4 ,francis ,talbot1 ,esq2 ,john ,ivory ,talbot1 ,esq2 ,mr ,george ,talbot1 ,m4 ,a4 ,mr ,talbot ,thomas ,tanner1 ,d4 ,d4 ,chancellor of ,norwich ,mr ,thomas ,tanner ,mr ,tateham of ,clapham ,mr ,henry ,tatham ,mr ,john ,tatnall ,mr ,arthur ,tayldeur ,mr ,john ,tayleur ,arthur ,taylor1 ,esq2 ,joseph ,taylor1 ,esq2 ,simon ,taylor1 ,esq2 ,rev4 ,mr ,abraham ,taylor ,brook ,taylor1 ,l4 ,l4 ,d4 ,william ,tempest1 ,esq2 ,william ,tenison1 ,esq2 ,dr4 ,tenison ,rev4 ,dr4 ,terry1 ,canon of ,christ ,church1 ,oxon ,mr ,theed1 ,attorney ,mr ,lewis ,theobald ,james ,theobalds1 ,esq2 ,robert ,thistlethwayte1 ,d4 ,d4 ,warden of ,wadham ,coll4 ,oxon ,rev4 ,mr ,thomlinson ,richard ,thompson ,coley1 ,esq2 ,rev4 ,mr ,william ,thompson ,mr ,william ,thompson1 ,a4 ,b4 of ,trinity ,coll4 ,cambridge ,mr ,thoncas ,mr ,thornbury1 ,vicar of ,thame ,sir ,james ,thornhill1 #c ,books ,mr ,thornhill ,william ,thornton1 ,esq2 ,mr ,catlyn ,thorowgood ,mr ,john ,thorpe ,william ,thorseby1 ,esq2 ,mr ,william ,thurlbourn1 ,bookseller in ,cambridge ,mark ,thurston1 ,esq2 ,master in ,chancery ,rev4 ,mr ,william ,tiffin1 of ,lynn ,edmund ,tigh1 ,esq2 ,right ,hon4 ,richard ,tighe1 ,esq2 ,mr ,abraham ,tilghman ,mr ,george ,tilson ,rev ,mr ,tilson ,mr ,william ,tims ,rev4 ,mr ,john ,tisser ,capt4 ,joseph ,tolson ,mr ,tomkins ,mr ,william ,tomlinson ,richard ,topham1 ,esq2 ,dr4 ,torey ,george ,torriano1 of ,west-,ham1 ,esq2 ,mr ,john ,torriano ,mr ,james le ,touch ,rev4 ,mr ,charles ,tough ,mr ,john ,towers ,rev4 ,mr ,nehemiah ,towgood ,mr ,edward ,town ,joseph ,townsend1 ,esq2 ,charles ,townshend1 of ,lincoln's ,inn1 ,esq2 ,hon4 ,thomas ,townshend1 ,esq2 ,mr ,townson ,john ,tracey1 of ,stanway in ,gloucester1 ,esq2 ,capt4 ,richard ,tracey ,mr ,samuel ,traverse1 ,merchant ,mr ,charles ,trelawny1 ,student of ,christ ,church ,fredric ,trench1 ,esq2 ,mr ,edmund ,trench ,mr ,samuel ,trench ,richard ,trevor1 ,esq2 ,hon4 ,thomas ,trevor1 ,esq2 ,hon4 ,mr ,john ,trevor ,mr ,trimble1 ,merch4 in ,rotterd4 ,rev4 ,dr4 ,trimnell1 ,dean of ,winchester ,thomas ,trotter1 ,l4 ,l4 ,d4 ,john ,trubshaw1 ,esq2 ,mr ,thomas ,truman ,dr4 ,daniel ,turner ,rev4 ,mr4 ,robert ,turner1 of ,colchester ,mr ,john ,turton ,mr ,william ,turton ,john ,twistleton1 near the ,city of ,york1 ,esq2 ,col4 ,tyrrell ,mr ,william ,tyson ,mr ,samuel ,tyssen ,capt4 ,edward ,tyzack ,v ,rt4 ,hon4 ,lord ,viscount ,vane ,rev4 ,mr ,thomas ,valentine ,mr ,vallack1 of ,plymouth ,mr ,john ,vanderbank ,mr ,daniel ,vandewall ,mr ,john ,vandewall1 ,merchant ,mr ,edward ,vaus ,hon4 ,john ,verney1 ,esq2 ,william ,vesey1 ,esq2 ,rev4 ,mr ,john ,vesey ,william ,vigor1 of ,westbury ,college near ,bristol ,mr ,george ,virgoe ,mr ,frederick ,voguel1 ,merchant ,mr ,thomas ,vickers ,robert ,viner1 ,esq2 ,w ,rt4 ,hon4 the ,earl of ,winchelsea ,rt4 ,rev4 ,lord ,bishop of ,winchester ,rev4 ,mr ,wade ,sir ,charles ,wager ,rev4 ,mr ,wagstaffe ,rev4 ,dr4 ,edward ,wake ,mr ,jasper ,wakefield ,mr ,samuel ,walbank ,mr ,walbridge ,mr ,waldron ,edmund ,waldrond4 ,m4 ,a4 ,mr ,walford1 of ,wadham ,coll4 ,oxon ,rev4 ,mr ,edward ,walker ,mr ,samuel ,walker1 of ,trinity ,college1 ,cambridge ,mr ,thomas ,walker ,henry ,waller1 ,esq2 ,william ,waller1 ,esq2 ,mrs ,waller ,mr ,john ,waller1 of ,lincoln's-,inn ,mr ,george ,wallis ,rev4 ,mr ,william ,wallis ,mr ,edward ,walmsley1 #b ,books ,edward ,walpole1 ,esq2 ,mr ,peter ,walter ,john ,walton1 ,esq2 ,peter ,warburton of ,ford in ,cheshire1 ,esq2 ,richard ,warburton1 ,esq2 ,john ,ward1 jun4 ,esq2 ,michael ,ward1 ,esq2 ,edward ,ward1 ,esq2 ,knox ,ward1 ,esq2 ,mr ,john ,ward1 ,professor of ,rhetoric in ,gresham ,college ,william ,ward1 ,l4 ,l4 ,d4 ,mr ,richard ,warring ,mr ,jacob ,warneck ,mr ,richard ,warner ,mr ,robert ,warner ,william ,wasey1 ,m4 ,d4 ,rev4 ,mr ,washington1 ,fellow of ,peterhouse1 ,cambridge ,mr ,edward ,wastfield ,mr ,watkins ,rev4 ,mr ,thomas ,watkis1 of ,knutsford ,robert ,watley1 ,esq2 ,mr ,joel ,watson ,mr ,john ,watson ,mr ,thomas ,watson ,richard ,watts1 ,m4 ,d4 #b ,books ,mr ,thomas ,watts ,rev4 ,mr ,isaac ,watts ,mr ,william ,weamen ,mr ,thomas ,wear ,mr ,william ,weathers ,edward ,weaver1 ,esq2 ,anthony ,weaver1 ,m4 ,d4 ,mr ,webb ,mr ,willam ,webb1 ,a4 ,b4 of ,trinity ,college1 ,cambridge ,mr ,humphrey ,webb1 ,m4 ,a4 ,rt4 ,hon4 ,edward ,webster1 ,esq2 ,william ,wenman1 of ,edwinstowe1 ,esq2 ,mr ,samuel ,wesley1 jun4 ,gilbert ,west1 ,esq2 ,rt4 ,hon ,richard ,west1 ,esq2 late ,lord high ,chancellor of ,ireland ,thomas ,west1 ,esq2 ,dr4 ,thomas ,west ,mrs ,anne ,west ,daniel ,westcomb1 ,esq2 ,herbert ,westfaling1 ,esq2 ,messieurs ,werstein and ,smith1 ,booksellers in ,amsterdam ,mr ,western1 in ,dover-,street ,mr ,matthew ,westly ,mr ,tho4 ,weston1 of ,greenwich ,matthew ,weymondefold1 ,esq2 ,mr ,edward ,wharton ,mr ,stephen ,whatley ,mr ,james ,whatman ,granville ,wheler1 ,esq2 ,rev4 ,mr ,william ,whiston ,dr4 ,william ,whitaker ,taylor ,white1 ,esq2 ,mr ,charles ,white ,mr ,edward ,white1 ,scholar of ,caius ,college1 ,cambridge ,mr ,john ,white ,mr ,joseph ,white ,mr ,nicholas ,white ,mr ,william ,whitehead ,rev4 ,mr ,whitehead1 ,fellow of ,emanuel ,college1 ,cambridge #f ,books ,john ,whitfield1 ,d4 ,d4 ,rector of ,dickleburgh ,rev4 ,mr ,whitfield ,mr ,nathaniel ,whitlock ,mr ,john ,whittering ,robert ,wild1 ,esq2 ,mr ,william ,wildman ,rev4 ,mr ,wilkes1 ,prebendary of ,westminster ,dr4 ,wilkin ,mr ,wilkins1 ,bookseller ,mr ,abel ,wilkinson ,mr ,william ,wilks ,john ,willes1 ,esq2 ,john ,willet1 ,esq2 of the ,island of ,st4 ,christophers ,john ,williams1 ,esq2 ,william ,peer ,williams1 jun4 ,esq2 ,rev4 ,mr ,philip ,williams1 ,b4 ,d4 ,mr ,williams1 ,b4 ,a4 of ,jesus ,college1 ,oxon ,mr ,francis ,williams ,hon4 ,coll4 ,adam ,williamson ,mr ,robert ,willimott ,john ,willis1 ,esq2 ,edward ,wilmot1 ,m4 ,d4 ,mr ,robert ,willmott ,mr ,joseph ,willoughby ,william ,willys1 ,esq2 ,mr ,john ,wilmer1 ,merchant ,mr ,john ,wilmer1 ,apothecary ,mr ,wilmott1 ,bookseller in ,oxford ,richard ,wilson1 of ,leeds1 ,esq2 ,rev4 ,mr ,daniel ,wilson1 ,prebendary of the ,church of ,hereford ,william ,winde1 ,esq2 ,mr ,samuel ,winder1 jun4 ,sir ,william ,windham ,bar4 ,mr ,john ,windsor ,library of ,windsor ,college ,mr ,winnington ,mr ,winnock ,mr ,abraham ,winterbottom ,will4 ,withers1 of ,gray's-,inn1 ,esq2 ,mr ,conway ,withorne1 of the ,inner-,temple ,rev4 ,mr ,john ,witter ,jacobus ,wittichius1 ,phil4 ,d4 in ,acad4 ,lugd4 ,bat4 ,prof4 ,mr ,john ,wittingham ,rev4 ,mr ,john ,witton1 ,rector of ,howton ,witton1 ,cambridge ,mr ,thomas ,wood ,thomas ,woodcock1 ,esq2 ,thomas ,woodford1 ,esq2 ,william ,woodford1 ,m4 ,d4 ,john ,woodhouse1 ,m4 ,d4 ,mr ,j4 ,woods1 of ,bramshot1 ,merch4 ,rev4 ,mr ,benjamin ,woodroof1 ,prebendary of ,worcester ,mr ,joseph ,woodward ,josiah ,woolaston1 ,esq2 ,mr ,woolball1 ,merchant ,francis ,woollaston1 ,esq2 ,charlton ,woollaston1 ,esq2 ,mr ,william ,woollaston ,wight ,woolly1 ,esq2 ,library of the ,cathed4 of ,worcester ,josias ,wordsworth1 jun4 ,esq2 ,mr ,john ,worster1 ,merchant ,rev4 ,dr4 ,william ,wotton ,mr ,john ,wowen ,edward ,wright1 of the ,middle-,temple1 ,esq2 ,henry ,wright1 of ,molberly1 in ,cheshire1 ,esq2 ,samual ,wright1 ,esq2 ,william ,wright1 of ,offerton1 in ,cheshire1 ,esq2 ,mr ,wright ,mr ,william ,wright1 of ,baldock1 ,hertfordshire ,rev4 ,mr ,wrigley1 ,fellow of ,st4 ,john's ,college1 ,cambridge ,rt4 ,hon4 ,thomas ,wyndham1 ,ld4 ,chief ,justice of the ,common ,pleas1 of ,ireland ,mr ,joseph ,wyeth ,thomas ,wyndham1 ,esq2 ,rev4 ,mr ,john ,wynne ,y ,mr ,john ,yardley1 ,surg4 in ,coven4 ,mr ,thomas ,yates ,mrs ,yeo1 of ,exeter1 ,bookseller ,sir ,william ,yonge ,lady ,york ,nicholas ,young1 of the ,inner-,temple1 ,esq2 ,hitch ,young1 ,esq2 ,rev4 ,edward ,young1 ,l4 ,l4 ,d4 ,illustration ,,introduction4 ,,the manner1 in which ,sir ,,isaac ,,newton has published his philosophical discoveries1 occasions them to lie very much concealed from all1 who have not made the mathematics particularly their study4 ,he once1 indeed1 intended to deliver1 in a more familiar way1 that part of his inventions1 which relates to the system of the world2 but upon farther consideration he altered his design4 ,for as the nature of those discoveries made it impossible to prove them upon any other than geometrical principles2 he apprehended1 that those1 who should not fully perceive the force of his arguments1 would hardly be prevailed on to exchange their former sentiments for new opinions1 so very different from what were commonly received#a4 ,he therefore chose rather to explain himself only to mathematical readers2 and declined the attempting to instruct such in any of his principles1 who1 by not comprehending his method of reasoning1 could not1 at the first appearance of his discoveries1 have been persuaded of their truth4 ,but now1 since ,sir ,,isaac ,,newton's doctrine has been fully established by the unanimous approbation of all1 who are qualified to understand the same2 it is without doubt to be wished1 that the whole of his improvements in philosophy might be universally known4 ,for this purpose therefore ,i drew up the following papers1 to give a general notion of our great philosopher's inventions to such1 as are not prepared to read his own works1 and yet might desire to be informed of the progress1 he has made in natural knowledge2 not doubting but there were many1 besides those1 whose turn of mind had led them into a course of mathematical studies1 that would take great pleasure in tasting of this delightful fountain of science4 #b4 ,,it is a just remark1 which has been made upon the human mind1 that nothing is more suitable to it1 than the contemplation of truth2 and that all men are moved with a strong desire after knowledge2 esteeming it honourable to excel therein2 and holding it1 on the contrary1 disgraceful to mistake1 err1 or be in any way deceived4 ,and this sentiment is by nothing more fully illustrated1 than by the inclination of men to gain an acquaintance with the operations of nature2 which disposition to enquire after the causes of things is so general1 that all men of letters1 ,i believe1 find themselves influenced by it4 ,nor is it difficult to assign a reason for this1 if we consider only1 that our desire after knowledge is an effect of that taste for the sublime and the beautiful in things1 which chiefly constitutes the difference between the human life1 and the life of brutes4 ,these inferior animals partake with us of the pleasures1 that immediately flow from the bodily senses and appetites2 but our minds are furnished with a superior sense1 by which we are capable of receiving various degrees of delight1 where the creatures below us perceive no difference4 ,hence arises that pursuit of grace and elegance in our thoughts and actions1 and in all things belonging to us1 which principally creates imployment for the active mind of man4 ,the thoughts of the human mind are too extensive to be confined only to the providing and enjoying of what is necessary for the support of our being4 ,it is this taste1 which has given rise to poetry1 oratory1 and every branch of literature and science4 ,from hence we feel great pleasure in conceiving strongly1 and in apprehending clearly1 even where the passions are not concerned4 ,perspicuous reasoning appears not only beautiful2 but1 when set forth in its full strength and dignity1 it partakes of the sublime1 and not only pleases1 but warms and elevates the soul4 ,this is the source of our strong desire of knowledge2 and the same taste for the sublime and the beautiful directs us to chuse particularly the productions of nature for the subject of our contemplation3 our creator having so adapted our minds to the condition1 wherein he has placed us1 that all his visible works1 before we inquire into their make1 strike us with the most lively ideas of beauty and magnificence4 #c4 ,,but if there be so strong a passion in contemplative minds for natural philosophy2 all such must certainly receive a particular pleasure in being informed of ,sir ,,isaac ,,newton's discoveries1 who alone has been able to make any great advancements in the true course leading to natural knowledge3 whereas this important subject had before been usually attempted with that negligence1 as cannot be reflected on without surprize4 ,excepting a very few1 who1 by pursuing a more rational method1 had gained a little true knowledge in some particular parts of nature2 the writers in this science had generally treated of it after such a manner1 as if they thought1 that no degree of certainty was ever to be hoped for4 ,the custom was to frame conjectures2 and if upon comparing them with things1 there appeared some kind of agreement1 though very imperfect1 it was held sufficient4 ,yet at the same time nothing less was undertaken than intire systems1 and fathoming at once the greatest depths of nature2 as if the secret causes of natural effects1 contrived and framed by infinite wisdom1 could be searched out by the slightest endeavours of our weak understandings4 ,whereas the only method1 that can afford us any prospect of success in this difficult work1 is to make our enquiries with the utmost caution1 and by very slow degrees4 ,and after our most diligent labour1 the greatest part of nature will1 no doubt1 for ever remain beyond our reach4 #d4 ,,this neglect of the proper means to enlarge our knowledge1 joined with the presumption to attempt1 what was quite out of the power of our limited faculties1 the ,lord ,,bacon judiciously observes to be the great obstruction to the progress of science#b4 ,indeed that excellent person was the first1 who expresly writ against this way of philosophizing2 and he has laid open at large the absurdity of it in his admirable treatise1 intitled ,,novum ,,organon ,,scientiarum2 and has there likewise described the true method1 which ought to be followed4 #e4 ,,there are1 saith he1 but two methods1 that can be taken in the pursuit of natural knowledge4 ,one is to make a hasty transition from our first and slight observations on things to general axioms1 and then to proceed upon those axioms1 as certain and uncontestable principles1 without farther examination4 ,the other method2 7which he observes to be the only true one1 but to his time unattempted27 is to proceed cautiously1 to advance step by step1 reserving the most general principles for the last result of our inquiries#c4 ,concerning the first of these two methods2 where objections1 which happen to appear against any such axioms taken up in haste1 are evaded by some frivolous distinction1 when the axiom it self ought rather to be corrected#d2 he affirms1 that the united endeavours of all ages cannot make it successful2 because this original error in the first digestion of the mind 7as he expresses himself7 cannot afterwards be remedied#e3 whereby he would signify to us1 that if we set out in a wrong way2 no diligence or art1 we can use1 while we follow so erroneous a course1 will ever bring us to our designed end4 ,and doubtless it cannot prove otherwise2 for in this spacious field of nature1 if once we forsake the true path1 we shall immediately lose our selves1 and must for ever wander with uncertainty4 #f4 ,,the impossibility of succeeding in so faulty a method of philosophizing his ,lordship endeavours to prove from the many false notions and prejudices1 to which the mind of man is exposed#f4 ,and since this judicious writer apprehends1 that men are so exceeding liable to fall into these wrong tracts of thinking1 as to incur great danger of being misled by them1 even while they enter on the true course in pursuit of nature#g2 ,i trust1 ,i shall be excused1 if1 by insisting a little particularly upon this argument1 ,i endeavour to remove whatever prejudice of this kind1 might possibly entangle the mind of any of my readers4 #g4 ,,his ,lordship has reduced these prejudices and false modes of conception under four distinct heads#h4 #h4 ,,the first head contains such1 as we are subject to from the very condition of humanity1 through the weakness both of our senses1 and of the faculties of the mind#i2 seeing1 as this author well observes1 the subtilty of nature far exceeds the greatest subtilty of our senses or acutest reasonings#aj4 ,one of the false modes of conception1 which he mentions under this head1 is the forming to our selves a fanciful simplicity and regularity in natural things4 ,this he illustrates by the following instances2 the conceiving the planets to move in perfect circles2 the adding an orb of fire to the other three elements1 and the supposing each of these to exceed the other in rarity1 just in a decuple proportion#aa4 ,and of the same nature is the assertion of ,,des ,,cartes1 without any proof1 that all things are made up of three kinds of matter only#ab4 ,as also this opinion of another philosopher2 that light1 in passing through different mediums1 was refracted1 so as to proceed by that way1 through which it would move more speedily1 than through any other#ac4 ,the second erroneous turn of mind1 taken notice of by his ,lordship under this head1 is1 that all men are in some degree prone to a fondness for any notions1 which they have once imbibed2 whereby they often wrest things to reconcile them to those notions1 and neglect the consideration of whatever will not be brought to an agreement with them2 just as those do1 who are addicted to judicial astrology1 to the observation of dreams1 and to such-like superstitions2 who carefully preserve the memory of every incident1 which serves to confirm their prejudices1 and let slip out of their minds all instances1 that make against them#ad4 ,there is also a farther impediment to true knowledge1 mentioned under the same head by this noble writer1 which is2 that whereas1 through the weakness and imperfection of our senses1 many things are concealed4 from us1 which have the greatest effect in producing natural appearances2 our minds are ordinarily most affected by that1 which makes the strongest impression on our organs of sense2 whereby we are apt to judge of the real importance of things in nature by a wrong measure#ae4 ,so1 because the figuration and the motion of bodies strike our senses more immediately than most of their other properties1 ,,des ,,cartes and his followers will not allow any other explication of natural appearances1 than from the figure and motion of the parts of matter4 ,by which example we see how justly his ,lordship observes this cause of error to be the greatest of any#af2 since it has given rise to a fundamental principle in a system of philosophy1 that not long ago obtained almost an universal reputation4 #i4 ,,these are the chief branches of those obstructions to knowledge1 which this author has reduced under his first head of false conceptions4 ,the second head contains the errors1 to which particular persons are more especially obnoxious#ag4 ,one of these is the consequence of a preceding observation3 that as we are exposed to be captivated by any opinions1 which have once taken possession of our minds2 so in particular1 natural knowledge has been much corrupted by the strong attachment of men to some one part of science1 of which they reputed themselves the inventers1 or about which they have spent much of their time2 and hence have been apt to conceive it to be of greater use in the study of natural philosophy than it was3 like ,,aristotle1 who reduced his physics to logical disputations2 and the chymists1 who thought1 that nature could be laid open only by the force of their fires#ah4 ,some again are wholly carried away by an excessive veneration for antiquity2 others1 by too great fondness for the moderns2 few having their minds so well balanced1 as neither to depreciate the merit of the ancients1 nor yet to despise the real improvements of later times#ai4 ,to this is added by his ,lordship a difference in the genius of men1 that some are most fitted to observe the similitude1 there is in things1 while others are more qualified to discern the particulars1 wherein they disagree2 both which dispositions of mind are useful3 but to the prejudice of philosophy men are apt to run into excess in each2 while one sort of genius dwells too much upon the gross and sum of things1 and the other upon trifling minutenesses and shadowy distinctions#bj4 #aj4 ,,under the third head of prejudices and false notions this writer considers such1 as follow from the lax and indefinite use of words in ordinary discourse2 which occasions great ambiguities and uncertainties in philosophical debates 7as another eminent philosopher has since shewn more at large#ba27 insomuch that this our author thinks a strict defining of terms to be scarce an infallible remedy against this inconvenience#bb4 ,and perhaps he has no small reason on his side3 for the common inaccurate sense of words1 notwithstanding the limitations given them by definitions1 will offer it self so constantly to the mind1 as to require great caution and circumspection for us not to be deceived thereby4 ,of this we have a very eminent instance in the great disputes1 that have been raised about the use of the word attraction in philosophy2 of which we shall be obliged hereafter to make particular mention#bc4 ,words thus to be guarded against are of two kinds4 ,some are names of things1 that are only imaginary#bd2 such words are wholly to be rejected4 ,but there are other terms1 that allude to what is real1 though their signification is confused#be4 ,and these latter must of necessity be continued in use2 but their sense cleared up1 and freed1 as much as possible1 from obscurity4 #aa4 ,,the last general head of these errors comprehends such1 as follow from the various sects of false philosophies2 which this author divides into three sorts1 the sophistical1 empirical1 and superstitious#bf4 ,by the first of these he means a philosophy built upon speculations only without experiments#bg2 by the second1 where experiments are blindly adhered to1 without proper reasoning upon them#bh2 and by the third1 wrong opinions of nature fixed in mens minds either through false religions1 or from misunderstanding the declarations of the true#bi4 #ab4 ,,these are the four principal canals1 by which this judicious author thinks1 that philosophical errors have flowed in upon us4 ,and he rightly observes1 that the faulty method of proceeding in philosophy1 against which he writes#cj1 is so far from assisting us towards overcoming these prejudices2 that he apprehends it rather suited to rivet them more firmly to the mind#ca4 ,how great reason then has his ,lordship to call this way of philosophizing the parent of error1 and the bane of all knowledge#cb8 ,for1 indeed1 what else but mistakes can so bold and presumptuous a treatment of nature produce8 have we the wisdom necessary to frame a world1 that we should think so easily1 and with so slight a search to enter into the most secret springs of nature1 and discover the original causes of things8 what chimeras1 what monsters has not this preposterous method brought forth8 what schemes1 or what hypothesis's of the subtilest wits has not a stricter enquiry into nature not only overthrown1 but manifested to be ridiculous and absurd8 ,every new improvement1 which we make in this science1 lets us see more and more the weakness of our guesses4 ,dr4 ,,harvey1 by that one discovery of the circulation of the blood1 has dissipated all the speculations and reasonings of many ages upon the animal oeconomy4 ,,asellius1 by detecting the lacteal veins1 shewed how little ground all physicians and philosophers had in conjecturing1 that the nutritive part of the aliment was absorbed by the mouths of the veins spread upon the bowels3 and then ,,pecquet1 by finding out the thoracic duct1 as evidently proved the vanity of the opinion1 which was persisted in after the lacteal vessels were known1 that the alimental juice was conveyed immediately to the liver1 and there converted into blood4 #ac4 ,,as these things set forth the great absurdity of proceeding in philosophy on conjectures1 by informing us how far the operations of nature are above our low conceptions2 so on the other hand1 such instances of success from a more judicious method shew us1 that our bountiful maker has not left us wholly without means of delighting our selves in the contemplation of his wisdom4 ,that by a just way of inquiry into nature1 we could not fail of arriving at discoveries very remote from our apprehensions2 the ,lord ,,bacon himself argues from the experience of mankind4 ,if1 says he1 the force of guns should be described to any one ignorant of them1 by their effects only1 he might reasonably suppose1 that those engines of destruction were only a more artificial composition1 than he knew1 of wheels and other mechanical powers3 but it could never enter his thoughts1 that their immense force should be owing to a peculiar substance1 which would enkindle into so violent an explosion1 as we experience in gunpowder3 since he would no where see the least example of any such operation2 except perhaps in earthquakes and thunder1 which he would doubtless look upon as exalted powers of nature1 greatly surpassing any art of man to imitate4 ,in the same manner1 if a stranger to the original of silk were shewn a garment made of it1 he would be very far from imagining so strong a substance to be spun out of the bowels of a small worm2 but must certainly believe it either a vegetable substance1 like flax or cotton2 or the natural covering of some animal1 as wool is of sheep4 ,or had we been told1 before the invention of the magnetic needle among us1 that another people was in possession of a certain contrivance1 by which they were inabled to discover the position of the heavens1 with vastly more ease1 than we could do2 what could have been imagined more1 than that they were provided with some fitter astronomical instrument for this purpose than we8 ,that any stone should have so amazing a property1 as we find in the magnet1 must have been the remotest from our thoughts#cc4 #ad4 ,,but what surprizing advancements in the knowledge of nature may be made by pursuing the true course in philosophical inquiries2 when those searches are conducted by a genius equal to so divine a work1 will be best understood by considering ,sir ,,isaac ,,newton discoveries4 ,that my's reader may apprehend as just a notion of these1 as can be conveyed to him1 by the brief account1 which ,i intend to lay before him2 ,i have set apart this introduction for explaining1 in the fullest manner ,i am able1 the principles1 whereon ,sir ,,isaac ,,newton proceeds4 ,for without a clear conception of these1 it is impossible to form any true idea of the singular excellence of the inventions of this great philosopher4 #ae4 ,,the principles then of this philosophy are2 upon no consideration to indulge conjectures concerning the powers and laws of nature1 but to make it our endeavour with all diligence to search out the real and true laws1 by which the constitution of things is regulated4 ,the philosopher's first care must be to distinguish1 what he sees to be within his power1 from what is beyond his reach2 to assume no greater degree of knowledge1 than what he finds himself possessed of2 but to advance by slow and cautious steps2 to search gradually into natural causes2 to secure to himself the knowledge of the most immediate cause of each appearance1 before he extends his views farther to causes more remote4 ,this is the method1 in which philosophy ought to be cultivated2 which does not pretend to so great things1 as the more airy speculations2 but will perform abundantly more3 we shall not perhaps seem to the unskilful to know so much1 but our real knowledge will be greater4 ,and certainly it is no objection against this method1 that some others promise1 what is nearer to the extent of our wishes3 since this1 if it will not teach us all we could desire to be informed of1 will however give us some true light into nature2 which no other can do4 ,nor has the philosopher any reason to think his labour lost1 when he finds himself stopt at the cause first discovered by him1 or at any other more remote cause1 short of the original3 for if he has but sufficiently proved any one cause1 he has entered so far into the real constitution of things1 has laid a safe foundation for others to work upon1 and has facilitated their endeavours in the search after yet more distant causes2 and besides1 in the mean time he may apply the knowledge of these intermediate causes to many useful purposes4 ,indeed the being able to make practical deductions from natural causes1 constitutes the great distinction between the true philosophy and the false4 ,causes assumed upon conjecture1 must be so loose and undefined1 that nothing particular can be collected from them4 ,but those causes1 which are brought to light by a strict examination of things1 will be more distinct4 ,hence it appears to have been no unuseful discovery1 that the ascent of water in pumps is owing to the pressure of the air by its weight or spring2 though the causes1 which make the air gravitate1 and render it elastic1 be unknown3 for notwithstanding we are ignorant of the original1 whence these powers of the air are derived2 yet we may receive much advantage from the bare knowledge of these powers4 ,if we are but certain of the degree of force1 wherewith they act1 we shall know the extent of what is to be expected from them2 we shall know the greatest height1 to which it is possible by pumps to raise water2 and shall thereby be prevented from making any useless efforts towards improving these instruments beyond the limits prescribed to them by nature2 whereas without so much knowledge as this1 we might probably have wasted in attempts of this kind much time and labour4 ,how long did philosophers busy themselves to no purpose in endeavouring to perfect telescopes1 by forming the glasses into some new figure2 till ,sir ,,isaac ,,newton demonstrated1 that the effects of telescopes were limited from another cause1 than was supposed2 which no alteration in the figure of the glasses could remedy8 ,what method ,sir ,,isaac ,,newton himself has found for the improvement of telescopes shall be explained hereafter#cd4 ,but at present ,i shall proceed to illustrate1 by some farther instances1 this distinguishing character of the true philosophy1 which we have now under consideration4 ,it was no trifling discovery1 that the contraction of the muscles of animals puts their limbs in motion1 though the original cause of that contraction remains a secret1 and perhaps may always do so2 for the knowledge of thus much only has given rise to many speculations upon the force and artificial disposition of the muscles1 and has opened no narrow prospect into the animal fabrick4 ,the finding out1 that the nerves are great agents in this action1 leads us yet nearer to the original cause1 and yields us a wider view of the subject4 ,and each of these steps affords us assistance towards restoring this animal motion1 when impaired in our selves1 by pointing out the seats of the injuries1 to which it is obnoxious4 ,to neglect all this1 because we can hitherto advance no farther1 is plainly ridiculous4 ,it is confessed by all1 that ,,galileo greatly improved philosophy1 by shewing1 as we shall relate hereafter1 that the power in bodies1 which we call gravity1 occasions them to move downwards with a velocity equably accelerated#ce2 and that when any body is thrown forwards1 the same power obliges it to describe in its motion that line1 which is called by geometers a parabola#cf3 yet we are ignorant of the cause1 which makes bodies gravitate4 ,but although we are unacquainted with the spring1 whence this power in nature is derived1 nevertheless we can estimate its effects4 ,when a body falls perpendicularly1 it is known1 how long time it takes in descending from any height whatever3 and if it be thrown forwards1 we know the real path1 which it describes2 we can determine in what direction1 and with what degree of swiftness it must be projected1 in order to its striking against any object desired2 and we can also ascertain the very force1 wherewith it will strike4 ,sir ,,isaac ,,newton has farther taught1 that this power of gravitation extends up to the moon1 and causes that planet to gravitate as much towards the earth1 as any of the bodies1 which are familiar to us1 would1 if placed at the same distance#cg3 he has proved likewise1 that all the planets gravitate towards the sun1 and towards one another2 and that their respective motions follow from this gravitation4 ,all this he has demonstrated upon indisputable geometrical principles1 which cannot be rendered precarious for want of knowing what it is1 which causes these bodies thus mutually to gravitate3 any more than we can doubt of the propensity in all the bodies about us1 to descend towards the earth2 or can call in question the forementioned propositions of ,,galileo1 which are built upon that principle4 ,and as ,,galileo has shewn more fully1 than was known before1 what effects were produced in the motion of bodies by their gravitation towards the earth2 so ,sir ,,isaac ,,newton1 by this his invention1 has much advanced our knowledge in the celestial motions4 ,by discovering that the moon gravitates towards the sun1 as well as towards the earth2 he has laid open those intricacies in the moon's motion1 which no astronomer1 from observations only1 could ever find out#ch3 and one kind of heavenly bodies1 the comets1 have their motion now clearly ascertained2 whereof we had before no true knowledge at all#ci4 #af4 ,,doubtless it might be expected1 that such surprizing success should have silenced1 at once1 every cavil4 ,but we have seen the contrary4 ,for because this philosophy professes modestly to keep within the extent of our faculties1 and is ready to confess its imperfections1 rather than to make any fruitless attempts to conceal them1 by seeking to cover the defects in our knowledge with the vain ostentation of rash and groundless conjectures2 hence has been taken an occasion to insinuate that we are led to miraculous causes1 and the occult qualities of the schools4 #ag4 ,,but the first of these accusations is very extraordinary4 ,if by calling these causes miraculous nothing more is meant than only1 that they often appear to us wonderful and surprizing1 it is not easy to see what difficulty can be raised from thence2 for the works of nature discover every where such proofs of the unbounded power1 and the consummate wisdom of their author1 that the more they are known1 the more they will excite our admiration3 and it is too manifest to be insisted on1 that the common sense of the word miraculous can have no place here1 when it implies what is above the ordinary course of things4 ,the other imputation1 that these causes are occult upon the account of our not perceiving what produces them1 contains in it great ambiguity4 ,that something relating to them lies hid1 the followers of this philosophy are ready to acknowledge1 nay desire it should be carefully remarked1 as pointing out proper subjects for future inquiry4 ,but this is very different from the proceeding of the schoolmen in the causes called by them occult4 ,for as their occult qualities were understood to operate in a manner occult1 and not apprehended by us2 so they were obtruded upon us for such original and essential properties in bodies1 as made it vain to seek any farther cause2 and a greater power was attributed to them1 than any natural appearances authorized4 ,for instance1 the rise of water in pumps was ascribed to a certain abhorrence of a vacuum1 which they thought fit to assign to nature4 ,and this was so far a true observation1 that the water does move1 contrary to its usual course1 into the space1 which otherwise would be left void of any sensible matter2 and1 that the procuring such a vacuity was the apparent cause of the water's ascent4 ,but while we were not in the least informed how this power1 called an abhorrence of a vacuum1 produced the visible effects2 instead of making any advancement in the knowledge of nature1 we only gave an artificial name to one of her operations3 and when the speculation was pushed so beyond what any appearances required1 as to have it concluded1 that this abhorrence of a vacuum was a power inherent in all matter1 and so unlimited as to render it impossible for a vacuum to exist at all2 it then became a much greater absurdity1 in being made the foundation of a most ridiculous manner of reasoning2 as at length evidently appeared1 when it came to be discovered1 that this rise of the water followed only from the pressure of the air1 and extended it self no farther1 than the power of that cause4 ,the scholastic stile in discoursing of these occult qualities1 as if they were essential differences in the very substances1 of which bodies consisted1 was certainly very absurd2 by reason it tended to discourage all farther inquiry4 ,but no such ill consequences can follow from the considering of any natural causes1 which confessedly are not traced up to their first original4 ,how shall we ever come to the knowledge of the several original causes of things1 otherwise than by storing up all intermediate causes which we can discover8 ,are all the original and essential properties of matter so very obvious1 that none of them can escape our first view8 ,this is not probable4 ,it is much more likely1 that1 if some of the essential properties are discovered by our first observations1 a stricter examination should bring more to light4 #ah4 ,,but in order to clear up this point concerning the essential properties of matter1 let us consider the subject a little distinctly4 ,we are to conceive1 that the matter1 out of which the universe of things is formed1 is furnished with certain qualities and powers1 whereby it is rendered fit to answer the purposes1 for which it was created4 ,but every property1 of which any particle of this matter is in it self possessed1 and which is not barely the consequence of the union of this particle with other portions of matter1 we may call an essential property3 whereas all other qualities or attributes belonging to bodies1 which depend on their particular frame and composition1 are not essential to the matter1 whereof such bodies are made2 because the matter of these bodies will be deprived of those qualities1 only by the dissolution of the body1 without working any change in the original constitution of one single particle of this mass of matter4 ,extension we apprehend to be one of these essential properties1 and impenetrability another4 ,these two belong universally to all matter2 and are the principal ingredients in the idea1 which this word matter usually excites in the mind4 ,yet as the idea1 marked by this name1 is not purely the creature of our own understandings1 but is taken for the representation of a certain substance without us2 if we should discover1 that every part of the substance1 in which we find these two properties1 should likewise be endowed universally with any other essential qualities2 all these1 from the time they come to our notice1 must be united under our general idea of matter4 ,how many such properties there are actually in all matter we know not2 those1 of which we are at present apprized1 have been found out only by our observations on things2 how many more a farther search may bring to light1 no one can say2 nor are we certain1 that we are provided with sufficient methods of perception to discern them all4 ,therefore1 since we have no other way of making discoveries in nature1 but by gradual inquiries into the properties of bodies2 our first step must be to admit without distinction all the properties1 which we observe2 and afterwards we must endeavour1 as far as we are able1 to distinguish between the qualities1 wherewith the very substances themselves are indued1 and those appearances1 which result from the structure only of compound bodies4 ,some of the properties1 which we observe in things1 are the attributes of particular bodies only2 others universally belong to all1 that fall under our notice4 ,whether some of the qualities and powers of particular bodies1 be derived from different kinds of matter entring their composition1 cannot1 in the present imperfect state of our knowledge1 absolutely be decided2 though we have not yet any reason to conclude1 but that all the bodies1 with which we converse1 are framed out of the very same kind of matter1 and that their distinct qualities are occasioned only by their structure2 through the variety whereof the general powers of matter are caused to produce different effects4 ,on the other hand1 we should not hastily conclude1 that whatever is found to appertain to all matter1 which falls under our examination1 must for that reason only be an essential property thereof1 and not be derived from some unseen disposition in the frame of nature4 ,sir ,,isaac ,,newton has found reason to conclude1 that gravity is a property universally belonging to all the perceptible bodies in the universe1 and to every particle of matter1 whereof they are composed4 ,but yet he no where asserts this property to be essential to matter4 ,and he was so far from having any design of establishing it as such1 that1 on the contrary1 he has given some hints worthy of himself at a cause for it#dj2 and expresly says1 that he proposed those hints to shew1 that he had no such intention#da4 #ai4 ,,it appears from hence1 that it is not easy to determine1 what properties of bodies are essentially inherent in the matter1 out of which they are made1 and what depend upon their frame and composition4 ,but certainly whatever properties are found to belong either to any particular systems of matter1 or universally to all1 must be considered in philosophy2 because philosophy will be otherwise imperfect4 ,whether those properties can be deduced from some other appertaining to matter1 either among those1 which are already known1 or among such as can be discovered by us1 is afterwards to be sought for the farther improvement of our knowledge4 ,but this inquiry cannot properly have place in the deliberation about admitting any property of matter or bodies into philosophy2 for that purpose it is only to be considered1 whether the existence of such a property has been justly proved or not4 ,therefore to decide what causes of things are rightly received into natural philosophy1 requires only a distinct and clear conception of what kind of reasoning is to be allowed of as convincing1 when we argue upon the works of nature4 #bj4 ,,the proofs in natural philosophy cannot be so absolutely conclusive1 as in the mathematics4 ,for the subjects of that science are purely the ideas of our own minds4 ,they may be represented to our senses by material objects1 but they are themselves the arbitrary productions of our own thoughts2 so that as the mind can have a full and adequate knowledge of its own ideas1 the reasoning in geometry can be rendered perfect4 ,but in natural knowledge the subject of our contemplation is without us1 and not so compleatly to be known3 therefore our method of arguing must fall a little short of absolute perfection4 ,it is only here required to steer a just course between the conjectural method of proceeding1 against which ,i have so largely spoke2 and demanding so rigorous a proof1 as will reduce all philosophy to mere scepticism1 and exclude all prospect of making any progress in the knowledge of nature4 #ba4 ,,the concessions1 which are to be allowed in this science1 are by ,sir ,,isaac ,,newton included under a very few simple precepts4 #bb4 ,,the first is1 that more causes are not to be received into philosophy1 than are sufficient to explain the appearances of nature4 ,that this rule is approved of unanimously1 is evident from those expressions so frequent among all philosophers1 that nature does nothing in vain2 and that a variety of means1 where fewer would suffice1 is needless4 ,and certainly there is the highest reason for complying with this rule4 ,for should we indulge the liberty of multiplying1 without necessity1 the causes of things1 it would reduce all philosophy to mere uncertainty2 since the only proof1 which we can have1 of the existence of a cause1 is the necessity of it for producing known effects4 ,therefore where one cause is sufficient1 if there really should in nature be two1 which is in the last degree improbable1 we can have no possible means of knowing it1 and consequently ought not to take the liberty of imagining1 that there are more than one4 #bc4 ,,the second precept is the direct consequence of the first1 that to like effects are to be ascribed the same causes4 ,for instance1 that respiration in men and in brutes is brought about by the same means2 that bodies descend to the earth here in ,,europe1 and in ,,america from the same principle2 that the light of a culinary fire1 and of the sun have the same manner of production2 that the reflection of light is effected in the earth1 and in the planets by the same power2 and the like4 #bd4 ,,the third of these precepts has equally evident reason for it4 ,it is only1 that those qualities1 which in the same body can neither be lessened nor increased1 and which belong to all bodies that are in our power to make trial upon1 ought to be accounted the universal properties of all bodies whatever4 #be4 ,,in this precept is founded that method of arguing by induction1 without which no progress could be made in natural philosophy4 ,for as the qualities of bodies become known to us by experiments only2 we have no other way of finding the properties of such bodies1 as are out of our reach to experiment upon1 but by drawing conclusions from those which fall under our examination4 ,the only caution here required is1 that the observations and experiments1 we argue upon1 be numerous enough1 and that due regard be paid to all objections1 that occur1 as the ,lord ,,bacon very judiciously directs#db4 ,and this admonition is sufficiently complied with1 when by virtue of this rule we ascribe impenetrability and extension to all bodies1 though we have no sensible experiment1 that affords a direct proof of any of the celestial bodies being impenetrable2 nor that the fixed stars are so much as extended4 ,for the more perfect our instruments are1 whereby we attempt to find their visible magnitude1 the less they appear2 insomuch that all the sensible magnitude1 which we observe in them1 seems only to be an optical deception by the scattering of their light4 ,however1 ,i suppose no one will imagine they are without any magnitude1 though their immense distance makes it undiscernable by us4 ,after the same manner1 if it can be proved1 that all bodies here gravitate towards the earth1 in proportion to the quantity of solid matter in each2 and that the moon gravitates to the earth likewise1 in proportion to the quantity of matter in it2 and that the sea gravitates towards the moon1 and all the planets towards each other2 and that the very comets have the same gravitating faculty2 we shall have as great reason to conclude by this rule1 that all bodies gravitate towards each other4 ,for indeed this rule will more strongly hold in this case1 than in that of the impenetrability of bodies2 because there will more instances be had of bodies gravitating1 than of their being impenetrable4 #be4 ,,this is that method of induction1 whereon all philosophy is founded2 which our author farther inforces by this additional precept1 that whatever is collected from this induction1 ought to be received1 notwithstanding any conjectural hypothesis to the contrary1 till such times as it shall be contradicted or limited by farther observations on nature4 ,illustration ,illustration ,,book ,i4 ,,concerning ,,the ,,motion of ,,bodies ,,in ,,general4 ,,chap4 ,i4 ,of the ,,laws of ,,motion4 ,,having thus explained ,sir ,,isaac ,,newton's method of reasoning in philosophy1 ,i shall now proceed to my intended account of his discoveries4 ,these are contained in two treatises4 ,in one of them1 the ,,mathematical ,,principles ,,of ,,natural ,,philosophy1 his chief design is to shew by what laws the heavenly motions are regulated2 in the other1 his ,,optics1 he discourses of the nature of light and colours1 and of the action between light and bodies4 ,this second treatise is wholly confined to the subject of light3 except some conjectures proposed at the end concerning other parts of nature1 which lie hitherto more concealed4 ,in the other treatise our author was obliged to smooth the way to his principal intention1 by explaining many things of a more general nature3 for even some of the most simple properties of matter were scarce well established at that time4 ,we may therefore reduce ,sir ,,isaac ,,newton's doctrine under three general heads2 and ,i shall accordingly divide my account into three books4 ,in the first ,i shall speak of what he has delivered concerning the motion of bodies1 without regard to any particular system of matter2 in the second ,i shall treat of the heavenly motions2 and the third shall be employed upon light4 #b4 ,,in the first part of my design1 we must begin with an account of the general laws of motion4 #c4 ,,these laws are some universal affections and properties of matter drawn from experience1 which are made use of as axioms and evident principles in all our arguings upon the motion of bodies4 ,for as it is the custom of geometers to assume in their demonstrations some propositions1 without exhibiting the proof of them2 so in philosophy1 all our reasoning must be built upon some properties of matter1 first admitted as principles whereon to argue4 ,in geometry these axioms are thus assumed1 on account of their being so evident as to make any proof in form needless4 ,but in philosophy no properties of bodies can be in this manner received for self-evident2 since it has been observed above1 that we can conclude nothing concerning matter by any reasonings upon its nature and essence1 but that we owe all the knowledge1 we have thereof1 to experience4 ,yet when our observations on matter have inform'd us of some of its properties1 we may securely reason upon them in our farther inquiries into nature4 ,and these laws of motion1 of which ,i am here to speak1 are found so universally to belong to bodies1 that there is no motion known1 which is not regulated by them4 ,these are by ,sir ,,isaac ,,newton reduced to three#dc4 #d4 ,,the first law is1 that all bodies have such an indifference to rest1 or motion1 that if once at rest they remain so1 till disturbed by some power acting upon them3 but if once put in motion1 they persist in it2 continuing to move right forwards perpetually1 after the power1 which gave the motion1 is removed2 and also preserving the same degree of velocity or quickness1 as was first communicated1 not stopping or remitting their course1 till interrupted or otherwise disturbed by some new power impressed4 #e4 ,,the second law of motion is1 that the alteration of the state of any body1 whether from rest to motion1 or from motion to rest1 or from one degree of motion to another1 is always proportional to the force impressed4 ,a body at rest1 when acted upon by any power1 yields to that power1 moving in the same line1 in which the power applied is directed2 and moves with a less or greater degree of velocity1 according to the degree of the power2 so that twice the power shall communicate a double velocity1 and three times the power a threefold velocity4 ,if the body be moving1 and the power impressed act upon the body in the direction of its motion1 the body shall receive an addition to its motion1 as great as the motion1 into which that power would have put it from a state of rest2 but if the power impressed upon a moving body act directly opposite to its former motion1 that power shall then take away from the body's motion1 as much as in the other case it would have added to it4 ,lastly1 if the power be impressed obliquely1 there will arise an oblique motion differing more or less from the former direction1 according as the new impression is greater or less4 ,for example1 if the body ,a 7in fig4 #a47 be moving in the direction ,a ,b1 and when it is at the point ,a1 a power be impressed upon it in the direction ,a ,c1 the body shall from henceforth neither move in its first direction ,a ,b1 nor in the direction of the adventitious power1 but shall take a course as ,a ,d between them3 and if the power last impressed be just equal to that1 which first gave to the body its motion2 the line ,a ,d shall pass in the middle between ,a ,b and ,a ,c1 dividing the angle under ,b ,a ,c into two equal parts2 but if the power last impressed be greater than the first1 the line ,a ,d shall incline most to ,a ,c2 whereas if the last impression be less than the first1 the line ,a ,d shall incline most to ,a ,b4 ,to be more particular1 the situation of the line ,a ,d is always to be determined after this manner4 ,let ,a ,e be the space1 which the body would have moved through in the line ,a ,b during any certain portion of time2 provided that body1 when at ,a1 had received no second impulse4 ,suppose likewise1 that ,a ,f is the part of the line ,a ,c1 through which the body would have moved during an equal portion of time1 if it had been at rest in ,a1 when it received the impulse in the direction ,a ,c3 then if from ,e be drawn a line parallel to1 or equidistant from ,a ,c1 and from ,f another line parallel to ,a ,b1 those two lines will meet in the line ,a ,d4 #f4 ,,the third and last of these laws of motion is1 that when any body acts upon another1 the action of that body upon the other is equalled by the contrary reaction of that other body upon the first4 #g4 ,,these laws of motion are abundantly confirmed by this1 that all the deductions made from them1 in relation to the motion of bodies1 how complicated soever1 are found to agree perfectly with observation4 ,this shall be shewn more at large in the next chapter4 ,but before we proceed to so diffusive a proof2 ,i chuse here to point out those appearances of bodies1 whereby the laws of motion are first suggested to us4 #h4 ,,daily observation makes it appear to us1 that any body1 which we once see at rest1 never puts it self into fresh motion2 but continues always in the same place1 till removed by some power applied to it4 #i4 ,,again1 whenever a body is once in motion1 it continues in that motion some time after the moving power has quitted it1 and it is left to it self4 ,now if the body continue to move but a single moment1 after the moving power has left it1 there can no reason be assigned1 why it should ever stop without some external force4 ,for it is plain1 that this continuance of the motion is caused only by the body's having already moved1 the sole operation of the power upon the body being the putting it in motion2 therefore that motion continued will equally be the cause of its farther motion1 and so on without end4 ,the only doubt that can remain1 is1 whether this motion communicated continues intire1 after the power1 that caused it1 ceases to act2 or whether it does not gradually languish and decrease4 ,and this suspicion cannot be removed by a transient and slight observation on bodies1 but will be fully cleared up by those more accurate proofs of the laws of motion1 which are to be considered in the next chapter4 #aj4 ,,lastly1 bodies in motion appear to affect a straight course without any deviation1 unless when disturbed by some adventitious power acting upon them4 ,if a body be thrown perpendicularly upwards or downwards1 it appears to continue in the same straight line during the whole time of its motion4 ,if a body be thrown in any other direction1 it is found to deviate from the line1 in which it began to move1 more and more continually towards the earth1 whither it is directed by its weight3 but since1 when the weight of a body does not alter the direction of its motion1 it always moves in a straight line1 without doubt in this other case the body's1 declining from its first course is no more1 than what is caused by its weight alone4 ,as this appears at first sight to be unquestionable1 so we shall have a very distinct proof thereof in the next chapter1 where the oblique motion of bodies will be particularly considered4 #aa4 ,,thus we see how the first of the laws of motion agrees with what appears to us in moving bodies4 ,but here occurs this farther consideration1 that the real and absolute motion of any body is not visible to us3 for we are our selves also in constant motion along with the earth whereon we dwell2 insomuch that we perceive bodies to move so far only1 as their motion is different from our own4 ,when a body appears to us to lie at rest1 in reality it only continues the motion1 it has received1 without putting forth any power to change that motion4 ,if we throw a body in the course or direction1 wherein we are carried our selves2 so much motion as we seem to have given to the body1 so much we have truly added to the motion1 it had1 while it appeared to us to be at rest4 ,but if we impel a body the contrary way1 although the body appears to us to have received by such an impulse as much motion1 as when impelled the other way2 yet in this case we have taken from the body so much real motion1 as we seem to have given it4 ,thus the motion1 which we see in bodies1 is not their real motion1 but only relative with respect to us2 and the forementioned observations only shew us1 that this first law of motion has place in this relative or apparent motion4 ,however1 though we cannot make any observation immediately on the absolute motion of bodies1 yet by reasoning upon what we observe in visible motion1 we can discover the properties and effects of real motion4 #ab4 ,,with regard to this first law of motion1 which is now under consideration1 we may from the foregoing observations most truly collect1 that bodies are disposed to continue in the absolute motion1 which they have once received1 without increasing or diminishing their velocity4 ,when a body appears to us to lie at rest1 it really preserves without change the motion1 which it has in common with our selves3 and when we put it into visible motion1 and we see it continue that motion2 this proves1 that the body retains that degree of its absolute motion1 into which it is put by our acting upon it3 if we give it such an apparent motion1 which adds to its real motion1 it preserves that addition2 and if our acting on the body takes off from its real motion1 it continues afterwards to move with no more real motion1 than we have left it4 #ac4 ,,again1 we do not observe in bodies any disposition or power within themselves to change the direction of their motion2 and if they had any such power1 it would easily be discovered4 ,for suppose a body by the structure or disposition of its parts1 or by any other circumstance in its make1 was indued with a power of moving it self2 this self-moving principle1 which should be thus inherent in the body1 and not depend on any thing external1 must change the direction wherein it would act1 as often as the position of the body was changed3 so that for instance1 if a body was lying before me in such a position1 that the direction1 wherein this principle disposes the body to move1 was pointed directly from me2 if ,i then gradually turned the body about1 the direction of this self-moving principle would no longer be pointed directly from me1 but would turn about along with the body4 ,now if any body1 which appears to us at rest1 were furnished with any such self-moving principle2 from the body's appearing without motion we must conclude1 that this self-moving principle lies directed the same way as the earth is carrying the body2 and such a body might immediately be put into visible motion only by turning it about in any degree1 that this self-moving principle might receive a different direction4 #ad4 ,,from these considerations it very plainly follows1 that if a body were once absolutely at rest2 not being furnished with any principle1 whereby it could put it self into motion1 it must for ever continue in the same place1 till acted upon by something external3 and also that when a body is put into motion1 it has no power within it self to make any change in the direction of that motion2 and consequently that the body must move on straight forward without declining any way whatever4 ,but it has before been shewn1 that bodies do not appear to have in themselves any power to change the velocity of their motion3 therefore this first law of motion has been illustrated and confirmed1 as much as can be from the transient observations1 which have here been discoursed upon2 and in the next chapter all this will be farther established by more correct observations4 #ae4 ,,but ,i shall now pass to the second law of motion2 wherein1 when it is asserted1 that the velocity1 with which any body is moved by the action of a power upon it1 is proportional to that power2 the degree of power is supposed to be measured by the greatness of the body1 which it can move with a given celerity4 ,so that the sense of this law is1 that if any body were put into motion with that degree of swiftness1 as to pass in one hour the length of a thousand yards2 the power1 which would give the same degree of velocity to a body twice as great1 would give this lesser body twice the velocity1 causing it to describe in the same space of an hour two thousand yards4 ,but by a body twice as great as another1 ,i do not here mean simply of twice the bulk1 but one that contains a double quantity of solid matter4 #af4 ,,why the power1 which can move a body twice as great as another with the same degree of velocity1 should be called twice as great as the power1 which can give the lesser body the same velocity1 is evident4 ,for if we should suppose the greater body to be divided into two equal parts1 each equal to the lesser body1 each of these halves will require the same degree of power to move them with the velocity of the lesser body1 as the lesser body it self requires2 and therefore both those halves1 or the whole greater body1 will require the moving power to be doubled4 #ag4 ,,that the moving power being in this sense doubled1 should just double likewise the velocity of the same body1 seems near as evident1 if we consider1 that the effect of the power applied must needs be the same1 whether that power be applied to the body at once1 or in parts4 ,suppose then the double power not applied to the body at once1 but half of it first1 and afterwards the other half2 it is not conceivable for what reason the half last applied should come to have a different effect upon the body1 from that which is applied first2 as it must have1 if the velocity of the body was not just doubled by the application of it4 ,so far as experience can determine1 we see nothing to favour such a supposition4 ,we cannot indeed 7by reason of the constant motion of the earth7 make trial upon any body perfectly at rest1 whereby to see whether a power applied in that case would have a different effect1 from what it has1 when the body is already moving2 but we find no alteration in the effect of the same power on account of any difference there may be in the motion of the body1 when the power is applied4 ,the earth does not always carry bodies with the same degree of velocity2 yet we find the visible effects of any power applied to the same body to be1 at all times the very same3 and a bale of goods1 or other moveable body lying in a ship is as easily removed from place to place1 while the ship is under sail1 if its motion be steady1 as when it is fixed at anchor4 #ah4 ,,now this experience is alone sufficient to shew to us the whole of this law of motion4 #ai4 ,,since we find1 that the same power will always produce the same change in the motion of any body1 whether that body were before moving with a swifter or slower motion2 the change wrought in the motion of a body depends only on the power applied to it1 without any regard to the body's former motion3 and therefore the degree of motion1 which the body already possesses1 having no influence on the power applied to disturb its operation1 the effects of the same power will not only be the same in all degrees of motion of the body2 but we have likewise no reason to doubt1 but that a body perfectly at rest would receive from any power as much motion1 as would be equivalent to the effect of the same power applied to that body already in motion4 #bj4 ,,again1 suppose a body being at rest1 any number of equal powers should be successively applied to it2 pushing it forward from time to time in the same course or direction4 ,upon the application of the first power the body would begin to move2 when the second power was applied1 it appears from what has been said1 that the motion of the body would become double2 the third power would treble the motion of the body2 and so on1 till after the operation of the last power the motion of the body would be as many times the motion1 which the first power gave it1 as there are powers in number4 and the effect of this number of powers will be always the same1 without any regard to the space of time taken up in applying them3 so that greater or lesser intervals between the application of each of these powers will produce no difference at all in their effects4 ,since therefore the distance of time between the action of each power is of no consequence2 without doubt the effect will still be the same1 though the powers should all be applied at the very same instant2 or although a single power should be applied equal in strength to the collective force of all these powers4 ,hence it plainly follows1 that the degree of motion1 into which any body will be put out of a state of rest by any power1 will be proportional to that power4 ,a double power will give twice the velocity1 a treble power three times the velocity1 and so on4 ,the foregoing reasoning will equally take place1 though the body were not supposed to be at rest1 when the powers began to be applied to it2 provided the direction1 in which the powers were applied1 either conspired with the action of the body1 or was directly opposite to it4 ,therefore if any power be applied to a moving body1 and act upon the body either in the direction wherewith the body moves1 so as to accelerate the body2 or if it act directly opposite to the motion of the body1 so as to retard it3 in both these cases the change of motion will be proportional to the power applied2 nay1 the augmentation of the motion in one case1 and the diminution thereof in the other1 will be equal to that degree of motion1 into which the same power would put the body1 had it been at rest1 when the power was applied4 #ba4 ,,farther1 a power may be so applied to a moving body1 as to act obliquely to the motion of the body4 ,and the effects of such an oblique motion may be deduced from this observation2 that as all bodies are continually moving along with the earth1 we see that the visible effects of the same power are always the same1 in whatever direction the power acts3 and therefore the visible effects of any power upon a body1 which seems only to be at rest1 is always to appearance the same as the real effect would be upon a body truly at rest4 ,now suppose a body were moving along the line ,a ,b 7in fig4 #b47 and the eye accompanied it with an equal motion in the line ,c ,d equidistant from ,a ,b2 so that when the body is at ,a1 the eye shall be at ,c1 and when the body is advanced to ,e in the line ,a ,b1 the eye shall be advanced to ,f in the line ,c ,d1 the distances ,a ,e and ,c ,f being equal4 ,it is evident1 that here the body will appear to the eye to be at rest2 and the line ,f ,e ,g drawn from the eye through the body shall seem to the eye to be immoveable2 though as the body and eye move forward together1 this line shall really also move2 so that when the body shall be advanced to ,h and the eye to ,k1 the line ,f ,e ,g shall be transferred into the situation ,k ,h ,l1 this line ,k ,h ,l being equidistant from ,f ,e ,g4 ,now if the body when at ,e were to receive an impulse in the direction of the line ,f ,e ,g2 while the eye is moving on from ,f to ,k and carrying along with it the line ,f ,e ,g1 the body will appear to the eye to move along this line ,f ,e ,g3 for this is what has just now been said2 that while bodies are moving along with the earth1 and the spectator's eye partakes of the same motion1 the effect of any power upon the body will appear to be what it would really have been1 had the body been truly at rest1 when the power was applied4 ,from hence it follows1 that when the eye is advanced to ,k1 the body will appear somewhere in the line ,k ,h ,l4 ,suppose it appear in ,m2 then it is manifest1 from what has been premised at the beginning of this paragraph1 that the distance ,h ,m is equal to what the body would have run upon the line ,e ,g1 during the time1 wherein the eye has passed from ,f to ,k1 provided that the body had been at rest1 when acted upon in ,e4 ,if it be farther asked1 after what manner the body has moved from ,e to ,m8 ,i answer1 through a straight line2 for it has been shewn above in the explication of the first law of motion1 that a moving body1 from the time it is left to it self1 will proceed on in one continued straight line4 #bb4 ,,if ,e ,n be taken equal to ,h ,m and ,n ,m be drawn2 since ,h ,m is equidistant from ,e ,n1 ,n ,m will be equidistant from ,e ,h4 ,therefore the effect of any power upon a moving body1 when that power acts obliquely to the motion of the body1 is to be determined in this manner4 ,suppose the body is moving along the straight line ,a ,e ,b1 if when the body is come to ,e1 a power gives it an impulse in the direction of the line ,e ,g1 to find what course the body will afterwards take we must proceed thus4 ,take in ,e ,b any length ,e ,h1 and in ,e ,g take such a length ,e ,n1 that if the body had been at rest in ,e1 the power applied to it would have caused it to move over ,e ,n in the same space of time1 as it would have employed in passing over ,e ,h1 if the power had not acted at all upon it4 ,then draw ,h ,l equidistant from ,e ,g1 and ,n ,m equidistant from ,e ,b4 ,after this1 if a line be drawn from ,e to the point ,m1 where these two lines meet1 the line ,e ,m will be the course into which the body will be put by the action of the power upon it at ,e4 #bc4 ,a ,,mathematical reader would here expect in some particulars more regular demonstrations2 but as ,i do not at present address my self to such1 so ,i hope1 what ,i have now written will render my meaning evident enough to those1 who are unacquainted with that kind of reasoning4 #bd4 ,,now as we have been shewing1 that some actual force is necessary either to put bodies out of a state of rest into motion1 or to change the motion1 which they have once received2 it is proper here to observe1 that this quality in bodies1 whereby they preserve their present state1 with regard to motion or rest1 till some active force disturb them1 is called the ,,vis ,,inertiae of matter3 and by this property1 matter1 sluggish and unactive of it self1 retains all the power impressed upon it1 and cannot be made to cease from action1 but by the opposition of as great a power1 as that which first moved it4 ,by the degree of this ,,vis ,,inertiae1 or power of inactivity1 as we shall henceforth call it1 we primarily judge of the quantity of solid matter in each body2 for as this quality is inherent in all the bodies1 upon which we can make any trial1 we conclude it to be a property essential to all matter2 and as we yet know no reason to suppose1 that bodies are composed of different kinds of matter1 we rather presume1 that the matter of all bodies is the same2 and that the degree of this power of inactivity is in every body proportional to the quantity of the solid matter in it4 ,but although we have no absolute proof1 that all the matter in the universe is uniform1 and possesses this power of inactivity in the same degree2 yet we can with certainty compare together the different degrees of this power of inactivity in different bodies4 ,particularly this power is proportional to the weight of bodies1 as ,sir ,,isaac ,,newton has demonstrated#dd4 ,however1 notwithstanding that this power of inactivity in any body can be more certainly known1 than the quantity of solid matter in it2 yet since there is no reason to suspect that one is not proportional to the other1 we shall hereafter speak without hesitation of the quantity of matter in bodies1 as the measure of the degree of their power of inactivity4 #be4 ,,this being established1 we may now compare the effects of the same power upon different bodies1 as hitherto we have shewn the effects of different powers upon the same body4 ,and here if we limit the word motion to the peculiar sense given to it in philosophy1 we may comprehend all that is to be said upon this head under one short precept2 that the same power1 to whatever body it is applied1 will always produce the same degree of motion4 ,but here motion does not signify the degree of celerity or velocity with which a body moves1 in which sense only we have hitherto used it2 but it is made use of particularly in philosophy to signify the force with which a body moves3 as if two bodies ,a and ,b being in motion1 twice the force would be required to stop ,a as to stop ,b1 the motion of ,a would be esteemed double the motion of ,b4 ,in moving bodies1 these two things are carefully to be distinguished2 their velocity or celerity1 which is measured by the space they pass through during any determinate portion of time2 and the quantity of their motion1 or the force1 with which they will press against any resistance4 ,which force1 when different bodies move with the same velocity1 is proportional to the quantity of solid matter in the bodies2 but if the bodies are equal1 this force is proportional to their respective velocities1 and in other cases it is proportional both to the quantity of solid matter in the body1 and also to its velocity4 ,to instance in two bodies ,a and ,b3 if ,a be twice as great as ,b1 and they have both the same velocity1 the motion of ,a shall be double the motion of ,b2 and if the bodies be equal1 and the velocity of ,a be twice that of ,b1 the motion of ,a shall likewise be double that of ,b2 but if ,a be twice as large as ,b1 and move twice as swift1 the motion of ,a will be four times the motion of ,b2 and lastly1 if ,a be twice as large as ,b1 and move but half as fast1 the degree of their motion shall be the same4 #bf4 ,,this is the particular sense given to the word motion by philosophers1 and in this sense of the word the same power always produces the same quantity or degree of motion4 ,if the same power act upon two bodies ,a and ,b1 the velocities1 it shall give to each of them1 shall be so adjusted to the respective bodies1 that the same degree of motion shall be produced in each4 ,if ,a be twice as great as ,b1 its velocity shall be half that of ,b2 if ,a has three times as much solid matter as ,b1 the velocity of ,a shall be one third of the velocity of ,b2 and generally the velocity given to ,a shall bear the same proportion to the velocity given to ,b1 as the quantity of solid matter contained in the body ,b bears to the quantity of solid matter contained in ,a4 #bg4 ,,the reason of all this is evident from what has gone before4 ,if a power were applied to ,b1 which should bear the same proportion to the power applied to ,a1 as the body ,b bears to ,a1 the bodies ,b and ,a would both receive the same velocity2 and the velocity1 which ,b will receive from this power1 will bear the same proportion to the velocity1 which it would receive from the action of the power applied to ,a1 as the former of these powers bears to the latter3 that is1 the velocity1 which ,a receives from the power applied to it1 will bear to the velocity1 which ,b would receive from the same power1 the same proportion as the body ,b bears to ,a4 #bh4 ,,from hence we may now pass to the third law of motion1 where this distinction between the velocity of a body and its whole motion is farther necessary to be regarded1 as shall immediately be shewn2 after having first illustrated the meaning of this law by a familiar instance4 ,if a stone or other load be drawn by a horse2 the load re-acts upon the horse1 as much as the horse acts upon the load2 for the harness1 which is strained between them1 presses against the horse as much as against the load2 and the progressive motion of the horse forward is hindred as much by the load1 as the motion of the load is promoted by the endeavour of the horse3 that is1 if the horse put forth the same strength1 when loosened from the load1 he would move himself forwards with greater swiftness in proportion to the difference between the weight of his own body and the weight of himself and load together4 #bi4 ,,this instance will afford some general notion of the meaning of this law4 ,but to proceed to a more philosophical explication3 if a body in motion strike against another at rest1 let the body striking be ever so small1 yet shall it communicate some degree of motion to the body it strikes against1 though the less that body be in comparison of that it impinges upon1 and the less the velocity is1 with which it moves1 the smaller will be the motion communicated4 ,but whatever degree of motion it gives to the resting body1 the same it shall lose it self4 ,this is the necessary consequence of the forementioned power of inactivity in matter4 ,for suppose the two bodies equal1 it is evident from the time they meet1 both the bodies are to be moved by the single motion of the first2 therefore the body in motion by means of its power of inactivity retaining the motion first given it1 strikes upon the other with the same force1 wherewith it was acted upon it self3 but now both the bodies being to be moved by that force1 which before moved one only1 the ensuing velocity will be the same1 as if the power1 which was applied to one of the bodies1 and put it into motion1 had been applied to both2 whence it appears1 that they will proceed forwards1 with half the velocity1 which the body first in motion had3 that is1 the body first moved will have lost half its motion1 and the other will have gained exactly as much4 ,this rule is just1 provided the bodies keep contiguous after meeting2 as they would always do1 if it were not for a certain cause that often intervenes1 and which must now be explained4 ,bodies upon striking against each other1 suffer an alteration in their figure1 having their parts pressed inwards by the stroke1 which for the most part recoil again afterwards1 the bodies endeavouring to recover their former shape4 ,this power1 whereby bodies are inabled to regain their first figure1 is usually called their elasticity1 and when it acts1 it forces the bodies from each other1 and causes them to separate4 ,now the effect of this elasticity in the present case is such1 that if the bodies are perfectly elastic1 so as to recoil with as great a force as they are bent with1 that they recover their figure in the same space of time1 as has been taken up in the alteration made in it by their compression together2 then this power will separate the bodies as swiftly1 as they before approached1 and acting upon both equally1 upon the body first in motion contrary to the direction in which it moves1 and upon the other as much in the direction of its motion1 it will take from the first1 and add to the other equal degrees of velocity3 so that the power being strong enough to separate them with as great a velocity1 as they approached with1 the first will be quite stopt1 and that which was at rest1 will receive all the motion of the other4 ,if the bodies are elastic in a less degree1 the first will not lose all its motion1 nor will the other acquire the motion of the first1 but fall as much short of it1 as the other retains4 ,for this rule is never deviated from1 that though the degree of elasticity determines how much more than half its velocity the body first in motion shall lose2 yet in every case the loss in the motion of this body shall be transferred to the other1 that other body always receiving by the stroke as much motion1 as is taken from the first4 #cj4 ,this is the case of a body striking directly against an equal body at rest1 and the reasoning here used is fully confirmed by experience4 ,there are many other cases of bodies impinging against one another3 but the mention of these shall be reserved to the next chapter1 where we intend to be more particular and diffusive in the proof of these laws of motion1 than we have been here4 ,,chap4 ,,ii4 ,farther proofs of the ,,laws ,,of ,,motion4 ,,having in the preceding chapter deduced the three laws of motion1 delivered by our great philosopher1 from the most obvious observations1 that suggest them to us2 ,i now intend to give more particular proofs of them1 by recounting some of the discoveries which have been made in philosophy before ,sir ,isaac ,newton4 ,for as they were all collected by reasoning upon those laws2 so the conformity of these discoveries to experience makes them so many proofs of the truth of the principles1 from which they were derived4 #b4 ,,let us begin with the subject1 which concluded the last chapter4 ,although the body in motion be not equal to the body at rest1 on which it strikes2 yet the motion after the stroke is to be estimated in the same manner as above4 ,let ,a 7in fig4 #c47 be a body in motion towards another body ,b lying at rest4 ,when ,a is arrived at ,b1 it cannot proceed farther without putting ,b into motion2 and what motion it gives to ,b1 it must lose it self1 that the whole degree of motion of ,a and ,b together1 if neither of the bodies be elastic1 shall be equal1 after the meeting of the bodies1 to the single motion of ,a before the stroke4 ,therefore1 from what has been said above1 it is manifest1 that as soon as the two bodies are met1 they will move on together with a velocity1 which will bear the same proportion to the original velocity of ,a1 as the body ,a bears to the sum of both the bodies4 #c4 ,,if the bodies are elastic1 so that they shall separate after the stroke1 ,a must lose a greater part of its motion1 and the subsequent motion of ,b will be augmented by this elasticity1 as much as the motion of ,a is diminished by it4 ,the elasticity acting equally between both the bodies1 it will communicate to each the same degree of motion2 that is1 it will separate the bodies by taking from the body ,a and adding to the body ,b different degrees of velocity1 so proportioned to their respective quantities of matter1 that the degree of motion1 wherewith ,a separates from ,b1 shall be equal to the degree of motion1 wherewith ,b separates from ,a4 ,it follows therefore1 that the velocity taken from ,a by the elasticity bears to the velocity1 which the same elasticity adds to ,b1 the same proportion1 as ,b bears to ,a3 consequently the velocity1 which the elasticity takes from ,a1 will bear the same proportion to the whole velocity1 wherewith this elasticity causes the two bodies to separate from each other1 as the body ,b bears to the sum of the two bodies ,a and ,b2 and the velocity1 which is added to ,b by the elasticity1 bears to the velocity1 wherewith the bodies separate1 the same proportion1 as the body ,a bears to the sum of the two bodies ,a and ,b4 ,thus is found1 how much the elasticity takes from the velocity of ,a1 and adds to the velocity of ,b2 provided the degree of elasticity be known1 whereby to determine the whole velocity wherewith the bodies separate from each other after the stroke#de4 #d4 ,,after this manner is determined in every case the result of a body in motion striking against another at rest4 ,the same principles will also determine the effects1 when both bodies are in motion4 #e4 ,,let two equal bodies move against each other with equal swiftness4 ,then the force1 with which each of them presses forwards1 being equal when they strike2 each pressing in its own direction with the same energy1 neither shall surmount the other1 but both be stopt1 if they be not elastic3 for if they be elastic1 they shall from thence recover new motion1 and recede from each other1 as swiftly as they met1 if they be perfectly elastic2 but more slowly1 if less so4 ,in the same manner1 if two bodies of unequal bigness strike against each other1 and their velocities be so related1 that the velocity of the lesser body shall exceed the velocity of the greater in the same proportion1 as the greater body exceeds the lesser 7for instance1 if one body contains twice the solid matter as the other1 and moves but half as fast7 two such bodies will entirely suppress each other's motion1 and remain from the time of their meeting fixed2 if1 as before1 they are not elastic3 but1 if they are so in the highest degree1 they shall recede again1 each with the same velocity1 wherewith they met4 ,for this elastic power1 as in the preceding case1 shall renew their motion1 and pressing equally upon both1 shall give the same motion to both2 that is1 shall cause the velocity1 which the lesser body receives1 to bear the same proportion to the velocity1 which the greater receives1 as the greater body bears to the lesser3 so that the velocities shall bear the same proportion to each other after the stroke1 as before4 ,therefore if the bodies1 by being perfectly elastic1 have the sum of their velocities after the stroke equal to the sum of their velocities before the stroke1 each body after the stroke will receive its first velocity4 ,and the same proportion will hold likewise between the velocities1 wherewith they go off1 though they are elastic but in a less degree2 only then the velocity of each will be less in proportion to the defect of elasticity4 #f4 ,,if the velocities1 wherewith the bodies meet1 are not in the proportion here supposed2 but if one of the bodies1 as ,a1 has a swifter velocity in comparison to the velocity of the other2 then the effect of this excess of velocity in the body ,a must be joined to the effect now mentioned1 after the manner of this following example4 ,let ,a be twice as great as ,b1 and move with the same swiftness as ,b4 ,here ,a moves with twice that degree of swiftness1 which would answer to the forementioned proportion4 ,for ,a being double to ,b1 if it moved but with half the swiftness1 wherewith ,b advances1 it has been just now shewn1 that the two bodies upon meeting would stop1 if they were not elastic2 and if they were elastic1 that they would each recoil1 so as to cause ,a to return with half the velocity1 wherewith ,b would return4 ,but it is evident from hence1 that ,b by encountring ,a will annul half its velocity1 if the bodies be not elastic2 and the future motion of the bodies will be the same1 as if ,a had advanced against ,b at rest with half the velocity here assigned to it4 ,if the bodies be elastic1 the velocity of ,a and ,b after the stroke may be thus discovered4 ,as the two bodies advance against each other1 the velocity1 with which they meet1 is made up of the velocities of both bodies added together4 ,after the stroke their elasticity will separate them again4 ,the degree of elasticity will determine what proportion the velocity1 wherewith they separate1 must bear to that1 wherewith they meet4 ,divide this velocity1 with which the bodies separate into two parts1 that one of the parts bear to the other the same proportion1 as the body ,a bears to ,b2 and ascribe the lesser part to the greater body ,a1 and the greater part of the velocity to the lesser body ,b4 ,then take the part ascribed to ,a from the common velocity1 which ,a and ,b would have had after the stroke1 if they had not been elastic2 and add the part ascribed to ,b to the same common velocity4 ,by this means the true velocities of ,a and ,b after the stroke will be made known4 #g4 ,,if the bodies are perfectly elastic1 the great ,,huygens has laid down this rule for finding their motion after concourse#df4 ,any straight line ,c ,d 7in fig4 #d1 #e47 being drawn1 let it be divided in ,e1 that ,c ,e bear the same proportion to ,e ,d1 as the swiftness of ,a bore to the swiftness of ,b before the stroke4 ,let the same line ,c ,d be also divided in ,f1 that ,c ,f bear the same proportion to ,f ,d1 as the body ,b bears to the body ,a4 ,then ,f ,g being taken equal to ,f ,e1 if the point ,g falls within the line ,c ,d1 both the bodies shall recoil after the stroke1 and the velocity1 wherewith the body ,a shall return1 will bear the same proportion to the velocity1 wherewith ,b shall return1 as ,g ,c bears to ,g ,d2 but if the point ,g falls without the line ,c ,d1 then the bodies after their concourse shall both proceed to move the same way1 and the velocity of ,a shall bear to the velocity of ,b the same proportion1 that ,g ,c bears to ,g ,d1 as before4 #h4 ,,if the body ,b had stood still1 and received the impulse of the other body ,a upon it2 the effect has been already explained in the case1 when the bodies are not elastic4 ,and when they are elastic1 the result of their collision is found by combining the effect of the elasticity with the other effect1 in the same manner as in the last case4 #i4 ,,when the bodies are perfectly elastic1 the rule of ,,huygens#dg here is to divide the line ,c ,d 7fig4 #f47 in ,e as before1 and to take ,e ,g equal to ,e ,d4 ,and by these points thus found1 the motion of each body after the stroke is determined1 as before4 #aj4 ,,in the next place1 suppose the bodies ,a and ,b were both moving the same way1 but ,a with a swifter motion1 so as to overtake ,b1 and strike against it4 ,the effect of the percussion or stroke1 when the bodies are not elastic1 is discovered by finding the common motion1 which the two bodies would have after the stroke1 if ,b were at rest1 and ,a were to advance against it with a velocity equal to the excess of the present velocity of ,a above the velocity of ,b2 and by adding to this common velocity thus found the velocity of ,b4 #aa4 ,,if the bodies are elastic1 the effect of the elasticity is to be united with this other1 as in the former cases4 #ab4 ,,when the bodies are perfectly elastic1 the rule of ,,huygens#dh in this case is to prolong ,c ,d 7fig4 #g47 and to take in it thus prolonged ,c ,e in the same proportion to ,e ,d1 as the greater velocity of ,a bears to the lesser velocity of ,b2 after which ,f ,g being taken equal to ,f ,e1 the velocities of the two bodies after the stroke will be determined1 as in the two preceding cases4 #ac4 ,,thus ,i have given the sum of what has been written concerning the effects of percussion1 when two bodies freely in motion strike directly against each other2 and the results here set down1 as the consequence of our reasoning from the laws of motion1 answer most exactly to experience4 ,a particular set of experiments has been invented to make trial of these effects of percussion with the greatest exactness4 ,but ,i must defer these experiments1 till ,i have explained the nature of pendulums#di4 ,i shall therefore now proceed to describe some of the appearances1 which are caused in bodies from the influence of the power of gravity united with the general laws of motion2 among which the motion of the pendulum will be included4 #ad4 ,,the most simple of these appearances is1 when bodies fall down merely by their weight4 ,in this case the body increases continually its velocity1 during the whole time of its fall1 and that in the very same proportion as the time increases4 ,for the power of gravity acts constantly on the body with the same degree of strength3 and it has been observed above in the first law of motion1 that a body being once in motion will perpetually preserve that motion without the continuance of any external influence upon it3 therefore1 after a body has been once put in motion by the force of gravity1 the body would continue that motion1 though the power of gravity should cease to act any farther upon it2 but1 if the power of gravity continues still to draw the body down1 fresh degrees of motion must continually be added to the body2 and the power of gravity acting at all times with the same strength1 equal degrees of motion will constantly be added in equal portions of time4 #ae4 ,,this conclusion is not indeed absolutely true3 for we shall find hereafter#ej1 that the power of gravity is not of the same strength at all distances from the center of the earth4 ,but nothing of this is in the least sensible in any distance1 to which we can convey bodies4 ,the weight of bodies is the very same to sense upon the highest towers or mountains1 as upon the level ground2 so that in all the observations we can make1 the forementioned proportion between the velocity of a falling body and the time1 in which it has been descending1 obtains without any the least perceptible difference4 #af4 ,,from hence it follows1 that the space1 through which a body falls1 is not proportional to the time of the fall2 for since the body increases its velocity1 a greater space will be passed over in the same portion of time at the latter part of the fall1 than at the beginning4 ,suppose a body let fall from the point ,a 7in fig4 #h47 were to descend from ,a to ,b in any portion of time2 then if in an equal portion of time it were to proceed from ,b to ,c2 ,i say1 the space ,b ,c is greater than ,a ,b2 so that the time of the fall from ,a to ,c being double the time of the fall from ,a to ,b1 ,a ,c shall be more than double of ,a ,b4 #ag4 ,,the geometers have proved1 that the spaces1 through which bodies fall thus by their weight1 are just in a duplicate or two-fold proportion of the times1 in which the body has been falling4 ,that is1 if we were to take the line ,d ,e in the same proportion to ,a ,b1 as the time1 which the body has imployed in falling from ,a to ,c1 bears to the time of the fall from ,a to ,b2 then ,a ,c will be to ,d ,e in the same proportion4 ,in particular1 if the time of the fall through ,a ,c be twice the time of the fall through ,a ,b2 then ,d ,e will be twice ,a ,b1 and ,a ,c twice ,d ,e2 or ,a ,c four times ,a ,b4 ,but if the time of the fall through ,a ,c had been thrice the time of the fall through ,a ,b2 ,d ,e would have been treble of ,a ,b1 and ,a ,c treble of ,d ,e2 that is1 ,a ,c would have been equal to nine times ,a ,b4 #ah4 ,,if a body fall obliquely1 it will approach the ground by slower degrees1 than when it falls perpendicularly4 ,suppose two lines ,a ,b1 ,a ,c 7in fig4 #i47 were drawn1 one perpendicular1 and the other oblique to the ground ,d ,e3 then if a body were to descend in the slanting line ,a ,c2 because the power of gravity draws the body directly downwards1 if the line ,a ,c supports the body from falling in that manner1 it must take off part of the effect of the power of gravity2 so that in the time1 which would have been sufficient for the body to have fallen through the whole perpendicular line ,a ,b1 the body shall not have passed in the line ,a ,c a length equal to ,a ,b2 consequently the line ,a ,c being longer than ,a ,b1 the body shall most certainly take up more time in passing through ,a ,c1 than it would have done in falling perpendicularly down through ,a ,b4 #ai4 ,,the geometers demonstrate1 that the time1 in which the body will descend through the oblique straight line ,a ,c1 bears the same proportion to the time of its descent through the perpendicular ,a ,b1 as the line it self ,a ,c bears to ,a ,b4 ,and in respect to the velocity1 which the body will have acquired in the point ,c1 they likewise prove1 that the length of the time imployed in the descent through ,a ,c so compensates the diminution of the influence of gravity from the obliquity of this line1 that though the force of the power of gravity on the body is opposed by the obliquity of the line ,a ,c1 yet the time of the body's descent shall be so much prolonged1 that the body shall acquire the very same velocity in the point ,c1 as it would have got at the point ,b by falling perpendicularly down4 #bj4 ,,if a body were to descend in a crooked line1 the time of its descent cannot be determined in so simple a manner2 but the same property1 in relation to the velocity1 is demonstrated to take place in all cases3 that is1 in whatever line the body descends1 the velocity will always be answerable to the perpendicular height1 from which the body has fell4 ,for instance1 suppose the body ,a 7in fig4 #aj47 were hung by a string to the pin ,b4 ,if this body were let fall1 till it came to the point ,c perpendicularly under ,b1 it will have moved from ,a to ,c in the arch of a circle4 ,then the horizontal line ,a ,d being drawn1 the velocity of the body in ,c will be the same1 as if it had fallen from the point ,d directly down to ,c4 #ba4 ,,if a body be thrown perpendicularly upward with any force1 the velocity1 wherewith the body ascends1 shall continually diminish1 till at length it be wholly taken away2 and from that time the body will begin to fall down again1 and pass over a second time in its descent the line1 wherein it ascended2 falling through this line with an increasing velocity in such a manner1 that in every point thereof1 through which it falls1 it shall have the very same velocity1 as it had in the same place1 when it ascended2 and consequently shall come down into the place1 whence it first ascended1 with the velocity which was at first given to it4 ,thus if a body were thrown perpendicularly up in the line ,a ,b 7in fig4 ,,ii47 with such a force1 as that it should stop at the point ,b1 and there begin to fall again2 when it shall have arrived in its descent to any point as ,c in this line1 it shall there have the same velocity1 as that wherewith it passed by this point ,c in its ascent2 and at the point ,a it shall have gained as great a velocity1 as that wherewith it was first thrown upwards4 ,as this is demonstrated by the geometrical writers2 so1 ,i think1 it will appear evident1 by considering only1 that while the body descends1 the power of gravity must act over again1 in an inverted order1 all the influence it had on the body in its ascent2 so as to give again to the body the same degrees of velocity1 which it had taken away before4 #bb4 ,,after the same manner1 if the body were thrown upwards in the oblique straight line ,c ,a 7in fig4 #i47 from the point ,c1 with such a degree of velocity as just to reach the point ,a2 it shall by its own weight return again through the line ,a ,c by the same degrees1 as it ascended4 #bc4 ,,and lastly1 if a body were thrown with any velocity in a line continually incurvated upwards1 the like effect will be produced upon its return to the point1 whence it was thrown4 ,suppose for instance1 the body ,a 7in fig4 #ab47 were hung by a string ,a ,b4 ,then if this body be impelled any way1 it must move in the arch of a circle4 ,let it receive such an impulse1 as shall cause it to move in the arch ,a ,c2 and let this impulse be of such strength1 that the body may be carried from ,a as far as ,d1 before its motion is overcome by its weight3 ,i say here1 that the body forthwith returning from ,d1 shall come again into the point ,a with the same velocity1 as that wherewith it began to move4 #bd4 ,,it will be proper in this place to observe concerning the power of gravity1 that its force upon any body does not at all depend upon the shape of the body2 but that it continues constantly the same without any variation in the same body1 whatever change be made in the figure of the body3 and if the body be divided into any number of pieces1 all those pieces shall weigh just the same1 as they did1 when united together in one body3 and if the body be of a uniform contexture1 the weight of each piece will be proportional to its bulk4 ,this has given reason to conclude1 that the power of gravity acts upon bodies in proportion to the quantity of matter in them4 ,whence it should follow1 that all bodies must fall from equal heights in the same space of time4 ,and as we evidently see the contrary in feathers and such like substances1 which fall very slowly in comparison of more solid bodies2 it is reasonable to suppose1 that some other cause concurs to make so manifest a difference4 ,this cause has been found by particular experiments to be the air4 ,the experiments for this purpose are made thus4 ,they set up a very tall hollow glass2 within which near the top they lodge a feather and some very ponderous body1 usually a piece of gold1 this metal being the most weighty of any body known to us4 ,this glass they empty of the air contained within it1 and by moving a wire1 which passes through the top of the glass1 they let the feather and the heavy body fall together2 and it is always found1 that as the two bodies begin to descend at the same time1 so they accompany each other in the fall1 and come to the bottom at the very same instant1 as near as the eye can judge4 ,thus1 as far as this experiment can be depended on1 it is certain1 that the effect of the power of gravity upon each body is proportional to the quantity of solid matter1 or to the power of inactivity in each body4 ,for in the limited sense1 which we have given above to the word motion1 it has been shown1 that the same force gives to all bodies the same degree of motion1 and different forces communicate different degrees of motion proportional to the respective powers#ea4 ,in this case1 if the power of gravity were to act equally upon the feather1 and upon the more solid body1 the solid body would descend so much slower than the feather1 as to have no greater degree of motion than the feather3 but as both bodies descend with equal swiftness1 the degree of motion in the solid body is greater than in the feather1 bearing the same proportion to it1 as the quantity of matter in the solid body to the quantity of matter in the feather4 ,therefore the effect of gravity on the solid body is greater than on the feather1 in proportion to the greater degree of motion communicated2 that is1 the effect of the power of gravity on the solid body bears the same proportion to its effect on the feather1 as the quantity of matter in the solid body bears to the quantity of matter in the feather4 ,thus it is the proper deduction from this experiment1 that the power of gravity acts not on the surface of bodies only1 but penetrates the bodies themselves most intimately1 and operates alike on every particle of matter in them4 ,but as the great quickness1 with which the bodies fall1 leaves it something uncertain1 whether they do descend absolutely in the same time1 or only so nearly together1 that the difference in their swift motion is not discernable to the eye2 this property of the power of gravity1 which has here been deduced from this experiment1 is farther confirmed by pendulums1 whose motion is such1 that a very minute difference would become sufficiently sensible4 ,this will be farther discoursed on in another place#eb2 but here ,i shall make use of the principle now laid down to explain the nature of what is called the center of gravity in bodies4 #be4 ,,the center of gravity is that point1 by which if a body be suspended1 it shall hang at rest in any situation4 ,in a globe of a uniform texture the center of gravity is the same with the center of the globe2 for as the parts of the globe on every side of its center are similarly disposed1 and the power of gravity acts alike on every part2 it is evident1 that the parts of the globe on each side of the center are drawn with equal force1 and therefore neither side can yield to the other2 but the globe1 if supported at its center1 must of necessity hang at rest4 ,in like manner1 if two equal bodies ,a and ,b 7in fig4 #ac47 be hung at the extremities of an inflexible rod ,c ,d1 which should have no weight2 these bodies1 if the rod be supported at its middle ,e1 shall equiponderate2 and the rod remain without motion4 ,for the bodies being equal and at the same distance from the point of support ,e1 the power of gravity will act upon each with equal strength1 and in all respects under the same circumstances2 therefore the weight of one cannot overcome the weight of the other4 ,the weight of ,a can no more surmount the weight of ,b1 than the weight of ,b can surmount the weight of ,a4 ,again1 suppose a body as ,a ,b 7in fig4 #ad47 of a uniform texture in the form of a roller1 or as it is more usually called a cylinder1 lying horizontally4 ,if a straight line be drawn between ,c and ,d1 the centers of the extreme circles of this cylinder2 and if this straight line1 commonly called the axis of the cylinder1 be divided into two equal parts in ,e3 this point ,e will be the center of gravity of the cylinder4 ,the cylinder being a uniform figure1 the parts on each side of the point ,e are equal1 and situated in a perfectly similar manner2 therefore this cylinder1 if supported at the point ,e1 must hang at rest1 for the same reason as the inflexible rod above-mentioned will remain without motion1 when suspended at its middle point4 ,and it is evident1 that the force applied to the point ,e1 which would uphold the cylinder1 must be equal to the cylinder's weight4 ,now suppose two cylinders of equal thickness ,a ,b and ,c ,d to be joined together at ,c ,b1 so that the two axis's ,e ,f1 and ,f ,g lie in one straight line4 ,let the axis ,e ,f be divided into two equal parts at ,h1 and the axis ,f ,g into two equal parts at ,i4 ,then because the cylinder ,a ,b would be upheld at rest by a power applied in ,h equal to the weight of this cylinder1 and the cylinder ,c ,d would likewise be upheld by a power applied in ,i equal to the weight of this cylinder2 the whole cylinder ,a ,d will be supported by these two powers3 but the whole cylinder may likewise be supported by a power applied to ,k1 the middle point of the whole axis ,e ,g1 provided that power be equal to the weight of the whole cylinder4 ,it is evident therefore1 that this power applied in ,k will produce the same effect1 as the two other powers applied in ,h and ,i4 ,it is farther to be observed1 that ,h ,k is equal to half ,f ,g1 and ,k ,i equal to half ,e ,f2 for ,e ,k being equal to half ,e ,g1 and ,e ,h equal to half ,e ,f1 the remainder ,h ,k must be equal to half the remainder ,f ,g2 so likewise ,g ,k being equal to half ,g ,e1 and ,g ,i equal to half ,g ,f1 the remainder ,i ,k must be equal to half the remainder ,e ,f4 ,it follows therefore1 that ,h ,k bears the same proportion to ,k ,i1 as ,f ,g bears to ,e ,f4 ,besides1 ,i believe1 my readers will perceive1 and it is demonstrated in form by the geometers1 that the whole body of the cylinder ,c ,d bears the same proportion to the whole body of the cylinder ,a ,b1 as the axis ,f ,g bears to the axis ,e ,f#ec4 ,but hence it follows1 that in the two powers applied at ,h and ,i1 the power applied at ,h bears the same proportion to the power applied at ,i1 as ,k ,i bears to ,k ,h4 ,now suppose two strings ,h ,l and ,i ,m extended upwards1 one from the point ,h and the other from ,i1 and to be laid hold on by two powers1 one strong enough to hold up the cylinder ,a ,b1 and the other of strength sufficient to support the cylinder ,c ,d4 ,here as these two powers uphold the whole cylinder1 and therefore produce an effect1 equal to what would have been produced by a power applied to the point ,k of sufficient force to sustain the whole cylinder3 it is manifest1 that if the cylinder be taken away1 the axis only being left1 and from the point ,k a string1 as ,k ,n1 be extended1 which shall be drawn down by a power equivalent to the weight of the cylinder1 this power shall act against the other two powers1 as much as the cylinder acted against them2 and consequently these three powers shall be upon a balance1 and hold the axis ,h ,i fixed between them4 ,but if these three powers preserve a mutual balance1 the two powers applied to the strings ,h ,l and ,i ,m are a balance to each other2 the power applied to the string ,h ,l bearing the same proportion to the power applied to the string ,i ,m1 as the distance ,i ,k bears to the distance ,k ,h4 ,hence it farther appears1 that if an inflexible rod ,a ,b 7in fig4 #ae47 be suspended by any point ,c not in the middle thereof2 and if at ,a the end of the shorter arm be hung a weight1 and at ,b the end of the longer arm be also hung a weight less than the other1 and that the greater of these weights bears to the lesser the same proportion1 as the longer arm of the rod bears to the shorter2 then these two weights will equiponderate3 for a power applied at ,c equal to both these weights will support without motion the rod thus charged2 since here nothing is changed from the preceding case but the situation of the powers1 which are now placed on the contrary sides of the line1 to which they are fixed4 ,also for the same reason1 if two weights ,a and ,b 7in fig4 #af47 were connected together by an inflexible rod ,c ,d1 drawn from ,c the center of gravity of ,a to ,d the center of gravity of ,b2 and if the rod ,c ,d were to be so divided in ,e1 that the part ,d ,e bear the same proportion to the other part ,c ,e1 as the weight ,a bears to the weight ,b3 then this rod being supported at ,e will uphold the weights1 and keep them at rest without motion4 ,this point ,e1 by which the two bodies ,a and ,b will be supported1 is called their common center of gravity4 ,and if a greater number of bodies were joined together1 the point1 by which they could all be supported1 is called the common center of gravity of them all4 ,suppose 7in fig4 #ag47 there were three bodies ,a1 ,b1 ,c1 whose respective centers of gravity were joined by the three lines ,d ,e1 ,d ,f1 ,e ,f3 the line ,d ,e being so divided in ,g1 that ,d ,g bear the same proportion to ,g ,e1 as ,b bears to ,a2 ,g is the center of gravity common to the two bodies ,a and ,b2 that is1 a power equal to the weight of both the bodies applied to ,g would support them1 and the point ,g is pressed as much by the two weights ,a and ,b1 as it would be1 if they were both hung together at that point4 ,therefore1 if a line be drawn from ,g to ,f1 and divided in ,h1 so that ,g ,h bear the same proportion to ,h ,f1 as the weight ,c bears to both the weights ,a and ,b1 the point ,h will be the common center of gravity of all the three weights2 for ,h would be their common center of gravity1 if both the weights ,a and ,b were hung together at ,g1 and the point ,g is pressed as much by them in their present situation1 as it would be in that case4 ,in the same manner from the common center of these three weights1 you might proceed to find the common center1 if a fourth weight were added1 and by a gradual progress might find the common center of gravity belonging to any number of weights whatever4 #bf4 ,,as all this is the obvious consequence of the proposition laid down for assigning the common center of gravity of any two weights1 by the same proposition the center of gravity of all figures is found4 ,in a triangle1 as ,a ,b ,c 7in fig4 #ah47 the center of gravity lies in the line drawn from the middle point of any one of the sides to the opposite angle1 as the line ,b ,d is drawn from ,d the middle of the line ,a ,c to the opposite angle ,b#ed2 so that if from the middle of either of the other sides1 as from the point ,e in the side ,a ,b1 a line be drawn1 as ,e ,c1 to the opposite angle2 the point ,f1 where this line crosses the other line ,b ,d1 will be the center of gravity of the triangle#ee4 ,likewise ,d ,f is equal to half ,f ,b1 and ,e ,f equal to half ,f ,c#ef4 ,in a hemisphere1 as ,a ,b ,c 7fig4 #ai47 if from ,d the center of the base the line ,d ,b be erected perpendicular to that base1 and this line be so divided in ,e1 that ,d ,e be equal to three fifths of ,b ,e1 the point ,e is the center of gravity of the hemisphere#eg4 #bg4 ,,it will be of use to observe concerning the center of gravity of bodies2 that since a power applied to this center alone can support a body against the power of gravity1 and hold it fixed at rest2 the effect of the power of gravity on a body is the same1 as if that whole power were to exert itself on the center of gravity only4 ,whence it follows1 that1 when the power of gravity acts on a body suspended by any point1 if the body is so suspended1 that the center of gravity of the body can descend2 the power of gravity will give motion to that body1 otherwise not3 or if a number of bodies are so connected together1 that1 when any one is put into motion1 the rest shall1 by the manner of their being joined1 receive such motion1 as shall keep their common center of gravity at rest2 then the power of gravity shall not be able to produce any motion in these bodies1 but in all other cases it will4 ,thus1 if the body ,a ,b 7in fig4 #bj1 #ba47 whose center of gravity is ,c1 be hung on the point ,a1 and the center ,c be perpendicularly under ,a 7as in fig4 #bj47 the weight of the body will hold it still without motion1 because the center ,c cannot descend any lower4 ,but if the body be removed into any other situation1 where the center ,c is not perpendicularly under ,a 7as in fig4 #ba47 the body by its weight will be put into motion towards the perpendicular situation of its center of gravity4 ,also if two bodies ,a1 ,b 7in fig4 #bb47 be joined together by the rod ,c ,d lying in an horizontal situation1 and be supported at the point ,e2 if this point be the center of gravity common to the two bodies1 their weight will not put them into motion2 but if this point ,e is not their common center of gravity1 the bodies will move2 that part of the rod ,c ,d descending1 in which the common center of gravity is found4 ,so in like manner1 if these two bodies were connected together by any more complex contrivance2 yet if one of the bodies cannot move without so moving the other1 that their common center of gravity shall rest1 the weight of the bodies will not put them in motion1 otherwise it will4 #bh4 ,i ,,shall proceed in the next place to speak of the mechanical powers4 ,these are certain instruments or machines1 contrived for the moving great weights with small force2 and their effects are all deducible from the observation we have just been making4 ,they are usually reckoned in number five2 the lever1 the wheel and axis1 the pulley1 the wedge1 and the screw2 to which some add the inclined plane4 ,as these instruments have been of very ancient use1 so the celebrated ,,archimedes seems to have been the first1 who discovered the true reason of their effects4 ,this1 ,i think1 may be collected from what is related of him1 that some expressions1 which he used to denote the unlimited force of these instruments1 were received as very extraordinary paradoxes3 whereas to those1 who had understood the cause of their great force1 no expressions of that kind could have appeared surprizing4 #bi4 ,,all the effects of these powers may be judged of by this one rule1 that1 when two weights are applied to any of these instruments1 the weights will equiponderate1 if1 when put into motion1 their velocities will be reciprocally proportional to their respective weights4 ,and what is said of weights1 must of necessity be equally understood of any other forces equivalent to weights1 such as the force of a man's arm1 a stream of water1 or the like4 #cj4 ,,but to comprehend the meaning of this rule1 the reader must know1 what is to be understood by reciprocal proportion2 which ,i shall now endeavour to explain1 as distinctly as ,i can2 for ,i shall be obliged very frequently to make use of this term4 ,when any two things are so related1 that one increases in the same proportion as the other1 they are directly proportional4 ,so if any number of men can perform in a determined space of time a certain quantity of any work1 suppose drain a fish-pond1 or the like2 and twice the number of men can perform twice the quantity of the same work1 in the same time2 and three times the number of men can perform as soon thrice the work2 here the number of men and the quantity of the work are directly proportional4 ,on the other hand1 when two things are so related1 that one decreases in the same proportion1 as the other increases1 they are said to be reciprocally proportional4 ,thus if twice the number of men can perform the same work in half the time1 and three times the number of men can finish the same in a third part of the time2 then the number of men and the time are reciprocally proportional4 ,we shewed above#eh how to find the common center of gravity of two bodies1 there the distances of that common center from the centers of gravity of the two bodies are reciprocally proportional to the respective bodies4 ,for ,c ,e in fig4 #af4 being in the same proportion to ,e ,d1 as ,b bears to ,a2 ,c ,e is so much greater in proportion than ,e ,d1 as ,a is less in proportion than ,b4 #ca4 ,,now this being understood1 the reason of the rule here stated will easily appear4 ,for if these two bodies were put in motion1 while the point ,e rested1 the velocity1 wherewith ,a would move1 would bear the same proportion to the velocity1 wherewith ,b would move1 as ,e ,c bears to ,e ,d4 ,the velocity therefore of each body1 when the common center of gravity rests1 is reciprocally proportional to the body4 ,but we have shewn above#ei1 that if two bodies are so connected together1 that the putting them in motion will not move their common center of gravity2 the weight of those bodies will not produce in them any motion4 ,therefore in any of these mechanical engines1 if1 when the bodies are put into motion1 their velocities are reciprocally proportional to their respective weights1 whereby the common center of gravity would remain at rest2 the bodies will not receive any motion from their weight1 that is1 they will equiponderate4 ,but this perhaps will be yet more clearly conceived by the particular description of each mechanical power4 #cb4 ,,the lever was first named above4 ,this is a bar made use of to sustain and move great weights4 ,the bar is applied in one part to some strong support2 as the bar ,a ,b 7in fig4 #bc1 #bd47 is applied at the point ,c to the support ,d4 ,in some other part of the bar1 as ,e1 is applied the weight to be sustained or moved2 and in a third place1 as ,f1 is applied another weight or equivalent force1 which is to sustain or move the weight at ,e4 ,now here1 if1 when the level should be put in motion1 and turned upon the point ,c1 the velocity1 wherewith the point ,f would move1 bears the same proportion to the velocity1 wherewith the point ,e would move1 as the weight at ,e bears to the weight or force at ,f2 then the lever thus charged will have no propensity to move either way4 ,if the weight or other force at ,f be not so great as to bear this proportion1 the weight at ,e will not be sustained2 but if the force at ,f be greater than this1 the weight at ,e will be surmounted4 ,this is evident from what has been said above#fj1 when the forces at ,e and ,f are placed 7as in fig4 #bc47 on different sides of the support ,d4 ,it will appear also equally manifest in the other case1 by continuing the bar ,b ,c in fig4 #bd4 on the other side of the support ,d1 till ,c ,g be equal to ,c ,f1 and by hanging at ,g a weight equivalent to the power at ,f2 for then1 if the power at ,f were removed1 the two weights at ,g and ,e would counterpoize each other1 as in the former case3 and it is evident1 that the point ,f will be lifted up by the weight at ,g with the same degree of force1 as by the other power applied to ,f2 since1 if the weight at ,e were removed1 a weight hung at ,f equal to that at ,g would balance the lever1 the distances ,c ,g and ,c ,f being equal4 #cc4 ,,if the two weights1 or other powers1 applied to the lever do not counterbalance each other2 a third power may be applied in any place proposed of the lever1 which shall hold the whole in a just counterpoize4 ,suppose 7in fig4 #be47 the two powers at ,e and ,f did not equiponderate1 and it were required to apply a third power to the point ,g1 that might be sufficient to balance the lever4 ,find what power in ,f would just counterbalance the power in ,e2 then if the difference between this power and that1 which is actually applied at ,f1 bear the same proportion to the third power to be applied at ,g1 as the distance ,c ,g bears to ,c ,f2 the lever will be counterpoized by the help of this third power1 if it be so applied as to act the same way with the power in ,f1 when that power is too small to counterbalance the power in ,e2 but otherwise the power in ,g must be so applied1 as to act against the power in ,f4 ,in like manner1 if a lever were charged with three1 or any greater number of weights or other powers1 which did not counterpoize each other1 another power might be applied in any place proposed1 which should bring the whole to a just balance4 ,and what is here said concerning a plurality of powers1 may be equally applied to all the following cases4 #cd4 ,,if the lever should consist of two arms making an angle at the point ,c 7as in fig4 #bf47 yet if the forces are applied perpendicularly to each arm1 the same proportion will hold between the forces applied1 and the distances of the center1 whereon the lever rests1 from the points to which they are applied4 ,that is1 the weight at ,e will be to the force in ,f in the same proportion1 as ,c ,f bears to ,c ,e4 #ce4 ,,but whenever the forces applied to the lever act obliquely to the arm1 to which they are applied 7as in fig4 #bg47 then the strength of the forces is to be estimated by lines let fall from the center of the lever to the directions1 wherein the forces act4 ,to balance the levers in fig4 #bg1 the weight or other force at ,f will bear the same proportion to the weight at ,e1 as the distance ,c ,e bears to ,c ,g the perpendicular let fall from the point ,c upon the line1 which denotes the direction wherein the force applied to ,f acts3 for here1 if the lever be put into motion1 the power applied to ,f will begin to move in the direction of the line ,f ,g2 and therefore its first motion will be the same1 as the motion of the point ,g4 #cf4 ,,when two weights hang upon a lever1 and the point1 by which the lever is supported1 is placed in the middle between the two weights1 that the arms of the lever are both of equal length2 then this lever is particularly called a balance2 and equal weights equiponderate as in common scales4 ,when the point of support is not equally distant from both weights1 it constitutes that instrument for weighing1 which is called a steelyard4 ,though both in common scales1 and the steelyard1 the point1 on which the beam is hung1 is not usually placed just in the same straight line with the points1 that hold the weights1 but rather a little above 7as in fig4 #bh47 where the lines drawn from the point ,c1 whereon the beam is suspended1 to the points ,e and ,f1 on which the weights are hung1 do not make absolutely one continued line4 ,if the three points ,e1 ,c1 and ,f were in one straight line1 those weights1 which equiponderated1 when the beam hung horizontally1 would also equiponderate in any other situation4 ,illustration ,but we see in these instruments1 when they are charged with weights1 which equiponderate with the beam hanging horizontally2 that1 if the beam be inclined either way1 the weight most elevated surmounts the other1 and descends1 causing the beam to swing1 till by degrees it recovers its horizontal position4 ,this effect arises from the forementioned structure3 for by this structure these instruments are levers composed of two arms1 which make an angle at the point of support 7as in fig4 #bi1 #cj47 the first of which represents the case of the common balance1 the second the case of the steelyard4 ,in the first1 where ,c ,e and ,c ,f are equal1 equal weights hung at ,e and ,f will equiponderate1 when the points ,e and ,f are in an horizontal situation4 ,suppose the lines ,e ,g and ,f ,h to be perpendicular to the horizon1 then they will denote the directions1 wherein the forces applied to ,e and ,f act4 ,therefore the proportion between the weights at ,e and ,f1 which shall equiponderate1 are to be judged of by perpendiculars1 as ,c ,i1 ,c ,k1 let fall from ,c upon ,e ,g and ,f ,h3 so that the weights being equal1 the lines ,c ,i1 ,c ,k1 must be equal also1 when the weights equiponderate4 ,but ,i believe my readers will easily see1 that since ,c ,e and ,c ,f are equal1 the lines ,c ,i and ,c ,k will be equal1 when the points ,e and ,f are horizontally situated4 #cg4 ,,if this lever be set into any other position 7as in fig4 #ca47 then the weight1 which is raised highest1 will outweigh the other4 ,here1 if the point ,f be raised higher than ,e1 the perpendicular ,c ,k will be longer than ,c ,i3 and therefore the weights would equiponderate1 if the weight at ,f were less than the weight at ,e4 ,but the weight at ,f is equal to that at ,e2 therefore is greater1 than is necessary to counterbalance the weight at ,e1 and consequently will outweigh it1 and draw the beam of the lever down4 #ch4 ,,in like manner in the case of the steelyard 7fig4 #cb47 if the weights at ,e and ,f are so proportioned1 as to equiponderate1 when the points ,e and ,f are horizontally situated2 then in any other situation of this lever the weight1 which is raised highest1 will preponderate4 ,that is1 if in the horizontal situation of the points ,e and ,f the weight at ,f bears the same proportion to the weight at ,e1 as ,c ,i bears to ,c ,k2 then1 if the point ,f be raised higher than ,e 7as in fig4 #cb47 the weight at ,f shall bear a greater proportion to the weight at ,e1 than ,c ,i bears to ,c ,k4 #ci4 ,,farther a lever may be hung upon an axis1 and then the two arms of the lever need not be continuous1 but fixed to different parts of this axis2 as in fig4 #cc1 where the axis ,a ,b is supported by its two extremities ,a and ,b4 ,to this axis one arm of the lever is fixed at the point ,c1 the other at the point ,d4 ,now here1 if a weight be hung at ,e1 the extremity of that arm1 which is fixed to the axis at the point ,c2 and another weight be hung at ,f1 the extremity of the arm1 which is fixed on the axis at ,d2 then these weights will equiponderate1 when the weight at ,e bears the same proportion to the weight at ,f1 as the arm ,d ,f bears to ,c ,e4 #dj4 ,,this is the case1 if both the arms are perpendicular to the axis1 and lie 7as the geometers express themselves7 in the same plane2 or1 in other words1 if the arms are so fixed perpendicularly upon the axis1 that1 when one of them lies horizontally1 the other shall also be horizontal4 ,if either arm stand not perpendicular to the axis2 then1 in determining the proportion between the weights1 instead of the length of that arm1 you must use the perpendicular let fall upon the axis from the extremity of that arm4 ,if the arms are not so fixed as to become horizontal1 at the same time2 the method of assigning the proportion between the weights is analogous to that made use of above in levers1 which make an angle at the point1 whereon they are supported4 #da4 ,,from this case of the lever hung on an axis1 it is easy to make a transition to another mechanical power1 the wheel and axis4 #db4 ,,this instrument is a wheel fixed on a roller1 the roller being supported at each extremity so as to turn round freely with the wheel1 in the manner represented in fig4 #cd1 where ,a ,b is the wheel1 ,c ,d the roller1 and ,e ,f its two supports4 ,now suppose a weight ,g hung by a cord wound round the roller1 and another weight ,h hung by a cord wound about the wheel the contrary way3 that these weights may support each other1 the weight ,h must bear the same proportion to the weight ,g1 as the thickness of the roller bears to the diameter of the wheel4 #dc4 ,,suppose the line k l to be drawn through the middle of the roller2 and from the place of the roller1 where the cord1 on which the weight ,g hangs1 begins to leave the roller1 as at m1 let the line m n be drawn perpendicularly to k l2 and from the point1 where the cord holding the weight ,h begins to leave the wheel1 as at o1 let the line o p be drawn perpendicular to k l4 ,this being done1 the two lines o p and m n represent two arms of a lever fixed on the axis k l2 consequently the weight ,h will bear to the weight ,g the same proportion1 as m n bears to o p4 ,but m n bears the same proportion to o p1 as the thickness of the roller bears to the diameter of the wheel2 for m n is half the thickness of the roller1 and o p half the diameter of the wheel4 #dd4 ,,if the wheel be put into motion1 and turned once round1 that the cord1 on which the weight ,g hangs1 be wound once more round the axis2 then at the same time the cord1 whereon the weight ,h hangs1 will be wound off from the wheel one circuit4 ,therefore the velocity of the weight ,g will bear the same proportion to the velocity of the weight ,h1 as the circumference of the roller to the circumference of the wheel4 ,but the circumference of the roller bears the same proportion to the circumference of the wheel1 as the thickness of the roller bears to the diameter of the wheel1 consequently the velocity of the weight ,g bears to the velocity of the weight ,h the same proportion1 as the thickness of the roller bears to the diameter of the wheel1 which is the proportion that the weight ,h bears to the weight ,g4 ,therefore as before in the lever1 so here also the general rule laid down above is verified1 that the weights equiponderate1 when their velocities would be reciprocally proportional to their respective weights4 #de4 ,,in like manner1 if on the same axis two wheels of different sizes are fixed 7as in fig4 #ce47 and a weight hung on each2 the weights will equiponderate1 if the weight hung on the greater wheel bear the same proportion to the weight hung on the lesser1 as the diameter of the lesser wheel bears to the diameter of the greater4 #df4 ,,it is usual to join many wheels together in the same frame1 which by the means of certain teeth1 formed in the circumference of each wheel1 shall communicate motion to each other4 ,a machine of this nature is represented in fig4 #cf4 ,here ,a ,b ,c is a winch1 upon which is fixed a small wheel ,d indented with teeth1 which move in the like teeth of a larger wheel ,e ,f fixed on the axis ,g ,h4 ,let this axis carry another wheel ,i1 which shall move in like manner a greater wheel ,k ,l fixed on the axis ,m ,n4 ,let this axis carry another small wheel ,o1 which after the same manner shall turn about a larger wheel ,p ,q fixed on the roller ,r ,s1 on which a cord shall be wound1 that holds a weight1 as ,t4 ,now the proportion required between the weight ,t and a power applied to the winch at ,a sufficient to support the weight1 will most easily be estimated1 by computing the proportion1 which the velocity of the point ,a would bear to the velocity of the weight4 ,if the winch be turned round1 the point ,a will describe a circle as ,a ,v4 ,suppose the wheel ,e ,f to have ten times the number of teeth1 as the wheel ,d2 then the winch must turn round ten times to carry the wheel ,e ,f once round4 ,if wheel ,k ,l has also ten times the number of teeth1 as ,i1 the wheel ,i must turn round ten times to carry the wheel ,k ,l once round2 and consequently the winch ,a ,b ,c must turn round an hundred times to turn the wheel ,k ,l once round4 ,lastly1 if the wheel ,p ,q has ten times the number of teeth1 as the wheel ,o1 the winch must turn about one thousand times in order to turn the wheel ,p ,q1 or the roller ,r ,s once round4 ,therefore here the point ,a must have gone over the circle ,a ,v a thousand times1 in order to lift the weight ,t through a space equal to the circumference of the roller ,r ,s3 whence it follows1 that the power applied at ,a will balance the weight ,t1 if it bear the same proportion to it1 as the circumference of the roller to one thousand times the circle ,a ,v2 or the same proportion as half the thickness of the roller bears to one thousand times ,a ,b4 #dg4 ,i ,,shall now explain the effect of the pulley4 ,let a weight hang by a pulley1 as in fig4 #cg4 ,here it is evident1 that the power ,a1 by which the weight ,b is supported1 must be equal to the weight2 for the cord ,c ,d is equally strained between them2 and if the weight ,b move1 the power ,a must move with equal velocity4 ,the pulley ,e has no other effect1 than to permit the power ,a to act in another direction1 than it must have done1 if it had been directly applied to support the weight without the intervention of any such instrument4 #dh4 ,,again1 let a weight be supported1 as in fig4 #ch2 where the weight ,a is fixed to the pulley ,b1 and the cord1 by which the weight is upheld1 is annexed by one extremity to a hook ,c1 and at the other end is held by the power ,d4 ,here the weight is supported by a cord doubled2 insomuch that although the cord were not strong enough to hold the weight single1 yet being thus doubled it might support it4 ,if the end of the cord held by the power ,d were hung on the hook ,c1 as well as the other end2 then1 when both ends of the cord were tied to the hook1 it is evident1 that the hook would bear the whole weight2 and each end of the string would bear against the hook with the force of half the weight only1 seeing both ends together bear with the force of the whole4 ,hence it is evident1 that1 when the power ,d holds one end of the weight1 the force1 which it must exert to support the weight1 must be equal to just half the weight4 ,and the same proportion between the weight and power might be collected from comparing the respective velocities1 with which they would move2 for it is evident1 that the power must move through a space equal to twice the distance of the pulley from the hook1 in order to lift the pulley up to the hook4 #di4 ,,it is equally easy to estimate the effect1 when many pulleys are combined together1 as in fig4 #ci1 #dj2 in the first of which the under set of pulleys1 and consequently the weight is held by six strings2 and in the latter figure by five3 therefore in the first of these figures the power to support the weight1 must be one sixth part only of the weight1 and in the latter figure the power must be one fifth part4 #ej4 ,,there are two other ways of supporting a weight by pulleys1 which ,i shall particularly consider4 #ea4 ,,one of these ways is represented in fig4 #da4 ,here the weight being connected to the pulley ,b1 a power equal to half the weight ,a would support the pulley ,c1 if applied immediately to it4 ,therefore the pulley ,c is drawn down with a force equal to half the weight ,a4 ,but if the pulley ,d were to be immediately supported by half the force1 with which the pulley ,c is drawn down1 this pulley ,d will uphold the pulley ,c2 so that if the pulley ,d be upheld with a force equal to one fourth part of the weight ,a1 that force will support the weight4 ,but1 for the same reason as before1 if the power in ,e be equal to half the force necessary to uphold the pulley ,d2 this pulley1 and consequently the weight ,a1 will be upheld3 therefore1 if the power in ,e be one eighth part of the weight ,a1 it will support the weight4 #eb4 ,,another way of applying pulleys to a weight is represented in fig4 #db4 ,to explain the effect of pulleys thus applied1 it will be proper to consider different weights hanging1 as in fig4 #dc4 ,here1 if the power and weights balance each other1 the power ,a is equal to the weight ,b2 the weight ,c is equal to twice the power ,a1 or the weight ,b2 and for the same reason the weight ,d is equal to twice the weight ,c1 or equal to four times the power ,a4 ,it is evident therefore1 that all the three weights ,b1 ,c1 ,d together are equal to seven times the power ,a4 ,but if these three weights were joined in one1 they would produce the case of fig4 #dj3 so that in that figure the weight ,a1 where there are three pulleys1 is seven times the power ,b4 ,if there had been but two pulleys1 the weight would have been three times the power2 and if there had been four pulleys1 the weight would have been fifteen times the power4 #ec4 ,,the wedge is next to be considered4 ,the form of this instrument is sufficiently known4 ,when it is put under any weight 7as in fig4 #dd47 the force1 with which the wedge will lift the weight1 when drove under it by a blow upon the end ,a ,b1 will bear the same proportion to the force1 wherewith the blow would act on the weight1 if directly applied to it2 as the velocity1 which the wedge receives from the blow1 bears to the velocity1 wherewith the weight is lifted by the wedge4 #ed4 ,,the screw is the fifth mechanical power4 ,there are two ways of applying this instrument4 ,sometimes it is screwed into a hole1 as in fig4 #de1 where the screw ,a ,b is screwed through the plank ,c ,d4 ,sometimes the screw is applied to the teeth of a wheel1 as in fig4 #df1 where the thread of the screw ,a ,b turns in the teeth of a wheel ,c ,d4 ,in both these cases1 if a bar1 as ,a ,e1 be fixed to the end ,a of the screw2 the force1 wherewith the end ,b of the screw in fig4 #de is forced down1 and the force1 wherewith the teeth of the wheel ,c ,d in fig4 #dd are held1 bears the same proportion to the power applied to the end ,e of the bar2 as the velocity1 wherewith the end ,e will move1 when the screw is turned1 bears to the velocity1 wherewith the end ,b of the screw in fig4 #dc1 or the teeth of the wheel ,c ,d in fig4 #df1 will be moved4 #ee4 ,,the inclined plane affords also a means of raising a weight with less force1 than what is equal to the weight it self4 ,suppose it were required to raise the globe ,a 7in fig4 #dg47 from the ground ,b ,c up to the point1 whose perpendicular height from the ground is ,e ,d4 ,if this globe be drawn along the slant ,d ,f1 less force will be required to raise it1 than if it were lifted directly up4 ,here if the force applied to the globe bear the same proportion only to its weight1 as ,e ,d bears to ,f ,d1 it will be sufficient to hold up the globe2 and therefore any addition to that force will put it in motion1 and draw it up2 unless the globe1 by pressing against the plane1 whereon it lies1 adhere in some degree to the plane4 ,this indeed it must always do more or less1 since no plane can be made so absolutely smooth as to have no inequalities at all2 nor yet so infinitely hard1 as not to yield in the least to the pressure of the weight4 ,therefore the globe cannot be laid on such a plane1 whereon it will slide with perfect freedom1 but they must in some measure rub against each other2 and this friction will make it necessary to imploy a certain degree of force more1 than what is necessary to support the globe1 in order to give it any motion4 ,but as all the mechanical powers are subject in some degree or other to the like impediment from friction2 ,i shall here only shew what force would be necessary to sustain the globe1 if it could lie upon the plane without causing any friction at all4 ,and ,i say1 that if the globe were drawn by the cord ,g ,h1 lying parallel to the plane ,d ,f2 and the force1 wherewith the cord is pulled1 bear the same proportion to the weight of the globe1 as ,e ,d bears to ,d ,f2 this force will sustain the globe4 ,in order to the making proof of this1 let the cord ,g ,h be continued on1 and turned over the pulley ,i1 and let the weight ,k be hung to it4 ,now ,i say1 if this weight bears the same proportion to the globe ,a1 as ,d ,e bears to ,d ,f1 the weight will support the globe4 ,i think it is very manifest1 that the center of the globe ,a will lie in one continued line with the cord ,h ,g4 ,let ,l be the center of the globe1 and ,m the center of gravity of the weight ,k4 ,in the first place let the weight hang so1 that a line drawn from ,l to ,m shall lie horizontally2 and ,i say1 if the globe be moved either up or down the plane ,d ,f1 the weight will so move along with it1 that the center of gravity common to both the weights shall continue in this line ,l ,m1 and therefore shall in no case descend4 ,to prove this more fully1 ,i shall depart a little from the method of this treatise1 and make use of a mathematical proportion or two3 but they are such1 as any person1 who has read ,,euclid',s ,,elements1 will fully comprehend2 and are in themselves so evident1 that1 ,i believe1 my readers1 who are wholly strangers to geometrical writings1 will make no difficulty of admitting them4 ,this being premised1 let the globe be moved up1 till its center be at ,g1 then will ,m the center of gravity of the weight ,k be sunk to ,n2 so that ,m ,n shall be equal to ,g ,l4 ,draw ,n ,g crossing the line ,m ,l in ,o2 then ,i say1 that ,o is the common center of gravity of the two weights in this their new situation4 ,let ,g ,p be drawn perpendicular to ,m ,l2 then ,g ,l will bear the same proportion to ,g ,p1 as ,d ,f bears to ,d ,e2 and ,m ,n being equal to ,g ,l1 ,m ,n will bear the same proportion to ,g ,p1 as ,d ,f bears to ,d ,e4 ,but ,n ,o bears the same proportion to ,o ,g1 as ,m ,n bears to ,g ,p2 consequently ,n ,o will bear the same proportion to ,o ,g1 as ,d ,f bears to ,d ,e4 ,in the last place1 the weight of the globe ,a bears the same proportion to the other weight ,k1 as ,d ,f bears to ,d ,e2 therefore ,n ,o bears the same proportion to ,o ,g1 as the weight of the globe ,a bears to the weight ,k4 ,whence it follows1 that1 when the center of the globe ,a is in ,g1 and the center of gravity of the weight ,k is in ,n1 ,o will be the center of gravity common to both the weights4 ,after the same manner1 if the globe had been caused to descend1 the common center of gravity would have been found in this line ,m ,l4 ,since therefore no motion of the globe either way will make the common center of gravity descend1 it is manifest1 from what has been said above1 that the weights ,a and ,k counterpoize each other4 #ef4 ,i ,,shall now consider the case of pendulums4 ,a pendulum is made by hanging a weight to a line1 so that it may swing backwards and forwards4 ,this motion the geometers have very carefully considered1 because it is the most commodious instrument of any for the exact measurement of time4 #eg4 ,i ,,have observed already#fa1 that if a body hanging perpendicularly by a string1 as the body ,a 7in fig4 #dh47 hangs by the string ,a ,b1 be put so into motion1 as to be made to ascend up the circular arch ,a ,c2 then as soon as it has arrived at the highest point1 to which the motion1 that the body has received1 will carry it2 it will immediately begin to descend1 and at ,a will receive again as great a degree of motion1 as it had at first4 ,this motion therefore will carry the body up the arch ,a ,d1 as high as it ascended before in the arch ,a ,c4 ,consequently in its return through the arch ,d ,a it will acquire again at ,a its original velocity1 and advance a second time up the arch ,a ,c as high as at first2 by this means continuing without end its reciprocal motion4 ,it is true indeed1 that in fact every pendulum1 which we can put in motion1 will gradually lessen its swing1 and at length stop1 unless there be some power constantly applied to it1 whereby its motion shall be renewed2 but this arises from the resistance1 which the body meets with both from the air1 and the string by which it is hung3 for as the air will give some obstruction to the progress of the body moving through it2 so also the string1 whereon the body hangs1 will be a farther impediment2 for this string must either slide on the pin1 whereon it hangs1 or it must bend to the motion of the weight2 in the first there must be some degree of friction1 and in the latter the string will make some resistance to its inflection4 ,however1 if all resistance could be removed1 the motion of a pendulum would be perpetual4 #eh4 ,,but to proceed1 the first property1 ,i shall take notice of in this motion1 is1 that the greater arch the pendulous body moves through1 the greater time it takes up3 though the length of time does not increase in so great a proportion as the arch4 ,thus if ,c ,d be a greater arch1 and ,e ,f a lesser1 where ,c ,a is equal to ,a ,d1 and ,e ,a equal to ,a ,f2 the body1 when it swings through the greater arch ,c ,d1 shall take up in its swing from ,c to ,d a longer time than in swinging from ,e to ,f1 when it moves only in that lesser arch2 or the time in which the body let fall from ,c will descend through the arch ,c ,a is greater than the time1 in which it will descend through the arch ,e ,a1 when let fall from ,e4 ,but the first of these times will not hold the same proportion to the latter1 as the first arch ,c ,a bears to the other arch ,e ,a2 which will appear thus4 ,let ,c ,g and ,e ,h be two horizontal lines4 ,it has been remarked above#fb1 that the body in falling through the arch ,c ,a will acquire as great a velocity at the point ,a1 as it would have gained by falling directly down through ,g ,a2 and in falling through the arch ,e ,a it will acquire in the point ,a only that velocity1 which it would have got in falling through ,h ,a4 ,therefore1 when the body descends through the greater arch ,c ,a1 it shall gain a greater velocity1 than when it passes only through the lesser2 so that this greater velocity will in some degree compensate the greater length of the arch4 #ei4 ,,the increase of velocity1 which the body acquires in falling from a greater height1 has such an effect1 that1 if straight lines be drawn from ,a to ,c and ,e1 the body would fall through the longer straight line ,c ,a just in the same time1 as through the shorter straight line ,e ,a4 ,this is demonstrated by the geometers1 who prove1 that if any circle1 as ,a ,b ,c ,d 7fig4 #di47 be placed in a perpendicular situation2 a body shall fall obliquely through every line1 as ,a ,b drawn from the lowest point ,a in the circle to any other point in the circumference just in the same time1 as would be imployed by the body in falling perpendicularly down through the diameter ,c ,a4 ,but the time in which the body will descend through the arch1 is different from the time1 which it would take up in falling through the line ,a ,b4 #fj4 ,,it has been thought by some1 that because in very small arches this correspondent straight line differs but little from the arch itself2 therefore the descent through this straight line would be performed in such small arches nearly in the same time as through the arches themselves3 so that if a pendulum were to swing in small arches1 half the time of a single swing would be nearly equal to the time1 in which a body would fall perpendicularly through twice the length of the pendulum4 ,that is1 the whole time of the swing1 according to this opinion1 will be four fold the time required for the body to fall through half the length of the pendulum2 because the time of the body's falling down twice the length of the pendulum is half the time required for the fall through one quarter of this space1 that is through half the pendulum's length4 ,however there is here a mistake2 for the whole time of the swing1 when the pendulum moves through small arches1 bears to the time required for a body to fall down through half the length of the pendulum very nearly the same proportion1 as the circumference of a circle bears to its diameter2 that is very nearly the proportion of #cee to #aac1 or little more than the proportion of #c to #a4 ,if the pendulum takes so great a swing1 as to pass over an arch equal to one sixth part of the whole circumference of the circle1 it will swing #aae times1 while it ought according to this proportion to have swung #aag times2 so that1 when it swings in so large an arch1 it loses something less than two swings in an hundred4 ,if it swing through #a_/aj only of the circle1 it shall not lose above one vibration in #afj4 ,if it swing in #a_/bj of the circle1 it shall lose about one vibration in #fij4 ,if its swing be confined to #a_/dj of the whole circle1 it shall lose very little more than one swing in #bfjj4 ,and if it take no greater a swing than through #a_/fj of the whole circle1 it shall not lose one swing in #ehjj4 #fa4 ,,now it follows from hence1 that1 when pendulums swing in small arches1 there is very nearly a constant proportion observed between the time of their swing1 and the time1 in which a body would fall perpendicularly down through half their length4 ,and we have declared above1 that the spaces1 through which bodies fall1 are in a two fold proportion of the times1 which they take up in falling#fc4 ,therefore in pendulums of different lengths1 swinging through small arches1 the lengths of the pendulums are in a two fold or duplicate proportion of the times1 they take in swinging2 so that a pendulum of four times the length of another shall take up twice the time in each swing1 one of nine times the length will make one swing only for three swings of the shorter1 and so on4 #fb4 ,,this proportion in the swings of different pendulums not only holds in small arches2 but in large ones also1 provided they be such1 as the geometers call similar2 that is1 if the arches bear the same proportion to the whole circumferences of their respective circles4 ,suppose 7in fig4 #dh47 ,a ,b1 ,c ,d to be two pendulums4 ,let the arch ,e ,f be described by the motion of the pendulum ,a ,b1 and the arch ,g ,h be described by the pendulum ,c ,d2 and let the arch ,e ,f bear the same proportion to the whole circumference1 which would be formed by turning the pendulum ,a ,b quite round about the point ,a1 as the arch ,g ,h bears to the whole circumference1 that would be formed by turning the pendulum ,c ,d quite round the point ,c4 ,then ,i say1 the proportion1 which the length of the pendulum ,a ,b bears to the length of the pendulum ,c ,d1 will be two fold of the proportion1 which the time taken up in the description of the arch ,e ,f bears to the time employed in the description of the arch ,g ,h4 #fc4 ,,thus pendulums1 which swing in very small arches1 are nearly an equal measure of time4 ,but as they are not such an equal measure to geometrical exactness2 the mathematicians have found out a method of causing a pendulum so to swing1 that1 if its motion were not obstructed by any resistance1 it would always perform each swing in the same time1 whether it moved through a greater1 or a lesser space4 ,this was first discovered by the great ,,huygens1 and is as follows4 ,upon the straight line ,a ,b 7in fig4 #di47 let the circle ,c ,d ,e be so placed1 as to touch the straight line in the point ,c4 ,then let this circle roll along upon the straight line ,a ,b1 as a coach-wheel rolls along upon the ground4 ,it is evident1 that1 as soon as ever the circle begins to move1 the point ,c in the circle will be lifted off from the straight line ,a ,b2 and in the motion of the circle will describe a crooked course1 which is represented by the line ,c ,f ,g ,h4 ,here the part ,c ,h of the straight line included between the two extremities ,c and ,h of the line ,c ,f ,g ,h will be equal to the whole circumference of the circle ,c ,d ,e2 and if ,c ,h be divided into two equal parts at the point ,i1 and the straight line ,i ,k be drawn perpendicular to ,c ,h1 this line ,i ,k will be equal to the diameter of the circle ,c ,d ,e4 ,now in this line if a body were to be let fall from the point ,h1 and were to be carried by its weight down the line ,h ,g ,k1 as far as the point ,k1 which is the lowest point of the line ,c ,f ,g ,h2 and if from any other point ,g a body were to be let fall in the same manner2 this body1 which falls from ,g1 will take just the same time in coming to ,k1 as the body takes up1 which falls from ,h4 ,therefore if a pendulum can be so hung1 that the ball shall move in the line ,a ,g ,f ,e1 all its swings1 whether long or short1 will be performed in the same time2 for the time1 in which the ball will descend to the point ,k1 is always half the time of the whole swing4 ,but the ball of a pendulum will be made to swing in this line by the following means4 ,let ,k ,i 7in fig4 #eb47 be prolonged upwards to ,l1 till ,i ,l is equal to ,i ,k4 ,then let the line ,l ,m ,h equal and like to ,k ,h be applied1 as in the figure between the points ,l and ,h1 so that the point which in this line ,l ,m ,h answers to the point ,h in the line ,k ,h shall be applied to the point ,l1 and the point answering to the point ,k shall be applied to the point ,h4 ,also let such another line ,l ,n ,c be applied between ,l and ,c in the same manner4 ,this preparation being made2 if a pendulum be hung at the point ,l of such a length1 that the ball thereof shall reach to ,k2 and if the string shall continually bend against the lines ,h ,m ,l and ,l ,n ,c1 as the pendulum swings to and fro2 by this means the ball shall constantly keep in the line ,c ,k ,h4 #fd4 ,,now in this pendulum1 as all the swings1 whether long or short1 will be performed in the same time2 so the time of each will exactly bear the same proportion to the time required for a body to fall perpendicularly down1 through half the length of the pendulum1 that is from ,i to ,k1 as the circumference of a circle bears to its diameter4 #fe4 ,,it may from hence be understood in some measure1 why1 when pendulums swing in circular arches1 the times of their swings are nearly equal1 if the arches are small1 though those arches be of very unequal lengths2 for if with the semidiameter ,l ,k the circular arch ,o ,k ,p be described1 this arch in the lower part of it will differ very little from the line ,c ,k ,h4 #ff4 ,,it may not be amiss here to remark1 that a body will fall in this line ,c ,k ,h 7fig4 #ec47 from ,c to any other point1 as ,q or ,r in a shorter space of time1 than if it moved through the straight line drawn from ,c to the other point2 or through any other line whatever1 that can be drawn between these two points4 #fg4 ,,but as ,i have observed1 that the time1 which a pendulum takes in swinging1 depends upon its length2 ,i shall now say something concerning the way1 in which this length of the pendulum is to be estimated4 ,if the whole ball of the pendulum could be crouded into one point1 this length1 by which the motion of the pendulum is to be computed1 would be the length of the string or rod4 ,but the ball of the pendulum must have a sensible magnitude1 and the several parts of this ball will not move with the same degree of swiftness2 for those parts1 which are farthest from the point1 whereon the pendulum is suspended1 must move with the greatest velocity4 ,therefore to know the time in which the pendulum swings1 it is necessary to find that point of the ball1 which moves with the same degree of velocity1 as if the whole ball were to be contracted into that point4 #fh4 ,,this point is not the center of gravity1 as ,i shall now endeavour to shew4 ,suppose the pendulum ,a ,b 7in fig4 #ed47 composed of an inflexible rod ,a ,c and ball ,c ,b1 to be fixed on the point ,a1 and lifted up into an horizontal situation4 ,here if the rod were not fixed to the point ,a1 the body ,c ,b would descend directly with the whole force of its weight2 and each part of the body would move down with the same degree of swiftness4 ,but when the rod is fixed at the point ,a1 the body must fall after another manner2 for the parts of the body must move with different degrees of velocity1 the parts more remote from ,a descending with a swifter motion1 than the parts nearer to ,a2 so that the body will receive a kind of rolling motion while it descends4 ,but it has been observed above1 that the effect of gravity upon any body is the same1 as if the whole force were exerted on the body's center of gravity#fd4 ,illustration ,since therefore the power of gravity in drawing down the body must also communicate to it the rolling motion just described2 it seems evident1 that the center of gravity of the body cannot be drawn down as swiftly1 as when the power of gravity has no other effect to produce on the body1 than merely to draw it downward4 ,if therefore the whole matter of the body ,c ,b could be crouded into its center of gravity1 so that being united into one point1 this rolling motion here mentioned might give no hindrance to its descent2 this center would descend faster1 than it can now do4 ,and the point1 which now descends as fast1 as if the whole matter or the body ,c ,b were crouded into it1 will be farther removed from the point ,a1 than the center of gravity of the body ,c ,b4 #fi4 ,,again1 suppose the pendulum ,a ,b 7in fig4 #ee47 to hang obliquely4 ,here the power of gravity will operate less upon the ball of the pendulum1 than before3 but the line ,d ,e being drawn so1 as to stand perpendicular to the rod ,a ,c of the pendulum2 the force of gravity upon the body ,c ,b1 now it is in this situation1 will produce the same effect1 as if the body were to glide down an inclined plane in the position of ,d ,e4 ,but here the motion of the body1 when the rod is fixed to the point ,a1 will not be equal to the uninterrupted descent of the body down this plane2 for the body will here also receive the same kind of rotation in its motion1 as before2 so that the motion of the center of gravity will in like manner be retarded2 and the point1 which here descends with that degree of swiftness1 which the body would have1 if not hindered by being fixed to the point ,a2 that is1 the point1 which descends as fast1 as if the whole body were crouded into it1 will be as far removed from the point ,a1 as before4 #gj4 ,,this point1 by which the length of the pendulum is to be estimated1 is called the center of oscillation4 ,and the mathematicians have laid down general directions1 whereby to find this center in all bodies4 ,if the globe ,a ,b 7in fig4 #ef47 be hung by the string ,c ,d1 whose weight need not be regarded1 the center of oscillation is found thus4 ,let the straight line drawn from ,c to ,d be continued through the globe to ,f4 ,that it will pass through the center of the globe is evident4 ,suppose ,e to be this center of the globe2 and take the line ,g of such a length1 that it shall bear the same proportion to ,e ,d1 as ,e ,d bears to ,e ,c4 ,then ,e ,h being made equal to of ,g1 the point ,h shall be the center of oscillation#fe4 ,if the weight of the rod ,c ,d is too considerable to be neglected1 divide ,c ,d 7fig4 #eg7 in ,i1 that ,d ,i be equal to 1 part of ,c ,d2 and take ,k in the same proportion to ,c ,i1 as the weight of the globe ,a ,b to the weight of the rod ,c ,d4 ,then having found ,h1 the center of oscillation of the globe1 as before1 divide ,i ,k in ,i1 so that ,i ,l shall bear the same proportion to ,l ,h1 as the line ,c ,h bears to ,k2 and ,l shall be the center of oscillation of the whole pendulum4 #ga4 ,,this computation is made upon supposition1 that the center of oscillation of the rod ,c ,d1 if that were to swing alone without any other weight annexed1 would be the point ,i4 ,and this point would be the true center of oscillation1 so far as the thickness of the rod is not to be regarded4 ,if any one chuses to take into consideration the thickness of the rod1 he must place the center of oscillation thereof so much below the point ,i1 that eight times the distance of the center from the point ,i shall bear the same proportion to the thickness of the rod1 as the thickness of the rod bears to its length ,c ,d#ff4 #gb4 ,,it has been observed above1 that when a pendulum swings in an arch of a circle1 as here in fig4 #eh1 the pendulum ,a ,b swings in the circular arch ,c ,d2 if you draw an horizontal line1 as ,e ,f1 from the place whence the pendulum is let fall1 to the line ,a ,g1 which is perpendicular to the horizon3 then the velocity1 which the pendulum will acquire in coming to the point ,g1 will be the same1 as any body would acquire in falling directly down from ,f to ,g4 ,now this is to be understood of the circular arch1 which is described by the center of oscillation of the pendulum4 ,i shall here farther observe1 that if the straight line ,e ,g be drawn from the point1 whence the pendulum falls1 to the lowest point of the arch2 in the same or in equal pendulums the velocity1 which the pendulum acquires in ,g1 is proportional to this line3 that is1 if the pendulum1 after it has descended from ,e to ,g1 be taken back to ,h1 and let fall from thence1 and the line ,h ,g be drawn2 the velocity1 which the pendulum shall acquire in ,g by its descent from ,h1 shall bear the same proportion to the velocity1 which it acquires in falling from ,e to ,g1 as the straight line ,h ,g bears to the straight line ,e ,g4 #gc4 ,,we may now proceed to those experiments upon the percussion of bodies1 which ,i observed above might be made with pendulums4 ,this expedient for examining the effects of percussion was first proposed by our late great architect ,sir ,,christopher ,,wren4 ,and it is as follows4 ,two balls1 as ,a and ,b 7in fig4 #ei47 either equal or unequal1 are hung by two strings from two points ,c and ,d1 so that1 when the balls hang down without motion1 they shall just touch each other1 and the strings be parallel4 ,here if one of these balls be removed to any distance from its perpendicular situation1 and then let fall to descend and strike against the other2 by the last preceding paragraph it will be known1 with what velocity this ball shall return into its first perpendicular situation1 and consequently with what force it shall strike against the other ball2 and by the height to which this other ball ascends after the stroke1 the velocity communicated to this ball will be discovered4 ,for instance1 let the ball ,a be taken up to ,e1 and from thence be let fall to strike against ,b1 passing over in its descent the circular arch ,e ,f4 ,by this impulse let ,b fly up to ,g1 moving through the circular arch ,h ,g4 ,then ,e ,i and ,g ,k being drawn horizontally1 the ball ,a will strike against ,b with the velocity1 which it would acquire in falling directly down from ,i2 and the ball ,b has received a velocity1 wherewith1 if it had been thrown directly upward1 it would have ascended up to ,k4 ,likewise if straight lines be drawn from ,e to ,f and from ,h to ,g1 the velocity of ,a1 wherewith it strikes1 will bear the same proportion to the velocity1 which ,b has received by the blow1 as the straight line ,e ,f bears to the straight line ,h ,g4 ,in the same manner by noting the place to which ,a ascends after the stroke1 its remaining velocity may be compared with that1 wherewith it struck against ,b4 ,thus may be experimented the effects of the body ,a striking against ,b at rest4 ,if both the bodies are lifted up1 and so let fall as to meet and impinge against each other just upon the coming of both into their perpendicular situation2 by observing the places into which they move after the stroke1 the effects of their percussion in all these cases may be found in the same manner as before4 #gd4 ,,sir ,,isaac ,,newton has described these experiments2 and has shewn how to improve them to a greater exactness by making allowance for the resistance1 which the air gives to the motion of the balls#fg4 ,but as this resistance is exceeding small1 and the manner of allowing for it is delivered by himself in very plain terms1 ,i need not enlarge upon it here4 ,i shall rather speak to a discovery1 which he made by these experiments upon the elasticity of bodies4 ,it has been explained above#fh1 that when two bodies strike1 if they be not elastic1 they remain contiguous after the stroke2 but that if they are elastic1 they separate1 and that the degree of their elasticity determines the proportion between the celerity wherewith they separate1 and the celerity wherewith they meet4 ,now our author found1 that the degree of elasticity appeared in the same bodies always the same1 with whatever degree of force they struck2 that is1 the celerity wherewith they separated1 always bore the same proportion to the celerity wherewith they met3 so that the elastic power in all the bodies1 he made trial upon1 exerted it self in one constant proportion to the compressing force4 ,our author made trial with balls of wool bound up very compact1 and found the celerity with which they receded1 to bear about the proportion of #e to #i to the celerity wherewith they met2 and in steel he found nearly the same proportion2 in cork the elasticity was something less2 but in glass much greater2 for the celerity1 wherewith balls of that material separated after percussion1 he found to bear the proportion of #ae to #af to the celerity wherewith they met#fi4 #ge4 ,i ,,shall finish my discourse on pendulums1 with this farther observation only1 that the center of oscillation is also the center of another force4 ,if a body be fixed to any point1 and being put in motion turns round it2 the body1 if uninterrupted by the power of gravity or any other means1 will continue perpetually to move about with the same equable motion4 ,now the force1 with which such a body moves1 is all united in the point1 which in relation to the power of gravity is called the center of oscillation4 ,let the cylinder ,a ,b ,c ,d 7in fig4 #fj47 whose axis is ,e ,f1 be fixed to the point ,e4 ,and supposing the point ,e to be that on which the cylinder is suspended1 let the center of oscillation be found in the axis ,e ,f1 as has been explained above#gj4 ,let ,g be that center3 then ,i say1 that the force1 wherewith this cylinder turns round the point ,e1 is so united in the point ,g1 that a sufficient force applied in that point shall stop the motion of the cylinder1 in such a manner1 that the cylinder should immediately remain without motion1 though it were to be loosened from the point ,e at the same instant1 that the impediment was applied to ,g3 whereas1 if this impediment had been applied to any other point of the axis1 the cylinder would turn upon the point1 where the impediment was applied4 ,if the impediment had been applied between ,e and ,g1 the cylinder would so turn on the point1 where the impediment was applied1 that the end ,b ,c would continue to move on the same way it moved before along with the whole cylinder2 but if the impediment were applied to the axis farther off from ,e than ,g1 the end ,a ,d of the cylinder would start out of its present place that way in which the cylinder moved4 ,from this property of the center of oscillation1 it is also called the center of percussion4 ,that excellent mathematician1 ,dr4 ,,brook ,,taylor1 has farther improved this doctrine concerning the center of percussion1 by shewing1 that if through this point ,g a line1 as ,g ,h ,i1 be drawn perpendicular to ,e ,f1 and lying in the course of the body's motion2 a sufficient power applied to any point of this line will have the same effect1 as the like power applied to ,g#ga3 so that as we before shewed the center of percussion within the body on its axis2 by this means we may find this center on the surface of the body also1 for it will be where this line ,h ,i crosses that surface4 #gf4 ,i ,,shall now proceed to the last kind of motion1 to be treated on in this place1 and shew what line the power of gravity will cause a body to describe1 when it is thrown forwards by any force4 ,this was first discovered by the great ,,galileo1 and is the principle1 upon which engineers should direct the shot of great guns4 ,but as in this case bodies describe in their motion one of those lines1 which in geometry are called conic sections2 it is necessary here to premise a description of those lines4 ,in which ,i shall be the more particular1 because the knowledge of them is not only necessary for the present purpose1 but will be also required hereafter in some of the principal parts of this treatise4 #gg4 ,,the first lines considered by the ancient geometers were the straight line and the circle4 ,of these they composed various figures1 of which they demonstrated many properties1 and resolved divers problems concerning them4 ,these problems they attempted always to resolve by the describing straight lines and circles4 ,for instance1 let a square ,a ,b ,c ,d 7fig4 #fa47 be proposed1 and let it be required to make another square in any assigned proportion to this4 ,prolong one side1 as ,d ,a1 of this square to ,e1 till ,a ,e bear the same proportion to ,a ,d1 as the new square is to bear to the square ,a ,c4 ,if the opposite side ,b ,c of the square ,a ,c be also prolonged to ,f1 till ,b ,f be equal to ,a ,e1 and ,e ,f be afterwards drawn1 ,i suppose my readers will easily conceive1 that the figure ,a ,b ,f ,e will bear to the square ,a ,b ,c ,d the same proportion1 as the line ,a ,e bears to the line ,a ,d4 ,therefore the figure ,a ,b ,f ,e will be equal to the new square1 which is to be found1 but is not it self a square1 because the side ,a ,e is not of the same length with the side ,e ,f4 ,but to find a square equal to the figure ,a ,b ,f ,e you must proceed thus4 ,divide the line ,d ,e into two equal parts in the point ,g1 and to the center ,g with the interval ,g ,d describe the circle ,d ,h ,e ,i2 then prolong the line ,a ,b1 till it meets the circle in ,k2 and make the square ,a ,k ,l ,m1 which square will be equal to the figure ,a ,b ,f ,e1 and bear to the square ,a ,b ,c ,d the same proportion1 as the line ,a ,e bears to ,a ,d4 #gh4 ,i ,,shall not proceed to the proof of this1 having only here set it down as a specimen of the method of resolving geometrical problems by the description of straight lines and circles4 ,but there are some problems1 which cannot be resolved by drawing straight lines or circles upon a plane4 ,for the management therefore of these they took into consideration solid figures1 and of the solid figures they found that1 which is called a cone1 to be the most useful4 #gi4 ,a ,,cone is thus defined by ,,euclide in his elements of geometry#gb4 ,if to the straight line ,a ,b 7in fig4 #fb47 another straight line1 as ,a ,c1 be drawn perpendicular1 and the two extremities ,b and ,c be joined by a third straight line composing the triangle ,a ,c ,b 7for so every figure is called1 which is included under three straight lines7 then the two points ,a and ,b being held fixed1 as two centers1 and the triangle ,a ,c ,b being turned round upon the line ,a ,b1 as on an axis2 the line ,a ,c will describe a circle1 and the figure ,a ,c ,b will describe a cone1 of the form represented by the figure ,b ,c ,d ,e ,f 7fig4 #fc47 in which the circle ,c ,d ,e ,f is usually called the base of the cone1 and ,b the vertex4 #hj4 ,,now by this figure may several problems be resolved1 which cannot by the simple description of straight lines and circles upon a plane4 ,suppose for instance1 it were required to make a cube1 which should bear any assigned proportion to some other cube named4 ,i need not here inform my readers1 that a cube is the figure of a dye4 ,this problem was much celebrated among the ancients1 and was once inforced by the command of an oracle4 ,this problem may be performed by a cone thus4 ,first make a cone from a triangle1 whose side ,a ,c shall be half the length of the side ,b ,c ,then on the plane ,a ,b ,c ,d 7fig4 #fd47 let the line ,e ,f be exhibited equal in length to the side of the cube proposed2 and let the line ,f ,g be drawn perpendicular to ,e ,f1 and of such a length1 that it bear the same proportion to ,e ,f1 as the cube to be sought is required to bear to the cube proposed4 ,through the points ,e1 ,f1 and ,g let the circle ,f ,h ,i be described4 ,then let the line ,e ,f be prolonged beyond ,f to ,k1 that ,f ,k be equal to ,f ,e1 and let the triangle ,f ,k ,l1 having all its sides ,f ,k1 ,k ,l1 ,l ,f equal to each other1 be hung down perpendicularly from the plane ,a ,b ,c ,d4 ,after this1 let another plane ,m ,n ,o ,p be extended through the point ,l1 so as to be equidistant from the former plane ,a ,b ,c ,d1 and in this plane let the line ,q ,l ,r be drawn so1 as to be equidistant from the line ,e ,f ,k4 ,all this being thus prepared1 let such a cone1 as was above directed to be made1 be so applied to the plane ,m ,n ,o ,p1 that it touch this plane upon the line ,q ,r1 and that the vertex of the cone be applied to the point ,l4 ,this cone1 by cutting through the first plane ,a ,b ,c ,d1 will cross the circle ,f ,h ,i before described4 ,and if from the point ,s1 where the surface of this cone intersects the circle1 the line ,s ,t be drawn so1 as to be equidistant from the line ,e ,f2 the line ,f ,t will be equal to the side of the cube sought3 that is1 if there be two cubes or dyes formed1 the side of one being equal to ,e ,f1 and the side of the other equal to ,f ,t2 the former of these cubes shall bear the same proportion to the latter1 as the line ,e ,f bears to ,f ,g4 #ha4 ,,indeed this placing a cone to cut through a plane is not a practicable method of resolving problems4 ,but when the geometers had discovered this use of the cone1 they applied themselves to consider the nature of the lines1 which will be produced by the intersection of the surface of a cone and a plane2 whereby they might be enabled both to reduce these kinds of solutions to practice1 and also to render their demonstrations concise and elegant4 #hb4 ,,whenever the plane1 which cuts the cone1 is equidistant from another plane1 that touches the cone on the side2 7which is the case of the present figure27 the line1 wherein the plane cuts the surface of the cone1 is called a parabola4 ,but if the plane1 which cuts the cone1 be so inclined to this other1 that it will pass quite through the cone 7as in fig4 #fe47 such a plane by cutting the cone produces the figure called an ellipsis1 in which we shall hereafter shew the earth and other planets to move round the sun4 ,if the plane1 which cuts the cone1 recline the other way 7as in fig4 #ff47 so as not to be parallel to any plane1 whereon the cone can lie1 nor yet to cut quite through the cone2 such a plane shall produce in the cone a third kind of line1 which is called an hyperbola4 ,but it is the first of these lines named the parabola1 wherein bodies1 that are thrown obliquely1 will be carried by the force of gravity2 as ,i shall here proceed to shew1 after having first directed my readers how to describe this sort of line upon a plane1 by which the form of it may be seen4 #hc4 ,,to any straight line ,a ,b 7fig4 #fg47 let a straight ruler ,c ,d be so applied1 as to stand against it perpendicularly4 ,upon the edge of this ruler let another ruler ,e ,f be so placed1 as to move along upon the edge of the first ruler ,c ,d1 and keep always perpendicular to it4 ,this being so disposed1 let any point1 as ,g1 be taken in the line ,a ,b1 and let a string equal in length to the ruler ,e ,f be fastened by one end to the point ,g1 and by the other to the extremity ,f of the ruler ,e ,f4 ,then if the string be held down to the ruler ,e ,f by a pin ,h1 as is represented in the figure2 the point of this pin1 while the ruler ,e ,f moves on the ruler ,c ,d1 shall describe the line ,i ,k ,l1 which will be one part of the curve line1 whose description we were here to teach3 and by applying the rulers in the like manner on the other side of the line ,a ,b1 we may describe the other part ,i ,m of this line4 ,if the distance ,c ,g be equal to half the line ,e ,f in fig4 #fd1 the line ,m ,i ,l will be that very line1 wherein the plane ,a ,b ,c ,d in that figure cuts the cone4 #hd4 ,,the line ,a ,i is called the axis of the parabola ,m ,i ,l1 and the point ,g is called the focus4 #he4 ,,now by comparing the effects of gravity upon falling bodies1 with what is demonstrated of this figure by the geometers1 it is proved1 that every body thrown obliquely is carried forward in one of these lines1 the axis whereof is perpendicular to the horizon4 #hf4 ,,the geometers demonstrate1 that if a line be drawn to touch a parabola in any point1 as the line ,a ,b 7in fig4 #fh47 touches the parabola ,c ,d1 whose axis is ,y ,z1 in the point ,e2 and several lines ,f ,g1 ,h ,i1 ,k ,l be drawn parallel to the axis of the parabola3 then the line ,f ,g will be to ,h ,i in the duplicate proportion of ,e ,f to ,e ,h1 and ,f ,g to ,k ,l in the duplicate proportion of ,e ,f to ,e ,k2 likewise ,h ,i to ,k ,l in the duplicate proportion of ,e ,h to ,e ,k4 ,what is to be understood by duplicate or two-fold proportion1 has been already explained#gc4 ,accordingly ,i mean here1 that if the line ,m be taken to bear the same proportion to ,e ,h1 as ,e ,h bears to ,e ,f1 ,h ,i will bear the same proportion to ,f ,g1 as ,m bears to ,e ,f2 and if the line ,n bears the same proportion to ,e ,k1 as ,e ,k bears to ,e ,f1 ,k ,l will bear the same proportion to ,f ,g1 as ,n bears to ,e ,f2 or if the line ,o bear the same proportion to ,e ,k1 as ,e ,k bears to ,e ,h1 ,k ,l will bear the same proportion to ,h ,i1 as ,o bears to ,e ,h4 #hg4 ,,this property is essential to the parabola1 being so connected with the nature of the figure1 that every line possessing this property is to be called by this name4 #hh4 ,,now suppose a body to be thrown from the point ,a 7in fig4 #fi47 towards ,b in the direction of the line ,a ,b4 ,this body1 if left to it self1 would move on with a uniform motion through this line ,a ,b4 ,suppose the eye of a spectator to be placed at the point ,c just under the point ,a2 and let us imagine the earth to be so put into motion along with the body1 as to carry the spectator's eye along the line ,c ,d parallel to ,a ,b2 and that the eye would move on with the same velocity1 wherewith the body would proceed in the line ,a ,b1 if it were to be left to move without any disturbance from its gravitation towards the earth4 ,in this case if the body moved on without being drawn towards the earth1 it would appear to the spectator to be at rest4 ,but if the power of gravity exerted it self on the body1 it would appear to the spectator to fall directly down4 ,suppose at the distance of time1 wherein the body by its own progressive motion would have moved from ,a to ,e1 it should appear to the spectator to have fallen through a length equal to ,e ,f3 then the body at the end of this time will actually have arrived at the point ,f4 ,if in the space of time1 wherein the body would have moved by its progressive motion from ,a to ,g1 it would have appeared to the spectator to have fallen down the space ,g ,h3 then the body at the end of this greater interval of time will be arrived at the point ,h4 ,now if the line ,a ,f ,h ,i be that1 through which the body actually passes2 from what has here been said1 it will follow1 that this line is one of those1 which ,i have been describing under the name of the parabola4 ,for the distances ,e ,f1 ,g ,h1 through which the body is seen to fall1 will increase in the duplicate proportion of the times#gd2 but the lines ,a ,e1 ,a ,g will be proportional to the times wherein they would have been described by the single progressive motion of the body3 therefore the lines ,e ,f1 ,g ,h will be in the duplicate proportion of the lines ,a ,f1 ,a ,g2 and the line ,a ,f ,h ,i possesses the property of the parabola4 #hi4 ,,if the earth be not supposed to move along with the body1 the case will be a little different4 ,for the body being constantly drawn directly towards the center of the earth1 the body in its motion will be drawn in a direction a little oblique to that1 wherein it would be drawn by the earth in motion1 as before supposed4 ,but the distance to the center of the earth bears so vast a proportion to the greatest length1 to which we can throw bodies1 that this obliquity does not merit any regard4 ,from the sequel of this discourse it may indeed be collected1 what line the body being thrown thus would be found to describe1 allowance being made for this obliquity of the earth's action#ge4 ,this is the discovery of ,sir ,,is4 ,,newton2 but has no use in this place4 ,here it is abundantly sufficient to consider the body as moving in a parabola4 #ij4 ,,the line1 which a projected body describes1 being thus known1 practical methods have been deduced from hence for directing the shot of great guns to strike any object desired4 ,this work was first attempted by ,,galileo1 and soon after farther improved by his scholar ,,torricelli2 but has lately been rendred more complete by the great ,mr4 ,,cotes1 whose immature death is an unspeakable loss to mathematical learning4 ,if it be required to throw a body from the point ,a 7in fig4 #gj47 so as to strike the point ,b2 through the points ,a1 ,b draw the straight line ,c ,d1 and erect the line ,a ,e perpendicular to the horizon1 and of four times the height1 from which a body must fall to acquire the velocity1 wherewith the body is intended to be thrown4 ,through the points ,a and ,e describe a circle1 that shall touch the line ,c ,d in the point ,a4 ,then from the point ,b draw the line ,b ,f perpendicular to the horizon1 intersecting the circle in the points ,g and ,h4 ,this being done1 if the body be projected directly towards either of these points ,g or ,h1 it shall fall upon the point ,b2 but with this difference1 that1 if it be thrown in the direction ,a ,g1 it shall sooner arrive at ,b1 than if it were projected in the direction ,a ,h4 ,when the body is projected in the direction ,a ,g2 the time1 it will take up in arriving at ,b1 will bear the same proportion to the time1 wherein it would fall down through one fourth part of ,a ,e1 as ,a ,g bears to half ,a ,e4 ,but when the body is thrown in the direction of ,a ,h1 the time of its passing to ,b will bear the same proportion to the time1 wherein it would fall through one fourth part of ,a ,e1 as ,a ,h bears to half ,a ,e4 #ia4 ,,if the line ,a ,i be drawn so as to divide the angle under ,e ,a ,d in the middle1 and the line ,i ,k be drawn perpendicular to the horizon2 this line will touch the circle in the point ,i1 and if the body be thrown in the direction ,a ,i1 it will fall upon the point ,k3 and this point ,k is the farthest point in the line ,a ,d1 which the body can be made to strike1 without increasing its velocity4 #ib4 ,,the velocity1 wherewith the body every where moves1 may be found thus4 ,suppose the body to move in the parabola ,a ,b 7fig4 #ga47 ,erect ,a ,c perpendicular to the horizon1 and equal to the height1 from which a body must fall to acquire the velocity1 wherewith the body sets out from ,a4 ,if you take any points as ,d and ,e in the parabola1 and draw ,d ,f and ,e ,g parallel to the horizon2 the velocity of the body in ,d will be equal to what a body will acquire in falling down by its own weight through ,c ,f1 and in ,e the velocity will be the same1 as would be acquired in falling through ,c ,g4 ,thus the body moves slowest at the highest point ,h of the parabola2 and at equal distances from this point will move with equal swiftness1 and descend from that highest point through the line ,h ,b altogether like to the line ,a ,h in which it ascended2 abating only the resistance of the air1 which is not here considered4 ,if the line ,h ,i be drawn from the highest point ,h parallel to the horizon1 ,a ,i will be equal to of ,b ,g in fig4 #gj1 when the body is projected in the direction ,a ,g1 and equal to of ,b ,h1 when the body is thrown in the direction ,a ,h provided ,a ,d be drawn horizontally4 #ic4 ,,thus ,i have recounted the principal discoveries1 which had been made concerning the motion of bodies by ,sir ,,isaac ,,newton',s predecessors2 all these discoveries1 by being found to agree with experience1 contributing to establish the laws of motion1 from whence they were deduced4 ,i shall therefore here finish what ,i had to say upon those laws2 and conclude this chapter with a few words concerning the distinction which ought to be made between absolute and relative motion4 ,for some have thought fit to confound them together2 because they observe the laws of motion to take place here on the earth1 which is in motion1 after the same manner as if it were at rest4 ,but ,sir ,,isaac ,,newton has been careful to distinguish between the relative and absolute consideration both of motion and time#gf4 ,the astronomers anciently found it necessary to make this distinction in time4 ,time considered in it self passes on equably without relation to any thing external1 being the proper measure of the continuance and duration of all things4 ,but it is most frequently conceived of by us under a relative view to some succession in sensible things1 of which we take cognizance4 ,the succession of the thoughts in our own minds is that1 from whence we receive our first idea of time1 but is a very uncertain measure thereof2 for the thoughts of some men flow on much more swiftly1 than the thoughts of others2 nor does the same person think equally quick at all times4 ,the motions of the heavenly bodies are more regular2 and the eminent division of time into night and day1 made by the sun1 leads us to measure our time by the motion of that luminary3 nor do we in the affairs of life concern our selves with any inequality1 which there may be in that motion2 but the space of time which comprehends a day and night is rather supposed to be always the same4 ,however astronomers anciently found these spaces of time not to be always of the same length1 and have taught how to compute their differences4 ,now the time1 when so equated as to be rendered perfectly equal1 is the true measure of duration1 the other not4 ,and therefore this latter1 which is absolutely true time1 differs from the other1 which is only apparent4 ,and as we ordinarily make no distinction between apparent time1 as measured by the sun1 and the true2 so we often do not distinguish in our usual discourse between the real1 and the apparent or relative motion of bodies2 but use the same words for one1 as we should for the other4 ,though all things about us are really in motion with the earth2 as this motion is not visible1 we speak of the motion of every thing we see1 as if our selves and the earth stood still4 ,and even in other cases1 where we discern the motion of bodies1 we often speak of them not in relation to the whole motion we see1 but with regard to other bodies1 to which they are contiguous4 ,if any body were lying on a table2 when that table shall be carried along1 we say the body rests upon the table1 or perhaps absolutely1 that the body is at rest4 ,however philosophers must not reject all distinction between true and apparent motions1 any more than astronomers do the distinction between true and vulgar time2 for there is as real a difference between them1 as will appear by the following consideration4 ,suppose all the bodies of the universe to have their courses stopped1 and reduced to perfect rest4 ,then suppose their present motions to be again restored2 this cannot be done without an actual impression made upon some of them at least4 ,if any of them be left untouched1 they will retain their former state1 that is1 still remain at rest2 but the other bodies1 which are wrought upon1 will have changed their former state of rest1 for the contrary state of motion4 ,let us now suppose the bodies left at rest to be annihilated1 this will make no alteration in the state of the moving bodies2 but the effect of the impression1 which was made upon them1 will still subsist4 ,this shews the motion they received to be an absolute thing1 and to have no necessary dependence upon the relation which the body said to be in motion has to any other body#gg4 #id4 ,,besides absolute and relative motion are distinguishable by their ,effects4 ,one effect of motion is1 that bodies1 when moved round any center or axis1 acquire a certain power1 by which they forcibly press themselves from that center or axis of motion4 ,as when a body is whirled about in a sling1 the body presses against the sling1 and is ready to fly out as soon as liberty is given it4 ,and this power is proportional to the true1 not relative motion of the body round such a center or axis4 ,of this ,sir ,,isaac ,,newton gives the following instance#gh4 ,if a pail or such like vessel near full of water be suspended by a string of sufficient length1 and be turned about till the string be hard twisted4 ,if then as soon as the vessel and water in it are become still and at rest1 the vessel be nimbly turned about the contrary way the string was twisted1 the vessel by the strings untwisting it self shall continue its motion a long time4 ,and when the vessel first begins to turn1 the water in it shall receive little or nothing of the motion of the vessel1 but by degrees shall receive a communication of motion1 till at last it shall move round as swiftly as the vessel it self4 ,now the definition of motion1 which ,,des ,,cartes has given us upon this principle of making all motion meerly relative1 is this3 that motion1 is a removal of any body from its vicinity to other bodies1 which were in immediate contact with it1 and are considered as at rest#gi4 ,and if this be compared with what he soon after says1 that there is nothing real or positive in the body moved1 for the sake of which we ascribe motion to it1 which is not to be found as well in the contiguous bodies1 which are considered as at rest#hj2 it will follow from thence1 that we may consider the vessel as at rest and the water as moving in it3 and the water in respect of the vessel has the greatest motion1 when the vessel first begins to turn1 and loses this relative motion more and more1 till at length it quite ceases4 ,but now1 when the vessel first begins to turn1 the surface of the water remains smooth and flat1 as before the vessel began to move2 but as the motion of the vessel communicates by degrees motion to the water1 the surface of the water will be observed to change1 the water subsiding in the middle and rising at the edges3 which elevation of the water is caused by the parts of it pressing from the axis1 they move about2 and therefore this force of receding from the axis of motion depends not upon the relative motion of the water within the vessel1 but on its absolute motion2 for it is least1 when that relative motion is greatest1 and greatest1 when that relative motion is least1 or none at all4 #ie4 ,,thus the true cause of what appears in the surface of this water cannot be assigned1 without considering the water's motion within the vessel4 ,so also in the system of the world1 in order to find out the cause of the planetary motions1 we must know more of the real motions1 which belong to each planet1 than is absolutely necessary for the uses of astronomy4 ,if the astronomer should suppose the earth to stand still1 he could ascribe such motions to the celestial bodies1 as should answer all the appearances2 though he would not account for them in so simple a manner1 as by attributing motion to the earth4 ,but the motion of the earth must of necessity be considered1 before the real causes1 which actuate the planetary system1 can be discovered4 ,,chap4 ,,iii4 ,of ,,centripetal ,,forces4 ,,we have just been describing in the preceding chapter the effects produced on a body in motion1 from its being continually acted upon by a power always equal in strength1 and operating in parallel directions#ha4 ,but bodies may be acted upon by powers1 which in different places shall have different degrees of force1 and whose several directions shall be variously inclined to each other4 ,the most simple of these in respect to direction is1 when the power is pointed constantly to one center4 ,this is truly the case of that power1 whose effects we described in the foregoing chapter2 though the center of that power is so far removed1 that the subject then before us is most conveniently to be considered in the light1 wherein we have placed it3 ,but ,sir ,,isaac ,,newton has considered very particularly this other case of powers1 which are constantly directed to the same center4 ,it is upon this foundation1 that all his discoveries in the system of the world are raised4 ,and therefore1 as this subject bears so very great a share in the philosophy1 of which ,i am discoursing1 ,i think it proper in this place to take a short view of some of the general effects of these powers1 before we come to apply them particularly to the system of the world4 #b4 ,,these powers or forces are by ,sir ,,isaac ,,newton called centripetal2 and their first effect is to cause the body1 on which they act1 to quit the straight course1 wherein it would proceed if undisturbed1 and to describe an incurvated line1 which shall always be bent towards the center of the force4 ,it is not necessary1 that such a power should cause the body to approach that center4 ,the body may continue to recede from the center of the power1 notwithstanding its being drawn by the power2 but this property must always belong to its motion1 that the line1 in which it moves1 will continually be concave towards the center1 to which the power is directed4 ,suppose ,a 7in fig4 #gb47 to be the center of a force4 ,let a body in ,b be moving in the direction of the straight line ,b ,c1 in which line it would continue to move1 if undisturbed2 but being attracted by the centripetal force towards ,a1 the body must necessarily depart from this line ,b ,c1 and being drawn into the curve line ,b ,d1 must pass between the lines ,a ,b and ,b ,c4 ,it is evident therefore1 that the body in ,b being gradually turned off from the straight line ,b ,c1 it will at first be convex toward the line ,b ,c1 and consequently concave towards the point ,a3 for these centripetal powers are supposed to be in strength proportional to the power of gravity1 and1 like that1 not to be able after the manner of an impulse to turn the body sensibly out of its course into a different one in an instant1 but to take up some space of time in producing a visible effect4 ,that the curve will always continue to have its concavity towards ,a may thus appear4 ,in the line ,b ,c near to ,b take any point as ,e1 from which the line ,e ,f ,g may be so drawn1 as to touch the curve line ,b ,d in some point as ,f4 ,now when the body is come to ,f1 if the centripetal power were immediately to be suspended1 the body would no longer continue to move in a curve line1 but being left to it self would forthwith reassume a straight course2 and that straight course would be in the line ,f ,g3 for that line is in the direction of the body's motion at the point ,f4 ,but the centripetal force continuing its energy1 the body will be gradually drawn from this line ,f ,g so as to keep in the line ,f ,d1 and make that line near the point ,f to be convex toward ,f ,g1 and concave toward ,a4 ,after the same manner the body may be followed on in its course through the line ,b ,d1 and every part of that line be shewn to be concave toward the point ,a4 #c4 ,,this then is the constant character belonging to those motions1 which are carried on by centripetal forces2 that the line1 wherein the body moves1 is throughout concave towards the center of the force4 ,in respect to the successive distances of the body from the center there is no general rule to be laid down2 for the distance of the body from the center may either increase1 or decrease1 or even keep always the same4 ,the point ,a 7in fig4 #gc47 being the center of a centripetal force1 let a body at ,b set out in the direction of the straight line ,b ,c perpendicular to the line ,a ,b drawn from ,a to ,b4 ,it will be easily conceived1 that there is no other point in the line ,b ,c so near to ,a1 as the point ,b2 that ,a ,b is the shortest of all the lines1 which can be drawn from ,a to any part of the line ,b ,c2 all other lines1 as ,a ,d1 or ,a ,e1 drawn from ,a to the line ,b ,c being longer than ,a ,b4 ,hence it follows1 that the body setting out from ,b1 if it moved in the line ,b ,c1 it would recede more and more from the point ,a4 ,now as the operation of a centripetal force is to draw a body towards the center of the force3 if such a force act upon a resting body1 it must necessarily put that body so into motion1 as to cause it to move towards the center of the force3 if the body were of it self moving towards that center1 the centripetal force would accelerate that motion1 and cause it to move faster down3 but if the body were in such a motion1 as being left to itself it would recede from this center1 it is not necessary1 that the action of a centripetal power upon it should immediately compel the body to approach the center1 from which it would otherwise have receded2 the centripetal power is not without effect1 if it cause the body to recede more slowly from that center1 than otherwise it would have done4 ,thus in the case before us1 the smallest centripetal power1 if it act on the body1 will force it out of the line ,b ,c1 and cause it to pass in a bent line between ,b ,c and the point ,a1 as has been before explained4 ,when the body1 for instance1 has advanced to the line ,a ,d1 the effect of the centripetal force discovers it self by having removed the body out of the line ,b ,c1 and brought it to cross the line ,a ,d somewhere between ,a and ,d3 suppose at ,f4 ,now ,a ,d being longer than ,a ,b1 ,a ,f may also be longer than ,a ,b4 ,the centripetal power may indeed be so strong1 that ,a ,f shall be shorter than ,a ,b2 or it may be so evenly balanced with the progressive motion of the body1 that ,a ,f and ,a ,b shall be just equal3 and in this last case1 when the centripetal force is of that strength1 as constantly to draw the body as much toward the center1 as the progressive motion would carry it off1 the body will describe a circle about the center ,a1 this center of the force being also the center of the circle4 #d4 ,,if the body1 instead of setting out in the line ,b ,c perpendicular to ,a ,b1 had set out in another line ,b ,g more inclined towards the line ,a ,b1 moving in the curve line ,b ,h2 then as the body1 if it were to continue its motion in the line ,b ,g1 would for some time approach the center ,a2 the centripetal force would cause it to make greater advances toward that center4 ,but if the body were to set out in the line ,b ,i reclined the other way from the perpendicular ,b ,c1 and were to be drawn by the centripetal force into the curve line ,b ,k2 the body1 notwithstanding any centripetal force1 would for some time recede from the center2 since some part at least of the curve line ,b ,k lies between the line ,b ,i and the perpendicular ,b ,c4 #e4 ,,thus far we have explained such effects1 as attend every centripetal force4 ,but as these forces may be very different in regard to the different degrees of strength1 wherewith they act upon bodies in different places2 ,i shall now proceed to make mention in general of some of the differences attending these centripetal motions4 #f4 ,,to reassume the consideration of the last mentioned case4 ,suppose a centripetal power directed toward the point ,a 7in fig4 #gd47 to act on a body in ,b1 which is moving in the direction of the straight line ,b ,c1 the line ,b ,c reclining off from ,a ,b4 ,if from ,a the straight lines ,a ,d1 ,a ,e1 ,a ,f are drawn at pleasure to the line ,c ,b2 the line ,c ,b being prolonged beyond ,b to ,g1 it appears that ,a ,d is inclined to the line ,g ,c more obliquely1 than ,a ,b is inclined to it1 ,a ,e is inclined more obliquely than ,a ,d1 and ,a ,f more than ,a ,e4 ,to speak more correctly1 the angle under ,a ,d ,g is less than that under ,a ,b ,g1 the angle under ,a ,e ,g less than that under ,a ,d ,g1 and the angle under ,a ,f ,g less than that under ,a ,e ,g4 ,now suppose the body to move in the curve line ,b ,h ,i ,k4 ,then it is here likewise evident1 that the line ,b ,h ,i ,k being concave towards ,a1 and convex towards the line ,b ,c1 it is more and more turned off from the line ,b ,c2 so that in the point ,h the line ,a ,h will be less obliquely inclined to the curve line ,b ,h ,i ,k1 than the same line ,a ,h ,d is inclined to ,b ,c at the point ,d2 at the point ,i the inclination of the line ,a ,i to the curve line will be more different from the inclination of the same line ,a ,i ,e to the line ,b ,c1 at the point ,e2 and in the points ,k and ,f the difference of inclination will be still greater2 and in both the inclination at the curve will be less oblique1 than at the straight line ,b ,c4 ,but the straight line ,a ,b is less obliquely inclined to ,b ,g1 than ,a ,d is inclined towards ,d ,g3 therefore although the line ,a ,h be less obliquely inclined towards the curve ,h ,b1 than the same line ,a ,h ,d is inclined towards ,d ,g2 yet it is possible1 that the inclination at ,h may be more oblique1 than the inclination at ,b4 ,the inclination at ,h may indeed be less oblique than the other1 or they may be both the same4 ,this depends upon the degree of strength1 wherewith the centripetal force exerts it self1 during the passage of the body from ,b to ,h4 ,after the same manner the inclinations at ,i and ,k depend entirely on the degree of strength1 wherewith the centripetal force acts on the body in its passage from ,h to ,k3 if the centripetal force be weak enough1 the lines ,a ,h and ,a ,i drawn from the center ,a to the body at ,h and at ,i shall be more obliquely inclined to the curve1 than the line ,a ,b is inclined towards ,b ,g4 ,the centripetal force may be of that strength as to render all these inclinations equal1 or if stronger1 the inclinations at ,i and ,k will be less oblique than at ,b4 ,sir ,,isaac ,,newton has particularly shewn1 that if the centripetal power decreases after a certain manner with the increase of distance1 a body may describe such a curve line1 that all the lines drawn from the center to the body shall be equally inclined to that curve line4#hb ,but ,i do not here enter into any particulars1 my present intention being only to shew1 that it is possible for a body to be acted upon by a force continually drawing it down towards a center1 and yet that the body shall continue to recede from that center2 for here as long as the lines ,a ,h1 ,a ,i1 c drawn from the center ,a to the body do not become less oblique to the curve1 in which the body moves2 so long shall those lines perpetually increase1 and consequently the body shall more and more recede from the center4 #g4 ,,but we may observe farther1 that if the centripetal power1 while the body increases its distance from the center1 retain sufficient strength to make the lines drawn from the center to the body to become at length less oblique to the curve2 then if this diminution of the obliquity continue1 till at last the line drawn from the center to the body shall cease to be obliquely inclined to the curve1 and shall become perpendicular thereto2 from this instant the body shall no longer recede from the center1 but in its following motion it shall again descend1 and shall describe a curve line in all respects like to that1 which it has described already2 provided the centripetal power1 every where at the same distance from the center1 acts with the same strength4 ,so we observed in the preceding chapter1 that1 when the motion of a projectile became parallel to the horizon1 the projectile no longer ascended1 but forthwith directed its course downwards1 descending in a line altogether like that1 wherein it had before ascended#hc4 #h4 ,,this return of the body may be proved by the following proposition3 that if the body in any place1 suppose at ,i1 were to be stopt1 and be thrown directly backward with the velocity1 wherewith it was moving forward in that point ,i2 then the body1 by the action of the centripetal force upon it1 would move back again over the path ,i ,h ,b1 in which it had before advanced forward1 and would arrive again at the point ,b in the same space of time1 as was taken up in its passage from ,b to ,i2 the velocity of the body at its return to the point ,b being the same1 as that wherewith it first set out from that point4 ,to give a full demonstration of this proposition1 would require that use of mathematics1 which ,i here purpose to avoid2 but1 ,i believe1 it will appear in great measure evident from the following considerations4 #i4 ,,suppose 7in fig4 #ge47 that a body were carried after the following manner through the bent figure ,a ,b ,c ,d ,e ,f1 composed of the straight lines ,a ,b1 ,b ,c1 ,c ,d1 ,d ,e1 ,e ,f4 ,first let it be moving in the line ,a ,b1 from ,a towards ,b1 with any uniform velocity4 ,at ,b let the body receive an impulse directed toward some point1 as ,g1 taken within the concavity of the figure4 ,now whereas this body1 when once moving in the straight line ,a ,b1 will continue to move on in this line1 so long as it shall be left to it self2 but being disturbed at the point ,b in its motion by the impulse1 which there acts upon it1 it will be turned out of this line ,a ,b into some other straight line1 wherein it will afterwards continue to move1 as long as it shall be left to itself4 ,therefore let this impulse have strength sufficient to turn the body into the line ,b ,c4 ,then let the body move on undisturbed from ,b to ,c1 but at ,c let it receive another impulse pointed toward the same point ,g1 and of sufficient strength to turn the body into the line ,c ,d4 ,at ,d let a third impulse1 directed like the rest to the point ,g1 turn the body into the line ,d ,e4 ,and at ,e let another impulse1 directed likewise to the point ,g1 turn the body into the line ,e ,f4 ,now1 ,i say1 if the body while moving in the line ,e ,f be stopt1 and turned back again in this line with the same velocity1 as that wherewith it was moving forward in this line2 then by the repetition of the former impulse at ,e the body will be turned into the line ,e ,d1 and move in it from ,e to ,d with the same velocity as before it moved with from ,d to ,e2 by the repetition of the impulse at ,d1 when the body shall have returned to that point1 it will be turned into the line ,d ,c2 and by the repetition of the other impulses at ,c and ,b the body will be brought back again into the line ,b ,a1 with the velocity1 wherewith it first moved in that line4 #aj4 ,,this ,i prove as follows4 ,let ,d ,e and ,f ,e be continued beyond ,e4 ,in ,d ,e thus continued take at pleasure the length ,e ,h1 and let ,h ,i be so drawn1 as to be equidistant from the line ,g ,e4 ,then1 by what has been written upon the second law of motion#hd1 it follows1 that after the impulse on the body in ,e it will move through ,e ,i in the same time1 as it would have imployed in moving from ,e to ,h1 with the velocity which it had in the line ,d ,e4 ,in ,f ,e prolonged take ,e ,k equal to ,e ,i1 and draw ,k ,l equidistant from ,g ,e4 ,then1 because the body is thrown back in the line ,f ,e with the same velocity as that wherewith it went forward in that line2 if1 when the body was returned to ,e1 it were permitted to go straight on1 it would pass through ,e ,k in the same time1 as it took up in passing through ,e ,i1 when it went forward in the line ,e ,f4 ,but1 if at the body's return to the point ,e1 such an impulse directed toward the point ,d were to be given it1 whereby it should be turned into the line ,d ,e2 ,i say1 that the impulse necessary to produce this effect must be equal to that1 which turned the body out of the line ,d ,e into ,e ,f2 and that the velocity1 with which the body will return into the line ,e ,d1 is the same1 as that wherewith it before moved through this line from ,d to ,e4 ,because ,e ,k is equal to ,e ,i1 and ,k ,l and ,h ,i1 being each equidistant from ,g ,e1 are by consequence equidistant from each other2 it follows1 that the two triangular figures ,i ,e ,h and ,k ,e ,l are altogether like and equal to each other4 ,if ,i were writing to mathematicians1 ,i might refer them to some proportions in the elements of ,,euclid for the proof of this#he but as ,i do not here address my self to such1 so ,i think this assertion will be evident enough without a proof in form2 at least ,i must desire my readers to receive it as a proposition true in geometry4 ,but these two triangular figures being altogether like each other and equal2 as ,e ,k is equal to ,e ,i1 so ,e ,l is equal to ,e ,h1 and ,k ,l equal to ,h ,i4 ,now the body after its return to ,e being turned out of the line ,f ,e into ,e ,d by an impulse acting upon it in ,e1 after the manner above expressed2 the body will receive such a velocity by this impulse1 as will carry it through ,e ,l in the same time1 as it would have imployed in passing through ,e ,k1 if it had gone on in that line undisturbed4 ,and it has already been observed1 that the time1 in which the body would pass over ,e ,k with the velocity wherewith it returns1 is equal to the time it took up in going forward from ,e to ,i2 that is1 equal to the time1 in which it would have gone through ,e ,h with the velocity1 wherewith it moved from ,d to ,e4 ,therefore the time1 in which the body will pass through ,e ,l after its return into the line ,e ,d1 is the same1 as would have been taken up by the body in passing through ,e ,h with the velocity1 wherewith the body first moved in the line ,d ,e4 ,since therefore ,e ,l and ,e ,h are equal1 the body returns into the line ,d ,e with the velocity1 which it had before in that line4 ,again ,i say1 the second impulse in ,e is equal to the first4 ,by what has been said on the second law of motion concerning the effect of oblique impulses#hf1 it will be understood1 that the impulse in ,e1 whereby the body was turned out of the line ,d ,e into the line ,e ,f1 is of such strength1 that if the body had been at rest1 when this impulse had acted upon it1 this impulse would have communicated so much motion to the body1 as would have carried it through a length equal to ,h ,i1 in the time wherein the body would have passed from ,e to ,h1 or in the time wherein it passed from ,e to ,i4 ,in the same manner1 on the return of the body1 the impulse in ,e1 whereby the body is turned out of the line ,f ,e into ,e ,d1 is of such strength1 that if it had acted on the body at rest1 it would have caused the body to move through a length equal to ,k ,l1 in the same time1 as the body would imploy in passing through ,e ,k with the velocity1 wherewith it returns in the line ,f ,e4 ,therefore the second impulse1 had it acted on the body at rest1 would have caused it to move through a length equal to ,k ,l in the same space of time1 as would be taken up by the body in passing through a length equal to ,h ,i1 were the first impulse to act on the body when at rest4 ,that is1 the effects of the first and second impulse on the body when at rest would be the same2 for ,k ,l and ,h ,i are equal3 consequently the second impulse is equal to the first4 #aa4 ,,thus if the body be returned through ,f ,e with the velocity1 wherewith it moved forward2 we have shewn how by the repetition of the impulse1 which acted on it at ,e1 the body will return again into the line ,d ,e with the velocity1 which it had before in that line4 ,by the same process of reasoning it may be proved1 that1 when the body is returned back to ,d1 the impulse1 which before acted on the body at that point1 will throw the body into the line ,d ,c with the velocity1 which it first had in that line2 and the other impulses being successively repeated1 the body will at length be brought back again into the line ,b ,a with the velocity1 wherewith it set out in that line4 #ab4 ,,thus these impulses1 by acting over again in an inverted order all their operation on the body1 bring it back again through the path1 in which it had proceeded forward4 ,and this obtains equally1 whatever be the number of the straight lines1 whereof this curve figure is composed4 ,now by a method of reasoning1 which ,sir ,,isaac ,,newton makes great use of1 and which he introduced into geometry1 thereby greatly inriching that science#hg2 we might make a transition from this figure composed of a number of straight lines to a figure of one continued curvature1 and from a number of separate impulses repeated at distinct intervals to a continual centripetal force1 and shew1 that1 because what has been here advanced holds universally true1 whatever be the number of straight lines1 whereof the curve figure ,a ,c ,f is composed1 and howsoever frequently the impulses at the angles of this figure are repeated2 therefore the same will still remain true1 although this figure should be converted into one of a continued curvature1 and these distinct impulses should be changed into a continual centripetal force4 ,but as the explaining this method of reasoning is foreign to my present design2 so ,i hope my readers1 after what has been said1 will find no difficulty in receiving the proposition laid down above3 that1 if the body1 which has moved through the curve line ,b ,h ,i 7in fig4 #gd47 from ,b to ,i1 when it is come to ,i1 be thrown directly back with the same velocity as that1 wherewith it proceeded forward1 the centripetal force1 by acting over again all its operation on the body1 shall bring the body back again in the line ,i ,h ,b3 and as the motion of the body in its course from ,b to ,i was every where in such a manner oblique to the line drawn from the center to the body1 that the centripetal power acted in some degree against the body's motion1 and gradually diminished it2 so in the return of the body1 the centripetal power will every where draw the body forward1 and accelerate its motion by the same degrees1 as before it retarded it4 #ac4 ,,this being agreed1 suppose the body in ,k to have the line ,a ,k no longer obliquely inclined to its motion4 ,in this case1 if the body be turned back1 in the manner we have been considering1 it must be directed back perpendicularly to ,a ,k4 ,but if it had proceeded forward1 it would likewise have moved in a direction perpendicular to ,a ,k2 consequently1 whether it move from this point ,k backward or forward1 it must describe the same kind of course4 ,therefore since by being turned back it will go over again the line ,k ,i ,h ,b2 if it be permitted to go forward1 the line ,k ,l1 which it shall describe1 will be altogether similar to the line ,k ,h ,b4 #ad4 ,,in like manner we may determine the nature of the motion1 if the line1 wherein the body sets out1 be inclined 7as in fig4 #gf47 down toward the line ,b ,a drawn between the body and the center4 ,if the centripetal power so much increases in strength1 as the body approaches1 that it can bend the path1 in which the body moves1 to that degree1 as to cause all the lines as ,a ,h1 ,a ,i1 ,a ,k to remain no less oblique to the motion of the body1 than ,a ,b is oblique to ,b ,c2 the body shall continually more and more approach the center4 ,but if the centripetal power increases in so much less a degree1 as to permit the line drawn from the center to the body1 as it accompanies the body in its motion1 at length to become more and more erect to the curve wherein the body moves1 and in the end1 suppose at ,k1 to become perpendicular thereto2 from that time the body shall rise again4 ,this is evident from what has been said above2 because for the very same reason here also the body shall proceed from the point ,k to describe a line altogether similar to the line1 in which it has moved from ,b to ,k4 ,thus1 as it was observed of the pendulum in the preceding chapter#hh1 that all the time it approaches towards being perpendicular to the horizon1 it more and more descends2 but1 as soon as it is come into that perpendicular situation1 it immediately rises again by the same degrees1 as it descended by before3 so here the body more and more approaches the center all the time it is moving from ,b to ,k2 but thence forward it rises from the center again by the same degrees1 as it approached by before4 #ae4 ,,if 7in fig4 #gg47 the line ,b ,c be perpendicular to ,a ,b2 then it has been observed above#hi1 that the centripetal power may be so balanced with the progressive motion of the body1 that the body may keep moving round the center ,a constantly at the same distance2 as a body does1 when whirled about any point1 to which it is tyed by a string4 ,if the centripetal power be too weak to produce this effect1 the motion of the body will presently become oblique to the line drawn from itself to the center1 after the manner of the first of the two cases1 which we have been considering4 ,if the centripetal power be stronger1 than what is required to carry the body in a circle1 the motion of the body will presently fall in with the second of the cases1 we have been considering4 #af4 ,,if the centripetal power so change with the change of distance1 that the body1 after its motion has become oblique to the line drawn from itself to the center1 shall again become perpendicular thereto2 which we have shewn to be possible in both the cases treated of above2 then the body shall in its subsequent motion return again to the distance of ,a ,b1 and from that distance take a course similar to the former3 and thus1 if the body move in a space free from all resistance1 which has been here all along supposed2 it shall continue in a perpetual motion about the center1 descending and ascending alternately therefrom4 ,if the body setting out from ,b 7in fig4 #gh47 in the line ,b ,c perpendicular to ,a ,b1 describe the line ,b ,d ,e1 which in ,d shall be oblique to the line ,a ,d1 but in ,e shall again become erect to ,a ,e drawn from the body in ,e to the center ,a2 then from this point ,e the body shall describe the line ,e ,f ,g altogether like to the line ,b ,d ,e1 and at ,g shall be at the same distance from ,a1 as it was at ,b4 ,but likewise the line ,a ,g shall be erect to the body's motion4 ,therefore the body shall proceed to describe from ,g the line ,g ,h ,i altogether similar to the line ,g ,f ,e1 and at ,i have the same distance from the center1 as it had at ,e2 and also have the line ,a ,i erect to its motion3 so that its following motion must be in the line ,i ,k ,l similar to ,i ,h ,g1 and the distance ,a ,l equal to ,a ,g4 ,thus the body will go on in a perpetual round without ceasing1 alternately inlarging and contracting its distance from the center4 ,illustration #ag4 ,,if it so happen1 that the point ,e fall upon the line ,b ,a continued beyond ,a2 then the point ,g will fall on ,b1 ,i on ,e1 and ,l also on ,b2 so that the body will describe in this case a simple curve line round the center ,a1 like the line ,b ,d ,e ,f in fig4 #gi1 in which it will continually revolve from ,b to ,e and from ,e to ,b without end4 #ah4 ,,if ,a ,e in fig4 #gh should happen to be perpendicular to ,a ,b1 in this case also a simple line will be described2 for the point ,g will fall on the line ,b ,a prolonged beyond ,a1 the point ,i on the line ,a ,e prolonged beyond ,a1 and the point ,l on ,b3 so that the body will describe a line like the curve line ,b ,e ,g ,i in fig4 #hj1 in which the opposite points ,b and ,g are equally distant from ,a1 and the opposite points ,e and ,i are also equally distant from the same point ,a4 #ai4 ,,in other cases the line described will have a more complex figure4 #bj4 ,,thus we have endeavoured to shew how a body1 while it is constantly attracted towards a center1 may notwithstanding by its progressive motion keep it self from falling down to that center2 but describe about it an endless circuit1 sometimes approaching toward that center1 and at other times as much receding from the same4 #ba4 ,,but here we have supposed1 that the centripetal power is of equal strength every where at the same distance from the center4 ,and this is the case of that centripetal power1 which will hereafter be shewn to be the cause1 that keeps the planets in their courses4 ,but a body may be kept on in a perpetual circuit round a center1 although the centripetal power have not this property4 ,indeed a body may by a centripetal force be kept moving in any curve line whatever1 that shall have its concavity turned every where towards the center of the force4 #bb4 ,,to make this evident ,i shall first propose the case of a body moving through the incurvated figure ,a ,b ,c ,d ,e 7in fig4 #ha47 which is composed of the straight lines ,a ,b1 ,b ,c1 ,c ,d1 ,d ,e1 and ,e ,a2 the motion being carried on in the following manner4 ,let the body first move in the line ,a ,b with any uniform velocity4 ,when it is arrived at the point ,b1 let it receive an impulse directed toward any point ,f taken within the figure2 and let the impulse be of that strength as to turn the body out of the line ,a ,b into the line ,b ,c4 ,the body after this impulse1 while left to itself1 will continue moving in the line ,b ,c4 ,at ,c let the body receive another impulse directed towards the same point ,f1 of such strength1 as to turn the body from the line ,b ,c into the line ,c ,d4 ,at ,d let the body by another impulse1 directed likewise to the point ,f1 be turned out of the line ,c ,d into ,d ,e4 ,and at ,e let another impulse1 directed toward the point ,f1 turn the body from the line ,d ,e into ,e ,a4 ,thus we see how a body may be carried through the figure ,a ,b ,c ,d ,e by certain impulses directed always toward the same center1 only by their acting on the body at proper intervals1 and with due degrees of strength4 #bc4 ,,but farther1 when the body is come to the point ,a1 if it there receive another impulse directed like the rest toward the point ,f1 and of such a degree of strength as to turn the body into the line ,a ,b1 wherein it first moved2 ,i say that the body shall return into this line with the same velocity1 as it had at first4 #bd4 ,,let ,a ,b be prolonged beyond ,b at pleasure1 suppose to ,g2 and from ,g let ,g ,h be drawn1 which if produced should always continue equidistant from ,b ,f1 or1 according to the more usual phrase1 let ,g ,h be drawn parallel to ,b ,f4 ,then it appears1 from what has been said upon the second law of motion#ij1 that in the time1 wherein the body would have moved from ,b to ,g1 had it not received a new impulse in ,b1 by the means of that impulse it will have acquired a velocity1 which will carry it from ,b to ,h4 ,after the same manner1 if ,c ,i be taken equal to ,b ,h1 and ,i ,k be drawn equidistant from or parallel to ,c ,f2 the body will have moved from ,c to ,k with the velocity1 which it has in the line ,c ,d1 in the same time1 as it would have employed in moving from ,c to ,i with the velocity1 it had in the line ,b ,c4 ,therefore since ,c ,i and ,b ,h are equal1 the body will move through ,c ,k in the same time1 as it would have taken up in moving from ,b to ,g with the original velocity1 wherewith it moved through the line ,a ,b4 ,again1 ,d ,l being taken equal to ,c ,k and ,l ,m drawn parallel to ,d ,f2 for the same reason as before the body will move through ,d ,m with the velocity1 which it has in the line ,d ,e1 in the same time1 as it would imploy in moving through ,b ,g with its original velocity4 ,in the last place1 if ,e ,n be taken equal to ,d ,m1 and ,n ,o be drawn parallel to ,e ,f2 likewise if ,a ,p be taken equal to ,e ,o1 and ,p ,q be drawn parallel to ,a ,f3 then the body with the velocity1 wherewith it returns into the line ,a ,b1 will pass through ,a ,q in the same time1 as it would have imployed in passing through ,b ,g with its original velocity4 ,now as all this follows directly from what has above been delivered1 concerning the effect of oblique impulses impressed upon bodies in motion2 so we must here observe farther1 that it can be proved by geometry1 that ,a ,q will always be equal to ,e ,g4 ,the proof of this ,i am obliged1 from the nature of my present design1 to omit2 but this geometrical proportion being granted1 it follows1 that the body has returned into the line ,a ,b with the velocity1 which it had1 when it first moved in that line2 for the velocity1 with which it returns into the line ,a ,b1 will carry it over the line ,a ,q in the same time1 as would have been taken up in its passing over an equal line ,b ,g with the original velocity4 #be4 ,,thus we have found1 how a body may be carried round the figure ,a ,b ,c ,d ,e by the action of certain impulses upon it which should all be pointed toward one center4 ,and we likewise see1 that when the body is brought back again to the point1 whence it first set out2 if it there meet with an impulse sufficient to turn it again into the line1 wherein it moved at first1 its original velocity will be again restored2 and by the repetition of the same impulses1 the body will be carried again in the same round4 ,therefore if these impulses1 which act on the body at the points ,b1 ,c1 ,d1 ,e1 and ,a1 continue always the same1 the body will make round this figure innumerable revolutions4 #bf4 ,,the proof1 which we have here made use of1 holds the same in any number of straight lines1 whereof the figure ,a ,b ,d should be composed2 and therefore by the method of reasoning referred to above#ia we are to conclude1 that what has here been said upon this rectilinear figure1 will remain true1 if this figure were changed into one of a continued curvature1 and instead of distinct impulses acting by intervals at the angles of this figure1 we had a continual centripetal force4 ,we have therefore shewn1 that a body may be carried round in any curve figure ,a ,b ,c 7 fig4 #hb47 which shall every where be concave towards any one point as ,d1 by the continual action of a centripetal power directed to that point1 and when it is returned to the point1 from whence it set out1 it shall recover again the velocity1 with which it departed from that point4 ,it is not indeed always necessary1 that it should return again into its first course2 for the curve line may have some such figure as the line ,a ,b ,c ,d ,b ,e in fig4 #hc4 ,in this curve line1 if the body set out from ,b in the direction ,b ,f1 and moved through the line ,b ,c ,d1 till it returned to ,b2 here the body would not enter again into the line ,b ,c ,d1 because the two parts ,b ,d and ,b ,c of the curve line make an angle at the point ,b3 so that the centripetal power1 which at the point ,b could turn the body from the line ,b ,f into the curve1 will not be able to turn the body into the line ,b ,c from the direction1 in which it returns to the point ,b2 a forceable impulse must be given the body in the point ,b to produce that effect4 #bg4 ,,if at the point ,b1 whence the body sets out1 the curve line return into it self 7as in fig4 #hb27 then the body1 upon its arrival again at ,b1 may return into its former course1 and thus make an endless circuit about the center of the centripetal power4 #bh4 ,,what has here been said1 ,i hope1 will in some measure enable my readers to form a just idea of the nature of these centripetal motions4 #bi4 ,i ,,have not attempted to shew1 how to find particularly1 what kind of centripetal force is necessary to carry a body in any curve line proposed4 ,this is to be deduced from the degree of curvature1 which the figure has in each point of it1 and requires a long and complex mathematical reasoning4 ,however ,i shall speak a little to the first proportion1 which ,sir ,,isaac ,,newton lays down for this purpose4 ,by this proposition1 when a body is found moving in a curve line1 it may be known1 whether the body be kept in its course by a power always pointed toward the same center2 and if it be so1 where that center is placed4 ,the proposition is this3 that if a line be drawn from some fixed point to the body1 and remaining by one extream united to that point1 it be carried round along with the body2 then1 if the power1 whereby the body is kept in its course1 be always pointed to this fixed point as a center1 this line will move over equal spaces in equal portions of time4 ,suppose a body were moving through the curve line ,a ,b ,c ,d 7in fig4 #hd47 and passed over the arches ,a ,b1 ,b ,c1 ,c ,d in equal portions of time2 then if a point1 as ,e1 can be found1 from whence the line ,e ,a being drawn to the body in ,a1 and accompanying the body in its motion1 it shall make the spaces ,e ,a ,b1 ,e ,b ,c1 and ,e ,c ,d equal1 over which it passes1 while the body describes the arches ,a ,b1 ,b ,c1 and ,c ,d3 and if this hold the same in all other arches1 both great and small1 of the curve line ,a ,b ,c ,d1 that these spaces are always equal1 where the times are equal2 then is the body kept in this line by a power always pointed to ,e as a center4 #cj4 ,,the principle1 upon which ,sir ,,isaac ,,newton has demonstrated this1 requires but small skill in geometry to comprehend4 ,i shall therefore take the liberty to close the present chapter with an explication of it2 because such an example will give the clearest notion of our author's method of applying mathematical reasoning to these philosophical subjects4 #ca4 ,,he reasons thus4 ,suppose a body set out from the point ,a 7in fig4 #he47 to move in the straight line ,a ,b2 and after it had moved for some time in that line1 it were to receive an impulse directed to some point as ,c4 ,let it receive that impulse at ,d2 and thereby be turned into the line ,d ,e2 and let the body after this impulse take the same length of time in passing from ,d to ,e1 as it imployed in the passing from ,a to ,d4 ,then the straight lines ,c ,a1 ,c ,d1 and ,c ,e being drawn1 ,sir ,,isaac ,,newton proves1 that the and triangular spaces ,c ,a ,d and ,c ,d ,e are equal4 ,this he does in the following manner4 #cb4 ,,let ,e ,f be drawn parallel to ,c ,d4 ,then1 from what has been said upon the second law of motion#ib1 it is evident1 that since the body was moving in the line ,a ,b1 when it received the impulse in the direction ,d ,c2 it will have moved after that impulse through the line ,d ,e in the same time1 as it would have taken up in moving through ,d ,f1 provided it had received no disturbance in ,d4 ,but the time of the body's moving from ,d to ,e is supposed to be equal to the time of its moving through ,a ,d2 therefore the time1 which the body would have imployed in moving through ,d ,f1 had it not been disturbed in ,d1 is equal to the time1 wherein it moved through ,a ,d3 consequently ,d ,f is equal in length to ,a ,d2 for if the body had gone on to move through the line ,a ,b without interruption1 it would have moved through all parts thereof with the same velocity1 and have passed over equal parts of that line in equal portions of time4 ,now ,c ,f being drawn1 since ,a ,d and ,d ,f are equal1 the triangular space ,c ,d ,f is equal to the triangular space ,c ,a ,d4 ,farther1 the line ,e ,f being parallel to ,c ,d1 it is proved by ,,euclid1 that the triangle ,c ,e ,d is equal to the triangle ,c ,f ,d#ic3 therefore the triangle ,c ,e ,d is equal to the triangle ,c ,a ,d4 #cc4 ,,after the same manner1 if the body receive at ,e another impulse directed toward the point ,c1 and be turned by that impulse into the line ,e ,g2 if it move afterwards from ,e to ,g in the same space of time1 as was taken up by its motion from ,d to ,e1 or from ,a to ,d2 then ,c ,g being drawn1 the triangle ,c ,e ,g is equal to ,c ,d ,e4 ,a third impulse at ,g directed as the two former to ,c1 whereby the body shall be turned into the line ,g ,h1 will have also the like effect with the rest4 ,if the body move over ,g ,h in the same time1 as it took up in moving over ,e ,g1 the triangle ,c ,g ,h will be equal to the triangle ,c ,e ,g4 ,lastly1 if the body at ,h be turned by a fresh impulse directed toward ,c into the line ,h ,i1 and at ,i by another impulse directed also to ,c be turned into the line ,i ,k2 and if the body move over each of the lines ,h ,i1 and ,i ,k in the same time1 as it imployed in moving over each of the preceding lines ,a ,d1 ,d ,e1 ,e ,g1 and ,g ,h3 then each of the triangles ,c ,h ,i1 and ,c ,i ,k will be equal to each of the preceding4 ,likewise as the time1 in which the body moves over ,a ,d ,e1 is equal to the time of its moving over ,e ,g ,h1 and to the time of its moving over ,h ,i ,k2 the space ,c ,a ,d ,e will be equal to the space ,c ,e ,g ,h1 and to the space ,c ,h ,i ,k4 ,in the same manner as the time1 in which the body moved over ,a ,d ,e ,g is equal to the time of its moving over ,g ,h ,i ,k1 so the space ,c ,a ,d ,e ,g will be equal to the space ,c ,g ,h ,i ,k4 #cd4 ,,from this principle ,sir ,,isaac ,,newton demonstrates the proposition mentioned above1 by that method of arguing introduced by him into geometry1 whereof we have before taken notice#id1 by making according to the principles of that method a transition from this incurvated figure composed of straight lines1 to a figure of continued curvature2 and by shewing1 that since equal spaces are described in equal times in this present figure composed of straight lines1 the same relation between the spaces described and the times of their description will also have place in a figure of one continued curvature4 ,he also deduces from this proposition the reverse of it2 and proves1 that whenever equal spaces are continually described2 the body is acted upon by a centripetal force directed to the center1 at which the spaces terminate4 ,,chap4 ,,iv4 ,of the ,,resistance of ,,fluids4 ,,before the cause can be discovered1 which keeps the planets in motion1 it is necessary first to know1 whether the space1 wherein they move1 is empty and void1 or filled with any quantity of matter4 ,it has been a prevailing opinion1 that all space contains in it matter of some kind or other2 so that where no sensible matter is found1 there was yet a subtle fluid substance by which the space was filled up2 even so as to make an absolute plenitude4 ,in order to examine this opinion1 ,sir ,,isaac ,,newton has largely considered the effects of fluids upon bodies moving in them4 #b4 ,,these effects he has reduced under these three heads4 ,in the first place he shews how to determine in what manner the resistance1 which bodies suffer1 when moving in a fluid1 gradually increases in proportion to the space1 they describe in any fluid2 to the velocity1 with which they describe it2 and to the time they have been in motion4 ,under the second head he considers what degree of resistance different bodies moving in the same fluid undergo1 according to the different proportion between the density of the fluid and the density of the body4 ,the densities of bodies1 whether fluid or solid1 are measured by the quantity of matter1 which is comprehended under the same magnitude2 that body being the most dense or compact1 which under the same bulk contains the greatest quantity of solid matter1 or which weighs most1 the weight of every body being observed above to be proportional to the quantity of matter in it#ie4 ,thus water is more dense than cork or wood1 iron more dense than water1 and gold than iron4 ,the third particular ,sir ,,is4 ,,newton considers concerning the resistance of fluids is the influence1 which the diversity of figure in the resisted body has upon its resistance4 #c4 ,,for the more perfect illustration of the first of these heads1 he distinctly shews the relation between all the particulars specified upon three different suppositions4 ,the first is1 that the same body be resisted more or less in the simple proportion to its velocity2 so that if its velocity be doubled1 its resistance shall become threefold4 ,the second is of the resistance increasing in the duplicate proportion of the velocity2 so that1 if the velocity of a body be doubled1 its resistance shall be rendered four times2 and if the velocity be trebled1 nine times as great as at first4 ,but what is to be understood by duplicate proportion has been already explained#if4 ,the third supposition is1 that the resistance increases partly in the single proportion of the velocity1 and partly in the duplicate proportion thereof4 #d4 ,,in all these suppositions1 bodies are considered under two respects1 either as moving1 and opposing themselves against the fluid by that power alone1 which is essential to them1 of resisting to the change of their state from rest to motion1 or from motion to rest1 which we have above called their power of inactivity2 or else1 as descending or ascending1 and so having the power of gravity combined with that other power4 ,thus our author has shewn in all those three suppositions1 in what manner bodies are resisted in an uniform fluid1 when they move with the aforesaid progressive motion#ig2 and what the resistance is1 when they ascend or descend perpendicularly#ih4 ,and if a body ascend or descend obliquely1 and the resistance be singly proportional to the velocity1 it is shewn how the body is resisted in a fluid of an uniform density1 and what line it will describe#ii1 which is determined by the measurement of the hyperbola1 and appears to be no other than that line1 first considered in particular by ,dr4 ,,barrow#ajj1 which is now commonly known by the name of the logarithmical curve4 ,in the supposition that the resistance increases in the duplicate proportion of the velocity1 our author has not given us the line which would be described in an uniform fluid2 but has instead thereof discussed a problem1 which is in some sort the reverse2 to find the density of the fluid at all altitudes1 by which any given curve line may be described2 which problem is so treated by him1 as to be applicable to any kind of resistance whatever#aja4 ,but here not unmindful of practice1 he shews that a body in a fluid of uniform density1 like the air1 will describe a line1 which approaches towards an hyperbola2 that is1 its motion will be nearer to that curve line than to the parabola4 ,and consequent upon this remark1 he shews how to determine this hyperbola by experiment1 and briefly resolves the chief of those problems relating to projectiles1 which are in use in the art of gunnery1 in this curve#ajb2 as ,,torricelli and others have done in the parabola#ajc1 whose inventions have been explained at large above#ajd4 #e4 ,,our author has also handled distinctly that particular sort of motion1 which is described by pendulums#aje2 and has likewise considered some few cases of bodies moving in resisting fluids round a center1 to which they are impelled by a centripetal force1 in order to give an idea of those kinds of motions#ajf4 #f4 ,,the treating of the resistance of pendulums has given him an opportunity of inserting into another part of his work some speculations upon the motions of them without resistance1 which have a very peculiar elegance2 where in he treats of them as moved by a gravitation acting in the law1 which he shews to belong to the earth below its surface#ajg2 performing in this kind of gravitation1 where the force is proportional to the distance from the center1 all that ,,huygens had before done in the common supposition of its being uniform1 and acting in parallel lines#ajh4 #g4 ,,huygens at the end of his treatise of the cause of gravity#aji informs us1 that he likewise had carried his speculations on the first of these suppositions1 of the resistance in fluids being proportional to the velocity of the body1 as far as our author4 ,but finding by experiment that the second was more conformable to nature1 he afterwards made some progress in that1 till he was stopt1 by not being able to execute to his wish what related to the perpendicular descent of bodies2 not observing that the measurement of the curve line1 he made use of to explain it by1 depended on the hyperbola4 ,which oversight may well be pardoned in that great man1 considering that our author had not been pleased at that time to communicate to the publick his admirable discourse of the ,,quadrature or ,,measurement ,,of ,,curve ,,lines1 with which he has since obliged the world3 for without the use of that treatise1 it is ,i think no injury even to our author's unparalleled abilities to believe1 it would not have been easy for himself to have succeeded so happily in this and many other parts of his writings4 #h4 ,,what ,,huygens found by experiment1 that bodies were in reality resisted in the duplicate proportion of their velocity1 agrees with the reasoning of our author#aaj1 who distinguishes the resistance1 which fluids give to bodies by the tenacity of their parts1 and the friction between them and the body1 from that1 which arises from the power of inactivity1 with which the constituent particles of fluids are endued like all other portions of matter1 by which power the particles of fluids like other bodies make resistance against being put into motion4 #i4 ,,the resistance1 which arises from the friction of the body against the parts of the fluid1 must be very inconsiderable2 and the resistance1 which follows from the tenacity of the parts of fluids1 is not usually very great1 and does not depend much upon the velocity of the body in the fluid2 for as the parts of the fluid adhere together with a certain degree of force1 the resistance1 which the body receives from thence1 cannot much depend upon the velocity1 with which the body moves2 but like the power of gravity1 its effect must be proportional to the time of its acting4 ,this the reader may find farther explained by ,sir ,,isaac ,,newton himself in the postscript to a discourse published by me in ,,the ,,philosophical ,,transactions1 ,no #cga4 ,the principal resistance1 which most fluids give to bodies1 arises from the power of inactivity in the parts of the fluids1 and this depends upon the velocity1 with which the body moves1 on a double account4 ,in the first place1 the quantity of the fluid moved out of place by the moving body in any determinate space of time is proportional to the velocity1 wherewith the body moves2 and in the next place1 the velocity with which each particle of the fluid is moved1 will also be proportional to the velocity of the body3 therefore since the resistance1 which any body makes against being put into motion1 is proportional both to the quantity of matter moved and the velocity it is moved with2 the resistance1 which a fluid gives on this account1 will be doubly increased with the increase of the velocity in the moving body2 that is1 the resistance will be in a two-fold or duplicate proportion of the velocity1 wherewith the body moves through the fluid4 #aj4 ,,farther it is most manifest1 that this latter kind of resistance increasing with the increase of velocity1 even in a greater degree than the velocity it self increases1 the swifter the body moves1 the less proportion the other species of resistance will bear to this3 nay that this part of the resistance may be so much augmented by a due increase of velocity1 till the former resistances shall bear a less proportion to this1 than any that might be assigned4 ,and indeed experience shews1 that no other resistance1 than what arises from the power of inactivity in the parts of the fluid1 is of moment1 when the body moves with any considerable swiftness4 #aa4 ,,there is besides these yet another species of resistance1 found only in such fluids1 as1 like our air1 are elastic4 ,elasticity belongs to no fluid known to us beside the air4 ,by this property any quantity of air may be contracted into a less space by a forcible pressure1 and as soon as the compressing power is removed1 it will spring out again to its former dimensions4 ,the air we breath is held to its present density by the weight of the air above us4 ,and as this incumbent weight1 by the motion of the winds1 or other causes1 is frequently varied 7which appears by the barometer27 so when this weight is greatest1 we breath a more dense air than at other times4 ,to what degree the air would expand it self by its spring1 if all pressure were removed1 is not known1 nor yet into how narrow a compass it is capable of being compressed4 ,mr4 ,,boyle found it by experiment capable both of expansion and compression to such a degree1 that he could cause a quantity of air to expand it self over a space some hundred thousand times greater1 than the space to which he could confine the same quantity#aaa4 ,but ,i shall treat more fully of this spring in the air hereafter#aab4 ,i am now only to consider what resistance to the motion of bodies arises from it4 #ab4 ,,but before our author shews in what manner this cause of resistance operates1 he proposes a method1 by which fluids may be rendered elastic1 demonstrating that if their particles be provided with a power of repelling each other1 which shall exert it self with degrees of strength reciprocally proportional to the distances between the centers of the particles2 that then such fluids will observe the same rule in being compressed1 as our air does1 which is this1 that the space1 into which it yields upon compression1 is reciprocally proportional to the compressing weight#aac4 ,the term reciprocally proportional has been explained above#aad4 ,and if the centrifugal force of the particles acted by other laws1 such fluids would yield in a different manner to compression#aae4 #ac4 ,,whether the particles of the air be endued with such a power1 by which they can act upon each other out of contact1 our author does not determine1 but leaves that to future examination1 and to be discussed by philosophers4 ,only he takes occasion from hence to consider the resistance in elastic fluids1 under this notion2 making remarks1 as he passes along1 upon the differences1 which will arise1 if their elasticity be derived from any other fountain#aaf4 ,and this1 ,i think1 must be confessed to be done by him with great judgment2 for this is far the most reasonable account1 which has been given of this surprizing power1 as must without doubt be freely acknowledged by any one1 who in the least considers the insufficiency of all the other conjectures1 which have been framed2 and also how little reason there is to deny to bodies other powers1 by which they may act upon each other at a distance1 as well as that of gravity2 which we shall hereafter shew to be a property universally belonging to all the bodies of the universe1 and to all their parts#aag4 ,nay we actually find in the loadstone a very apparent repelling1 as well as an attractive power4 ,but of this more in the conclusion of this discourse4 #ad4 ,,by these steps our author leads the way to explain the resistance1 which the air and such like fluids will give to bodies by their elasticity2 which resistance he explains thus4 ,if the elastic power of the fluid were to be varied so1 as to be always in the duplicate proportion of the velocity of the resisted body1 it is shewn that then the resistance derived from the elasticity1 would increase in the duplicate proportion of the velocity2 in so much that the whole resistance would be in that proportion1 excepting only that small part1 which arises from the friction between the body and the parts of the fluid4 ,from whence it follows1 that because the elastic power of the same fluid does in truth continue the same1 if the velocity of the moving body be diminished1 the resistance from the elasticity1 and therefore the whole resistance1 will decrease in a less proportion1 than the duplicate of the velocity2 and if the velocity be increased1 the resistance from the elasticity will increase in a less proportion1 than the duplicate of the velocity1 that is in a less proportion1 than the resistance made by the power of inactivity of the parts of the fluid4 ,and from this foundation is raised the proof of a property of this resistance1 given by the elasticity in common with the others from the tenacity and friction of the parts of the fluid2 that the velocity may be increased1 till this resistance from the fluid's elasticity shall bear no considerable proportion to that1 which is produced by the power of inactivity thereof#aah4 ,from whence our author draws this conclusion2 that the resistance of a body1 which moves very swiftly in an elastic fluid1 is near the same1 as if the fluid were not elastic2 provided the elasticity arises from the centrifugal power of the parts of the medium1 as before explained1 especially if the velocity be so great1 that this centrifugal power shall want time to exert it self#aai4 ,but it is to be observed1 that in the proof of all this our author proceeds upon the supposition of this centrifugal power in the parts of the fluid2 but if the elasticity be caused by the expansion of the parts in the manner of wool compressed1 and such like bodies1 by which the parts of the fluid will be in some measure entangled together1 and their motion be obstructed1 the fluid will be in a manner tenacious1 and give a resistance upon that account over and above what depends upon its elasticity only#abj2 and the resistance derived from that cause is to be judged of in the manner before set down4 #ae4 ,,it is now time to pass to the second part of this theory2 which is to assign the measure of resistance1 according to the proportion between the density of the body and the density of the fluid4 ,what is here to be understood by the word density has been explained above#aba4 ,for this purpose as our author before considered two distinct cases of bodies moving in mediums2 one when they opposed themselves to the fluid by their power of inactivity only1 and another when by ascending or descending their weight was combined with that other power3 so likewise1 the fluids themselves are to be regarded under a double capacity2 either as having their parts at rest1 and disposed freely without restraint1 or as being compressed together by their own weight1 or any other cause4 #af4 ,,in the first case1 if the parts of the fluid be wholly disingaged from one another1 so that each particle is at liberty to move all ways without any impediment1 it is shewn1 that if a globe move in such a fluid1 and the globe and particles of the fluid are endued with perfect elasticity2 so that as the globe impinges upon the particles of it1 they shall bound off and separate themselves from the globe1 with the same velocity1 with which the globe strikes upon them2 then the resistance1 which the globe moving with any known velocity suffers1 is to be thus determined4 ,from the velocity of the globe1 the time1 wherein it would move over two third parts of its own diameter with that velocity1 will be known4 ,and such proportion as the density of the fluid bears to the density of the globe1 the same the resistance given to the globe will bear to the force1 which acting1 like the power of gravity1 on the globe without intermission during the space of time now mentioned1 would generate in the globe the same degree of motion1 as that wherewith it moves in the fluid#abb4 ,but if neither the globe nor the particles of the fluid be elastic1 so that the particles1 when the globe strikes against them1 do not rebound from it1 then the resistance will be but half so much#abc4 ,again1 if the particles of the fluid and the globe are imperfectly elastic1 so that the particles will spring from the globe with part only of that velocity wherewith the globe impinges upon them2 then the resistance will be a mean between the two preceding cases1 approaching nearer to the first or second1 according as the elasticity is more or less#abd4 #ag4 ,,the elasticity1 which is here ascribed to the particles of the fluid1 is not that power of repelling one another1 when out of contact1 by which1 as has before been mentioned1 the whole fluid may be rendred elastic2 but such an elasticity only1 as many solid bodies have of recovering their figure1 whenever any forcible change is made in it1 by the impulse of another body or otherwise4 ,which elasticity has been explained above at large#abe4 #ah4 ,,this is the case of discontinued fluids1 where the body1 by pressing against their particles1 drives them before itself1 while the space behind the body is left empty4 ,but in fluids which are compressed1 so that the parts of them removed out of place by the body resisted immediately retire behind the body1 and fill that space1 which in the other case is left vacant1 the resistance is still less2 for a globe in such a fluid which shall be free from all elasticity1 will be resisted but half as much as the least resistance in the former case#abf4 ,but by elasticity ,i now mean that power1 which renders the whole fluid so2 of which if the compressed fluid be possessed1 in the manner of the air1 then the resistance will be greater than by the foregoing rule2 for the fluid being capable in some degree of condensation1 it will resemble so far the case of uncompressed fluids#abg4 ,but1 as has been before related1 this difference is most considerable in slow motions4 #ai4 ,,in the next place our author is particular in determining the degrees of resistance accompanying bodies of different figures2 which is the last of the three heads1 we divided the whole discourse of resistance into4 ,and in this disquisition he finds a very surprizing and unthought of difference1 between free and compressed fluids4 ,he proves1 that in the former kind1 a globe suffers but half the resistance1 which the cylinder1 that circumscribes the globe1 will do1 if it move in the direction of its axis#abh4 ,but in the latter he proves1 that the globe and cylinder are resisted alike#abi4 ,and in general1 that let the shape of bodies be ever so different1 yet if the greatest sections of the bodies perpendicular to the axis of their motion be equal1 the bodies will be resisted equally#acj4 #bj4 ,,pursuant to the difference found between the resistance of the globe and cylinder in rare and uncompressed fluids1 our author gives us the result of some other inquiries of the same nature4 ,thus of all the frustums of a cone1 that can be described upon the same base and with the same altitude1 he shews how to find that1 which of all others will be the least resisted1 when moving in the direction of its axis#aca4 ,and from hence he draws an easy method of altering the figure of any spheroidical solid1 so that its capacity may be enlarged1 and yet the resistance of it diminished#acb3 a note which he thinks may not be useless to ship-wrights4 ,he concludes with determining the solid1 which will be resisted the least that is possible1 in these discontinued fluids#acc4 #ba4 ,,that ,i may here be understood by readers unacquainted with mathematical terms1 ,i shall explain what ,i mean by a frustum of a cone1 and a spheroidical solid4 ,a cone has been defined above4 ,a frustum is what remains1 when part of the cone next the vertex is cut away by a section parallel to the base of the cone1 as in fig4 #hf4 ,a spheroid is produced from an ellipsis1 as a sphere or globe is made from a circle4 ,if a circle turn round on its diameter1 it describes by its motion a sphere2 so if an ellipsis 7which figure has been defined above1 and will be more fully explained hereafter#acd7 be turned round either upon the longest or shortest line1 that can be drawn through the middle of it1 there will be described a kind of oblong or flat sphere1 as in fig4 #hg4 ,both these figures are called spheroids1 and any solid resembling these ,i here call spheroidical4 #bb4 ,,if it should be asked1 how the method of altering spheroidical bodies1 here mentioned1 can contribute to the facilitating a ship's motion1 when ,i just above affirmed1 that the figure of bodies1 which move in a compressed fluid not elastic1 has no relation to the augmentation or diminution of the resistance2 the reply is1 that what was there spoken relates to bodies deep immerged into such fluids1 but not of those1 which swim upon the surface of them2 for in this latter case the fluid1 by the appulse of the anterior parts of the body1 is raised above the level of the surface1 and behind the body is sunk somewhat below2 so that by this inequality in the superficies of the fluid1 that part of it1 which at the head of the body is higher than the fluid behind1 will resist in some measure after the manner of discontinued fluids#ace1 analogous to what was before observed to happen in the air through its elasticity1 though the body be surrounded on every side by it#acf4 ,and as far as the power of these causes extends1 the figure of the moving body affects its resistance2 for it is evident1 that the figure1 which presses least directly against the parts of the fluid1 and so raises least the surface of a fluid not elastic1 and least compresses one that is elastic1 will be least resisted4 #bc4 ,,the way of collecting the difference of the resistance in rare fluids1 which arises from the diversity of figure1 is by considering the different effect of the particles of the fluid upon the body moving against them1 according to the different obliquity of the several parts of the body upon which they respectively strike2 as it is known1 that any body impinging against a plane obliquely1 strikes with a less force1 than if it fell upon it perpendicularly2 and the greater the obliquity is1 the weaker is the force4 ,and it is the same thing1 if the body be at rest1 and the plane move against it#acg4 #bd4 ,,that there is no connexion between the figure of a body and its resistance in compressed fluids1 is proved thus4 ,suppose ,a ,b ,c ,d 7in fig4 #hh47 to be a canal1 having such a fluid1 water for instance1 running through it with an equable velocity2 and let any body ,e1 by being placed in the axis of the canal1 hinder the passage of the water4 ,it is evident1 that the figure of the fore part of this body will have little influence in obstructing the water's motion1 but the whole impediment will arise from the space taken up by the body1 by which it diminishes the bore of the canal1 and straightens the passage of the water#ach4 ,but proportional to the obstruction of the water's motion1 will be the force of the water upon the body ,e#aci4 ,now suppose both orifices of the canal to be closed1 and the water in it to remain at rest2 the body ,e to move1 so that the parts of the water may pass by it with the same degree of velocity1 as they did before2 it is beyond contradiction1 that the pressure of the water upon the body1 that is1 the resistance it gives to its motion1 will remain the same2 and therefore will have little connexion with the figure of the body#adj4 #be4 ,,by a method of reasoning drawn from the same fountain is determined the measure of resistance these compressed fluids give to bodies1 in reference to the proportion between the density of the body and that of the fluid4 ,this shall be explained particularly in my comment on ,sir ,,is4 ,,newton's mathematical principles of natural philosophy2 but is not a proper subject to be insisted on farther in this place4 #bf4 ,,we have now gone through all the parts of this theory4 ,there remains nothing more1 but in few words to mention the experiments1 which our author has made1 both with bodies falling perpendicularly through water1 and the air#ada1 and with pendulums#adb3 all which agree with the theory4 ,in the case of falling bodies1 the times of their fall determined by the theory come out the same1 as by observation1 to a surprizing exactness2 in the pendulums1 the rod1 by which the ball of the pendulum hangs1 suffers resistance as well as the ball1 and the motion of the ball being reciprocal1 it communicates such a motion to the fluid1 as increases the resistance1 but the deviation from the theory is no more1 than what may reasonably follow from these causes4 #bg4 ,,by this theory of the resistance of fluids1 and these experiments1 our author decides the question so long agitated among natural philosophers1 whether all space is absolutely full of matter4 ,the ,aristotelians and ,cartesians both assert this plenitude2 the ,atomists have maintained the contrary4 ,our author has chose to determine this question by his theory of resistance1 as shall be explained in the following chapter4 ,illustration ,illustration ,,book ,,ii4 ,,concerning ,,the ,,system of the ,,world4 ,,chap4 ,i4 ,that the ,planets move in a space empty of all sensible matter4 ,i ,,have now gone through the first part of my design1 and have explained1 as far as the nature of my undertaking would permit1 what ,sir ,,isaac ,,newton has delivered in general concerning the motion of bodies4 ,it follows now to speak of the discoveries1 he has made in the system of the world2 and to shew from him what cause keeps the heavenly bodies in their courses4 ,but it will be necessary for the use of such1 as are not skilled in astronomy1 to premise a brief description of the planetary system4 #b4 ,,this system is disposed in the following manner4 ,in the middle is placed the sun4 ,about him six globes continually roll4 ,these are the primary planets2 that which is nearest to the sun is called ,mercury1 the next ,venus1 next to this is our earth1 the next beyond is ,mars1 after him ,jupiter1 and the outermost of all ,saturn4 ,besides these there are discovered in this system ten other bodies1 which move about some of these primary planets in the same manner1 as they move round the sun4 ,these are called secondary planets4 ,the most conspicuous of them is the moon1 which moves round our earth2 four bodies move in like manner round ,jupiter2 and five round ,saturn4 ,those which move about ,jupiter and ,saturn1 are usually called satellites2 and cannot any of them be seen without a telescope4 ,it is not impossible1 but there may be more secondary planets1 beside these2 though our instruments have not yet discovered any other4 ,this disposition of the planetary or solar system is represented in fig4 #hi4 #c4 ,,the same planet is not always equally distant from the sun4 ,but the middle distance of ,mercury is between and of the distance of the earth from the sun2 ,venus is distant from the sun almost of the distance of the earth2 the middle distance of ,mars is something more than half as much again1 as the distance of the earth2 ,jupiter's middle distance exceeds five times the distance of the earth1 by between and #a_/f part of this distance2 ,saturn's middle distance is scarce more than #i times the distance between the earth and sun2 but the middle distance between the earth and sun is about #bag times the sun's semidiameter4 ,illustration #d4 ,,all these planets move one way1 from west to east2 and of the primary planets the most remote is longest in finishing its course round the sun4 ,the period of ,saturn falls short only sixteen days of #bi years and a half4 ,the period of ,jupiter is twelve years wanting about #ej days4 ,the period of ,mars falls short of two years by about #dc days4 ,the revolution of the earth constitutes the year4 ,venus performs her period in about #bbd days1 and ,mercury in about #hh days4 #e4 ,,the course of each planet lies throughout in one plane or flat surface1 in which the sun is placed2 but they do not all move in the same plane1 though the different planes1 in which they move1 cross each other in very small angles4 ,they all cross each other in lines1 which pass through the sun2 because the sun lies in the plane of each orbit4 ,this inclination of the several orbits to each other is represented in fig4 #ij4 ,the line1 in which the plane of any orbit crosses the plane of the earth's motion1 is called the line of the nodes of that orbit4 #f4 ,,each planet moves round the sun in the line1 which we have mentioned above#adc under the name of ellipsis2 which ,i shall here shew more particularly how to describe4 ,i have there said how it is produced in the cone4 ,i shall now shew how to form it upon a plane4 ,fix upon any plane two pins1 as at ,a and ,b in fig4 #ia4 ,to these tye a string ,a ,c ,b of any length4 ,then apply a third pin ,d so to the string1 as to hold it strained2 and in that manner carrying this pin about1 the point of it will describe an ellipsis4 ,if through the points ,a1 ,b the straight line ,e ,a ,b ,f be drawn1 to be terminated at the ellipsis in the points ,e and ,f1 this is the longest line of any1 that can be drawn within the figure1 and is called the greater axis of the ellipsis4 ,the line ,g ,h1 drawn perpendicular to this axis ,e ,f1 so as to pass through the middle of it1 is called the lesser axis4 ,the two points ,a and ,b are called focus's4 ,now each planet moves round the sun in a line of this kind1 so that the sun is found in one focus4 ,suppose ,a to be the place of the sun4 ,then ,e is the point1 wherein the planet will be nearest of all to the sun1 and at ,f it will be most remote4 ,the point ,e is called the perihelion of the planet1 and ,f the aphelion4 ,in ,g and ,h the planet is said to be in its middle or mean distance2 because the distance ,a ,g or ,a ,h is truly the middle between ,a ,e the least1 and ,a ,f the greatest distance4 ,in fig4 #ib4 is represented how the greater axis of each orbit is situated in respect of the rest4 ,the proportion between the greatest and least distances of the planet from the sun is very different in the different planets4 ,illustration ,in ,saturn the proportion of the greatest distance to the least is something less1 than the proportion of #i to #h1 but much nearer to this1 than to the proportion of #aj to #i4 ,in ,jupiter this proportion is a little greater1 than that of #aa to #aj4 ,in ,mars it exceeds the proportion of #f to #e4 ,in the earth it is about the proportion of #cj to #bi4 ,in ,venus it is near to that of #gj to #fi4 ,and in ,mercury it comes not a great deal short of the proportion of #c to #b4 ,illustration #g4 ,,each of these planets so moves through its ellipsis1 that the line drawn from the sun to the planet1 by accompanying the planet in its motion1 will describe about the sun equal spaces in equal times1 after the manner spoke of in the chapter of centripetal forces#add4 ,there is also a certain relation between the greater axis's of these ellipsis's1 and the times1 in which the planets perform their revolutions through them4 ,which relation may be expressed thus4 ,let the period of one planet be denoted by the letter ,a1 the greater axis of its orbit by ,d2 let the period of another planet be denoted by ,b1 and the greater axis of this planet's orbit by ,e4 ,then if ,c be taken to bear the same proportion to ,b1 as ,b bears to ,a2 likewise if ,f be taken to bear the same proportion to ,e1 as ,e bears to ,d2 and ,g taken to bear the same proportion likewise to ,f1 as ,e bears to ,d2 then ,a shall bear the same proportion to ,c1 as ,d bears to ,g4 #h4 ,,the secondary planets move round their respective primary1 much in the same manner as the primary do round the sun4 ,but the motions of these shall be more fully explained hereafter#ade4 ,and there is1 besides the planets1 another sort of bodies1 which in all probability move round the sun2 ,i mean the comets4 ,the farther description of which bodies ,i also leave to the place1 where they are to be particularly treated on#adf4 #i4 ,,far without this system the fixed stars are placed4 ,these are all so remote from us1 that we seem almost incapable of contriving any means to estimate their distance4 ,their number is exceeding great4 ,besides two or three thousand1 which we see with the naked eye1 telescopes open to our view vast numbers2 and the farther improved these instruments are1 we still discover more and more4 ,without doubt these are luminous globes1 like our sun1 and ranged through the wide extent of space2 each of which1 it is to be supposed1 perform the same office1 as our sun1 affording light and heat to certain planets moving about them4 ,but these conjectures are not to be pursued in this place4 #aj4 ,i ,,shall therefore now proceed to the particular design of this chapter1 and shew1 that there is no sensible matter lodged in the space where the planets move4 #aa4 ,,that they suffer no sensible resistance from any such matter1 is evident from the agreement between the observations of astronomers in different ages1 with regard to the time1 in which the planets have been found to perform their periods4 ,but it was the opinion of ,,des ,,cartes#adg1 that the planets might be kept in their courses by the means of a fluid matter1 which continually circulating round should carry the planets along with it4 ,there is one appearance that may seem to favour this opinion2 which is1 that the sun turns round its own axis the same way1 as the planets move4 ,the earth also turns round its axis the same way1 as the moon moves round the earth4 ,and the planet ,jupiter turns upon its axis the same way1 as his satellites revolve round him4 ,it might therefore be supposed1 that if the whole planetary region were filled with a fluid matter1 the sun1 by turning round on its own axis1 might communicate motion first to that part of the fluid1 which was contiguous1 and by degrees propagate the like motion to the parts more remote4 ,after the same manner the earth might communicate motion to this fluid1 to a distance sufficient to carry round the moon1 and ,jupiter communicate the like to the distance of its satellites4 ,sir ,,isaac ,,newton has particularly examined what might be the result of such a motion as this#adh2 and he finds1 that the velocities1 with which the parts of this fluid will move in different distances from the center of the motion1 will not agree with the motion observed in different planets3 for instance1 that the time of one intire circulation of the fluid1 wherein ,jupiter should swim1 would bear a greater proportion to the time of one intire circulation of the fluid1 where the earth is2 than the period of ,jupiter bears to the period of the earth4 ,but he also proves#adi1 that the planet cannot circulate in such a fluid1 so as to keep long in the same course1 unless the planet and the contiguous fluid are of the same density1 and the planet be carried along with the same degree of motion1 as the fluid4 ,there is also another remark made upon this motion by our author2 which is1 that some vivifying force will be continually necessary at the center of the motion#aej4 ,the sun in particular1 by communicating motion to the ambient fluid1 will lose from it self as much motion1 as it imparts to the fluid2 unless some acting principle reside in the sun to renew its motion continually4 ,if the fluid be infinite1 this gradual loss of motion would continue till the whole should stop#aea2 and if the fluid were limited1 this loss of motion would continue1 till there would remain no swifter a revolution in the sun1 than in the utmost part of the fluid2 so that the whole would turn together about the axis of the sun1 like one solid globe#aeb4 #ab4 ,,it is farther to be observed1 that as the planets do not move in perfect circles round the sun2 there is a greater distance between their orbits in some places1 than in others4 ,for instance1 the distance between the orbit of ,mars and ,venus is near half as great again in one part of their orbits1 as in the opposite place4 ,now here the fluid1 in which the earth should swim1 must move with a less rapid motion1 where there is this greater interval between the contiguous orbits2 but on the contrary1 where the space is straitest1 the earth moves more slowly1 than where it is widest#aec4 #ac4 ,,farther1 if this our globe of earth swam in a fluid of equal density with the earth it self1 that is1 in a fluid more dense than water2 all bodies put in motion here upon the earth's surface must suffer a great resistance from it2 where as1 by ,sir ,,isaac ,,newton's experiments mentioned in the preceding chapter1 bodies1 that fell perpendicularly down through the air1 felt about #a_/hfj part only of the resistance1 which bodies suffered that fell in like manner through water4 #ad4 ,sir ,,isaac ,,newton applies these experiments yet farther1 and examines by them the general question concerning the absolute plenitude of space4 ,according to the ,aristotelians1 all space was full without any the least vacuities whatever4 ,,descartes embraced the same opinion1 and therefore supposed a subtile fluid matter1 which should pervade all bodies1 and adequately fill up their pores4 ,the ,atomical philosophers1 who suppose all bodies both fluid and solid to be composed of very minute but solid atoms1 assert that no fluid1 how subtile soever the particles or atoms whereof it is composed should be1 can ever cause an absolute plenitude2 because it is impossible that any body can pass through the fluid without putting the particles of it into such a motion1 as to separate them1 at least in part1 from one another1 and so perpetually to cause small vacuities2 by which these ,atomists endeavour to prove1 that a vacuum1 or some space empty of all matter1 is absolutely necessary to be in nature4 ,sir ,,isaac ,,newton objects against the filling of space with such a subtile fluid1 that all bodies in motion must be unmeasurably resisted by a fluid so dense1 as absolutely to fill up all the space1 through which it is spread4 ,and lest it should be thought1 that this objection might be evaded by ascribing to this fluid such very minute and smooth parts1 as might remove all adhesion or friction between them1 whereby all resistance would be lost1 which this fluid might otherwise give to bodies moving in it2 ,sir ,,isaac ,,newton proves1 in the manner above related1 that fluids resist from the power of inactivity of their particles2 and that water and the air resist almost entirely on this account3 so that in this subtile fluid1 however minute and lubricated the particles1 which compose it1 might be2 yet if the whole fluid was as dense as water1 it would resist very near as much as water does2 and whereas such a fluid1 whose parts are absolutely close together without any intervening spaces1 must be a great deal more dense than water1 it must resist more than water in proportion to its greater density2 unless we will suppose the matter1 of which this fluid is composed1 not to be endued with the same degree of inactivity as other matter4 ,but if you deprive any substance of the property so universally belonging to all other matter1 without impropriety of speech it can scarce be called by this name4 #ae4 ,sir ,,isaac ,,newton made also an experiment to try in particular1 whether the internal parts of bodies suffered any resistance4 ,and the result did indeed appear to favour some small degree of resistance2 but so very little1 as to leave it doubtful1 whether the effect did not arise from some other latent cause#aed4 ,,chap4 ,,ii4 ,concerning the cause1 which keeps in motion the primary planets4 ,,since the planets move in a void space and are free from resistance2 they1 like all other bodies1 when once in motion1 would move on in a straight line without end1 if left to themselves4 ,and it is now to be explained what kind of action upon them carries them round the sun4 ,here ,i shall treat of the primary planets only1 and discourse of the secondary apart in the next chapter4 ,it has been just now declared1 that these primary planets move so about the sun1 that a line extended from the sun to the planet1 will1 by accompanying the planet in its motion1 pass over equal spaces in equal portions of time#aee4 ,and this one property in the motion of the planets proves1 that they are continually acted on by a power directed perpetually to the sun as a center4 ,this therefore is one property of the cause1 which keeps the planets in their courses1 that it is a centripetal power1 whose center is the sun4 #b4 ,,again1 in the chapter upon centripetal forces#aef it was observ'd1 that if the strength of the centripetal power was suitably accommodated every where to the motion of any body round a center1 the body might be carried in any bent line whatever1 whose concavity should be every where turned towards the center of the force4 ,it was farther remarked1 that the strength of the centripetal force1 in each place1 was to be collected from the nature of the line1 wherein the body moved#aeg4 ,now since each planet moves in an ellipsis1 and the sun is placed in one focus2 ,sir ,,isaac ,,newton deduces from hence1 that the strength of this power is reciprocally in the duplicate proportion of the distance from the sun4 ,this is deduced from the properties1 which the geometers have discovered in the ellipsis4 ,the process of the reasoning is not proper to be enlarged upon here2 but ,i shall endeavour to explain what is meant by the reciprocal duplicate proportion4 ,each of the terms reciprocal proportion1 and duplicate proportion1 has been already defined#aeh4 ,their sense when thus united is as follows4 ,suppose the planet moved in the orbit ,a ,b ,c 7in fig4 #ic47 about the sun in ,s4 ,then1 when it is said1 that the centripetal power1 which acts on the planet in ,a1 bears to the power acting on it in ,b a proportion1 which is the reciprocal of the duplicate proportion of the distance ,s ,a to the distance ,s ,b2 it is meant that the power in ,a bears to the power in ,b the duplicate of the proportion of the distance ,s ,b to the distance ,s ,a4 ,the reciprocal duplicate proportion may be explained also by numbers as follows4 ,suppose several distances to bear to each other proportions expressed by the numbers #a1 #b1 #c1 #d1 #e2 that is1 let the second distance be double the first1 the third be three times1 the fourth four times1 and the fifth five times as great as the first4 ,multiply each of these numbers by it self1 and #a multiplied by #a produces still #a1 #b multiplied by #b produces #d1 #c by #c makes #i1 #d by #d makes #af1 and #e by #e gives #be4 ,this being done1 the fractions 1 #a_/i1 #a_/af1 #a_/be1 will respectively express the proportion1 which the centripetal power in each of the following distances bears to the power at the first distance3 for in the second distance1 which is double the first1 the centripetal power will be one fourth part only of the power at the first distance2 at the third distance the power will be one ninth part only of the first power2 at the fourth distance1 the power will be but one sixteenth part of the first2 and at the fifth distance1 one twenty fifth part of the first power4 #c4 ,,thus is found the proportion1 in which this centripetal power decreases1 as the distance from the sun increases1 within the compass of one planet's motion4 ,how it comes to pass1 that the planet can be carried about the sun by this centripetal power in a continual round1 sometimes rising from the sun1 then descending again as low1 and from thence be carried up again as far remote as before1 alternately rising and falling without end2 appears from what has been written above concerning centripetal forces3 for the orbits of the planets resemble in shape the curve line proposed in #ag of the chapter on these forces#aei4 #d4 ,,but farther1 in order to know whether this centripetal force extends in the same proportion throughout1 and consequently whether all the planets are influenced by the very same power1 our author proceeds thus4 ,he inquires what relation there ought to be between the periods of the different planets1 provided they were acted upon by the same power decreasing throughout in the forementioned proportion2 and he finds1 that the period of each in this case would have that very relation to the greater axis of its orbit1 as ,i have declared above#afj to be found in the planets by the observations of astronomers4 ,and this puts it beyond question1 that the different planets are pressed towards the sun1 in the same proportion to their distances1 as one planet is in its several distances4 ,and thence in the last place it is justly concluded1 that there is such a power acting towards the sun in the foresaid proportion at all distances from it4 #e4 ,,this power1 when referred to the planets1 our author calls centripetal1 when to the sun attractive2 he gives it likewise the name of gravity1 because he finds it to be of the same nature with that power of gravity1 which is observed in our earth1 as will appear hereafter#afa4 ,by all these names he designs only to signify a power endued with the properties before mentioned2 but by no means would he have it understood1 as if these names referred any way to the cause of it4 ,in particular in one place where he uses the name of attraction1 he cautions us expressly against implying any thing but a power directing a body to a center without any reference to the cause of it1 whether residing in that center1 or arising from any external impulse#afb4 #f4 ,,but now1 in these demonstrations some very minute inequalities in the motion of the planets are neglected2 which is done with a great deal of judgment2 for whatever be their cause1 the effects are very inconsiderable1 they being so exceeding small1 that some astronomers have thought fit wholly to pass them by#afc4 ,however the excellency of this philosophy1 when in the hands of so great a geometer as our author1 is such1 that it is able to trace the least variations of things up to their causes4 ,the only inequalities1 which have been observed common to all the planets1 are the motion of the aphelion and the nodes4 ,the transverse axis of each orbit does not always remain fixed1 but moves about the sun with a very slow progressive motion3 nor do the planets keep constantly the same plane1 but change them1 and the lines in which those planes intersect each other by insensible degrees4 ,the first of these inequalities1 which is the motion of the aphelion1 may be accounted for1 by supposing the gravitation of the planets towards the sun to differ a little from the forementioned reciprocal duplicate proportion of the distances2 but the second1 which is the motion of the nodes1 cannot be accounted for by any power directed towards the sun2 for no such can give the planet any lateral impulse to divert it from the plane of its motion into any new plane1 but of necessity must be derived from some other center4 ,where that power is lodged1 remains to be discovered4 ,now it is proved1 as shall be explained in the following chapter1 that the three primary planets ,saturn1 ,jupiter1 and the earth1 which have satellites revolving about them1 are endued with a power of causing bodies1 in particular those satellites1 to gravitate towards them with a force1 which is reciprocally in the duplicate proportion of their distances2 and the planets are in all respects1 in which they come under our examination1 so similar and alike1 that there is no reason to question1 but they have all the same property4 ,though it be sufficient for the present purpose to have it proved of ,jupiter and ,saturn only2 for these planets contain much greater quantities of matter than the rest1 and proportionally exceed the others in power#afd4 ,but the influence of these two planets being allowed1 it is evident how the planets come to shift continually their planes3 for each of the planets moving in a different plane1 the action of ,jupiter and ,saturn upon the rest will be oblique to the planes of their motion2 and therefore will gradually draw them into new ones4 ,the same action of these two planets upon the rest will cause likewise a progressive motion of the aphelion2 so that there will be no necessity of having recourse to the other cause for this motion1 which was before hinted at#afe2 viz1 the gravitation of the planets towards the sun differing from the exact reciprocal duplicate proportion of the distances4 ,and in the last place1 the action of ,jupiter and ,saturn upon each other will produce in their motions the same inequalities1 as their joint action produces in the rest4 ,all this is effected in the same manner1 as the sun produces the same kind of inequalities and many others in the motion of the moon and the other secondary planets2 and therefore will be best apprehended by what shall be said in the next chapter4 ,those other irregularities in the motion of the secondary planets have place likewise here2 but are too minute to be observable3 because they are produced and rectified alternately1 for the most part in the time of a single revolution2 whereas the motion of the aphelion and nodes1 which continually increase1 become sensible in a long series of years4 ,yet some of these other inequalities are discernible in ,jupiter and ,saturn1 in ,saturn chiefly2 for when ,jupiter1 who moves faster than ,saturn1 approaches near to a conjunction with him1 his action upon ,saturn will a little retard the motion of that planet1 and by the reciprocal action of ,saturn he will himself be accelerated4 ,after conjunction1 ,jupiter will again accelerate ,saturn1 and be likewise retarded in the same degree1 as before the first was retarded and the latter accelerated4 ,whatever inequalities besides are produced in the motion of ,saturn by the action of ,jupiter upon that planet1 will be sufficiently rectified1 by placing the focus of ,saturn's ellipsis1 which should otherwise be in the sun1 in the common center of gravity of the sun and ,jupiter4 ,and all the inequalities in the motion of ,jupiter1 caused by ,saturn's action upon him1 are much less considerable than the irregularities of ,saturn's motion#aff4 #g4 ,,this one principle therefore of the planets having a power1 as well as the sun1 to cause bodies to gravitate towards them1 which is proved by the motion of the secondary planets to obtain in fact1 explains all the irregularities relating to the planets ever observed by astronomers4 #h4 ,sir ,,isaac ,,newton after this proceeds to make an improvement in astronomy by applying this theory to the farther correction of their motions4 ,for as we have here observed the planets to possess a principle of gravitation1 as well as the sun2 so it will be explained at large hereafter1 that the third law of motion1 which makes action and reaction equal1 is to be applied in this case#afg2 and that the sun does not only attract each planet1 but is it self also attracted by them2 the force1 wherewith the planet is acted on1 bearing to the force1 wherewith the sun it self is acted on at the same time1 the proportion1 which the quantity of matter in the sun bears to the quantity of matter in the planet4 ,from the action between the sun and planet being thus mutual ,sir ,,isaac ,,newton proves that the sun and planet will describe about their common center of gravity similar ellipsis's2 and then that the transverse axis of the ellipsis described thus about the moveable sun1 will bear to the transverse axis of the ellipsis1 which would be described about the sun at rest in the same time1 the same proportion as the quantity of solid matter in the sun and planet together bears to the first of two mean proportionals between this quantity and the quantity of matter in the sun only#afh4 #i4 ,,above1 where ,i shewed how to find a cube1 that should bear any proportion to another cube#afi1 the lines ,f ,t and ,t ,s are two mean proportionals between ,e ,f and ,f ,g2 and counting from ,e ,f1 ,f ,t is called the first1 and ,f ,s the second of those means4 ,in numbers these mean proportionals are thus found4 ,illustration ,suppose ,a and ,b two numbers1 and it be required to find ,c the first1 and ,d the second of the two mean proportionals between them4 ,first multiply ,a by it self1 and the product multiply by ,b2 then ,c will be the number which in arithmetic is called the cubic root of this last product2 that is1 the number ,c being multiplied by it self1 and the product again multiplied by the same number ,c1 will produce the product above mentioned4 ,in like manner ,d is the cubic root of the product of ,b multiplied by it self1 and the produce of that multiplication multiplied again by ,a4 #aj4 ,,it will be asked1 perhaps1 how this correction can be admitted1 when the cause of the motions of the planets was before found by supposing the sun the center of the power1 which acted upon them3 for according to the present correction this power appears rather to be directed to their common center of gravity4 ,but whereas the sun was at first concluded to be the center1 to which the power acting on the planets was directed1 because the spaces described round the sun in equal times were found to be equal2 so ,sir ,,isaac ,,newton proves1 that if the sun and planet move round their common center of gravity1 yet to an eye placed in the planet1 the spaces1 which will appear to be described about the sun1 will have the same relation to the times of their description1 as the real spaces would have1 if the sun were at rest#agj4 ,i farther asserted1 that1 supposing the planets to move round the sun at rest1 and to be attracted by a power1 which every where should act with degrees of strength reciprocally in the duplicate proportion of the distances2 then the periods of the planets must observe the same relation to their distances1 as astronomers find them to do4 ,but here it must not be supposed1 that the observations of astronomers absolutely agree without any the least difference2 and the present correction will not cause a deviation from any one astronomer's observations1 so much as they differ from one another4 ,for in ,jupiter1 where this correction is greatest1 it hardly amounts to the #cjjjth part of the whole axis4 #aa4 ,,upon this head ,i think it not improper to mention a reflection made by our excellent author upon these small inequalities in the planets motions2 which contains under it a very strong philosophical argument against the eternity of the world4 ,it is this1 that these inequalities must continually increase by slow degrees1 till they render at length the present frame of nature unfit for the purposes1 it now serves#aga4 ,and a more convincing proof cannot be desired against the present constitution's having existed from eternity than this1 that a certain period of years will bring it to an end4 ,i am aware this thought of our author has been represented even as impious1 and as no less than casting a reflection upon the wisdom of the author of nature1 for framing a perishable work4 ,but ,i think so bold an assertion ought to have been made with singular caution4 ,for if this remark upon the increasing irregularities of the heavenly motions be true in fact1 as it really is1 the imputation must return upon the asserter1 that this does detract from the divine wisdom4 ,certainly we cannot pretend to know all the omniscient ,creator's purposes in making this world1 and therefore cannot undertake to determine how long he designed it should last4 ,and it is sufficient1 if it endure the time intended by the author4 ,the body of every animal shews the unlimited wisdom of its author no less1 nay in many respects more1 than the larger frame of nature2 and yet we see1 they are all designed to last but a small space of time4 #ab4 ,,there need nothing more be said of the primary planets2 the motions of the secondary shall be next considered4 ,,chap4 ,,iii4 ,of the motion of the ,,moon and the other ,,secondary ,,planets4 ,,the excellency of this philosophy sufficiently appears from its extending in the manner1 which has been related1 to the minutest circumstances of the primary planets motions2 which nevertheless bears no proportion to the vast success of it in the motions of the secondary2 for it not only accounts for all the irregularities1 by which their motions were known to be disturbed1 but has discovered others so complicated1 that astronomers were never able to distinguish them1 and reduce them under proper heads2 but these were only to be found out from their causes1 which this philosophy has brought to light1 and has shewn the dependence of these inequalities upon such causes in so perfect a manner1 that we not only learn from thence in general1 what those inequalities are1 but are able to compute the degree of them4 ,of this ,sir ,,is4 ,,newton has given several specimens1 and has moreover found means to reduce the moon's motion so completely to rule1 that he has framed a theory1 from which the place of that planet may at all times be computed1 very nearly or altogether as exactly1 as the places of the primary planets themselves1 which is much beyond what the greatest astronomers could ever effect4 #b4 ,,the first thing demonstrated of these secondary planets is1 that they are drawn towards their respective primary in the same manner as the primary planets are attracted by the sun4 ,that each secondary planet is kept in its orbit by a power pointed towards the center of the primary planet1 about which the secondary revolves2 and that the power1 by which the secondaries of the same primary are influenced1 bears the same relation to the distance from the primary1 as the power1 by which the primary planets are guided1 does in regard to the distance from the sun#agb4 ,this is proved in the satellites of ,jupiter and ,saturn1 because they move in circles1 as far as we can observe1 about their respective primary with an equable course1 the respective primary being the center of each orbit3 and by comparing the times1 in which the different satellites of the same primary perform their periods1 they are found to observe the same relation to the distances from their primary1 as the primary planets observe in respect of their mean distances from the sun#agc4 ,here these bodies moving in circles with an equable motion1 each satellite passes over equal parts of its orbit in equal portions of time2 consequently the line drawn from the center of the orbit1 that is1 from the primary planet1 to the satellite1 will pass over equal spaces along with the satellite in equal portions of time2 which proves the power1 by which each satellite is held in its orbit1 to be pointed towards the primary as a center#agd4 ,it is also manifest that the centripetal power1 which carries a body in a circle concentrical with the power1 acts upon the body at all times with the same strength4 ,but ,sir ,,isaac ,,newton demonstrates that1 when bodies are carried in different circles by centripetal powers directed to the centers of those circles1 then1 the degrees of strength of those powers are to be compared by considering the relation between the times1 in which the bodies perform their periods through those circles#age2 and in particular he shews1 that if the periodical times bear that relation1 which ,i have just now asserted the satellites of the same primary to observe2 then the centripetal powers are reciprocally in the duplicate proportion of the semidiameters of the circles1 or in that proportion to the distances of the bodies from the centers#agf4 ,hence it follows that in the planets ,jupiter and ,saturn1 the centripetal power in each decreases with the increase of distance1 in the same proportion as the centripetal power appertaining to the sun decreases with the increase of distance4 ,i do not here mean that this proportion of the centripetal powers holds between the power of ,jupiter at any distance compared with the power of ,saturn at any other distance2 but only in the change of strength of the power belonging to the same planet at different distances from him4 ,moreover what is here discovered of the planets ,jupiter and ,saturn by means of the different satellites1 which revolve round each of them1 appears in the earth by the moon alone2 because she is found to move round the earth in an ellipsis after the same manner as the primary planets do about the sun2 excepting only some small irregularities in her motion1 the cause of which will be particularly explained in what follows1 whereby it will appear1 that they are no objection against the earth's acting on the moon in the same manner as the sun acts on the primary planets2 that is1 as the other primary planets ,jupiter and ,saturn act upon their satellites4 ,certainly since these irregularities can be otherwise accounted for1 we ought not to depart from that rule of induction so necessary in philosophy1 that to like bodies like properties are to be attributed1 where no reason to the contrary appears4 ,we cannot therefore but ascribe to the earth the same kind of action upon the moon1 as the other primary planets ,jupiter and ,saturn have upon their satellites2 which is known to be very exactly in the proportion assigned by the method of comparing the periodical times and distances of all the satellites which move about the same planet2 this abundantly compensating our not being near enough to observe the exact figure of their orbits4 ,for if the little deviation of the moon's orbit orbit from a true permanent ellipsis arose from the action of the earth upon the moon not being in the exact reciprocal duplicate proportion of the distance1 were another moon to revolve about the earth1 the proportion between the periodical times of this new moon1 and the present1 would discover the deviation from the mentioned proportion much more manifestly4 #c4 ,,by the number of satellites1 which move round ,jupiter and ,saturn1 the power of each of these planets is measured in a great diversity of distance2 for the distance of the outermost satellite in each of these planets exceeds several times the distance of the innermost4 ,in ,jupiter the astronomers have usually placed the innermost satellite at a distance from the center of that planet equal to about #e of the semidiameters of ,jupiter's body1 and this satellite performs its revolution in about #a day #ah hours4 ,the next satellite1 which revolves round ,jupiter in about #c days #ac hours1 they place at the distance from ,jupiter of about #i of that planet's semidiameters4 ,to the third satellite1 which performs its period nearly in #g days #c hours1 they assign the distance of about #ad semidiameters4 ,but the outermost satellite they remove to #be semidiameters1 and this satellite makes its period in about #af days #af hours#agg4 ,in ,saturn there is still a greater diversity in the distance of the several satellites4 ,by the observations of the late ,,cassini1 a celebrated astronomer in ,france1 who first discovered all these satellites1 except one known before1 the innermost is distant about #d of ,saturn's semidiameters from his center1 and revolves round in about #a day #ba hours4 ,the next satellite is distant about #e semidiameters1 and makes its period in about #b days #ag hours4 ,the third is removed to the distance of about #h semidiameters1 and performs its revolution in near #d days #ab hours4 ,the fourth satellite discovered first by the great ,,huygens1 is near #ah semidiameters1 and moves round ,saturn in about #ae days #bb hours4 ,the outermost is distant #ef semidiameters1 and makes its revolution in about #gi days #g hours#agh4 ,besides these satellites1 there belongs to the planet ,saturn another body of a very singular kind4 ,this is a shining1 broad1 and flat ring1 which encompasses the planet round4 ,the diameter of the outermost verge of this ring is more than double the diameter of ,saturn4 ,,huygens1 who first described this ring1 makes the whole diameter thereof to bear to the diameter of ,saturn the proportion of #i to #d4 ,the late reverend ,mr4 ,,pound makes the proportion something greater1 viz4 that of #g to #c4 ,the distances of the satellites of this planet ,saturn are compared by ,,cassini to the diameter of the ring4 ,his numbers ,i have reduced to those above1 according to ,mr4 ,,pound's proportion between the diameters of ,saturn and of his ring4 ,as this ring appears to adhere no where to ,saturn1 so the distance of ,saturn from the inner edge of the ring seems rather greater than the breadth of the ring4 ,the distances1 which have here been given1 of the several satellites1 both for ,jupiter and ,saturn1 may be more depended on in relation to the proportion1 which those belonging to the same primary planet bear one to another1 than in respect to the very numbers1 that have been here set down1 by reason of the difficulty there is in measuring to the greatest exactness the diameters of the primary planets2 as will be explained hereafter1 when we come to treat of telescopes#agi4 ,by the observations of the forementioned ,mr4 ,,pound1 in ,jupiter the distance of the innermost satellite should rather be about #f semidiameters1 of the second #i-1 of the third #ae1 and of the outermost #bfahj2 and in ,saturn the distance of the innermost satellite #d semidiameters1 of the next #f1 of the third #h1 of the fourth #bj1 and of the fifth #eiaha4 ,however the proportion between the distances of the satellites in the same primary is the only thing necessary to the point we are here upon4 #d4 ,,but moreover the force1 wherewith the earth acts in different distances1 is confirmed from the following consideration1 yet more expresly than by the preceding analogical reasoning4 ,it will appear1 that if the power of the earth1 by which it retains the moon in her orbit1 be supposed to act at all distances between the earth and moon1 according to the forementioned rule2 this power will be sufficient to produce upon bodies1 near the surface of the earth1 all the effects ascribed to the principle of gravity4 ,this is discovered by the following method4 ,let ,a 7in fig4 #id47 represent the earth1 ,b the moon1 ,b ,c ,d the moon's orbit1 which differs little from a circle1 of which ,a is the center4 ,if the moon in ,b were left to it self to move with the velocity1 it has in the point ,b1 it would leave the orbit1 and proceed right forward in the line ,b ,e1 which touches the orbit in ,b4 ,suppose the moon would upon this condition move from ,b to ,e in the space of one minute of time4 ,by the action of the earth upon the moon1 whereby it is retained in its orbit1 the moon will really be found at the end of this minute in the point ,f1 from whence a straight line drawn to ,a shall make the space ,b ,f ,a in the circle equal to the triangular space ,b ,e ,a2 so that the moon in the time wherein it would have moved from ,b to ,e1 if left to it self1 has been impelled towards the earth from ,e to ,f4 ,and when the time of the moon's passing from ,b to ,f is small1 as here it is only one minute1 the distance between ,e and ,f scarce differs from the space1 through which the moon would descend in the same time1 if it were to fall directly down from ,b toward ,a without any other motion4 ,a ,b the distance of the earth and moon is about #fj of the earth's semidiameters1 and the moon completes her revolution round the earth in about #bg days #g hours and #dc minutes3 therefore the space ,e ,f will here be found by computation to be about #af feet4 ,consequently1 if the power1 by which the moon is retained in its orbit1 be near the surface of the earth greater1 than at the distance of the moon in the duplicate proportion of that distance2 the number of feet1 a body would descend near the surface of the earth by the action of this power upon it in one minute of time1 would be equal to #af multiplied twice into the number #fj1 that is1 equal to #ehjej4 ,but how fast bodies fall near the surface of the earth may be known by the pendulum#ahb2 and by the exactest experiments they are found to descend the space of #af feet in a second of time2 and the spaces described by falling bodies being in the duplicate proportion of the times of their fall#ahc1 the number of feet1 a body would describe in its fall near the surface of the earth in one minute of time1 will be equal to #af twice multiplied by #fj1 the same as would be caused by the power which acts upon the moon4 #e4 ,,in this computation the earth is supposed to be at rest1 whereas it would have been more exact to have supposed it to move1 as well as the moon1 about their common center of gravity2 as will easily be understood1 by what has been said in the preceding chapter1 where it was shewn1 that the sun is subjected to the like motion about the common center of gravity of it self and the planets4 ,the action of the sun upon the moon1 which is to be explain'd in what follows1 is likewise here neglected3 and ,sir ,,isaac ,,newton shews1 if you take in both these considerations1 the present computation will best agree to a somewhat greater distance of the moon and earth1 viz4 to #fj semidiameters of the earth1 which distance is more conformable to astronomical observations4 #f4 ,,these computations afford an additional proof1 that the action of the earth observes the same proportion to the distance1 which is here contended for4 ,before ,i said1 it was reasonable to conclude so by induction from the planets ,jupiter and ,saturn2 because they act in that manner4 ,but now the same thing will be evident by drawing no other consequence from what is seen in those planets1 than that the power1 by which the primary planets act on their secondary1 is extended from the primary through the whole interval between1 so that it would act in every part of the intermediate space4 ,in ,jupiter and ,saturn this power is so far from being confined to a small extent of distance1 that it not only reaches to several satellites at very different distances1 but also from one planet to the other1 nay even through the whole planetary system#ahd4 ,consequently there is no appearance of reason1 why this power should not act at all distances1 even at the very surfaces of these planets as well as farther off4 ,but from hence it follows1 that the power1 which retains the moon in her orbit1 is the same1 as causes bodies near the surface of the earth to gravitate4 ,for since the power1 by which the earth acts on the moon1 will cause bodies near the surface of the earth to descend with all the velocity they are found to do1 it is certain no other power can act upon them besides2 because if it did1 they must of necessity descend swifter4 ,now from all this it is at length very evident1 that the power in the earth1 which we call gravity1 extends up to the moon1 and decreases in the duplicate proportion of the increase of the distance from the earth4 #g4 ,,this finishes the discoveries made in the action of the primary planets upon their secondary4 ,the next thing to be shewn is1 that the sun acts upon them likewise3 for this purpose it is to be observed1 that if to the motion of the satellite1 whereby it would be carried round its primary at rest1 be superadded the same motion both in regard to velocity and direction1 as the primary it self has1 it will describe about the primary the same orbit1 with as great regularity1 as if the primary was indeed at rest4 ,the cause of this is that law of motion1 which makes a body near the surface of the earth1 when let fall1 to descend perpendicularly1 though the earth be in so swift a motion1 that if the falling body did not partake of it1 its descent would be remarkably oblique2 and that a body projected describes in the most regular manner the same parabola1 whether projected in the direction1 in which the earth moves1 or in the opposite direction1 if the projecting force be the same#ahe4 ,from this we learn1 that if the satellite moved about its primary with perfect regularity1 besides its motion about the primary1 it would participate of all the motion of its primary2 have the same progressive velocity1 with which the primary is carried about the sun2 and be impelled with the same velocity as the primary towards the sun1 in a direction parallel to that impulse of its primary4 ,and on the contrary1 the want of either of these1 in particular of the impulse towards the sun1 will occasion great inequalities in the motion of the secondary planet4 ,the inequalities1 which would arise from the absence of this impulse towards the sun are so great1 that by the regularity1 which appears in the motion of the secondary planets1 it is proved1 that the sun communicates1 the same velocity to them by its action1 as it gives to their primary at the same distance4 ,for ,sir ,,isaac ,,newton informs us1 that upon examination he found1 that if any of the satellites of ,jupiter were attracted by the sun more or less1 than ,jupiter himself at the same distance1 the orbit of that satellite1 instead of being concentrical to ,jupiter1 must have its center at a greater or less distance1 than the center of ,jupiter from the sun1 nearly in the subduplicate proportion of the difference between the sun's action upon the satellite1 and upon ,jupiter2 and therefore if any satellite were attracted by the sun but #a_/ajjj part more or less1 than ,jupiter is at the same distance1 the center of the orbit of that satellite would be distant from the center of ,jupiter no less than a fifth part of the distance of the outermost satellite from ,jupiter#ahf2 which is almost the whole distance of the innermost satellite4 ,by the like argument the satellites of ,saturn gravitate towards the sun1 as much as ,saturn it self at the same distance2 and the moon as much as the earth4 #h4 ,,thus is proved1 that the sun acts upon the secondary planets1 as much as upon the primary at the same distance3 but it was found in the last chapter1 that the action of the sun upon bodies is reciprocally in the duplicate proportion of the distance2 therefore the secondary planets being sometimes nearer to the sun than the primary1 and sometimes more remote1 they are not alway acted upon in the same degree with their primary1 but when nearer to the sun1 are attracted more1 and when farther distant1 are attracted less4 ,hence arise various inequalities in the motion of the secondary planets#ahg4 #i4 ,,some of these inequalities would take place1 though the moon1 if undisturbed by the sun1 would have moved in a circle concentrical to the earth1 and in the plane of the earth's motion2 others depend on the elliptical figure1 and the oblique situation of the moon's orbit4 ,one of the first kind is1 that the moon is caused so to move1 as not to describe equal spaces in equal times1 but is continually accelerated1 as she passes from the quarter to the new or full1 and is retarded again by the like degrees in returning from the new and full to the next quarter4 ,here we consider not so much the absolute1 as the apparent motion of the moon in respect to us4 #aj4 ,,the principles of astronomy teach how to distinguish these two motions4 ,let ,s 7in fig4 #ie47 represent the sun1 ,a the earth moving in its orbit ,b ,c1 ,d ,e ,f ,g the moon's orbit1 the place of the moon ,h4 ,suppose the earth to have moved from ,a to ,i4 ,because it has been shewn1 that the moon partakes of all the progressive motion of the earth2 and likewise that the sun attracts both the earth and moon equally1 when they are at the same distance from it1 or that the mean action of the sun upon the moon is equal to its action upon the earth3 we must therefore consider the earth as carrying about with it the moon's orbit2 so that when the earth is removed from ,a to ,i1 the moon's orbit shall likewise be removed from its former situation into that denoted by ,k ,l ,m ,n4 ,but now the earth being in ,i1 if the moon were found in ,o1 so that ,o ,i should be parallel to ,h ,a1 though the moon would really have moved from ,h to ,o1 yet it would not have appeared to a spectator upon the earth to have moved at all1 because the earth has moved as much it self2 so that the moon would still appear in the same place with respect to the fixed stars4 ,but if the moon be observed in ,p1 it will then appear to have moved1 its apparent motion being measured by the angle under ,o ,i ,p4 ,and if the angle under ,p ,i ,s be less than the angle under ,h ,a ,s1 the moon will have approached nearer to its conjunction with the sun4 #aa4 ,,to come now to the explication of the mentioned inequality in the moon's motion3 let ,s 7in fig4 #if47 represent the sun1 ,a the earth1 ,b ,c ,d ,e the moon's orbit1 ,c the place of the moon1 when in the latter quarter4 ,here it will be nearly at the same distance from the sun1 as the earth is4 ,in this case therefore they will both be equally attracted1 the earth in the direction ,a ,s1 and the moon in the direction ,c ,s4 ,whence as the earth in moving round the sun is continually descending toward it1 so the moon in this situation must in any equal portion of time descend as much2 and therefore the position of the line ,a ,c in respect of ,a ,s1 and the change1 which the moon's motion produces in the angle under ,c ,a ,s1 will not be altered by the sun4 #ab4 ,,but now as soon as ever the moon is advanced from the quarter toward the new or conjunction1 suppose to ,g1 the action of the sun upon it will have a different effect4 ,here1 were the sun's action upon the moon to be applied in the direction ,g ,h parallel to ,a ,s1 if its action on the moon were equal to its action on the earth1 no change would be wrought by the sun on the apparent motion of the moon round the earth4 ,but the moon receiving a greater impulse in ,g than the earth receives in ,a1 were the sun to act in the direction ,g ,h1 yet it would accelerate the description of the space ,d ,a ,g1 and cause the angle under ,g ,a ,d to decrease faster1 than otherwise it would4 ,the sun's action will have this effect upon account of the obliquity of its direction to that1 in which the earth attracts the moon4 ,for the moon by this means is drawn by two forces oblique to each other1 one drawing from ,g toward ,a1 the other from ,g toward ,h1 therefore the moon must necessarily be impelled toward ,d4 ,again1 because the sun does not act in the direction ,g ,h parallel to ,s ,a1 but in the direction ,g ,s oblique to it1 the sun's action on the moon will by reason of this obliquity farther contribute to the moon's acceleration4 ,suppose the earth in any short space of time would have moved from ,a to ,i1 if not attracted by the sun2 the point ,i being in the straight line ,c ,e1 which touches the earth's orbit in ,a4 ,suppose the moon in the same time would have moved in her orbit from ,g to ,k1 and besides have partook of all the progressive motion of the earth4 ,then if ,k ,l be drawn parallel to ,a ,i1 and taken equal to it1 the moon1 if not attracted by the sun1 would be found in ,l4 ,but the earth by the sun's action is removed from ,i4 ,suppose it were moved down to ,m in the line ,i ,m ,n parallel to ,s ,a1 and if the moon were attracted but as much1 and in the same direction1 as the earth is here supposed to be attracted1 so as to have descended during the same time in the line ,l ,o1 parallel also to ,a ,s1 down as far as ,p1 till ,l ,p were equal to ,i ,m2 the angle under ,p ,m ,n would be equal to that under ,l ,i ,n1 that is1 the moon will appear advanced no farther forward1 than if neither it nor the earth had been subject to the sun's action4 ,but this is upon the supposition1 that the action of the sun upon the moon and earth were equal2 whereas the moon being acted upon more than the earth1 did the sun's action draw the moon in the line ,l ,o parallel to ,a ,s1 it would draw it down so far as to make ,l ,p greater than ,i ,m2 whereby the angle under ,p ,m ,n will be rendred less1 than that under ,l ,i ,n4 ,but moreover1 as the sun draws the earth in a direction oblique to ,i ,n1 the earth will be found in its orbit somewhat short of the point ,m2 however the moon is attracted by the sun still more out of the line ,l ,o1 than the earth is out of the line ,i ,n2 therefore this obliquity of the sun's action will yet farther diminish the angle under ,p ,m ,n4 #ac4 ,,thus the moon at the point ,g receives an impulse from the sun1 whereby her motion is accelerated4 ,and the sun producing this effect in every place between the quarter and the conjunction1 the moon will move from the quarter with a motion continually more and more accelerated2 and therefore by acquiring from time to time additional degrees of velocity in its orbit1 the spaces1 which are described in equal times by the line drawn from the earth to the moon1 will not be every where equal1 but those toward the conjunction will be greater1 than those toward the quarter4 ,but now in the moon's passage from the conjunction ,d to the next quarter the sun's action will again retard the moon1 till at the next quarter in ,e it be restored to the first velocity1 which it had in ,c4 #ad4 ,,again as the moon moves from ,e to the full or opposition to the sun in ,b1 it is again accelerated1 the deficiency of the sun's action upon the moon1 from what it has upon the earth1 producing here the same effect as before the excess of its action4 ,consider the moon in ,q1 moving from ,e towards ,b4 ,here if the moon were attracted by the sun in a direction parallel to ,a ,s1 yet being acted on less than the earth1 as the earth descends toward the sun1 the moon will in some measure be left behind4 ,therefore ,q ,f being drawn parallel to ,s ,b1 a spectator on the earth would see the moon move1 as if attracted from the point ,q in the direction ,q ,f with a degree of force equal to that1 whereby the sun's action on the moon falls short of its action on the earth4 ,but the obliquity of the sun's action has also here an effect4 ,in the time the earth would have moved from ,a to ,i without the influence of the sun1 let the moon have moved in its orbit from ,q to ,r4 ,drawing therefore ,r ,t parallel to ,a ,i1 and equal to the same1 for the like reason as before1 the moon by the motion of its orbit1 if not at all attracted by the sun1 must be found in ,t2 and therefore1 if attracted in a direction parallel to ,s ,a1 would be in the line ,t ,v parallel to ,a ,s2 suppose in ,w4 ,but the moon in ,q being farther off the sun than the earth1 it will be less attracted1 that is1 ,t ,w will be less than ,i ,m1 and if the line ,s ,m be prolonged toward ,x1 the angle under ,x ,m ,w will be less than that under ,x ,i ,t4 ,thus by the sun's action the moon's passage from the quarter to the full would be accelerated1 if the sun were to act on the earth and moon in a direction parallel to ,a ,s3 and the obliquity of the sun's action will still more increase this acceleration4 ,for the action of the sun on the moon is oblique to the line ,s ,a the whole time of the moon's passage from ,q to ,t1 and will carry the moon out of the line ,t ,v toward the earth4 ,here ,i suppose the time of the moon's passage from ,q to ,t so short1 that it shall not pass beyond the line ,s ,a4 ,the earth also will come a little short of the line ,i ,n1 as was said before4 ,from these causes the angle under ,x ,m ,w will be still farther lessened4 #ae4 ,,the moon in passing from the opposition ,b to the next quarter will be retarded again by the same degrees1 as it is accelerated before its appulse to the opposition4 ,because this action of the sun1 which in the moon's passage from the quarter to the opposition causes it to be extraordinarily accelerated1 and diminishes the angle1 which measures its distance from the opposition2 will make the moon slacken its pace afterwards1 and retard the augmentation of the same angle in its passage from the opposition to the following quarter2 that is1 will prevent that angle from increasing so fast1 as otherwise it would4 ,and thus the moon1 by the sun's action upon it1 is twice accelerated and twice restored to its first velocity1 every circuit it makes round the earth4 ,this inequality of the moon's motion about the earth is called by astronomers its variation4 #af4 ,,the next effect of the sun upon the moon is1 that it gives the orbit of the moon in the quarters a greater degree of curvature1 than it would receive from the action of the earth alone2 and on the contrary in the conjunction and opposition the orbit is less inflected4 #ag4 ,,when the moon is in conjunction with the sun in the point ,d1 the sun attracting the moon more forcibly than it does the earth1 the moon by that means is impelled less toward the earth1 than otherwise it would be1 and so the orbit is less incurvated2 for the power1 by which the moon is impelled toward the earth1 being that1 by which it is inflected from a rectilinear course1 the less that power is1 the less it will be inflected4 ,again1 when the moon is in the opposition in ,b1 farther removed from the sun than the earth is2 it follows then1 though the earth and moon are both continually descending to the sun1 that is1 are drawn by the sun toward it self out of the place they would otherwise move into1 yet the moon descends with less velocity than the earth2 insomuch that the moon in any given space of time from its passing the point of opposition will have less approached the earth1 than otherwise it would have done1 that is1 its orbit in respect of the earth will approach nearer to a straight line4 ,in the last place1 when the moon is in the quarter in ,f1 and equally distant from the sun as the earth1 we observed before1 that the earth and moon would descend with equal pace toward the sun1 so as to make no change by that descent in the angle under ,f ,a ,s2 but the length of the line ,f ,a must of necessity be shortned4 ,therefore the moon in moving from ,f toward the conjunction with the sun will be impelled more toward the earth by the sun's action1 than it would have been by the earth alone1 if neither the earth nor moon had been acted on by the sun2 so that by this additional impulse the orbit is rendred more curve1 than it would otherwise be4 ,the same effect will also be produced in the other quarter4 #ah4 ,,another effect of the sun's action1 consequent upon this we have now explained1 is1 that though the moon undisturbed by the sun might move in a circle having the earth for its center2 by the sun's action1 if the earth were to be in the very middle or center of the moon's orbit1 yet the moon would be nearer the earth at the new and full1 than in the quarters4 ,in this probably will at first appear some difficulty1 that the moon should come nearest to the earth1 where it is least attracted to it1 and be farthest off when most attracted4 ,which yet will appear evidently to follow from that very cause1 by considering what was last shewn1 that the orbit of the moon in the conjunction and opposition is rendred less curve2 for the less curve the orbit of the moon is1 the less will the moon have descended from the place it would move into1 without the action of the earth4 ,now if the moon were to move from any place without farther disturbance from that action1 since it would proceed in the line1 which would touch its orbit in that place1 it would recede continually from the earth2 and therefore if the power of the earth upon the moon1 be sufficient to retain it at the same distance1 this diminution of that power will cause the distance to increase1 though in a less degree4 ,but on the other hand in the quarters1 the moon1 being pressed more towards the earth than by the earth's single action1 will be made to approach it2 so that in passing from the conjunction or opposition to the quarters the moon ascends from the earth1 and in passing from the quarters to the conjunction and opposition it descends again1 becoming nearer in these last mentioned places than in the other4 #ai4 ,,all these forementioned inequalities are of different degrees1 according as the sun is more or less distant from the earth2 greater when the earth is nearest the sun1 and less when it is farthest off4 ,for in the quarters1 the nearer the moon is to the sun1 the greater is the addition to the earth's action upon it by the power of the sun2 and in the conjunction and opposition1 the difference between the sun's action upon the earth and upon the moon is likewise so much the greater4 #bj4 ,this difference in the distance between the earth and the sun produces a farther effect upon the moon's motion2 causing the orbit to dilate when less remote from the sun1 and become greater1 than when at a farther distance4 ,for it is proved by ,sir ,,isaac ,,newton1 that the action of the sun1 by which it diminishes the earth's power over the moon1 in the conjunction or opposition1 is about twice as great1 as the addition to the earth's action by the sun in the quarters#ahh2 so that upon the whole1 the power of the earth upon the moon is diminished by the sun1 and therefore is most diminished1 when the action of the sun is strongest3 but as the earth by its approach to the sun has its influence lessened1 the moon being less attracted will gradually recede from the earth2 and as the earth in its recess from the sun recovers by degrees its former power1 the orbit of the moon must again contract4 ,two consequences follow from hence3 the moon will be most remote from the earth1 when the earth is nearest the sun2 and also will take up a longer time in performing its revolution through the dilated orbit1 than through the more contracted4 #ba4 ,,these irregularities the sun would produce in the moon1 if the moon1 without being acted on unequally by the sun1 would describe a perfect circle about the earth1 and in the plane of the earth's motion2 but though neither of these suppositions obtain in the motion of the moon1 yet the forementioned inequalities will take place1 only with some difference in respect to the degree of them2 but the moon by not moving in this manner is subject to some other inequalities also4 ,for as the moon describes1 instead of a circle concentrical to the earth1 an ellipsis1 with the earth in one focus1 that ellipsis will be subjected to various changes4 ,it can neither preserve constantly the same position1 nor yet the same figure2 and because the plane of this ellipsis is not the same with that of the earth's orbit1 the situation of the plane1 wherein the moon moves1 will continually change2 neither the line in which it intersects the plane of the earth's orbit1 nor the inclination of the planes to each other1 will remain for any time the same4 ,all these alterations offer themselves now to be explained4 #bb4 ,i ,,shall first consider the changes which are made in the plane of the moon's orbit4 ,the moon not moving in the same plane with the earth1 the sun is seldom in the plane of the moon's orbit1 viz4 only when the line made by the common intersection of the two planes1 if produced1 will pass through the sun1 as is represented in fig4 #ig4 where ,s denotes the sun2 ,t the earth2 ,a ,t ,b the earth's orbit described upon the plane of this scheme2 ,c ,d ,e ,f the moon's orbit1 the part ,c ,d ,e being raised above1 and the part ,c ,f ,e depressed under the plane of this scheme4 ,here the line ,c ,e1 in which the plane of this scheme1 that is1 the plane of the earth's orbit and the plane of the moon's orbit intersect each other1 being continued passes through the sun in ,s4 ,when this happens1 the action of the sun is directed in the plane of the moon's orbit1 and cannot draw the moon out of this plane1 as will evidently appear to any one that shall consider the present scheme3 for suppose the moon in ,g1 and let a straight line be drawn from ,g to ,s1 the sun draws the moon in the direction of this line from ,g toward ,s3 but this line lies in the plane of the orbit2 and if it be prolonged from ,s beyond ,g1 the continuation of it will lie on the plane ,c ,d ,e2 for the plane itself1 if sufficiently extended1 will pass through the sun4 ,but in other cases the obliquity of the sun's action to the plane of the orbit will cause this plane continually to change4 #bc4 ,,suppose in the first place1 the line1 in which the two planes intersect each other1 to be perpendicular to the line which joins the earth and sun4 ,let ,t 7in fig4 #ih1 #ii1 #ajj1 #aja47 represent the earth2 ,s the sun2 the plane of this scheme the plane of the earth's motion1 in which both the sun and earth are placed4 ,let ,a ,c be perpendicular to ,s ,t1 which joins the earth and sun2 and let the line ,a ,c be that1 in which the plane of the moon's orbit intersects the plane of the earth's motion4 ,to the center ,t describe in the plane of the earth's motion the circle ,a ,b ,c ,d4 ,and in the plane of the moon's orbit describe the circle ,a ,e ,c ,f1 one half of which ,a ,e ,c will be elevated above the plane of this scheme1 the other half ,a ,f ,c as much depressed below it4 #bd4 ,,now suppose the moon to set forth from the point ,a 7in fig4 #ih47 in the direction of the plane ,a ,e ,c4 ,here she will be continually drawn out of this plane by the action of the sun3 for this plane ,a ,e ,c1 if extended1 will not pass through the sun1 but above it2 so that the sun1 by drawing the moon directly toward it self1 will force it continually more and more from that plane towards the plane of the earth's motion1 in which it self is2 causing it to describe the line ,a ,k ,g ,h ,i1 which will be convex to the plane ,a ,e ,c1 and concave to the plane of the earth's motion4 ,but here this power of the sun1 which is said to draw the moon toward the plane of the earth's motion1 must be understood principally of so much only of the sun's action upon the moon1 as it exceeds the action of the same upon the earth4 ,for suppose the preceding figure to be viewed by the eye1 placed in the plane of that scheme1 and in the line ,c ,t ,a on the side of ,a1 the plane ,a ,b ,c ,d will appear as the straight line ,d ,t ,b1 7in fig4 #ajb47 and the plane ,a ,e ,c ,f as another straight line ,f ,e2 and the curve line ,a ,k ,g ,h ,i under the form of the line ,t ,k ,g ,h ,i4 ,illustration ,now it is plain1 that the earth and moon being both attracted by the sun1 if the sun's action upon both was equally strong1 the earth ,t1 and with it the plane ,a ,e ,c ,f or line ,f ,t ,e in this scheme1 would be carried toward the sun with as great a pace as the moon1 and therefore the moon not drawn out of it by the sun's action1 excepting only from the small obliquity of the direction of this action upon the moon to that of the sun's action upon the earth1 which arises from the moon's being out of the plane of the earth's motion1 and is not very considerable2 but the action of the sun upon the moon being greater than upon the earth1 all the time the moon is nearer to the sun than the earth is1 it will be drawn from the plane ,a ,e ,c or the line ,t ,e by that excess1 and made to describe the curve line ,a ,g ,i or ,t ,g ,i4 ,but it is the custom of astronomers1 instead of considering the moon as moving in such a curve line1 to refer its motion continually to the plane1 which touches the true line wherein it moves1 at the point where at any time the moon is4 ,thus when the moon is in the point ,a1 its motion is considered as being in the plane ,a ,e ,c1 in whose direction it then essaies to move2 and when in the point ,k 7in fig4 #ii47 its motion is referred to the plane1 which passes through the earth1 and touches the line ,a ,k ,g ,h ,i in the point ,k4 ,thus the moon in passing from ,a to ,i will continually change the plane of her motion4 ,in what manner this change proceeds1 ,i shall now particularly explain4 #be4 ,,let the plane1 which touches the line ,a ,k ,i in the point ,k 7in fig4 #ii47 intersect the plane of the earth's orbit in the line ,l ,t ,m4 ,then1 because the line ,a ,k ,i is concave to the plane ,a ,b ,c1 it falls wholly between that plane1 and the plane which touches it in ,k2 so that the plane ,m ,k ,l will cut the plane ,a ,e ,c1 before it meets with the plane of the earth's motion2 suppose in the line ,y ,t1 and the point ,a will fall between ,k and ,l4 ,with a semidiameter equal to ,t ,y or ,t ,l describe the semicircle ,l ,y ,m4 ,now to a spectator on the earth the moon1 when in ,a1 will appear to move in the circle ,a ,e ,c ,f1 and1 when in ,k1 will appear to be moving in the semicircle ,l ,y ,m4 ,the earth's motion is performed in the plane of this scheme1 and to a spectator on the earth the sun will appear always moving in that plane4 ,we may therefore refer the apparent motion of the sun to the circle ,a ,b ,c ,d1 described in this plane about the earth4 ,but the points where this circle1 in which the sun seems to move1 intersects the circle in which the moon is seen at any time to move1 are called the nodes of the moon's orbit at that time4 ,when the moon is seen moving in the circle ,a ,e ,c ,d1 the points ,a and ,c are the nodes of the orbit2 when she appears in the semicircle ,l ,y ,m1 then ,l and ,m are the nodes4 ,now here it appears1 from what has been said1 that while the moon has moved from ,a to ,k1 one of the nodes has been carried from ,a to ,l1 and the other as much from ,c to ,m4 ,but the motion from ,a to ,l1 and from ,c to ,m1 is backward in regard to the motion of the moon1 which is the other way from ,a to ,k1 and from thence toward ,c4 #bf4 ,,farther the angle1 which the plane1 wherein the moon at any time appears1 makes with the plane of the earth's motion1 is called the inclination of the moon's orbit at that time4 ,and ,i shall now proceed to shew1 that this inclination of the orbit1 when the moon is in ,k1 is less than when she was in ,a2 or1 that the plane ,l ,y ,m1 which touches the line of the moon's motion in ,k1 makes a less angle with the plane of the earth's motion or with the circle ,a ,b ,c ,d1 than the plane ,a ,e ,c makes with the same4 ,the semicircle ,l ,y ,m intersects the semicircle ,a ,e ,c in ,y2 and the arch ,a ,y is less than ,l ,y1 and both together less than half a circle4 ,but it is demonstrated by the writers on that part of astronomy1 which is called the doctrine of the sphere1 that when a triangle is made1 as here1 by three arches of circles ,a ,l1 ,a ,y1 and ,y ,l1 the angle under ,y ,a ,b without the triangle is greater than the angle under ,y ,l ,a within1 if the two arches ,a ,y1 ,y ,l taken together do not amount to a semicircle2 if the two arches make a complete semicircle1 the two angles will be equal2 but if the two arches taken together exceed a semicircle1 the inner angle under ,y ,l ,a is greater than the other#ahi4 ,here therefore the two arches ,a ,y and ,l ,y together being less than a semicircle1 the angle under ,a ,l ,y is less1 than the angle under ,b ,a ,e4 ,but from the doctrine of the sphere it is also evident1 that the angle under ,a ,l ,y is equal to that1 in which the plane of the circle ,l ,y ,k ,m1 that is1 the plane which touches the line ,a ,k ,g ,h ,i in ,k1 is inclined to the plane of the earth's motion ,a ,b ,c2 and the angle under ,b ,a ,e is equal to that1 in which the plane ,a ,e ,c is inclined to the same plane4 ,therefore the inclination of the former plane is less than the inclination of the latter4 #bg4 ,,suppose now the moon to be advanced to the point ,g 7in fig4 #ajj47 and in this point to be distant from its node a quarter part of the whole circle2 or in other words1 to be in the midway between its two nodes4 ,and in this case the nodes will have receded yet more1 and the inclination of the orbit be still more diminished3 for suppose the line ,a ,k ,g ,h ,i to be touched in the point ,g by a plane passing through the earth ,t3 let the intersection of this plane with the plane of the earth's motion be the line ,w ,t ,o1 and the line ,t ,p its intersection with the plane ,l ,k ,m4 ,in this plane let the circle ,n ,g ,o be described with the semidiameter ,t ,p or ,n ,t cutting the other circle ,l ,k ,m in ,p4 ,now the line ,a ,k ,g ,i is convex to the plane ,l ,k ,m1 which touches it in ,k2 and therefore the plane ,n ,g ,o1 which touches it in ,g1 will intersect the other touching plane between ,g and ,k2 that is1 the point ,p will fall between those two points1 and the plane continued to the plane of the earth's motion will pass beyond ,l2 so that the points ,n and ,o1 or the places of the nodes1 when the moon is in ,g1 will be farther from ,a and ,c than ,l and ,m1 that is1 will have moved farther backward4 ,besides1 the inclination of the plane ,n ,g ,o to the plane of the earth's motion ,a ,b ,c is less1 than the inclination of the plane ,l ,k ,m to the same2 for here also the two arches ,l ,p and ,n ,p taken together are less than a semicircle1 each of these arches being less than a quarter of a circle2 as appears1 because ,g ,n1 the distance of the moon in ,g from its node ,n1 is here supposed to be a quarter part of a circle4 #bh4 ,,after the moon is passed beyond ,g1 the case is altered2 for then these arches will be greater than quarters of the circle1 by which means the inclination will be again increased1 tho' the nodes still go on to move the same way4 ,suppose the moon in ,h1 7in fig4 #aja47 and that the plane1 which touches the line ,a ,k ,g ,i in ,h1 intersects the plane of the earth's motion in the line ,q ,t ,r1 and the plane ,n ,g ,o in the line ,t ,v1 and besides that the circle ,q ,h ,r be described in that plane2 then1 for the same reason as before1 the point ,v will fall between ,h and ,g1 and the plane ,r ,v ,q will pass beyond the last plane ,o ,v ,n1 causing the points ,q and ,r to fall farther from ,a and ,c than ,n and ,o4 ,but the arches ,n ,v1 ,v ,q are each greater than a quarter of a circle1 ,n ,v the least of them being greater than ,g ,n1 which is a quarter of a circle2 and therefore the two arches ,n ,v and ,v ,q together exceed a semicircle2 consequently the angle under ,b ,q ,v will be greater1 than that under ,b ,n ,v4 #bi4 ,,in the last place1 when the moon is by this attraction of the sun1 drawn at length into the plane of the earth's motion1 the node will have receded yet more1 and the inclination be so much increased1 as to become somewhat more than at first3 for the line ,a ,k ,g ,h ,i being convex to all the planes1 which touch it1 the part ,h ,i will wholly fall between the plane ,q ,v ,r and the plane ,a ,b ,c2 so that the point ,i will fall between ,b and ,r2 and drawing ,i ,t ,w1 the point ,w will be farther remov'd from ,a than ,q4 ,but it is evident1 that the plane1 which passes through the earth ,t1 and touches the line ,a ,g ,i in the point ,i1 will cut the plane of the earth's motion ,a ,b ,c ,d in the line ,i ,t ,w1 and be inclined to the same in the angle under ,h ,i ,b2 so that the node1 which was first in ,a1 after having passed into ,l1 ,n and ,q1 comes at last into the point ,w2 as the node which was at first in ,c has passed successively from thence through the points ,m1 ,o and ,r to ,i3 but the angle under ,h ,i ,b1 which is now the inclination of the orbit to the plane of the ecliptic1 is manifestly not less than the angle under ,e ,c ,b or ,e ,a ,b1 but rather something greater4 #cj4 ,,thus the moon in the case before us1 while it passes from the plane of the earth's motion in the quarter1 till it comes again into the same plane1 has the nodes of its orbit continually moved backward1 and the inclination of its orbit is at first diminished1 viz4 till it comes to ,g in fig4 #ajj1 which is near to its conjunction with the sun1 but afterwards is increased again almost by the same degrees1 till upon the moon's arrival again to the plane of the earth's motion1 the inclination of the orbit is restored to something more than its first magnitude1 though the difference is not very great1 because the points ,i and ,c are not far distant from each other#aij4 #ca4 ,,after the same manner1 if the moon had departed from the quarter in ,c1 it should have described the curve line ,c ,x ,w 7in fig4 #ih47 between the planes ,a ,f ,c and ,a ,d ,c1 which would be convex to the former of those planes1 and concave to the latter2 so that1 here also1 the nodes should continually recede1 and the inclination of the orbit gradually diminish more and more1 till the moon arrived near its opposition to the sun in ,x2 but from that time the inclination should again increase1 till it became a little greater than at first4 ,this will easily appear1 by considering1 that as the action of the sun upon the moon1 by exceeding its action upon the earth1 drew it out of the plane ,a ,e ,c towards the sun1 while the moon passed from ,a to ,i2 so1 during its passage from ,c to ,w1 the moon being all that time farther from the sun than the earth1 it will be attracted less2 and the earth1 together with the plane ,a ,e ,c ,f1 will as it were be drawn from the moon1 in such sort1 that the path the moon describes shall appear from the earth1 as it did in the former case by the moon's being drawn away4 #cb4 ,,these are the changes1 which the nodes and the inclination of the moon's orbit undergo1 when the nodes are in the quarters2 but when the nodes by their motion1 and the motion of the sun together1 come to be situated between the quarter and conjunction or opposition1 their motion and the change made in the inclination of the orbit are somewhat different4 #cc4 ,,let ,a ,g ,c ,h 7in fig4 #ajc47 be a circle described in the plane of the earth's motion1 having the earth in ,t for its center4 ,let the point opposite to the sun be ,a1 and the point ,g a fourth part of the circle distant from ,a4 ,let the nodes of the moon's orbit be situated in the line ,b ,t ,d1 and ,b the node1 falling between ,a1 the place where the moon would be in the full1 and ,g the place where the moon would be in the quarter4 ,suppose ,b ,e ,d ,f to be the plane1 in which the moon essays to move1 when it proceeds from the point ,b4 ,because the moon in ,b is more distant from the sun than the earth1 it shall be less attracted by the sun1 and shall not descend towards the sun so fast as the earth3 consequently it shall quit the plane ,b ,e ,d ,f1 which we suppose to accompany the earth1 and describe the line ,b ,i ,k convex thereto1 till such time as it comes to the point ,k1 where it will be in the quarter3 but from thenceforth being more attracted than the earth1 the moon shall change its course1 and the following part of the path it describes shall be concave to the plane ,b ,e ,d or ,b ,g ,d1 and shall continue concave to the plane ,b ,g ,d1 till it crosses that plane in ,l1 just as in the preceding case4 ,now ,i say1 while the moon is passing from ,b to ,k1 the nodes1 contrary to what was found in the foregoing case1 will proceed forward1 or move the same way with the moon#aia2 and at the same time the inclination of the orbit will increase#aib4 ,illustration #cd4 ,,when the moon is in the point ,i1 let the plane ,m ,i ,n pass through the earth ,t1 and touch the path of the moon in ,i1 cutting the plane of the earth's motion1 in the line ,m ,t ,n1 and the plane ,b ,e ,d in the line ,t ,o4 ,because the line ,b ,i ,k is convex to the plane ,b ,e ,d1 which touches it in ,b1 the plane ,n ,i ,m must cross the plane ,d ,e ,b1 before it meets the plane ,c ,g ,b2 and therefore the point ,m will fall from ,b towards ,g1 and the node of the moon's orbit being translated from ,b to ,m is moved forward4 #ce4 ,i ,,say farther1 the angle under ,o ,m ,g1 which the plane ,m ,o ,n makes with the plane ,b ,g ,c1 is greater than the angle under ,o ,b ,g1 which the plane ,b ,o ,d makes with the same4 ,this appears from what has been already explained2 because the arches ,b ,o1 ,o ,m are each less than the quarter of a circle1 and therefore taken both together are less than a semicircle4 #cf4 ,,again1 when the moon is come to the point ,k in its quarter1 the nodes will be advanced yet farther forward1 and the inclination of the orbit also more augmented4 ,hitherto the moon's motion has been referred to the plane1 which passing through the earth touches the path of the moon in the point1 where the moon is1 according to what was asserted at the beginning of this discourse upon the nodes1 that it is the custom of astronomers so to do4 ,but here in the point ,k no such plane can be found2 on the contrary1 seeing the line of the moon's motion on one side the point ,k is convex to the plane ,b ,e ,d1 and on the other side concave to the same1 no plane can pass through the points ,t and ,k but will cut the line ,b ,k ,l in that point4 ,therefore instead of such a touching plane1 we must here make use of what is equivalent1 the plane ,p ,k ,q1 with which the line ,b ,k ,l shall make a less angle than with any other plane2 for this plane does as it were touch the line ,b ,k in the point ,k1 since it so cuts it1 that no other plane can be drawn so1 as to pass between the line ,b ,k and the plane ,p ,k ,q4 ,but now it is evident1 that the point ,p1 or the node1 is removed from ,m towards ,g1 that is1 has moved yet farther forward2 and it is likewise as manifest1 that the angle under ,k ,p ,g1 or the inclination of the moon's orbit in the point ,k1 is greater than the angle under ,i ,m ,g1 for the reason so often assigned4 #cg4 ,,after the moon has passed the quarter1 the path of the moon being concave to the plane ,a ,g ,c ,h1 the nodes1 as in the preceding case1 shall recede1 till the moon arrives at the point ,l2 which shews1 that considering the whole time of the moon's passing from ,b to ,l1 at the end of that time the nodes shall be found to have receded1 or to be placed backwarder1 when the moon is in ,l1 than when it was in ,b4 ,for the moon takes a longer time in passing from ,k to ,l1 than in passing from ,b to ,k2 and therefore the nodes continue to recede a longer time1 than they moved forwards2 so that their recess must surmount their advance4 #ch4 ,,in the same manner1 while the moon is in its passage from ,k to ,l1 the inclination of the orbit shall diminish1 till the moon comes to the point1 in which it is one quarter part of a circle distant from its node2 suppose in the point ,r2 and from that time the inclination shall again increase4 ,since therefore the inclination of the orbit increases1 while the moon is passing from ,b to ,k1 and diminishes itself again only1 while the moon is passing from ,k to ,r1 and then augments again1 till the moon arrive in ,l2 while the moon is passing from ,b to ,l1 the inclination of the orbit is much more increased than diminished1 and will be distinguishably greater1 when the moon is come to ,l1 than when it set out from ,b4 #ci4 ,,in like manner1 while the moon is passing from ,l on the other side the plane ,a ,g ,c ,h1 the node shall advance forward1 as long as the moon is between the point ,l and the next quarter2 but afterwards it shall recede1 till the moon come to pass the plane ,a ,g ,c ,h again in the point ,v1 between ,b and ,a3 and because the time between the moon's passing from ,l to the next quarter is less1 than the time between that quarter and the moon's coming to the point ,v1 the node shall have more receded than advanced2 so that the point ,v will be nearer to ,a1 than ,l is to ,c4 ,so also the inclination of the orbit1 when the moon is in ,v1 will be greater1 than when the moon was at ,l2 for this inclination increases all the time the moon is between ,l and the next quarter2 it decreases only while the moon is passing from this quarter to the mid way between the two nodes1 and from thence increases again during the whole passage through the other half of the way to the next node4 #dj4 ,,thus we have traced the moon from her node in the quarter1 and shewn1 that at every period of the moon the nodes will have receded1 and thereby will have approached toward a conjunction with the sun4 ,but this conjunction will be much forwarded by the visible motion of the sun itself4 ,in the last scheme the sun will appear to move from ,s toward ,w4 ,suppose it appeared to have moved from ,s to ,w1 while the moon's node has receded from ,b to ,v1 then drawing the line ,w ,t ,x1 the arch ,v ,x will represent the distance of the line drawn between the nodes from the sun1 when the moon is in ,v2 whereas the arch ,b ,a represented that distance1 when the moon was in ,b4 ,this visible motion of the sun is much greater1 than that of the node2 for the sun appears to revolve quite round each year1 and the node is near #ai years in making one revolution4 ,we have also seen1 that when the node was in the quadrature1 the inclination of the moon's orbit decreased1 till the moon came to the conjunction1 or opposition1 according to which node it set out from2 but that afterwards it again increased1 till it became at the next node rather greater than at the former4 ,when the node is once removed from the quarter nearer to a conjunction with the sun1 the inclination of the moon's orbit1 when the moon comes into the node1 is more sensibly greater1 than it was in the node preceding2 the inclination of the orbit by this means more and more increasing till the node comes into conjunction with the sun2 at which time it has been shewn above1 that the sun has no power to change the plane of the moon's motion2 and consequently has no effect either on the nodes1 or on the inclination of the orbit4 #da4 ,,as soon as the nodes1 by the action of the sun1 are got out of conjunction toward the other quarters1 they begin again to recede as before2 but the inclination of the orbit in the appulse of the moon to each succeeding node is less than at the preceding1 till the nodes come again into the quarters4 ,this will appear as follows4 ,let ,a 7in fig4 #ajd47 represent one of the moon's nodes placed between the point of opposition ,b and the quarter ,c4 ,let the plane ,a ,d ,e pass through the earth ,t1 and touch the path of the moon in ,a4 ,let the line ,a ,f ,g ,h be the path of the moon in her passage from ,a to ,h1 where she crosses again the plane of the earth's motion4 ,this line will be convex toward the plane ,a ,d ,e1 till the moon comes to ,g1 where she is in the quarter2 and after this1 between ,g and ,h1 the same line will be concave toward this plane4 ,all the time this line is convex toward the plane ,a ,d ,e1 the nodes will recede2 and on the contrary proceed1 while it is concave to that plane4 ,all this will easily be conceived from what has been before so largely explained4 ,but the moon is longer in passing from ,a to ,g1 than from ,g to ,h2 therefore the nodes recede a longer time1 than they proceed2 consequently upon the whole1 when the moon is arrived at ,h1 the nodes will have receded1 that is1 the point ,h will fall between ,b and ,e4 ,the inclination of the orbit will decrease1 till the moon is arrived to the point ,f1 in the middle between ,a and ,h4 ,through the passage between ,f and ,g the inclination will increase1 but decrease again in the remaining part of the passage from ,g to ,h1 and consequently at ,h must be less than at ,a4 ,the like effects1 both in respect to the nodes and inclination of the orbit1 will take place in the following passage of the moon on the other side of the plane ,a ,b ,e ,c1 from ,h1 till it comes over that plane again in ,i4 #db4 ,,thus the inclination of the orbit is greatest1 when the line drawn between the moon's nodes will pass through the sun2 and least1 when this line lies in the quarters1 especially if the moon at the same time be in conjunction with the sun1 or in the opposition4 ,in the first of these cases the nodes have no motion1 in all others1 the nodes will each month have receded3 and this regressive motion will be greatest1 when the nodes are in the quarters2 for in that case the nodes have no progressive motion during the whole month1 but in all other cases the nodes do at some times proceed forward1 viz4 whenever the moon is between either quarter1 and the node which is less distant from that quarter than a fourth part of a circle4 #dc4 ,,it now remains only to explain the irregularities in the moon's motion1 which follow from the elliptical figure of the orbit4 ,by what has been said at the beginning of this chapter it appears1 that the power of the earth on the moon acts in the reciprocal duplicate proportion of the distance3 therefore the moon1 if undisturbed by the sun1 would move round the earth in a true ellipsis1 and the line drawn from the earth to the moon would pass over equal spaces in equal portions of time4 ,that this description of the spaces is altered by the sun1 has been already declared4 ,it has also been shown1 that the figure of the orbit is changed each month2 that the moon is nearer the earth at the new and full1 and more remote in the quarters1 than it would be without the sun4 ,now we must pass by these monthly changes1 and consider the effect1 which the sun will have in the different situations of the axis of the orbit in respect of that luminary4 #dd4 ,,the action of the sun varies the force1 wherewith the moon is drawn toward the earth2 in the quarters the force of the earth is directly increased by the sun2 at the new and full the same is diminished2 and in the intermediate places the influence of the earth is sometimes aided1 and sometimes lessened by the sun4 ,in these intermediate places between the quarters and the conjunction or opposition1 the sun's action is so oblique to the action of the earth on the moon1 as to produce that alternate acceleration and retardment of the moon's motion1 which ,i observed above to be stiled the variation4 ,but besides this effect1 the power1 by which the earth attracts the moon toward itself1 will not be at full liberty to act with the same force1 as if the sun acted not at all on the moon4 ,and this effect of the sun's action1 whereby it corroborates or weakens the action of the earth1 is here only to be considered4 ,and by this influence of the sun it comes to pass1 that the power1 by which the moon is impelled toward the earth1 is not perfectly in the reciprocal duplicate proportion of the distance4 ,consequently the moon will not describe a perfect ellipsis4 ,one particular1 wherein the moon's orbit will differ from an ellipsis1 consists in the places1 where the motion of the moon is perpendicular to the line drawn from itself to the earth4 ,in an ellipsis1 after the moon should have set out in the direction perpendicular to this line drawn from itself to the earth1 and at its greatest distance from the earth1 its motion would again become perpendicular to this line drawn between itself and the earth1 and the moon be at its nearest distance from the earth1 when it should have performed half its period2 after performing the other half of its period its motion would again become perpendicular to the forementioned line1 and the moon return into the place whence it set out1 and have recovered again its greatest distance4 ,but the moon in its real motion1 after setting out as before1 sometimes makes more than half a revolution1 before its motion comes again to be perpendicular to the line drawn from itself to the earth1 and the moon is at its nearest distance2 and then performs more than another half of an intire revolution before its motion can a second time recover its perpendicular direction to the line drawn from the moon to the earth1 and the moon arrive again to its greatest distance from the earth4 ,at other times the moon will descend to its nearest distance1 before it has made half a revolution1 and recover again its greatest distance1 before it has made an intire revolution4 ,the place1 where the moon is at its greatest distance from the earth1 is called the moon's apogeon1 and the place of the least distance the perigeon4 ,this change of the place1 where the moon successively comes to its greatest distance from the earth1 is called the motion of the apogeon4 ,in what manner the sun causes the apogeon to move1 ,i shall now endeavour to explain4 #de4 ,,our author shews1 that if the moon were attracted toward the earth by a composition of two powers1 one of which were reciprocally in the duplicate proportion of the distance from the earth1 and the other reciprocally in the triplicate proportion of the same distance2 then1 though the line described by the moon would not be in reality an ellipsis1 yet the moon's motion might be perfectly explained by an ellipsis1 whose axis should be made to move round the earth2 this motion being in consequence1 as astronomers express themselves1 that is1 the same way as the moon itself moves1 if the moon be attracted by the sum of the two powers2 but the axis must move in antecedence1 or the contrary way1 if the moon be acted on by the difference of these powers4 ,what is meant by duplicate proportion has been often explained2 namely1 that if three magnitudes1 as ,a1 ,b1 and ,c1 are so related1 that the second ,b bears the same proportion to the third ,c1 as the first ,a bears to the second ,b1 then the proportion of the first ,a to the third ,c1 is the duplicate of the proportion of the first ,a to the second ,b4 ,now if a fourth magnitude1 as ,d1 be assumed1 to which ,c shall bear the same proportion as ,a bears to ,b1 and ,b to ,c1 then the proportion of ,a to ,d is the triplicate of the proportion of ,a to ,b4 #df4 ,,the way of representing the moon's motion in this case is thus4 ,t denoting the earth 7in fig4 #aje1 #ajf47 suppose the moon in the point ,a1 its apogeon1 or greatest distance from the earth1 moving in the direction ,a ,f perpendicular to ,a ,b1 and acted upon from the earth by two such forces as have been named4 ,by that power alone1 which is reciprocally in the duplicate proportion of the distance1 if the moon let out from the point ,a with a proper degree of velocity1 the ellipsis ,a ,m ,b may be described4 ,but if the moon be acted upon by the sum of the forementioned powers1 and the velocity of the moon in the point ,a be augmented in a certain proportion#aic2 or if that velocity be diminished in a certain proportion1 and the moon be acted upon by the difference of those powers2 in both these cases the line ,a ,e1 which shall be described by the moon1 is thus to be determined4 ,let the point ,m be that1 into which the moon would have arrived in any given space of time1 had it moved in the ellipsis ,a ,m ,b4 ,draw ,m ,t1 and likewise ,c ,t ,d in such sort1 that the angle under ,a ,t ,m shall bear the same proportion to the angle under ,a ,t ,c1 as the velocity1 with which the ellipsis ,a ,m ,b must have been described1 bears to the difference between this velocity1 and the velocity1 with which the moon must set out from the point ,a in order to describe the path ,a ,e4 ,let the angle ,a ,t ,c be taken toward the moon 7as in fig4 #aje47 if the moon be attracted by the sum of the powers2 but the contrary way 7as in fig4 #ajf47 if by their difference4 ,then let the line ,a ,b be moved into the position ,c ,d1 and the ellipsis ,a ,m ,b into the situation ,c ,n ,d1 so that the point ,m be translated to ,l3 then the point ,l shall fall upon the path of the moon ,a ,e4 #dg4 ,,the angular motion of the line ,a ,t1 wereby it is removed into the situation ,c ,t1 represents the motion of the apogeon2 by the means of which the motion of the moon might be fully explicated by the ellipsis ,a ,m ,b1 if the action of the sun upon it was directed to the center of the earth1 and reciprocally in the triplicate proportion of the moon's distance from it4 ,but that not being so1 the apogeon will not move in the regular manner now described4 ,however1 it is to be observed here1 that in the first of the two preceding cases1 where the apogeon moves forward1 the whole centripetal power increases faster1 with the decrease of distance1 than if the intire power were reciprocally in the duplicate proportion of the distance2 because one part only is in that proportion1 and the other part1 which is added to this to make up the whole power1 increases faster with the decrease of distance4 ,on the other hand1 when the centripetal power is the difference between these two1 it increases less with the decrease of the distance1 than if it were simply in the reciprocal duplicate proportion of the distance4 ,therefore if we chuse to explain the moon's motion by an ellipsis 7as is most convenient for astronomical uses to be done1 and by reason of the small effect of the sun's power1 the doing so will not be attended with any sensible error27 we may collect in general1 that when the power1 by which the moon is attracted to the earth1 by varying the distance1 increases in a greater than in the duplicate proportion of the distance diminished1 a motion in consequence must be ascribed to the apogeon2 but that when the attraction increases in a less proportion than that named1 the apogeon must have given to it a motion in antecedence#aid4 ,it is then observed by ,sir ,,is4 ,,newton1 that the first of these cases obtains1 when the moon is in the conjunction and opposition2 and the latter1 when the moon is in the quarters3 so that in the first the apogeon moves according to the order of the signs2 in the other1 the contrary way#aie4 ,but1 as was said before1 the disturbance given to the action of the earth by the sun in the conjunction and opposition being near twice as great as in the quarters#aif1 the apogeon will advance with a greater velocity than recede1 and in the compass of a whole revolution of the moon will be carried in consequence#aig4 #dh4 ,,it is shewn in the next place by our author1 that when the line ,a ,b coincides with that1 which joins the earth and the sun1 the progressive motion of the apogeon1 when the moon is in the conjunction or opposition1 exceeds the regressive in the quadratures more than in any other situation of the line ,a ,b#aih4 ,on the contrary1 when the line ,a ,b makes right angles with that1 which joins the earth and sun1 the retrograde motion will be more considerable#aii1 nay is found so great as to exceed the progressive2 so that in this case the apogeon in the compass of an intire revolution of the moon is carried in antecedence4 ,yet from the considerations in the last paragraph the progressive motion exceeds the other2 so that in the whole the mean motion of the apogeon is in consequence1 according as astronomers find4 ,moreover1 the line ,a ,b changes its situation with that1 which joins the earth and sun1 by such slow degrees1 that the inequalities in the motion of the apogeon arising from this last consideration1 are much greater than what arises from the other#bjj4 #di4 ,,farther1 this unsteady motion in the apogeon is attended with another inequality in the motion of the moon1 that it cannot be explained at all times by the same ellipsis4 ,the ellipsis in general is called by astronomers an eccentric orbit4 ,the point1 in which the two axis's cross1 is called the center of the figure2 because all lines drawn through this point within the ellipsis1 from side to side1 are divided in the middle by this point4 ,but the center1 about which the heavenly bodies revolve1 lying out of this center of the figure in one focus1 these orbits are said to be eccentric2 and where the distance of the focus from this center bears the greatest proportion to the whole axis1 that orbit is called the most eccentric3 and in such an orbit the distance from the focus to the remoter extremity of the axis bears the greatest proportion to the distance of the nearer extremity4 ,now whenever the apogeon of the moon moves in consequence1 the moon's motion must be referred to an orbit more eccentric1 than what the moon would describe1 if the whole power1 by which the moon was acted on in its passing from the apogeon1 changed according to the reciprocal duplicate proportion of the distance from the earth1 and by that means the moon did describe an immoveable ellipsis2 and when the apogeon moves in antecedence1 the moon's motion must be referred to an orbit less eccentric4 ,in the first of the two figures last referred to1 the true place of the moon ,l falls without the orbit ,a ,m ,b1 to which its motion is referred3 whence the orbit ,a ,l ,e1 truly described by the moon1 is less incurvated in the point ,a1 than is the orbit ,a ,m ,b2 therefore the orbit ,a ,m ,b is more oblong1 and differs farther from a circle1 than the ellipsis would1 whose curvature in ,a were equal to that of the line ,a ,l ,b1 that is1 the proportion of the distance of the earth ,t from the center of the ellipsis to its axis will be greater in the ellipsis ,a ,m ,b1 than in the other2 but that other is the ellipsis1 which the moon would describe1 if the power acting upon it in the point ,a were altered in the reciprocal duplicate proportion of the distance4 ,in the second figure1 when the apogeon recedes1 the place of the moon ,l falls within the orbit ,a ,m ,b1 and therefore that orbit is less eccentric1 than the immoveable orbit which the moon should describe4 ,the truth of this is evident2 for1 when the apogeon moves forward1 the power1 by which the moon is influenced in its descent from the apogeon1 increases faster with the decrease of distance1 than in the duplicate proportion of the distance2 and consequently the moon being drawn more forcibly toward the earth1 it will descend nearer to it4 ,on the other hand1 when the apogeon recedes1 the power acting on the moon increases with the decrease of distance in less than the duplicate proportion of the distance2 and therefore the moon is less impelled toward the earth1 and will not descend so low4 #ej4 ,,now suppose in the first of these figures1 that the apogeon ,a is in the situation1 where it is approaching toward the conjunction or opposition of the sun4 ,in this case the progressive motion of the apogeon is more and more accelerated4 ,here suppose that the moon1 after having descended from ,a through the orbit ,a ,e as far as ,f1 where it is come to its nearest distance from the earth1 ascends again up the line ,f ,g4 ,because the motion of the apogeon is here continually more and more accelerating1 the cause of its motion is constantly upon the increase2 that is1 the power1 whereby the moon is drawn to the earth1 will decrease with the increase of distance1 in the moon's ascent from ,f1 in a greater proportion than that wherewith it increased with the decrease of distance in the moon's descent to ,f4 ,consequently the moon will ascend higher than to the distance ,a ,t1 from whence it descended2 therefore the proportion of the greatest distance of the moon to the least is increased4 ,and when the moon descends again1 the power will yet more increase with the decrease of distance1 than in the last ascent it decreased with the augmentation of distance2 the moon therefore must descend nearer to the earth than it did before1 and the proportion of the greatest distance to the least yet be more increased4 ,thus as long as the apogeon is advancing toward the conjunction or opposition1 the proportion of the greatest distance of the moon from the earth to the least will continually increase2 and the elliptical orbit1 to which the moon's motion is referred1 will be rendered more and more eccentric4 #ea4 ,,as soon as the apogeon is passed the conjunction with the sun or the opposition1 the progressive motion thereof abates1 and with it the proportion of the greatest distance of the moon from the earth to the least distance will also diminish2 and when the apogeon becomes regressive1 the diminution of this proportion will be still farther continued on1 till the apogeon comes into the quarter2 from thence this proportion1 and the eccentricity of the orbit will increase again4 ,thus the orbit of the moon is most eccentric1 when the apogeon is in conjunction with the sun1 or in opposition to it1 and least of all when the apogeon is in the quarters4 #eb4 ,,these changes in the nodes1 in the inclination of the orbit to the plane of the earth's motion1 in the apogeon1 and in the eccentricity1 are varied like the other inequalities in the motion of the moon1 by the different distance of the earth from the sun2 being greatest1 when their cause is greatest1 that is1 when the earth is nearest to the sun4 #ec4 ,i ,,said at the beginning of this chapter1 that ,sir ,,isaac ,,newton has computed the very quantity of many of the moon's inequalities4 ,that acceleration of the moon's motion1 which is called the variation1 when greatest1 removes the moon out of the place1 in which it would otherwise be found1 something more than half a degree#bja4 ,in the phrase of astronomers1 a degree is #a_/cfj part of the whole circuit of the moon or any planet4 ,if the moon1 without disturbance from the sun1 would have described a circle concentrical to the earth1 the sun will cause the moon to approach nearer to the earth in the conjunction and opposition1 than in the quarters1 nearly in the proportion of #fi to #gjbjb4 ,we had occasion to mention above1 that the nodes perform their period in almost #ai years4 ,this the astronomers found by observation2 and our author's computations assign to them the same period#bjc4 ,the inclination of the moon's orbit when least1 is an angle about #a_/ah part of that angle1 which constitutes a perpendicular2 and the difference between the greatest and least inclination of the orbit is determined by our author's computation to be about #a_/ah of the least inclination#bjd4 ,and this also is agreeable to the observations of astronomers4 ,the motion of the apogeon1 and the changes in the eccentricity1 ,sir ,,isaac ,,newton has not computed4 ,the apogeon performs its revolution in about eight years and ten months4 ,when the moon's orbit is most eccentric1 the greatest distance of the moon from the earth bears to the least distance nearly the proportion of #h to #g2 when the orbit is least eccentric1 this proportion is hardly so great as that of #ab to #aa4 #ed4 ,,sir ,,isaac ,,newton shews farther1 how1 by comparing the periods of the motion of the satellites1 which revolve round ,jupiter and ,saturn1 with the period of our moon round the earth1 and the periods of those planets round the sun with the period of our earth's motion1 the inequalities in the motion of those satellites may be derived from the inequalities in the moon's motion2 excepting only in regard to that motion of the axis of the orbit1 which in the moon makes the motion of the apogeon2 for the orbits of those satellites1 as far as can be discerned by us at this distance1 appearing little or nothing eccentric1 this motion1 as deduced from the moon1 must be diminished4 ,,chap4 ,,iv4 ,of ,comets4 ,,in the former of the two preceding chapters the powers have been explained1 which keep in motion those celestial bodies1 whose courses had been well determined by the astronomers4 ,in the last chapter we have shewn1 how those powers have been applied by our author to the making a more perfect discovery of the motion of those bodies1 the courses of which were but imperfectly understood2 for some of the inequalities1 which we have been describing in the moon's motion1 were unknown to the astronomers4 ,in this chapter we are to treat of a third species of the heavenly bodies1 the true motion of which was not at all apprehended before our author writ2 in so much1 that here ,sir ,,isaac ,,newton has not only explained the causes of the motion of these bodies1 but has performed also the part of an astronomer1 by discovering what their motions are4 #b4 ,,that these bodies are not meteors in our air1 is manifest2 because they rise and set in the same manner1 as the sun and stars4 ,the astronomers had gone so far in their inquiries concerning them1 as to prove by their observations1 that they moved in the etherial spaces far beyond the moon2 but they had no true notion at all of the path1 which they described4 ,the most prevailing opinion before our author was1 that they moved in straight lines2 but in what part of the heavens was not determined4 ,,descartes#bje removed them far beyond the sphere of ,saturn1 as finding the straight motion attributed to them1 inconsistent with the vortical fluid1 by which he explains the motions of the planets1 as we have above related#bjf4 ,but ,sir ,,isaac ,,newton distinctly proves from astronomical observation1 that the comets pass through the region of the planets1 and are mostly invisible at a less distance1 than that of ,jupiter#bjg4 #c4 ,,and from hence finding the comets to be evidently within the sphere of the sun's action1 he concludes they must1 necessarily move about the sun1 as the planets do#bjh4 ,the planets move in ellipsis's2 but it is not necessary that every body1 which is influenced by the sun1 should move in that particular kind of line4 ,however our author proves1 that the power of the sun being reciprocally in the duplicate proportion of the distance1 every body acted on by the sun must either fall directly down1 or move in some conic section2 of which lines ,i have above observed1 that there are three species1 the ellipsis1 parabola1 and hyperbola#bji4 ,if a body1 which descends toward the sun as low as the orbit of any planet1 move with a swifter motion than the planet does1 that body will describe an orbit of a more oblong figure1 than that of the planet1 and have a longer axis at least4 ,the velocity of the body may be so great1 that it shall move in a parabola1 and having once passed about the sun1 shall ascend for ever without returning any more3 but the sun will be placed in the focus of this parabola4 ,with a velocity still greater the body will move in an hyperbola4 ,but it is most probable1 that the comets move in elliptical orbits1 though of a very oblong1 or in the phrase of astronomers1 of a very eccentric form1 such as is represented in fig4 #ajg1 where ,s is the sun1 ,c the comet1 and ,a ,b ,d ,e its orbit1 wherein the distance of ,s and ,d far exceeds that of ,s and ,a4 ,whence it is1 that they sometimes are found at a moderate distance from the sun1 and appear within the planetary regions2 at other times they ascend to vast distances1 far beyond the very orbit of ,saturn1 and so become invisible4 ,that the comets do move in this manner is proved by our author1 from computations built upon the observations1 which astronomers had made on many comets4 ,these computations were performed by ,sir ,,isaac ,,newton himself upon the comet1 which appeared toward the latter end of the year #afhj1 and at the beginning of the year following#baj2 but the learned ,dr4 ,,halley prosecuted the like computations more at large in this1 and also in many other comets#baa4 ,which computations are made upon propositions highly worthy of our author's unparallel'd genius1 such as could scarce have been discovered by any one not possessed of the utmost force of invention2 #d4 ,,those computations depend upon this principle1 that the eccentricity of the orbits of the comets is so great1 that if they are really elliptical1 yet they approach so near to parabolas in that part of them1 where they come under our view1 that they may be taken for such without sensible error#bab3 as in the preceding figure the parabola ,f ,a ,g differs in the lower part of it about ,a very little from the ellipsis ,d ,e ,a ,b4 ,upon which ground our great author teaches a method of finding by three observations made upon any comet the parabola1 which nearest agrees with its orbit#bac4 #e4 ,,now what confirms this whole theory beyond the least room for doubt is1 that the places of the comets computed in the orbits1 which the method here mentioned assigns them1 agree to the observations of astronomers with the same degree of exactness1 as the computations of the primary planets places usually do2 and this in comets1 whose motions are very extraordinary#bad4 #f4 ,,our author afterwards shews how to make use of any small deviation from the parabola1 that shall be observed1 to determine whether the orbits of the comets are elliptical or not1 and so to discover if the same comet returns at certain periods#bae4 ,and upon examining the comet in #afhj1 by the rule laid down for this purpose1 he finds its orbit to agree more exactly to an ellipsis than to a parabola1 though the ellipsis be so very eccentric1 that the comet cannot perform its period through it in the space of #ejj years#baf4 ,upon this ,dr4 ,,halley observed1 that mention is made in history of a comet1 with the like eminent tail as this1 having appeared three several times before2 the first of which appearances was at the death of ,,julius ,,cesar1 and each appearance was at the distance of #ege years from the next preceding4 ,he therefore computed the motion of this comet in such an elliptic orbit1 as would require this number of years for the body to revolve through it2 and these computations agree yet more perfectly with the observations made on this comet1 than any parabolical orbit will do#bag4 #g4 ,,the comparing together different appearances of the same comet1 is the only way to discover certainly the true form of the orbit3 for it is impossible to determine with exactness the figure of an orbit so exceedingly eccentric1 from single observations taken in one part of it2 and therefore ,sir ,,isaac ,,newton#bah proposes to compare the orbits1 upon the supposition that they are parabolical1 of such comets as appear at different times2 for if the same orbit be found to be described by a comet at different times1 in all probability it will be the same comet which describes it4 ,and here he remarks from ,dr4 ,,halley1 that the same orbit very nearly agrees to two appearances of a comet about the space of #ge years distance#bai2 so that if those two appearances were really of the same comet1 the transverse axis of the orbit of the comet would be near #ah times the axis of the earth's orbit2 and the comet1 when at its greatest distance from the sun1 will be removed not less than #ce times as far as the middle distance of the earth4 #h4 ,,and this seems to be the shortest period of any of the comets4 ,but it will be farther confirmed1 if the same comet should return a third time after another period of #ge years4 ,however it is not to be expected1 that comets should preserve the same regularity in their periods1 as the planets2 because the great eccentricity of their orbits makes them liable to suffer very considerable alterations from the action of the planets1 and other comets1 upon them4 #i4 ,,it is therefore to prevent too great disturbances in their motions from these causes1 as our author observes1 that while the planets revolve all of them nearly in the same plane1 the comets are disposed in very different ones2 and distributed over all parts of the heavens2 that1 when in their greatest distance from the sun1 and moving slowest1 they might be removed as far as possible out of the reach of each other's action#bbj4 ,the same end is likewise farther answered in those comets1 which by moving slowest in the aphelion1 or remotest distance from the sun1 descend nearest to it1 by placing the aphelion of these at the greatest height from the sun#bba4 #aj4 ,,our philosopher being led by his principles to explain the motions of the comets1 in the manner now related1 takes occasion from thence to give us his thoughts upon their nature and use4 ,for which end he proves in the first place1 that they must necessarily be solid and compact bodies1 and by no means any sort of vapour or light substance exhaled from the planets or stars3 because at the near distance1 to which some comets approach the sun1 it could not be1 but the immense heat1 to which they are exposed1 should instantaneously disperse and scatter any such light volatile substance#bbb4 ,in particular the forementioned comet of #afhj descended so near the sun1 as to come within a sixth part of the sun's diameter from the surface of it4 ,in which situation it must have been exposed1 as appears by computation1 to a degree of heat exceeding the heat of the sun upon our earth no less than #bhjjj times2 and therefore might have contracted a degree of heat #bjjj times greater1 than that of red hot iron#bbc4 ,now a substance1 which could endure so intense a heat1 without being dispersed in vapor1 must needs be firm and solid4 #aa4 ,,it is shewn likewise1 that the comets are opake substances1 shining by a reflected light1 borrowed from the sun#bbd4 ,this is proved from the observation1 that comets1 though they are approaching the earth1 yet diminish in lustre1 if at the same time they recede from the sun2 and on the contrary1 are found to encrease daily in brightness1 when they advance towards the sun1 though at the same time they move from the earth#bbe4 #ab4 ,,the comets therefore in these respects resemble the planets2 that both are durable opake bodies1 and both revolve about the sun in conic sections4 ,but farther the comets1 like our earth1 are surrounded by an atmosphere4 ,the air we breath is called the earth's atmosphere2 and it is most probable1 that all the other planets are invested with the like fluid4 ,indeed here a difference is found between the planets and comets4 ,the atmospheres of the planets are of so fine and subtile a substance1 as hardly to be discerned at any distance1 by reason of the small quantity of light which they reflect1 except only in the planet ,mars4 ,in him there is some little appearance of such a substance surrounding him1 as stars which have been covered by him are said to look somewhat dim a small space before his body comes under them1 as if their light1 when he is near1 were obstructed by his atmosphere4 ,but the atmospheres which surround the comets are so gross and thick1 as to reflect light very copiously4 ,they are also much greater in proportion to the body they surround1 than those of the planets1 if we may judge of the rest from our air2 for it has been observed of comets1 that the bright light appearing in the middle of them1 which is reflected from the solid body1 is scarce a ninth or tenth part of the whole comet1 #ac4 ,i ,,speak only of the heads of the comets1 the most lucid part of which is surrounded by a fainter light1 the most lucid part being usually not above a ninth or tenth part of the whole in breadth#bbf4 ,their tails are an appearance very peculiar1 nothing of the same nature appertaining in the least degree to any other of the celestial bodies4 ,of that appearance there are several opinions2 our author reduces them to three#bbg4 ,the two first1 which he proposes1 are rejected by him2 but the third he approves4 ,the first is1 that they arise from a beam of light transmitted through the head of the comet1 in like manner as a stream of light is discerned1 when the sun shines into a darkened room through a small hole4 ,this opinion1 as ,sir ,,isaac ,,newton observes1 implies the authors of it wholly unskilled in the principles of optics2 for that stream of light1 seen in a darkened room1 arises from the reflection of the sun beams by the dust and motes floating in the air3 for the rays of light themselves are not seen1 but by their being reflected to the eye from some substance1 upon which they fall#bbh4 ,the next opinion examined by our author is that of the celebrated ,,descartes1 who imagins these tails to be the light of the comet refracted in its passage to us1 and thence affording an oblong representation2 as the light of the sun does1 when refracted by the prism in that noted experiment1 which will have a great share in the third book of this discourse#bbi4 ,but this opinion is at once overturned from this consideration only1 that the planets could be no more free from this refraction than the comets2 nay ought to have larger or brighter tails1 than they1 because the light of the planets is strongest4 ,however our author has thought proper to add some farther objections against this opinion3 for instance1 that these tails are not variegated with colours1 as is the image produced by the prism1 and which is inseparable from that unequal refraction1 which produces that disproportioned length of the image4 ,and besides1 when the light in its passage from different comets to the earth describes the same path through the heavens1 the refraction of it should of necessity be in all respects the same4 ,but this is contrary to observation2 for the comet in #afhj1 the #bhth day of ,december1 and a former comet in the year #aegg1 the #bith day of ,december1 appear'd in the same place of the heavens1 that is1 were seen adjacent to the same fixed stars1 the earth likewise being in the same place at both times2 yet the tail of the latter comet deviated from the opposition to the sun a little to the northward1 and the tail of the former comet declined from the opposition of the sun five times as much southward#bcj4 #ad4 ,,there are some other false opinions1 though less regarded than these1 which have been advanced upon this argument4 ,these our excellent author passes over1 hastening to explain1 what he takes to be the true cause of this appearance4 ,he thinks it is certainly owing to steams and vapours exhaled from the body1 and gross atmosphere of the comets1 by the heat of the sun2 because all the appearances agree perfectly to this sentiment4 ,the tails are but small1 while the comet is descending to the sun1 but enlarge themselves to an immense degree1 as soon as ever the comet has passed its perihelion2 which shews the tail to depend upon the degree of heat1 which the comet receives from the sun4 ,and that the intense heat to which comets1 when nearest the sun1 are exposed1 should exhale from them a very copious vapour1 is a most reasonable supposition2 especially if we consider1 that in those free and empty regions steams will more easily ascend1 than here upon the surface of the earth1 where they are suppressed and hindered from rising by the weight of the incumbent air3 as we find by experiments made in vessels exhausted of the air1 where upon removal of the air several substances will fume and discharge steams plentifully1 which emit none in the open air4 ,the tails of comets1 like such a vapour1 are always in the plane of the comet's orbit1 and opposite to the sun1 except that the upper part thereof inclines towards the parts1 which the comet has left by its motion2 resembling perfectly the smoak of a burning coal1 which1 if the coal remain fixed1 ascends from it perpendicularly2 but1 if the coal be in motion1 ascends obliquely1 inclining from the motion of the coal4 ,and besides1 the tails of comets may be compared to this smoak in another respect1 that both of them are denser and more compact on the convex side1 than on the concave4 ,the different appearance of the head of the comet1 after it has past its perihelion1 from what it had before1 confirms greatly this opinion of their tails3 for smoke raised by a strong heat is blacker and grosser1 than when raised by a less2 and accordingly the heads of comets1 at the same distance from the sun1 are observed less bright and shining after the perihelion1 than before1 as if obscured by such a gross smoke4 #ae4 ,,the observations of ,,hevelius upon the atmospheres of comets still farther illustrate the same2 who relates1 that the atmospheres1 especially that part of them next the sun1 are remarkably contracted when near the sun1 and dilated again afterwards4 #af4 ,,to give a more full idea of these tails1 a rule is laid down by our author1 whereby to determine at any time1 when the vapour in the extremity of the tail first rose from the head of the comet4 ,by this rule it is found1 that the tail does not consist of a fleeting vapour1 dissipated soon after it is raised1 but is of long continuance2 that almost all the vapour1 which rose about the time of the perihelion from the comet of #afhj1 continued to accompany it1 ascending by degrees1 being succeeded constantly by fresh matter1 which rendered the tail contiguous to the comet4 ,from this computation the tails are found to participate of another property of ascending vapours1 that1 when they ascend with the greatest velocity1 they are least incurvated4 #ag4 ,,the only objection that can be made against this opinion is the difficulty of explaining1 how a sufficient quantity of vapour can be raised from the atmosphere of a comet to fill those vast spaces1 through which their tails are sometimes extended4 ,this our author removes by the following computation3 our air being an elastic fluid1 as has been said before#bca1 is more dense here near the surface of the earth1 where it is pressed upon by the whole air above2 than it is at a distance from the earth1 where it has a less weight incumbent4 ,i have observed1 that the density of the air is reciprocally proportional to the compressing weight4 ,from hence our author computes to what degree of rarity the air must be expanded1 according to this rule1 at an height equal to a semidiameter of the earth3 and he finds1 that a globe of such air1 as we breath here on the surface of the earth1 which shall be one inch only in diameter1 if it were expanded to the degree of rarity1 which the air must have at the height now mentioned1 would fill all the planetary regions even to the very sphere of ,saturn1 and far beyond4 ,now since the air at a greater height will be still immensly more rarified1 and the surface of the atmospheres of comets is usually about ten times the distance from the center of the comet1 as the surface of the comet it self1 and the tails are yet vastly farther removed from the center of the comet2 the vapour1 which composes those tails1 may very well be allowed to be so expanded1 as that a moderate quantity of matter may fill all that space1 they are seen to take up4 ,though indeed the atmospheres of comets being very gross1 they will hardly be rarified in their tails to so great a degree1 as our air under the same circumstances2 especially since they may be something condensed1 as well by their gravitation to the sun1 as that the parts will gravitate to one another2 which will hereafter be shewn to be the universal property of all matter#bcb4 ,the only scruple left is1 how so much light can be reflected from a vapour so rare1 as this computation implies4 ,for the removal of which our author observes1 that the most refulgent of these tails hardly appear brighter1 than a beam of the sun's light transmitted into a darkened room through a hole of a single inch diameter2 and that the smallest fixed stars are visible through them without any sensible diminution of their lustre4 #ah4 ,,all these considerations put it beyond doubt1 what is the true nature of the tails of comets4 ,there has indeed nothing been said1 which will account for the irregular figures1 in which those tails are sometimes reported to have appeared2 but since none of those appearances have ever been recorded by astronomers1 who on the contrary ascribe the same likeness to the tails of all comets1 our author with great judgment refers all those to accidental refractions by intervening clouds1 or to parts of the milky way contiguous to the comets#bcc4 #ai4 ,,the discussion of this appearance in comets has led ,sir ,,isaac ,,newton into some speculations relating to their use1 which ,i cannot but extreamly admire1 as representing in the strongest light imaginable the extensive providence of the great author of nature1 who1 besides the furnishing this globe of earth1 and without doubt the rest of the planets1 so abundantly with every thing necessary for the support and continuance of the numerous races of plants and animals1 they are stocked with1 has over and above provided a numerous train of comets1 far exceeding the number of the planets1 to rectify continually1 and restore their gradual decay1 which is our author's opinion concerning them#bcd4 ,for since the comets are subject to such unequal degrees of heat1 being sometimes burnt with the most intense degree of it1 at other times scarce receiving any sensible influence from the sun2 it can hardly be supposed1 they are designed for any such constant use1 as the planets4 ,now the tails1 which they emit1 like all other kinds of vapour1 dilate themselves as they ascend1 and by consequence are gradually dispersed and scattered through all the planetary regions1 and thence cannot but be gathered up by the planets1 as they pass through their orbs3 for the planets having a power to cause all bodies to gravitate towards them1 as will in the sequel of this discourse be shewn#bce2 these vapours will be drawn in process of time into this or the other planet1 which happens to act strongest upon them4 ,and by entering the atmospheres of the earth and other planets1 they may well be supposed to contribute to the renovation of the face of things1 in particular to supply the diminution caused in the humid parts by vegetation and putrefaction4 ,for vegetables are nourished by moisture1 and by putrefaction are turned in great part into dry earth2 and an earthy substance always subsides in fermenting liquors2 by which means the dry parts of the planets must continually increase1 and the fluids diminish1 nay in a sufficient length of time be exhausted1 if not supplied by some such means4 ,it is farther our great author's opinion1 that the most subtile and active parts of our air1 upon which the life of things chiefly depends1 is derived to us1 and supplied by the comets4 ,so far are they from portending any hurt or mischief to us1 which the natural fears of men are so apt to suggest from the appearance of any thing uncommon and astonishing4 #bj4 ,,that the tails of comets have some such important use seems reasonable1 if we consider1 that those bodies do not send out those fumes merely by their near approach to the sun2 but are framed of a texture1 which disposes them in a particular manner to fume in that sort3 for the earth1 without emitting any such steam1 is more than half the year at a less distance from the sun1 than the comet of #affd and #affe approached it1 when nearest2 likewise the comets of #afhb and #afhc never approached the sun much above a seventh part nearer than ,venus1 and were more than half as far again from the sun as ,mercury2 yet all these emitted tails4 #ba4 ,,from the very near approach of the comet of #afhj our author draws another speculation2 for if the sun have an atmosphere about it1 the comet mentioned seems to have descended near enough to the sun to enter within it4 ,if so1 it must have been something retarded by the resistance it would meet with1 and consequently in its next descent to the sun will fall nearer than now2 by which means it will meet with a greater resistance1 and be again more retarded4 ,the event of which must be1 that at length it will impinge upon the sun's surface1 and thereby supply any decrease1 which may have happened by so long an emission of light1 or otherwise4 ,and something like this our author conjectures may be the case of those fixed stars which by an additional increase of their lustre have for a certain time become visible to us1 though usually they are out of sight4 ,there is indeed a kind of fixed stars1 which appear and disappear at regular and equal intervals3 here some more steady cause must be sought for2 perhaps these stars turn round their own axis's1 as our sun does#bcf1 and have some part of their body more luminous than the other1 whereby they are seen1 when the most lucid part is next to us1 and when the darker part is turned toward us1 they vanish out of sight4 #bb4 ,,whether the sun does really diminish1 as has been here suggested1 is difficult to prove2 yet that it either does so1 or that the earth increases1 if not both1 is rendered probable from ,dr4 ,,halley's observation#bcg1 that by comparing the proportion1 which the periodical time of the moon bore to that of the sun in former times1 with the proportion between them at present1 the moon is found to be something accelerated in respect of the sun4 ,but if the sun diminish1 the periods of the primary planets will be lengthened2 and if the earth be encreased1 the period of the moon will be shortened3 as will appear by the next chapter1 wherein it shall be shewn1 that the power of the sun and earth is the result of the same power being lodg'd in all their parts1 and that this principle of producing gravitation in other bodies is proportional to the solid matter in each body4 ,,chap4 ,v4 ,of the ,,bodies of the ,,sun and ,,planets4 ,,our author1 after having discovered that the celestial motions are performed by a force extended from the sun and primary planets1 follows this power into the deepest recesses of those bodies themselves1 and proves the same to accompany the smallest particle1 of which they are composed4 #b4 ,,preparative hereto he shews first1 that each of the heavenly bodies attracts the rest1 and all bodies1 with such different degrees of force1 as that the force of the same attracting body is exerted on others exactly in proportion to the quantity of matter in the body attracted#bch4 #c4 ,,of this the first proof he brings is from experiments made here upon the earth4 ,the power by which the moon is influenced was above shewn to be the same1 with that power here on the surface of the earth1 which we call gravity#bci4 ,now one of the effects of the principle of gravity is1 that all bodies descend by this force from the same height in equal times4 ,which has been long taken notice of2 particular methods having been invented to shew that the only cause1 why some bodies were observed to fall from the same height sooner than others1 was the resistance of the air4 ,this we have above related#bdj2 and proved from hence1 that since bodies resist to any change of their state from rest to motion1 or from motion to rest1 in proportion to the quantity of matter contained in them2 the power that can move different quantities of matter equally1 must be proportional to the quantity4 ,the only objection here is1 that it can hardly be made certain1 whether this proportion in the effect of gravity on different bodies holds perfectly exact or not from these experiments2 by reason that the great swiftness1 with which bodies fall1 prevents our being able to determine the times of their descent with all the exactness requisite4 ,therefore to remedy this inconvenience1 our author substitutes another more certain experiment in the room of these made upon falling bodies4 ,pendulums are caused to vibrate by the same principle1 as makes bodies descend2 the power of gravity putting them in motion1 as well as the other4 ,but if the ball of any pendulum1 of the same length with another1 were more or less attracted in proportion to the quantity of solid matter in the ball1 that pendulum must accordingly move faster or slower than the other4 ,now the vibrations of pendulums continue for a great length of time1 and the number of vibrations they make may easily be determined without suspicion of error2 so that this experiment may be extended to what exactness one pleases3 and our author assures us1 that he examined in this way several substances1 as gold1 silver1 lead1 glass1 sand1 common salt1 wood1 water1 and wheat2 in all which he found not the least deviation from the proportion mentioned1 though he made the experiment in such a manner1 that in bodies of the same weight a difference in the quantity of their matter less than a thousandth part of the whole would have discovered it self#bda4 ,it appears therefore1 that all bodies are made to descend by the power of gravity here1 near the surface of the earth1 with the same degree of swiftness4 ,we have above observed this descent to be after the rate of #af feet in the first second of time from the beginning of their fall4 ,moreover it was also observed1 that if any body1 which fell here at the surface of the earth after this rate1 were to be conveyed up to the height of the moon1 it would descend from thence just with the same degree of velocity1 as that with which the moon is attracted toward the earth2 and therefore the power of the earth upon the moon bears the same proportion to the power it would have upon those bodies at the same distance1 as the quantity of matter in the moon bears to the quantity in those bodies4 #d4 ,,thus the assertion laid down is proved in the earth1 that the power of the earth on every body it attracts is1 at the same distance from the earth1 proportional to the quantity of solid matter in the body acted on4 ,as to the sun1 it has been shewn1 that the power of the sun's action upon the same primary planet is reciprocally in the duplicate proportion of the distance2 and that the power of the sun decreases throughout in the same proportion1 the motion of comets traversing the whole planetary region testifies4 ,this proves1 that if any planet were removed from the sun to any other distance whatever1 the degree of its acceleration toward the sun would yet remain reciprocally in the duplicate proportion of its distance4 ,but it has likewise been shewn1 that the degree of acceleration1 which the sun gives to every one of the planets1 is reciprocally in the duplicate proportion of their respective distances4 ,all which compared together puts it out of doubt1 that the power of the sun upon any planet1 removed into the place of any ether1 would give it the same velocity of descent1 as it gives that other2 and consequently1 that the sun's action upon different planets at the same distance would be proportional to the quantity of matter in each4 ,it has farther been shewn1 that the sun attracts the primary planets1 and their respective secondary1 when at the same distance1 so as to communicate to both the same degree of velocity2 and therefore the force1 wherewith the sun acts on the secondary planet1 bears the same proportion to the force1 wherewith at the same distance it attracts the primary1 as the quantity of solid matter in the secondary planet bears to the quantity of matter in the primary4 #e4 ,,this property therefore is proved of both kinds of planets1 in respect of the sun4 ,therefore the sun possesses the quality found in the earth1 of acting on bodies with a degree of force proportional to the quantity of matter in the body1 which receives the influence4 #f4 ,,that the power of attraction1 with which the other planets are endued1 should differ from that of the earth1 can hardly be supposed1 if we consider the similitude between those bodies2 and that it does not in this respect1 is farther proved from the satellites of ,saturn and ,jupiter1 which are attracted by their respective primary according to the same law1 that is1 in the same proportion to their distances1 as the primary are attracted by the sun3 so that what has been concluded of the sun in relation to the primary planets1 may be justly concluded of these primary in respect of their secondary1 and in consequence of that1 in regard likewise to all other bodies1 viz4 that they will attract every body in proportion to the quantity of solid matter it contains4 #g4 ,,hence it follows1 that this attraction extends itself to every particle of matter in the attracted body3 and that no portion of matter whatever is exempted from the influence of those bodies1 to which we have proved this attractive power to belong4 #h4 ,,before we proceed farther1 we may here remark1 that this attractive power both of the sun and planets now appears to be quite of the same nature in all2 for it acts in each in the same proportion to the distance1 and in the same manner acts alike upon every particle of matter4 ,this power therefore in the sun and other planets is not of a different nature from this power in the earth2 which has been already shewn to be the same with that1 which we call gravity#bdb4 #i4 ,,and this lays open the way to prove1 that the attracting power lodged in the sun and planets1 belongs likewise to every part of them3 and that their respective powers upon the same body are proportional to the quantity of matter1 of which they are composed2 for instance1 that the force with which the earth attracts the moon1 is to the force1 with which the sun would attract it at the same distance1 as the quantity of solid matter contained in the earth1 to the quantity contained in the sun#bdc4 #aj4 ,,the first of these assertions is a very evident consequence from the latter4 ,and before we proceed to the proof1 it must first be shewn1 that the third law of motion1 which makes action and reaction equal1 holds in these attractive powers4 ,the most remarkable attractive force1 next to the power of gravity1 is that1 by which the loadstone attracts iron4 ,now if a loadstone were laid upon water1 and supported by some proper substance1 as wood or cork1 so that it might swim2 and if a piece of iron were caused to swim upon the water in like manner3 as soon as the loadstone begins to attract the iron1 the iron shall move toward the stone1 and the stone shall also move toward the iron2 when they meet1 they shall stop each other1 and remain fixed together without any motion4 ,this shews1 that the velocities1 wherewith they meet1 are reciprocally proportional to the quantities of solid matter in each2 and that by the stone's attracting the iron1 the stone itself receives as much motion1 in the strict philosophic sense of that word#bdd1 as it communicates to the iron3 for it has been declared above to be an effect of the percussion of two bodies1 that if they meet with velocities reciprocally proportional to the respective bodies1 they shall be stopped by the concourse1 unless their elasticity put them into fresh motion2 but if they meet with any other velocities1 they shall retain some motion after meeting#bde4 ,amber1 glass1 sealing-wax1 and many other substances acquire by rubbing a power1 which from its having been remarkable1 particularly in amber1 is called electrical4 ,by this power they will for some time after rubbing attract light bodies1 that shall be brought within the sphere of their activity4 ,on the other hand ,mr4 ,,boyle found1 that if a piece of amber be hung in a perpendicular position by a string1 it shall be drawn itself toward the body whereon it was rubbed1 if that body be brought near it4 ,both in the loadstone and in electrical bodies we usually ascribe the power to the particular body1 whose presence we find necessary for producing the effect4 ,the loadstone and any piece of iron will draw each other1 but in two pieces of iron no such effect is ordinarily observed2 therefore we call this attractive power the power of the loadstone3 though near a loadstone two pieces of iron will also draw each other4 ,in like manner the rubbing of amber1 glass1 or any such body1 till it is grown warm1 being necessary to cause any action between those bodies and other substances1 we ascribe the electrical power to those bodies4 ,but in all these cases if we would speak more correctly1 and not extend the sense of our expressions beyond what we see2 we can only say that the neighbourhood of a loadstone and a piece of iron is attended with a power1 whereby the loadstone and the iron are drawn toward each other2 and the rubbing of electrical bodies gives rise to a power1 whereby those bodies and other substances are mutually attracted4 ,thus we must also understand in the power of gravity1 that the two bodies are mutually made to approach by the action of that power4 ,when the sun draws any planet1 that planet also draws the sun2 and the motion1 which the planet receives from the sun1 bears the same proportion to the motion1 which the sun it self receives1 as the quantity of solid matter in the sun bears to the quantity of solid matter in the planet4 ,hitherto1 for brevity sake in speaking of these forces1 we have generally ascribed them to the body1 which is least moved2 as when we called the power1 which exerts itself between the sun and any planet1 the attractive power of the sun2 but to speak more correctly1 we should rather call this power in any case the force1 which acts between the sun and earth1 between the sun and ,jupiter1 between the earth and moon1 c4 for both the bodies are moved by the power acting between them1 in the same manner1 as when two bodies are tied together by a rope1 if that rope shrink by being wet1 or otherwise1 and thereby cause the bodies to approach1 by drawing both1 it will communicate to both the same degree of motion1 and cause them to approach with velocities reciprocally proportional to the respective bodies4 ,from this mutual action between the sun and planet it follows1 as has been observed above#bdf1 that the sun and planet do each move about their common center of gravity4 ,let ,a 7in fig4 #ajh47 represent the sun1 ,b a planet1 ,c their common center of gravity4 ,if these bodies were once at rest1 by their mutual attraction they would directly approach each other with such velocities1 that their common center of gravity would remain at rest1 and the two bodies would at length meet in that point4 ,if the planet ,b were to receive an impulse1 as in the direction of the line ,d ,e1 this would prevent the two bodies from falling together2 but their common center of gravity would be put into motion in the direction of the line ,c ,f equidistant from ,b ,e4 ,in this case ,sir ,,isaac ,,newton proves#bdg1 that the sun and planet would describe round their common center of gravity similar orbits1 while that center would proceed with an uniform motion in the line ,c ,f2 and so the system of the two bodies would move on with the center of gravity without end4 ,in order to keep the system in the same place1 it is necessary1 that when the planet received its impulse in the direction ,b ,e1 the sun should also receive such an impulse the contrary way1 as might keep the center of gravity ,c without motion2 for if these began once to move without giving any motion to their common center of gravity1 that center would always remain fixed4 #aa4 ,,by this may be understood in what manner the action between the sun and planets is mutual4 ,but farther1 we have shewn above#bdh1 that the power1 which acts between the sun and primary planets1 is altogether of the same nature with that1 which acts between the earth and the bodies at its surface1 or between the earth and its parts1 and with that which acts between the primary planets and their secondary2 therefore all these actions must be ascribed to the same cause#bdi4 ,again1 it has been already proved1 that in different planets the force of the sun's action upon each at the same distance would be proportional to the quantity of solid matter in the planet#bej2 therefore the reaction of each planet on the sun at the same distance1 or the motion1 which the sun would receive from each planet1 would also be proportional to the quantity of matter in the planet2 that is1 these planets at the same distance would act on the same body with degrees of strength proportional to the quantity of solid matter in each4 ,illustration #ab4 ,,in the next place1 from what has been now proved1 our great author has deduced this farther consequence1 no less surprizing than elegant2 that each of the particles1 out of which the bodies of the sun and planets are framed1 exert their power of gravitation by the same law1 and in the same proportion to the distance1 as the great bodies which they compose4 ,for this purpose he first demonstrates1 that if a globe were compounded of particles1 which will attract the particles of any other body reciprocally in the duplicate proportion of their distances1 the whole globe will attract the same in the reciprocal duplicate proportion of their distances from the center of the globe2 provided the globe be of uniform density throughout#bea4 ,and from this our author deduces the reverse1 that if a globe acts upon distant bodies by the law just now specified1 and the power of the globe is derived from its being composed of attractive particles2 each of those particles will attract after the same proportion#beb4 ,the manner of deducing this is not set down at large by our author1 but is as follows4 ,the globe is supposed to act upon the particles of a body without it constantly in the reciprocal duplicate proportion of their distances from its center2 and therefore at the same distance from the globe1 on which side soever the body be placed1 the globe will act equally upon it4 ,now because1 if the particles1 of which the globe is composed1 acted upon those without in the reciprocal duplicate proportion of their distances1 the whole globe would act upon them in the same manner as it does2 therefore1 if the particles of the globe have not all of them that property1 some must act stronger than in that proportion1 while others act weaker3 and if this be the condition of the globe1 it is plain1 that when the body attracted is in such a situation in respect of the globe1 that the greater number of the strongest particles are nearest to it1 the body will be more forcibly attracted2 than when by turning the globe about1 the greater quantity of weak particles should be nearest1 though the distance of the body should remain the same from the center of the globe4 ,which is contrary to what was at first remarked1 that the globe on all sides of it acts with the same strength at the same distance4 ,whence it appears1 that no other constitution of the globe can agree to it4 #ac4 ,,from these propositions it is farther collected1 that if all the particles of one globe attract all the particles of another in the proportion so often mentioned1 the attracting globe will act upon the other in the same proportion to the distance between the center of the globe which attracts1 and the center of that which is attracted#bec3 and farther1 that this proportion holds true1 though either or both the globes be composed of dissimilar parts1 some rarer and some more dense2 provided only1 that all the parts in the same globe equally distant from the center be homogeneous#bed4 ,and also1 if both the globes attract each other#bee4 ,all which place it beyond contradiction1 that this proportion obtains with as much exactness near and contiguous to the surface of attracting globes1 as at greater distances from them4 #ad4 ,,thus our author1 without the pompous pretence of explaining the cause of gravity1 has made one very important step toward it1 by shewing that this power in the great bodies of the universe1 is derived from the same power being lodged in every particle of the matter which composes them3 and consequently1 that this property is no less than universal to all matter whatever1 though the power be too minute to produce any visible effects on the small bodies1 wherewith we converse1 by their action on each other#bef4 ,in the fixed stars indeed we have no particular proof that they have this power2 for we find no apperance to demonstrate that they either act1 or are acted upon by it4 ,but since this power is found to belong to all bodies1 whereon we can make observation2 and we see that it is not to be altered by any change in the form of bodies1 but always accompanies them in every shape without diminution1 remaining ever proportional to the quantity of solid matter in each2 such a power must without doubt belong universally to all matter4 #ae4 ,,this therefore is the universal law of matter2 which recommends it self no less for its great plainness and simplicity1 than for the surprizing discoveries it leads us to4 ,by this principle we learn the different weight1 which the same body will have upon the surfaces of the sun and of diverse planets2 and by the same we can judge of the composition of those celestial bodies1 and know the density of each2 which is formed of the most compact1 and which of the most rare substance4 ,let the adversaries of this philosophy reflect here1 whether loading this principle with the appellation of an occult quality1 or perpetual miracle1 or any other reproachful name1 be sufficient to dissuade us from cultivating it2 since this quality1 which they call occult1 leads to the knowledge of such things1 that it would have been reputed no less than madness for any one1 before they had been discovered1 even to have conjectured that our faculties should ever have reached so far4 #af4 ,,see how all this naturally follows from the foregoing principles in those planets1 which have satellites moving about them4 ,by the times1 in which these satellites perform their revolutions1 compared with their distances from their respective primary1 the proportion between the power1 with which one primary attracts his satellites1 and the force with which any other attracts his will be known2 and the proportion of the power with which any planet attracts its secondary1 to the power with which it attracts a body at its surface is found1 by comparing the distance of the secondary planet from the center of the primary1 to the distance of the primary planet's surface from the same3 and from hence is deduced the proportion between the power of gravity upon the surface of one planet1 to the gravity upon the surface of another4 ,by the like method of comparing the periodical time of a primary planet about the sun1 with the revolution of a satellite about its primary1 may be found the proportion of gravity1 or of the weight of any body upon the surface of the sun1 to the gravity1 or to the weight of the same body upon the surface of the planet1 which carries about the satellite4 #ag4 ,,by these kinds of computation it is found1 that the weight of the same body upon the surface of the sun will be about #bc times as great1 as here upon the surface of the earth2 about #aj times as great1 as upon the surface of ,jupiter2 and near #ai times as great1 as upon the surface of ,saturn#beg4 #ah4 ,,the quantity of matter1 which composes each of these bodies1 is proportional to the power it has upon a body at a given distance4 ,by this means it is found1 that the sun contains #ajfg times as much matter as ,jupiter2 ,jupiter #aeh times as much as the earth1 and #b-e_/f times as much as ,saturn#beh4 ,the diameter of the sun is about #ib times1 that of ,jupiter about #i times1 and that of ,saturn about #g times the diameter of the earth4 #ai4 ,,by making a comparison between the quantity of matter in these bodies and their magnitudes1 to be found from their diameters1 their respective densities are readily deduced2 the density of every body being measured by the quantity of matter contained under the same bulk1 as has been above remarked#bei4 ,thus the earth is found #d times more dense than ,jupiter2 ,saturn has between and of the density of ,jupiter2 but the sun has one fourth part only of the density of the earth#bfj4 ,from which this observation is drawn by our author2 that the sun is rarified by its great heat1 and that of the three planets named1 the more dense is nearer the sun than the more rare2 as was highly reasonable to expect1 the densest bodies requiring the greatest heat to agitate and put their parts in motion2 as on the contrary1 the planets which are more rare1 would be rendered unfit for their office1 by the intense heat to which the denser are exposed4 ,thus the waters of our seas1 if removed to the distance of ,saturn from the sun1 would remain perpetually frozen2 and if as near the sun as ,mercury1 would constantly boil#bfa4 #bj4 ,,the densities of the three planets ,mercury1 ,venus1 and ,mars1 which have no satellites1 cannot be expresly assigned2 but from what is found in the others1 it is very probable1 that they also are of such different degrees of density1 that universally the planet which is nearest to the sun1 is formed of the most compact substance4 ,,chap4 ,,vi4 ,of the ,,fluid ,,parts of the ,,planets4 ,,this globe1 that we inhabit1 is composed of two parts2 the solid earth1 which affords us a foundation to dwell upon2 and the seas and other waters1 that furnish rains and vapours necessary to render the earth fruitful1 and productive of what is requisite for the support of life4 ,and that the moon1 though but a secondary planet1 is composed in like manner1 is generally thought1 from the different degrees of light which appear on its surface2 the parts of that planet1 which reflect a dim light1 being supposed to be fluid1 and to imbibe the sun's rays1 while the solid parts reflect them more copiously4 ,some indeed do not allow this to be a conclusive argument3 but whether we can distinguish the fluid part of the moon's surface from the rest or not2 yet it is most probable that there are two such different parts1 and with still greater reason we may ascribe the like to the other primary planets1 which yet more nearly resemble our earth4 ,the earth is also encompassed by another fluid the air1 and we have before remarked1 that probably the rest of the planets are surrounded by the like4 ,these fluid parts in particular engage our author's attention1 both by reason of some remarkable appearances peculiar to them1 and likewise of some effects they have upon the whole bodies to which they belong4 #b4 ,,fluids have been already treated of in general1 with respect to the effect they have upon solid bodies moving in them#bfb2 now we must consider them in reference to the operation of the power of gravity upon them4 ,by this power they are rendered weighty1 like all other bodies1 in proportion to the quantity of matter1 which is contained in them4 ,and in any quantity of a fluid the upper parts press upon the lower as much1 as any solid body would press on another1 whereon it should lie4 ,but there is an effect of the pressure of fluids on the bottom of the vessel1 wherein they are contained1 which ,i shall particularly explain4 ,the force supported by the bottom of such a vessel is not simply the weight of the quantity of the fluid in the vessel1 but is equal to the weight of that quantity of the fluid1 which would be contained in a vessel of the same bottom and of equal width throughout1 when this vessel is filled up to the same height1 as that to which the vessel proposed is filled4 ,suppose water were contained in the vessel ,a ,b ,c ,d 7in fig4 #aji47 filled up to ,e ,f4 ,here it is evident1 that if a part of the bottom1 as ,g ,h1 which is directly under any part of the space ,e ,f1 be considered separately2 it will appear at once1 that this part sustains the weight of as much of the fluid1 as stands perpendicularly over it up to the height of ,e ,f2 that is1 the two perpendiculars ,g ,i and ,h ,k being drawn1 the part ,g ,h of the bottom will sustain the whole weight of the fluid included between these two perpendiculars4 ,again1 ,i say1 every other part of the bottom equally broad with this1 will sustain as great a pressure4 ,let the part ,l ,m be of the same breadth with ,g ,h4 ,here the perpendiculars ,l ,o and ,m ,n being drawn1 the quantity of water contained between these perpendiculars is not so great1 as that contained between the perpendiculars ,g ,i and ,h ,k2 yet1 ,i say1 the pressure on ,l ,m will be equal to that on ,g ,h4 ,this will appear by the following considerations4 ,it is evident1 that if the part of the vessel between ,o and ,n were removed1 the water would immediately flow out1 and the surface ,e ,f would subside2 for all parts of the water being equally heavy1 it must soon form itself to a level surface1 if the form of the vessel1 which contains it1 does not prevent4 ,therefore since the water is prevented from rising by the side ,n ,o of the vessel1 it is manifest1 that it must press against ,n ,o with some degree of force4 ,in other words1 the water between the perpendiculars ,l ,o and ,m ,n endeavours to extend itself with a certain degree of force2 or more correctly1 the ambient water presses upon this1 and endeavours to force this pillar or column of water into a greater length4 ,but since this column of water is sustained between ,n ,o and ,l ,m1 each of these parts of the vessel will be equally pressed against by the power1 wherewith this column endeavours to extend4 ,consequently ,l ,m bears this force over and above the weight of the column of water between ,l ,o and ,m ,n4 ,to know what this expansive force is1 let the part ,o ,n of the vessel be removed1 and the perpendiculars ,l ,o and ,m ,n be prolonged2 then by means of some pipe fixed over ,n ,o let water be filled between these perpendiculars up to ,p ,q an equal height with ,e ,f4 ,here the water between the perpendiculars ,l ,p and ,m ,q is of an equal height with the highest part of the water in the vessel2 therefore the water in the vessel cannot by its pressure force it up higher1 nor can the water in this column subside2 because1 if it should1 it would raise the water in the vessel to a greater height than itself4 ,but it follows from hence1 that the weight of water contained between ,p ,o and ,q ,n is a just balance to the force1 wherewith the column between ,l ,o and ,m ,n endeavours to extend4 ,so the part ,l ,m of the bottom1 which sustains both this force and the weight of the water between ,l ,o and ,m ,n1 is pressed upon by a force equal to the united weight of the water between ,l ,o and ,m ,n1 and the weight of the water between ,p ,o and ,q ,n2 that is1 it is pressed on by a force equal to the weight of all the water contained between ,l ,p and ,m ,q4 ,and this weight is equal to that of the water contained between ,g ,i and ,h ,k1 which is the weight sustained by the part ,g ,h of the bottom4 ,now this being true of every part of the bottom ,b ,c1 it is evident1 that if another vessel ,r ,s ,t ,v be formed with a bottom ,r ,v equal to the bottom ,b ,c1 and be throughout its whole height of one and the same breadth2 when this vessel is filled with water to the same height1 as the vessel ,a ,b ,c ,d is filled1 the bottoms of these two vessels shall be pressed upon with equal force4 ,if the vessel be broader at the top than at the bottom1 it is evident1 that the bottom will bear the pressure of so much of the fluid1 as is perpendicularly over it1 and the sides of the vessel will support the rest4 ,this property of fluids is a corollary from a proposition of our author#bfc2 from whence also he deduces the effects of the pressure of fluids on bodies resting in them4 ,these are1 that any body heavier than a fluid will sink to the bottom of the vessel1 wherein the fluid is contained1 and in the fluid will weigh as much as its own weight exceeds the weight of an equal quantity of the fluid2 any body uncompressible of the same density with the fluid1 will rest any where in the fluid without suffering the least change either in its place or figure from the pressure of such a fluid1 but will remain as undisturbed as the parts of the fluid themselves2 but every body of less density than the fluid will swim on its surface1 a part only being received within the fluid4 ,which part will be equal in bulk to a quantity of the fluid1 whose weight is equal to the weight of the whole body2 for by this means the parts of the fluid under the body will suffer as great a pressure as any other parts of the fluid as much below the surface as these4 #c4 ,,in the next place1 in relation to the air1 we have above made mention1 that the air surrounding the earth being an elastic fluid1 the power of gravity will have this effect on it1 to make the lower parts near the surface of the earth more compact and compressed together by the weight of the air incumbent1 than the higher parts1 which are pressed upon by a less quantity of the air1 and therefore sustain a less weight#bfd4 ,it has been also observed1 that our author has laid down a rule for computing the exact degree of density in the air at all heights from the earth#bfe4 ,but there is a farther effect from the air's being compressed by the power of gravity1 which he has distinctly considered4 ,the air being elastic and in a state of compression1 any tremulous body will propagate its motion to the air1 and excite therein vibrations1 which will spread from the body that occasions them to a great distance4 ,this is the efficient cause of sound3 for that sensation is produced by the air1 which1 as it vibrates1 strikes against the organ of hearing4 ,as this subject was extremely difficult1 so our great author's success is surprizing4 #d4 ,,our author's doctrine upon this head ,i shall endeavour to explain somewhat at large4 ,but preliminary thereto must be shewn1 what he has delivered in general of pressure propagated through fluids2 and also what he has set down relating to that wave-like motion1 which appears upon the surface of water1 when agitated by throwing any thing into it1 or by the reciprocal motion of the finger1 c4 #e4 ,,concerning the first1 it is proved1 that pressure is spread through fluids1 not only right forward in a streight line1 but also laterally1 with almost the same ease and force4 ,of which a very obvious exemplification by experiment is proposed3 that is1 to agitate the surface of water by the reciprocal motion of the finger forwards and backwards only2 for though the finger have no circular motion given it1 yet the waves excited in the water will diffuse themselves on each hand of the direction of the motion1 and soon surround the finger4 ,nor is what we observe in sounds unlike to this1 which do not proceed in straight lines only1 but are heard though a mountain intervene1 and when they enter a room in any part of it1 they spread themselves into every corner2 not by reflection from the walls1 as some have imagined1 but as far as the sense can judge1 directly from the place where they enter4 #f4 ,,how the waves are excited in the surface of stagnant water1 may be thus conceived4 ,suppose in any place1 the water raised above the rest in form of a small hillock2 that water will immediately subside1 and raise the circumambient water above the level of the parts more remote1 to which the motion cannot be communicated under longer time4 ,and again1 the water in subsiding will acquire1 like all falling bodies1 a force1 which will carry it below the level surface1 till at length the pressure of the ambient water prevailing1 it will rise again1 and even with a force like to that wherewith it descended1 which will carry it again above the level4 ,but in the mean time the ambient water before raised will subside1 as this did1 sinking below the level2 and in so doing1 will not only raise the water1 which first subsided1 but also the water next without itself4 ,so that now beside the first hillock1 we shall have a ring investing it1 at some distance raised above the plain surface likewise2 and between them the water will be sunk below the rest of the surface4 ,after this1 the first hillock1 and the new made annular rising1 will descend2 raising the water between them1 which was before depressed1 and likewise the adjacent part of the surface without4 ,thus will these annular waves be successively spread more and more4 ,for1 as the hillock subsiding produces one ring1 and that ring subsiding raises again the hillock1 and a second ring2 so the hillock and second ring subsiding together raise the first ring1 and a third2 then this first and third ring subsiding together raise the first hillock1 the second ring1 and a fourth2 and so on continually1 till the motion by degrees ceases4 ,now it is demonstrated1 that these rings ascend and descend in the manner of a pendulum2 descending with a motion continually accelerated1 till they become even with the plain surface of the fluid1 which is half the space they descend2 and then being retarded again by the same degrees as those1 whereby they were accelerated1 till they are depressed below the plain surface1 as much as they were before raised above it3 and that this augmentation and diminution of their velocity proceeds by the same degrees1 as that of a pendulum vibrating in a cycloid1 and whose length should be a fourth part of the distance between any two adjacent waves3 and farther1 that a new ring is produced every time a pendulum1 whose length is four times the former1 that is1 equal to the interval between the summits of two waves1 makes one oscillation or swing#bff4 #g4 ,,this now opens the way for understanding the motion consequent upon the tremors of the air1 excited by the vibrations of sonorous bodies3 which we must conceive to be performed in the following manner4 #h4 ,,let ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 ,g1 ,h 7in fig4 #aaj47 represent a series of the particles of the air1 at equal distances from each other4 ,i ,k ,l a musical chord1 which ,i shall use for the tremulous and sonorous body1 to make the conception as simple as may be4 ,suppose this chord stretched upon the points ,i and ,l1 and forcibly drawn into the situation ,i ,k ,l1 so that it become contiguous to the particle ,a in its middle point ,k3 and let the chord from this situation begin to recoil1 pressing against the particle ,a1 which will thereby be put into motion towards ,b3 but the particles ,a1 ,b1 ,c being equidistant1 the elastic power1 by which ,b avoids ,a1 is equal to1 and balanced by the power1 by which it avoids ,c2 therefore the elastic force1 by which ,b is repelled from ,a1 will not put ,b into any degree of motion1 till ,a is by the motion of the chord brought nearer to ,b1 than ,b is to ,c3 but as soon as that is done1 the particle ,b will be moved towards ,c2 and being made to approach ,c1 will in the next place move that2 which will upon that advance1 put ,d likewise into motion1 and so on3 therefore the particle ,a being moved by the chord1 the following particles of the air ,b1 ,c1 ,d1 c4 will successively be moved4 ,farther1 if the point ,k of the chord moves forward with an accelerated velocity1 so that the particle ,a shall move against ,b with an advancing pace1 and gain ground of it1 approaching nearer and nearer continually2 ,a by approaching will press more upon ,b1 and give it a greater velocity likewise1 by reason that as the distance between the particles diminishes1 the elastic power1 by which they fly each other1 increases4 ,hence the particle ,b1 as well as ,a1 will have its motion gradually accelerated1 and by that means will more and more approach to ,c4 ,and from the same cause ,c will more and more approach ,d2 and so of the rest4 ,suppose now1 since the agitation of these particles has been shewn to be successive1 and to follow one another1 that ,e be the remotest particle moved1 while the chord is moving from its curve situation ,i ,k ,l into that of a streight line1 as ,i k ,l2 and ,f the first which remains unaffected1 though just upon the point of being put into motion4 ,then shall the particles ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 ,g1 when the point ,k is moved into k1 have acquired the rangement represented by the adjacent points a1 b1 c1 d1 e1 f1 g3 in which a is nearer to b than b to c1 and b nearer to c than c to d1 and c nearer to d than d to e and d nearer to e than e to f1 and lastly e nearer to f than f to g4 #i4 ,,but now the chord having recovered its rectilinear situation ,i k ,l1 the following motion will be changed1 for the point ,k1 which before advanced with a motion more and more accelerated1 though by the force it has acquired it will go on to move the same way as before1 till it has advanced near as far forwards1 as it was at first drawn backwards2 yet the motion of it will henceforth be gradually lessened4 ,the effect of which upon the particles a1 b1 c1 d1 e1 f1 g will be1 that by the time the chord has made its utmost advance1 and is upon the return1 these particles will be put into a contrary rangement2 so that f shall be nearer to g1 than e to f1 and e nearer to f than d to e2 and the like of the rest1 till you come to the first particles a1 b1 whose distance will then be nearly or quite what it was at first4 ,all which will appear as follows4 ,the present distance between a and b is such1 that the elastic power1 by which a repels b1 is strong enough to maintain that distance1 though a advance with the velocity1 with which the string resumes its rectilinear figure2 and the motion of the particle a being afterwards slower1 the present elasticity between a and b will be more than sufficient to preserve the distance between them4 ,therefore while it accelerates b it will retard a4 ,the distance b c will still diminish1 till b come about as near to c1 as it is from a at present2 for after the distances a b and b c are become equal1 the particle b will continue its velocity superior to that of c by its own power of inactivity1 till such time as the increase of elasticity between b and c more than shall be between a and b shall suppress its motion3 for as the power of inactivity in b made a greater elasticity necessary on the side of a than on the side of c to push b forward1 so what motion b has acquired it will retain by the same power of inactivity1 till it be suppressed by a greater elasticity on the side of c1 than on the side of a4 ,but as soon as b begins to slacken its pace the distance of b from c will widen as the distance a b has already done4 ,now as a acts on b1 so will b on c1 c on d1 c4 so that the distances between all the particles b1 c1 d1 e1 f1 g will be successively contracted into the distance of a from b1 and then dilated again4 ,now because the time1 in which the chord describes this present half of its vibration1 is about equal to that it took up in describing the former2 the particles a1 b will be as long in dilating their distance1 as before in contracting it1 and will return nearly to their original distance4 ,and farther1 the particles b1 c1 which did not begin to approach so soon as a1 b1 are now about as much longer1 before they begin to recede2 and likewise the particles c1 d1 which began to approach after b1 c1 begin to separate later4 ,whence it appears that the particles1 whose distance began to be lessened1 when that of a1 b was first enlarged1 viz4 the particles f1 g1 should be about their nearest distance1 when a and b have recovered their prime interval4 ,thus will the particles a1 b1 c1 d1 e1 f1 g have changed their situation in the manner asserted4 ,but farther1 as the particles f1 g or ,f1 ,g gradually approach each other1 they will move by degrees the succeeding particles to as great a length1 as the particles ,a1 ,b did by a like approach4 ,so that1 when the chord has made its greatest advance1 being arrived into the situation ,i ,l1 the particles moved by it will have the rangement noted by the points 1 1 1 1 1 1 1 1 1 1 1 4 ,where 1 are at the original distance of the particles in the line ,a ,h2 1 are the nearest of all1 and the distance is equal to that between and 4 #aj4 ,,by this time the chord ,i ,l begins to return1 and the distance between the particles and being enlarged to its original magnitude1 has lost all that force it had acquired by its motion1 being now at rest2 and therefore will return with the chord1 making the distance between and greater than the natural2 for will not return so soon1 because its motion forward is not yet quite suppressed1 the distance not being already enlarged to its prime dimension3 but the recess of 1 by diminishing the pressure upon by its elasticity1 will occasion the motion of to be stopt in a little time by the action of 1 and then shall begin to return3 at which time the distance between and shall by the superior action of above be enlarged to the dimension of the distance 1 and therefore soon after to that of 4 ,thus it appears1 that each of these particles goes on to move forward1 till its distance from the preceding one be equal to its original distance2 the whole chain 1 1 1 1 1 1 1 having an undulating motion forward1 which is stopt gradually by the excess of the expansive power of the preceding parts above that of the hinder4 ,thus are these parts successively stopt1 as before they were moved2 so that when the chord has regained its rectilinear situation1 the expansion of the parts of the air will have advanced so far1 that the interval between 1 which at present is most contracted1 will then be restored to its natural size3 the distances between and 1 and 1 and 1 and 1 and 1 being successively contracted into the present distance of from 1 and again enlarged2 so that the same effect shall be produced upon the parts beyond 1 by the enlargement of the distance between those two particles1 as was occasioned upon the particles 1 1 1 1 1 1 1 1 1 1 1 1 by the enlargement of the distance to its natural extent4 ,and therefore the motion in the air will be extended half as much farther as at present1 and the distance between and contracted into that1 which is at present between and 1 all the particles of the air in motion taking the rangement expressed in figure #aaa4 by the points 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 wherein the particles from to have their distances from each other gradually diminished1 the distances between the particles 1 being contracted the most from the natural distance between those particles1 and the distance between 1 as much augmented1 and the distance between the middle particles 1 becoming equal to the natural4 ,the particles 1 1 1 which follow 1 have their distances gradually greater and greater1 the particles 1 1 1 1 1 1 being ranged like the particles a1 b1 c1 d1 e1 f1 g1 or like the particles 1 1 1 1 1 1 in the former figure4 ,here it will be understood1 by what has been before explained1 that the particles 1 being at their natural distance from each other1 the particle is at rest1 the particles 1 1 1 1 between them and the string being in motion backward1 and the rest of the particles 1 1 1 1 1 1 1 1 1 in motion forward3 each of the particles between and moving faster than that1 which immediately follows it2 but of the particles from to 1 on the contrary1 those behind moving on faster than those1 which precede4 #aa4 ,,but now the string having recovered its rectilinear figure1 though it shall go on recoiling1 till it return near to its first situation ,i ,k ,l1 yet there will be a change in its motion2 so that whereas it returned from the situation ,i ,l with an accelerated motion1 its motion shall from hence be retarded again by the same degrees1 as accelerated before4 ,the effect of which change upon the particles of the air will be this4 ,as by the accelerated motion of the chord contiguous to it moved faster than 1 1 so as to make the interval greater than the interval 1 and from thence was made likewise to move faster than 1 and the distance between and rendered greater than the distance between and 1 and so of the rest2 now the motion of being diminished1 shall overtake it1 and the distance between and be reduced into that1 which is at present between and 1 the interval between and being inlarged into the present distance between and 2 but when the interval is increased to that1 which is at present between and the distance between and shall be enlarged to the present distance between and 1 and the distance between and inlarged into the present distance between and 2 and the same of the rest4 ,but the chord more and more slackening its pace1 the distance between and shall be more and more diminished2 and in consequence of that the distance between and shall be again contracted1 first into its present dimension1 and afterwards into a narrower space2 while the interval shall dilate into that at present between and 1 and as soon as it is so much enlarged1 it shall contract again4 ,thus by the reciprocal expansion and contraction of the air between and 1 by that time the chord is got into the situation ,i ,k ,l1 the interval shall be expanded into the present distance between and 2 and by that time likewise the present distance of from will be contracted into their natural interval3 for this distance will be about the same time in contracting it self1 as has been taken up in its dilatation2 seeing the string will be as long in returning from its rectilinear figure1 as it has been in recovering it from its situation ,i ,l4 ,this is the change which will be made in the particles between and 4 ,as for those between and 1 because each preceding particle advances faster than that1 which immediately follows it1 their distances will successively be dilated into that1 which is at present between and 4 ,and as soon as any two particles are arrived at their natural distance1 the hindermost of them shall be stopt1 and immediately after return1 the distances between the returning particles being greater than the natural4 ,and this dilatation of these distances shall extend so far1 by that time the chord is returned into its first situation ,i ,k ,l1 that the particles shall be removed to their natural distance4 ,but the dilatation of shall contract the interval into that at present between and 1 and the contraction of the distance between those two particles and will agitate a part of the air beyond2 so that when the chord is returned into the situation ,i ,k ,l1 having made an intire vibration1 the moved particles of the air will take the rangement expressed by the points1 l1 m1 n1 o1 p1 q1 r1 s1 t1 u1 w1 x1 y1 z1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h3 in which l m1 are at the natural distance of the particles1 the distance m n greater than l m and n o greater than m n1 and so on1 till you come to q r1 the widest of all3 and then the distances gradually diminish not only to the natural distance1 as w x1 but till they are contracted as much as was before2 which falls out in the points #b1 #c1 from whence the distances augment again1 till you come to the part of the air untouched4 #ab4 ,,this is the motion1 into which the air is put1 while the chord makes one vibration1 and the whole length of air thus agitated in the time of one vibration of the chord our author calls the length of one pulse4 ,when the chord goes on to make another vibration1 it will not only continue to agitate the air at present in motion1 but spread the pulsation of the air as much farther1 and by the same degrees1 as before4 ,for when the chord returns into its rectilinear situation ,i k ,l1 l m shall be brought into its most contracted state1 q r now in the state of greatest dilatation shall be reduced to its natural distance1 the points w1 x now at their natural distance shall be at their greatest distance1 the points #b1 #c now most contracted enlarged to their natural distance1 and the points #g1 #h reduced to their most contracted state3 and the contraction of them will carry the agitation of the air as far beyond them1 as that motion was carried from the chord1 when it first moved out of the situation ,i ,k ,l into its rectilinear figure4 ,when the chord is got into the situation ,i ,l1 l m shall recover its natural dimensions1 q r be reduced to its state of greatest contraction1 w x brought to its natural dimension1 the distance #b #c enlarged to the utmost1 and the points #g1 #h shall have recovered their natural distance2 and by thus recovering themselves they shall agitate the air to as great a length beyond them1 as it was moved beyond the chord1 when it first came into the situation ,i ,l4 ,when the chord is returned back again into its rectilinear situation1 l m shall be in its utmost dilatation1 q r restored again to its natural distance1 w x reduced into its state of greatest contraction1 #b #c shall recover its natural dimension1 and #g #h be in its state of greatest dilatation4 ,by which means the air shall be moved as far beyond the points #g1 #h1 as it was moved beyond the chord1 when it before made its return back to its rectilinear situation2 for the particles #g1 #h have been changed from their state of rest and their natural distance into a state of contraction1 and then have proceeded to the recovery of their natural distance1 and after that to a dilatation of it1 in the same manner as the particles contiguous to the chord were agitated before4 ,in the last place1 when the chord is returned into the situation ,i ,k ,l1 the particles of air from l to shall acquire their present rangement1 and the motion of the air be extended as much farther4 ,and the like will happen after every compleat vibration of the string4 #ac4 ,,concerning this motion of sound1 our author shews how to compute the velocity thereof1 or in what time it will reach to any proposed distance from the sonorous body4 ,for this he requires to know the height of air1 having the same density with the parts here at the surface of the earth1 which we breath1 that would be equivalent in weight to the whole incumbent atmosphere4 ,this is to be found by the barometer1 or common weatherglass4 ,in that instrument quicksilver is included in a hollow glass cane firmly closed at the top4 ,the bottom is open1 but immerged into quicksilver contained in a vessel open to the air4 ,care is taken when the lower end of the cane is immerged1 that the whole cane be full of quicksilver1 and that no air insinuate itself4 ,when the instrument is thus fixed1 the quicksilver in the cane being higher than that in the vessel1 if the top of the cane were open1 the fluid would soon sink out of the glass cane1 till it came to a level with that in the vessel4 ,but the top of the cane being closed up1 so that the air1 which has free liberty to press on the quicksilver in the vessel1 cannot bear at all on that1 which is within the cane1 the quicksilver in the cane will be suspended to such a height1 as to balance the pressure of the air on the quicksilver in the vessel4 ,here it is evident1 that the weight of the quicksilver in the glass cane is equivalent to the pressure of so much of the air1 as is perpendicularly over the hollow of the cane2 for if the cane be opened that the air may enter1 there will be no farther use of the quicksilver to sustain the pressure of the air without2 for the quicksilver in the cane1 as has already been observed1 will then subside to a level with that without4 ,hence therefore if the proportion between the density of quicksilver and of the air we breath be known1 we may know what height of such air would form a column equal in weight to the column of quicksilver within the glass cane4 ,when the quicksilver is sustained in the barometer at the height of #cj inches1 the height of such a column of air will be about #bigbe feet2 for in this case the air has about #a_/hgj of the density of water1 and the density of quicksilver exceeds that of water about #ac times1 so that the density of quicksilver exceeds that of the air about #aahij times2 and so many times #cj inches make #bigbe feet4 ,now ,sir ,,isaac ,,newton determines1 that while a pendulum of the length of this column should make one vibration or swing1 the space1 which any sound will have moved1 shall bear to this length the same proportion1 as the circumference of a circle bears to the diameter thereof2 that is1 about the proportion of #cee to #aacbfg4 ,only our author here considers singly the gradual progress of sound in the air from particle to particle in the manner we have explained1 without taking into consideration the magnitude of those particles4 ,and though there requires time for the motion to be propagated from one particle to another1 yet it is communicated to the whole of the same particle in an instant3 therefore whatever proportion the thickness of these particles bears to their distance from each other1 in the same proportion will the motion of sound be swifter4 ,again the air we breath is not simply composed of the elastic part1 by which sound is conveyed1 but partly of vapours1 which are of a different nature2 and in the computation of the motion of sound we ought to find the height of a column of this pure air only1 whose weight should be equal to the weight of the quicksilver in the cane of the barometer1 and this pure air being a part only of that we breath1 the column of this pure air will be higher than #bigbe feet4 ,on both these accounts the motion of sound is found to be about #aadb feet in one second of time1 or near #ac miles in a minute1 whereas by the computation proposed above1 it should move but #igi feet in one second4 #ad4 ,,we may observe here1 that from these demonstrations of our author it follows1 that all sounds whether acute or grave move equally swift1 and that sound is swiftest1 when the quicksilver stands highest in the barometer4 #ae4 ,,thus much of the appearances1 which are caused in these fluids from their gravitation toward the earth4 ,they also gravitate toward the moon2 for in the last chapter it has been proved1 that the gravitation between the earth and moon is mutual1 and that this gravitation of the whole bodies arises from that power acting in all their parts2 so that every particle of the moon gravitates toward the earth1 and every particle of the earth toward the moon4 ,but this gravitation of these fluids toward the moon produces no sensible effect1 except only in the sea1 where it causes the tides4 #af4 ,,that the tides depend upon the influence of the moon has been the receiv'd opinion of all antiquity2 nor is there indeed the least shadow of reason to suppose otherwise1 considering how steadily they accompany the moon's course4 ,though how the moon caused them1 and by what principle it was enabled to produce so distinguish'd an appearance1 was a secret left for this philosophy to unfold3 which teaches1 that the moon is not here alone concerned1 but that the sun likewise has a considerable share in their production2 though they have been generally ascribed to the other luminary1 because its effect is greatest1 and by that means the tides more immediately suit themselves to its motion2 the sun discovering its influence more by enlarging or restraining the moon's power1 than by any distinct effects4 ,our author finds the power of the moon to bear to the power of the sun about the proportion of #d to #a4 ,this he deduces from the observations made at the mouth of the river ,avon1 three miles from ,bristol1 by ,captain ,,sturmey1 and at ,plymouth by ,mr4 ,,colepresse1 of the height to which the water is raised in the conjunction and opposition of the luminaries1 compared with the elevation of it1 when the moon is in either quarter2 the first being caused by the united actions of the sun and moon1 and the other by the difference of them1 as shall hereafter be shewn4 #ag4 ,,that the sun should have a like effect on the sea1 as the moon1 is very manifest2 since the sun likewise attracts every single particle1 of which this earth is composed4 ,and in both luminaries since the power of gravity is reciprocally in the duplicate proportion of the distance1 they will not draw all the parts of the waters in the same manner2 but must act upon the nearest parts stronger1 than upon the remotest1 producing by this inequality an irregular motion4 ,we shall now attempt to shew how the actions of the sun and moon on the waters1 by being combined together1 produce all the appearances observed in the tides4 #ah4 ,,to begin therefore1 the reader will remember what has been said above1 that if the moon without the sun would have described an orbit concentrical to the earth1 the action of the sun would make the orbit oval1 and bring the moon nearer to the earth at the new and full1 than at the quarters#bfh4 ,now our excellent author observes1 that if instead of one moon1 we suppose a ring of moons1 contiguous and occupying the whole orbit of the moon1 his demonstration would still take place1 and prove that the parts of this ring in passing from the quarter to the conjunction or opposition would be accelerated1 and be retarded again in passing from the conjunction or opposition to the next quarter4 ,and as this effect does not depend on the magnitude of the bodies1 whereof the ring is composed1 the same would hold1 though the magnitude of these moons were so far to be diminished1 and their number increased1 till they should form a fluid#bfi4 ,now the earth turns round continually upon its own center1 causing thereby the alternate change of day and night1 while by this revolution each part of the earth is successively brought toward the sun1 and carried off again in the space of #bd hours4 ,and as the sea revolves round along with the earth itself in this diurnal motion1 it will represent in some sort such a fluid ring4 #ai4 ,,but as the water of the sea does not move round with so much swiftness1 as would carry it about the center of the earth in the circle it now describes1 without being supported by the body of the earth2 it will be necessary to consider the water under three distinct cases4 ,the first case shall suppose the water to move with the degree of swiftness1 required to carry a body round the center of the earth disingaged from it in a circle at the distance of the earth's semidiameter1 like another moon4 ,the second case is1 that the waters make but one turn about the axis of the earth in the space of a month1 keeping pace with the moon2 so that all parts of the water should preserve continually the same situation in respect of the moon4 ,the third case shall be the real one of the waters moving with a velocity between these two1 neither so swift as the first case requires1 nor so slow as the second4 #bj4 ,,in the first case the waters1 like the body which they equalled in velocity1 by the action of the moon would be brought nearer the center under and opposite to the moon1 than in the parts in the middle between these eastward or westward4 ,that such a body would so alter its distance by the moon's action upon it1 is clear from what has been mentioned of the like changes in the moon's motion caused by the sun#bgj4 ,and computation shews1 that the difference between the greatest and least distance of such a body would not be much above #d feet4 ,but in the second case1 where all the parts of the water preserve the same situation continually in respect of the moon1 the weight of those parts under and opposite to the moon will be diminished by the moon's action1 and the parts in the middle between these will have their weight increased3 this being effected just in the same manner1 as the sun diminishes the attraction of the moon towards the earth in the conjunction and opposition1 but increases that attraction in the quarters4 ,for as the first of these consequences from the sun's action on the moon is occasioned by the moon's being attracted by the sun in the conjunction more than the earth1 and in the opposition less than it1 and therefore in the common motion of the earth and moon1 the moon is made to advance toward the sun in one case too fast1 and in the other is left as it were behind2 so the earth will not have its middle parts drawn towards the moon so strongly as the nearer parts1 and yet more forcibly than the remotest3 and therefore since the earth and moon move each month round their common center of gravity#bga1 while the earth moves round this center1 the same effect will be produced1 on the parts of the water nearest to that center or to the moon1 as the moon feels from the sun when in conjunction1 and the water on the contrary side of the earth will be affected by the moon1 as the moon is by the sun1 when in opposition#bgb2 that is1 in both cases the weight of the water1 or its propensity towards the center of the earth1 will be diminished4 ,the parts in the middle between these will have their weight increased1 by being pressed towards the center of the earth through the obliquity of the moon's action upon them to its action upon the earth's center1 just as the sun increases the gravitation of the moon in the quarters from the same cause#bgc4 ,but now it is manifest1 that where the weight of the same quantity of water is least1 there it will be accumulated2 while the parts1 which have the greatest weight1 will subside4 ,therefore in this case there would be no tide or alternate rising and falling of the water1 but the water would form it self into an oblong figure1 whose axis prolonged would pass through the moon4 ,by ,sir ,,isaac ,,newton's computation the excess of this axis above the diameters perpendicular to it1 that is1 the height of the waters under and opposite to the moon above their height in the middle between these places eastward or westward caused by the moon1 is about #h feet4 #ba4 ,,thus the difference of height in this latter supposition is little short of twice that difference in the preceding4 ,but the case of the sea is a middle between these two3 for a body1 which should revolve round the center of the earth at the distance of a semidiameter without pressing on the earth's surface1 must perform its period in less than an hour and half1 whereas the earth turns round but once in a day2 and in the case of the waters keeping pace with the moon it should turn round but once in a month3 so that the real motion of the water is between the motions required in these two cases4 ,again1 if the waters moved round as swiftly as the first case required1 their weight would be wholly taken off by their motion2 for this case supposes the body to move so1 as to be kept revolving in a circle round the earth by the power of gravity without pressing on the earth at all1 so that its motion just supports its weight4 ,but if the power of gravity had been only #a_/bhi part of what it is1 the body could have moved thus without pressing on the earth1 and have been as long in moving round1 as the earth it self is4 ,consequently the motion of the earth takes off from the weight of the water in the middle between the poles1 where its motion is swiftest1 #a_/bhi part of its weight and no more4 ,since therefore in the first case the weight of the waters must be intirely taken off by their motion1 and by the real motion of the earth they lose only #a_/bhi part thereof1 the motion of the water will so little diminish their weight1 that their figure will much nearer resemble the case of their keeping pace with the moon than the other4 ,upon the whole1 if the waters moved with the velocity necessary to carry a body round the center of the earth at the distance of the earth's semidiameter without bearing on its surface1 the water would be lowest under the moon1 and rise gradually as it moved on with the earth eastward1 till it came half way toward the place opposite to the moon2 from thence it would subside again1 till it came to the opposition1 where it would become as low as at first2 afterwards it would rise again1 till it came half way to the place under the moon2 and from hence it would subside1 till it came a second time under the moon4 ,but in case the water kept pace with the moon1 it would be highest where in the other case it is lowest1 and lowest where in the other it is highest2 therefore the diurnal motion of the earth being between the motions of these two cases1 it will cause the highest place of the water to fall between the places of the greatest height in these two cases4 ,the water as it passes from under the moon shall for some time rise1 but descend again before it arrives half way to the opposite place1 and shall come to its least height before it becomes opposite to the moon2 then it shall rise again1 continuing so to do till it has passed the place opposite to the moon1 but subside before it comes to the middle between the places opposite to and under the moon2 and lastly it shall come to its lowest1 before it comes a second time under the moon4 ,if ,a 7in fig4 #aab1 #aac1 #aad47 represent the moon1 ,b the center of the earth1 the oval ,c ,d ,e ,f in fig4 #aab4 will represent the situation of the water in the first case2 but if the water kept pace with the moon1 the line ,c ,d ,e ,f in fig4 #aac4 would represent the situation of the water2 but the line ,c ,d ,e ,f in fig4 #aad4 will represent the same in the real motion of the water1 as it accompanies the earth in its diurnal rotation3 in all these figures ,c and ,e being the places where the water is lowest1 and ,d and ,f the places where it is highest4 ,pursuant to this determination it is found1 that on the shores1 which lie exposed to the open sea1 the high water usually falls out about three hours after the moon has passed the meridian of each place4 #bb4 ,,let this suffice in general for explaining the manner1 in which the moon acts upon the seas4 ,it is farther to be noted1 that these effects are greatest1 when the moon is over the earth's equator#bgd1 that is1 when it shines perpendicularly upon the parts of the earth in the middle between the poles4 ,for if the moon were placed over either of the poles1 it could have no effect upon the water to make it ascend and descend4 ,so that when the moon declines from the equator toward either pole1 it's action must be something diminished1 and that the more1 the farther it declines4 ,the tides likewise will be greatest1 when the moon is nearest to the earth1 it's action being then the strongest4 #bc4 ,,thus much of the action of the moon4 ,that the sun should produce the very same effects1 though in a less degree1 is too obvious to require a particular explanation3 but as was remarked before1 this action of the sun being weaker than that of the moon1 will cause the tides to follow more nearly the moon's course1 and principally shew it self by heightening or diminishing the effects of the other luminary4 ,which is the occasion1 that the highest tides are found about the conjunction and opposition of the luminaries1 being then produced by their united action1 and the weakest tides about the quarters of the moon2 because the moon in this case raising the water where the sun depresses it1 and depressing it where the sun raises it1 the stronger action of the moon is in part retunded and weakened by that of the sun4 ,our author computes that the sun will add near two feet to the height of the water in the first case1 and in the other take from it as much4 ,however the tides in both comply with the same hour of the moon4 ,but at other times1 between the conjunction or opposition and quarters1 the time deviates from that forementioned1 towards the hour in which the sun would make high water1 though still it keeps much nearer to the moon's hour than to the sun's4 #bd4 ,,again the tides have some farther varieties from the situation of the places where they happen northward or southward4 ,let p ,p 7in fig4 #aae47 represent the axis1 on which the earth daily revolves1 let h p ,h ,p represent the figure of the water1 and let n ,b ,n ,d be a globe inscribed within this figure4 ,suppose the moon to be advanced from the equator toward the north pole1 so that h ,h the axis of the figure of the water p ,a ,h ,p ,e h shall decline towards the north pole ,n2 take any place ,g nearer to the north pole than to the south1 and from the center of the earth ,c draw ,c ,g ,f2 then will ,g ,f denote the altitude to which the water is raised by the tide1 when the moon is above the horizon3 in the space of twelve hours1 the earth having turned half round its axis1 the place ,g will be removed to g2 but the axis h ,h will have kept its place preserving its situation in respect of the moon1 at least will have moved no more than the moon has done in that time1 which it is not necessary here to take into consideration4 ,now in this case the height of the water will be equal to g f1 which is not so great as ,g ,f4 ,but whereas ,g ,f is the altitude at high water1 when the moon is above the horizon1 g f will be the altitude of the same1 when the moon is under the horizon4 ,the contrary happens toward the south pole1 for ,k ,l is less than k l4 ,hence is proved1 that when the moon declines from the equator1 in those places1 which are on the same side of the equator as the moon1 the tides are greater1 when the moon is above the horizon1 than when under it2 and the contrary happens on the other side of the equator4 #be4 ,,now from these principles may be explained all the known appearances in the tides2 only by the assistance of this additional remark1 that the fluctuating motion1 which the water has in flowing and ebbing1 is of a durable nature1 and would continue for some time1 though the action of the luminaries should cease2 for this prevents the difference between the tide when the moon is above the horizon1 and the tide when the moon is below it from being so great1 as the rule laid down requires4 ,this likewise makes the greatest tides not exactly upon the new and full moon1 but to be a tide or two after2 as at ,bristol and ,plymouth they are found the third after4 #bf4 ,,this doctrine farther shews us1 why not only the spring tides fall out about the new and full moon1 and the neap tides about the quarters2 but likewise how it comes to pass1 that the greatest spring tides happen about the equinoxes2 because the luminaries are then one of them over the equator1 and the other not far from it4 ,it appears too1 why the neap tides1 which accompany these1 are the least of all1 for the sun still continuing over the equator continues to have the greatest power of lessening the moon's action1 and the moon in the quarters being far removed toward one of the poles1 has its power thereby weakned4 #bg4 ,,moreover the action of the moon being stronger1 when near the earth1 than when more remote2 if the moon1 when new suppose1 be at its nearest distance from the earth1 it shall when at the full be farthest off2 whence it is1 that two of the very largest spring tides do never immediately succeed each other4 #bh4 ,,because the sun in its passage from the winter solstice to the summer recedes from the earth1 and passing from the summer solstice to the winter approaches it1 and is therefore nearer the earth before the vernal equinox than after1 but nearer after the autumnal equinox than before2 the greatest tides oftner precede the vernal equinox than follow it1 and in the autumnal equinox on the contrary they oftner follow it than come before it4 #bi4 ,,the altitude1 to which the water is raised in the open ocean1 corresponds very well to the forementioned calculations2 for as it was shewn1 that the water in spring tides should rise to the height of #aj or #aa feet1 and the neap tides to #f or #g2 accordingly in the ,pacific1 ,atlantic and ,ethiopic oceans in the parts without the tropics1 the water is observed to rise about #f1 #i1 #ab or #ae feet4 ,in the ,pacific ocean this elevation is said to be greater than in the other1 as it ought to be by reason of the wide extent of that sea4 ,for the same reason in the ,ethiopic ocean between the tropics the ascent of the water is less than without1 by reason of the narrowness of the sea between the coasts of ,africa and the southern parts of ,america4 ,and islands in such narrow seas1 if far from shore1 have less tides than the coasts4 ,but now in those ports where the water flows in with great violence upon fords and shoals1 the force it acquires by that means will carry it to a much greater height1 so as to make it ascend and descend to #cj1 #dj or even #ej feet and more2 instances of which we have at ,plymouth1 and in the ,severn near ,chepstow2 at ,st4 ,michael's and ,auranches in ,normandy2 at ,cambay and ,pegu in the ,east ,indies4 #cj4 ,,again the tides take a considerable time in passing through long straits1 and shallow places4 ,thus the tide1 which is made on the west coast of ,ireland and on the coast of ,spain at the third hour after the moon's coming to the meridian1 in the ports eastward toward the ,british channel falls out later1 and as the flood passes up that channel still later and later1 so that the tide takes up full twelve hours in coming up to ,london bridge4 #ca4 ,,in the last place tides may come to the same port from different seas1 and as they may interfere with each other1 they will produce particular effects4 ,suppose the tide from one sea come to a port at the third hour after the moon's passing the meridian of the place1 but from another sea to take up six hours more in its passage4 ,here one tide will make high water1 when by the other it should be lowest2 so that when the moon is over the equator1 and the two tides are equal1 there will be no rising and falling of the water at all2 for as much as the water is carried off by one tide1 it will be supplied by the other4 ,but when the moon declines from the equator1 the same way as the port is situated1 we have shewn that of the two tides of the ocean1 which are made each day1 that tide1 which is made when the moon is above the horizon1 is greater than the other4 ,therefore in this case1 as four tides come to this port each day the two greatest will come on the third1 and on the ninth hour after the moon's passing the meridian1 and the two least at the fifteenth and at the twenty first hour4 ,thus from the third to the ninth hour more water will be in this port by the two greatest tides than from the ninth to the fifteenth1 or from the twenty first to the following third hour1 where the water is brought by one great and one small tide2 but yet there will be more water brought by these tides1 than what will be found between the two least tides1 that is1 between the fifteenth and twenty first hour4 ,therefore in the middle between the third and ninth hour1 or about the moon's setting1 the water will be at its greatest height2 in the middle between the ninth and fifteenth1 as also between the twenty first and following third hour it will have its mean height2 and be lowest in the middle between the fifteenth and twenty first hour1 that is1 at the moon's rising4 ,thus here the water will have but one flood and one ebb each day4 ,when the moon is on the other side of the equator1 the flood will be turned into ebb1 and the ebb into flood2 the high water falling out at the rising of the moon1 and the low water at the setting4 ,now this is the case of the port of ,batsham in the kingdom of ,tunquin in the ,east ,indies2 to which port there are two inlets1 one between the continent and the islands which are called the ,manillas1 and the other between the continent and ,borneo4 #cb4 ,,the next thing to be considered is the effect1 which these fluids of the planets have upon the solid part of the bodies to which they belong4 ,and in the first place ,i shall shew1 that it was necessary upon account of these fluid parts to form the bodies of the planets into a figure something different from that of a perfect globe4 ,because the diurnal rotation1 which our earth performs about its axis1 and the like motion we see in some of the other planets1 7which is an ample conviction that they all do the like7 will diminish the force1 with which bodies are attracted upon all the parts of their surfaces1 except at the very poles1 upon which they turn4 ,thus a stone or other weighty substance resting upon the surface of the earth1 by the force which it receives from the motion communicated to it by the earth1 if its weight prevented not1 would continue that motion in a straight line from the point where it received it1 and according to the direction1 in which it was given1 that is1 in a line which touches the surface at that point2 insomuch that it would move off from the earth in the same manner1 as a weight fasten'd to a string and whirled about endeavours continually to recede from the center of motion1 and would forthwith remove it self to a greater distance from it1 if loosed from the string which retains it4 ,and farther1 as the centrifugal force1 with which such a weight presses from the center of its motion1 is greater1 by how much greater the velocity is1 with which it moves2 so such a body1 as ,i have been supposing to lie on the earth1 would recede from it with the greater force1 the greater the velocity is1 with which the part of the earth's surface it rests upon is moved1 that is1 the farther distant it is from the poles4 ,but now the power of gravity is great enough to prevent bodies in any part of the earth from being carried off from it by this means2 however it is plain that bodies having an effort contrary to that of gravity1 though much weaker than it1 their weight1 that is1 the degree of force1 with which they are pressed to the earth1 will be diminished thereby1 and be the more diminished1 the greater this contrary effort is2 or in other words1 the same body will weigh heavier at either of the poles1 than upon any other part of the earth2 and if any body be removed from the pole towards the equator1 it will lose of its weight more and more1 and be lightest of all at the equator1 that is1 in the middle between the poles4 #cc4 ,,this now is easily applied to the waters of the seas1 and shews that the water under the poles will press more forcibly to the earth1 than at or near the equator3 and consequently that which presses least1 must give place1 till by ascending it makes room for receiving a greater quantity1 which by its additional weight may place the whole upon a ballance4 ,to illustrate this more particularly ,i shall make use of fig4 #aaf ,in which let ,a ,c ,b ,d be a circle1 by whose revolution about the diameter ,a ,b a globe should be formed1 representing a globe of solid earth4 ,suppose this globe covered on all sides with water to the same height1 suppose that of ,e ,a or ,b ,f1 at which distance the circle ,e ,g ,f ,h surrounds the circle ,a ,c ,b ,d2 then it is evident1 if the globe of earth be at rest1 the water which surrounds it will rest in that situation4 ,but if the globe be turned incessantly about its axis ,a ,b1 and the water have likewise the same motion1 it is also evident1 from what has been explained1 that the water between the circles ,e ,h ,f ,g and ,a ,d ,b ,c will remain no longer in the present situation1 the parts of it between ,h and ,d1 and between ,g and ,c being by this rotation become lighter1 than the parts between ,e and ,a and between ,b and ,f2 so that the water over the poles ,a and ,b must of necessity subside1 and the water be accumulated over ,d and ,c1 till the greater quantity in these latter places supply the defect of its weight4 ,this would be the case1 were the globe all covered with water4 ,and the same figure of the surface would also be preserved1 if some part of the water adjoining to the globe in any part of it were turned into solid earth1 as is too evident to need any proof2 because the parts of the water remaining at rest1 it is the same thing1 whether they continue in the state of being easily separable1 which denominates them fluid1 or were to be consolidated together1 so as to make a hard body3 and this1 though the water should in some places be thus consolidated1 even to the surface of it4 ,which shews that the form of the solid part of the earth makes no alteration in the figure the water will take3 and by consequence in order to the preventing some parts of the earth from being entirely overflowed1 and other parts quite deserted1 the solid parts of the earth must have given them much the same figure1 as if the whole earth were covered on all sides with water4 #cd4 ,,farther1 ,i say1 this figure of the earth is the same1 as it would receive1 were it entirely a globe of water1 provided that water were of the same density as the substance of the globe4 ,for suppose the globe ,a ,c ,b ,d to be liquified1 and that the globe ,e ,h ,f ,g1 now entirely water1 by its rotation about its axis should receive such a figure as we have been describing1 and then the globe ,a ,c ,b ,d should be consolidated again1 the figure of the water would plainly not be altered1 by such a consolidation4 #ce4 ,,but from this last observation our author is enabled to determine the proportion between the axis of the earth drawn from pole to pole1 and the diameter of the equator1 upon the supposition that all the parts of the earth are of equal density2 which he does by computing in the first place the proportion of the centrifugal force of the parts under the equator to the power of gravity2 and then by considering the earth as a spheroid1 made by the revolution of an ellipsis about its lesser axis1 that is1 supposing the line ,m ,i ,l ,k to be an exact ellipsis1 from which it can differ but little1 by reason that the difference between the lesser axis ,m ,l and the greater ,i ,k is but very small4 ,from this supposition1 and what was proved before1 that all the particles which compose the earth have the attracting power explained in the preceding chapter1 he finds at what distance the parts under the equator ought to be removed from the center1 that the force1 with which they shall be attracted to the center1 diminished by their centrifugal force1 shall be sufficient to keep those parts in a ballance with those which lie under the poles4 ,and upon the supposition of all the parts of the earth having the same degree of density1 the earth's surface at the equator must be above #ag miles more distant from the center1 than at the poles#bge4 #cf4 ,,after this it is shewn1 from the proportion of the equatorial diameter of the earth to its axis1 how the same may be determined of any other planet1 whose density in comparison of the density of the earth1 and the time of its revolution about its axis1 are known4 ,and by the rule delivered for this1 it is found1 that the diameter of the equator in ,jupiter should bear to its axis about the proportion of #aj to #ibgf1 and accordingly this planet appears of an oval form to the astronomers4 ,the most considerable effects of this spheroidical figure our author takes likewise into consideration2 one of which is that bodies are not equally heavy in all distances from the poles2 but near the equator1 where the distance from the center is greatest1 they are lighter than towards the poles3 and nearly in this proportion1 that the actual power1 by which they are drawn to the center1 resulting from the difference between their absolute gravity and centrifugal force1 is reciprocally as the distance from the center4 ,that this may not appear to contradict what has before been said of the alteration of the power of gravity1 in proportion to the change of the distance from the center1 it is proper carefully to remark1 that our author has demonstrated three things relating hereto3 the first is1 that decrease of the power of gravity as we recede from the center1 which has been fully explained in the last chapter1 upon supposition that the earth and planets are perfect spheres1 from which their difference is by many degrees too little to require notice for the purposes there intended3 the next is1 that whether they be perfect spheres1 or exactly such spheroids as have now been mentioned1 the power of gravity1 as we descend in the same line to the center1 is at all distances as the distance from the center1 the parts of the earth above the body by drawing the body towards them lessening its gravitation towards the center#bgg2 and both these assertions relate to gravity alone3 the third is what we mentioned in this place1 that the actual force on different parts of the surface1 with which bodies are drawn to the center1 is in the proportion here assigned#bgh4 #ch4 ,,the next effect of this figure of the earth is an obvious consequence of the former3 that pendulums of the same length do not in different distances from the pole make their vibrations in the same time2 but towards the poles1 where the gravity is strongest1 they move quicker than near the equator1 where they are less impelled to the center2 and accordingly pendulums1 that measure the same time by their vibrations1 must be shorter near the poles than at a greater distance4 ,both which deductions are found true in fact2 of which our author has recounted particularly several experiments1 in which it was found1 that clocks exactly adjusted to the true measure of time at ,paris1 when transported nearer to the equator1 became erroneous and moved too slow1 but were reduced to their true motion by contracting their pendulums4 ,our author is particular in remarking1 how much they lost of their motion1 while the pendulums remained unaltered2 and what length the observers are said to have shortened them1 to bring them to time4 ,and the experiments1 which appear to be most carefully made1 shew the earth to be raised in the middle between the poles1 as much as our author found it by his computation#bgi4 #ci4 ,,these experiments on the pendulum our author has been very exact in examining1 inquiring particularly how much the extension of the rod of the pendulum by the great heats in the torrid zone might make it necessary to shorten it4 ,for by an experiment made by ,,picart1 and another made by ,,de ,,la ,,hire1 heat1 though not very intense1 was found to increase the length of rods of iron4 ,the experiment of ,,picart was made with a rod one foot long1 which in winter1 at the time of frost1 was found to increase in length by being heated at the fire4 ,in the experiment of ,,de ,,la ,,hire a rod of six foot in length was found1 when heated by the summer sun only1 to grow to a greater length1 than it had in the aforesaid cold season4 ,from which observations a doubt has been raised1 whether the rod of the pendulums in the aforementioned experiments was not extended by the heat of those warm climates to all that excess of length1 the observers found themselves obliged to lessen them by4 ,but the experiments now mentioned shew the contrary4 ,for in the first of them the rod of a foot long was lengthened no more than #a_/i part of what the pendulum under the equator must be diminished2 and therefore a rod of the length of the pendulum would not have been extended above of that length4 ,in the experiment of ,,de ,,la ,,hire1 where the heat was less1 the rod of six foot long was extended no more than #c_/aj of what the pendulum must be shortened2 so that a rod of the length of the pendulum would not have gained above #c_/bj or #a_/g of that length4 ,and the heat in this latter experiment1 though less than in the former1 was yet greater than the rod of a pendulum can ordinarily contract in the hottest country2 for metals receive a great heat when exposed to the open sun1 certainly much greater than that of a human body4 ,but pendulums are not usually so exposed1 and without doubt in these experiments were kept cool enough to appear so to the touch2 which they would do in the hottest place1 if lodged in the shade4 ,our author therefore thinks it enough to allow about #a_/aj of the difference observed upon account of the greater warmth of the pendulum4 #dj4 ,,there is a third effect1 which the water has on the earth by changing its figure1 that is taken notice of by our author2 for the explaining of which we shall first prove1 that bodies descend perpendicularly to the surface of the earth in all places4 ,the manner of collecting this from observation1 is as follows4 ,the surfaces of all fluids rest parallel to that part of the surface of the sea1 which is in the same place with them1 to the figure of which1 as has been particularly shewn1 the figure of the whole earth is formed4 ,for if any hollow vessel1 open at the bottom1 be immersed into the sea2 it is evident1 that the surface of the sea within the vessel will retain the same figure it had1 before the vessel inclosed it2 since its communication with the external water is not cut off by the vessel4 ,but all the parts of the water being at rest1 it is as clear1 that if the bottom of the vessel were closed1 the figure of the water could receive no change thereby1 even though the vessel were raised out of the sea2 any more than from the insensible alteration of the power of gravity1 consequent upon the augmentation of the distance from the center4 ,but now it is clear1 that bodies descend in lines perpendicular to the surfaces of quiescent fluids2 for if the power of gravity did not act perpendicularly to the surface of fluids1 bodies which swim on them could not rest1 as they are seen to do2 because1 if the power of gravity drew such bodies in a direction oblique to the surface whereon they lay1 they would certainly be put in motion1 and be carried to the side of the vessel1 in which the fluid was contained1 that way the action of gravity inclined4 #da4 ,,hence it follows1 that as we stand1 our bodies are perpendicular to the surface of the earth4 ,therefore in going from north to south our bodies do not keep in a parallel direction4 ,now in all distances from the pole the same length gone on the earth will not make the same change in the position of our bodies1 but the nearer we are to the poles1 we must go greater length to cause the same variation herein4 ,let ,m ,i ,l ,k 7in fig4 #aag7 represent the figure of the earth1 ,m1 ,l the poles1 ,i1 ,k two opposite points in the middle between these poles4 ,let ,t ,v and ,p ,o be two arches1 ,t ,v being most remote from the pole ,l2 draw ,t ,w1 ,v ,x1 ,p ,q1 ,o ,r1 each perpendicular to the surface of the earth1 and let ,t ,w1 ,v ,x meet in ,y1 and ,p ,q1 ,o ,r in ,s4 ,here it is evident1 that in passing from ,v to ,t the position of a man's body would be changed by the angle under ,t ,y ,v1 for at ,v he would stand in the line ,y ,v continued upward1 and at ,t in the line ,y ,t2 but in passing from ,o to ,p the position of his body would be changed by the angle under ,o ,s ,p4 ,now ,i say1 if these two angles are equal the arch ,o ,p is longer than ,t ,v3 for the figure ,m ,i ,l ,k being oblong1 and ,i ,k longer than ,m ,l1 the figure will be more incurvated toward ,i than toward ,l2 so that the lines ,t ,w and ,v ,x will meet in ,y before they are drawn out to so great a length as the lines ,p ,q and ,o ,r must be continued to1 before they will meet in ,s4 ,since therefore ,y ,t and ,y ,v are shorter than ,p ,s and ,s ,v1 ,t ,v must be less than ,o ,p4 ,if these angles under ,t ,y ,v and ,o ,s ,p are each #a_/ij part of the angle made by a perpendicular line1 they are said each to contain one degree4 ,and the unequal length of these arches ,o ,p and ,v ,t gives occasion to the assertion1 that in passing from north to south the degrees on the earth's surface are not of an equal length1 but those near the pole longer than those toward the equator4 ,for the length of the arch on the earth lying between the two perpendiculars1 which make an angle of a degree with each other1 is called the length of a degree on the earth's surface4 #db4 ,,this figure of the earth has some effect on eclipses4 ,it has been observed above1 that sometimes the nodes of the moon's orbit lie in a straight line drawn from the sun to the earth2 in which case the moon will cross the plane of the earth's motion at the new and full4 ,but whenever the moon passes near the plane at the full1 some part of the earth will intercept the sun's light1 and the moon shining only with light borrow'd from the sun1 when that light is prevented from falling on any part of the moon1 so much of her body will be darkened4 ,also when the moon at the new is near the plane of the earth's motion1 the inhabitants on some part of the earth will see the moon come under the sun1 and the sun thereby be covered from them either wholly or in part4 ,now the figure1 which we have shewn to belong to the earth1 will occasion the shadow of the earth on the moon not to be perfectly round1 but cause the diameter from east to west to be somewhat longer than the diameter from north to south4 ,in eclipse of the sun this figure of the earth will make some little difference in the place1 where the sun shall appear wholly or in any given part covered4 ,let ,a ,b ,c ,d 7in fig4 #aah47 represent the earth1 ,a ,c the axis whereon it turns daily1 ,e the center4 ,let ,f ,a ,g ,c represent a perfect globe inscribed within the earth4 ,let ,h ,i be a line drawn through the centers of the sun and moon1 crossing the surface of the earth in ,k1 and the surface of the globe inscribed in ,l4 ,draw ,e ,l1 which will be perpendicular to the surface of the globe in ,l3 and draw likewise ,k ,m1 so that it shall be perpendicular to the surface of the earth in ,k4 ,now whereas the eclipse would appear central at ,l1 if the earth were the globe ,a ,g ,c ,f1 and does really appear so at ,k2 ,i say1 the latitude of the place ,k on the real earth is different from the latitude of the place ,l on the globe ,f ,a ,g ,c4 ,what is called the latitude of any place is determined by the angle which the line perpendicular to the surface of the earth at that place makes with the axis2 the difference between this angle1 and that made by a perpendicular line or square being called the latitude of each place4 ,but it might here be proved1 that the angle which ,k ,m makes with ,m ,c is less1 than the angle made between ,l ,e and ,e ,c3 consequently the latitude of the place ,k is greater1 than the latitude1 which the place ,l would have4 #dc4 ,,the next effect1 which follows from this figure of the earth1 is that gradual change in the distance of the fixed stars from the equinoctial points1 which astronomers observe4 ,but before this can be explained1 it is necessary to say something more particular1 than has yet been done1 concerning the manner of the earth's motion round the sun4 #dd4 ,,it has already been said1 that the earth turns round each day on its own axis1 while its whole body is carried round the sun once in a year4 ,how these two motions are joined together may be conceived in some degree by the motion of a bowl on the ground1 where the bowl in rouling on continually turns upon its axis1 and at the same time the whole body thereof is carried straight on4 ,but to be more express let ,a 7in fig4 #aai7 represent the sun ,b ,c ,d ,e four different situations of the earth in its orbit moving about the sun4 ,in all these let ,f ,g represent the axis1 about which the earth daily turns4 ,the points ,f1 ,g are called the poles of the earth2 and this axis is supposed to keep always parallel to it self in every situation of the earth2 at least that it would do so1 were it not for a minute deviation1 the cause whereof will be explained in what follows4 ,when the earth is in ,b1 the half ,h ,i ,k will be illuminated by the sun1 and the other half ,h ,l ,k will be in darkness4 ,now if on the globe any point be taken in the middle between the poles1 this point shall describe by the motion of the globe the circle ,m ,n1 half of which is in the enlightened part of the globe1 and half in the dark part4 ,but the earth is supposed to move round its axis with an equable motion2 therefore on this point of the globe the sun will be seen just half the day1 and be invisible the other half4 ,and the same will happen to every point of this circle1 in all situations of the earth during its whole revolution round the sun4 ,this circle ,m ,n is called the equator1 of which we have before made mention4 #de4 ,,now suppose any other point taken on the surface of the globe toward the pole ,f1 which in the diurnal revolution of the globe shall describe the circle ,o ,p4 ,here it appears that more than half this circle is enlightned by the sun1 and consequently that in any particular point of this circle the sun will be longer seen than lie hid1 that is the day will be longer than the night4 ,again if we consider the same circle ,o ,p on the globe situated in ,d the opposite part of the orbit from ,b1 we shall see1 that here in any place of this circle the night will be as much longer than the day4 #df4 ,,in these situations of the globe of earth a line drawn from the sun to the center of the earth will be obliquely inclined toward the axis ,f ,g4 ,now suppose1 that such a line drawn from the sun to the center of the earth1 when in ,c or ,e1 would be perpendicular to the axis ,f ,g2 in which cases the sun will shine perpendicularly upon the equator1 and consequently the line drawn from the center of the earth to the sun will cross the equator1 as it passes through the surface of the earth2 whereas in all other situations of the globe this line will pass through the surface of the globe at a distance from the equator either northward or southward4 ,now in both these cases half the circle ,o ,p will be in the light1 and half in the dark2 and therefore to every place in this circle the day will be equal to the night4 ,thus it appears1 that in these two opposite situations of the earth the day is equal to the night in all parts of the globe2 but in all other situations this equality will only be found in places situated in the very middle between the poles1 that is1 on the equator4 #dg4 ,,the times1 wherein this universal equality between the day and night happens1 are called the equinoxes4 ,now it has been long observed by astronomers1 that after the earth hath set out from either equinox1 suppose from ,e 7which will be the spring equinox1 if ,f be the north pole7 the same equinox shall again return a little before the earth has made a compleat revolution round the sun4 ,this return of the equinox preceding the intire revolution of the earth is called the precession of the equinox1 and is caused by the protuberant figure of the earth4 #di4 ,,since the sun shines perpendicularly upon the equator1 when the line drawn from the sun to the center of the earth is perpendicular to the earth's axis1 in this case the plane1 which should cut through the earth at the equator1 may be extended to pass through the sun2 but it will not do so in any other position of the earth4 ,now let us consider the prominent part of the earth about the equator1 as a solid ring moving with the earth round the sun4 ,at the time of the equinoxes1 this ring will have the same kind of situation in respect of the sun1 as the orbit of the moon has1 when the line of the nodes is directed to the sun2 and at all other times will resemble the moon's orbit in other situations4 ,consequently this ring1 which otherwise would keep throughout its motion parallel to it self1 will receive some change in its position from the action of the sun upon it1 except only at the time of the equinox4 ,the manner of this change may be understood as follows4 ,let ,a ,b ,c ,d 7in fig4 #abj7 represent this ring1 ,e the center of the earth1 ,s the sun1 ,a ,f ,c ,g a circle described in the plane of the earth's motion to the center ,e4 ,here ,a and ,c are the two points1 in which the earth's equator crosses the plane of the earth's motion2 and the time of the equinox falls out1 when the straight line ,a ,c continued would pass through the sun4 ,now let us recollect what was said above concerning the moon1 when her orbit was in the same situation with this ring4 ,from thence it will be understood1 if a body were supposed to be moving in any part of this circle ,a ,b ,c ,d1 what effect the action of the sun on the body would have toward changing the position of the line ,a ,c4 ,in particular ,h ,i being drawn perpendicular to ,s ,e1 if the body be in any part of this circle between ,a and ,h1 or between ,c and ,i1 the line ,a ,c would be so turned1 that the point ,a shall move toward ,b1 and the point ,c toward ,d2 but if it were in any other part of the circle1 either between ,h and ,c1 or between ,i and ,a1 the line ,a ,c would be turned the contrary way4 ,hence it follows1 that as this solid ring turns round the center of the earth1 the parts of this ring between ,a and ,h1 and between ,c and ,i1 are so influenced by the sun1 that they will endeavour1 so to change the situation of the line ,a ,c as to cause the point ,a to move toward ,b1 and the point ,c to move toward ,d2 but all the parts of the ring between ,h and ,c1 and between ,i and ,a1 will have the opposite tendency1 and dispose the line ,a ,c to move the contrary way4 ,and since these last named parts are larger than the other1 they will prevail over the other1 so that by the action of the sun upon this ring1 the line ,a ,c will be so turned1 that ,a shall continually be more and more moving toward ,d1 and ,c toward ,b4 ,thus no sooner shall the sun in its visible motion have departed from ,a1 but the motion of the line ,a ,c shall hasten its meeting with ,c1 and from thence the motion of this line shall again hasten the sun's second conjunction with ,a2 for as this line so turns1 that ,a is continually moving toward ,d1 so the sun's visible motion is the same way as from ,s toward ,t4 #di4 ,,the moon will have on this ring the like effect as the sun1 and operate on it more strongly1 in the same proportion as its force on the sea exceeded that of the sun on the same4 ,but the effect of the action of both luminaries will be greatly diminished by reason of this ring's being connected to the rest of the earth2 for by this means the sun and moon have not only this ring to move1 but likewise the whole globe of the earth1 upon whose spherical part they have no immediate influence4 ,beside the effect is also rendred less1 by reason that the prominent part of the earth is not collected all under the equator1 but spreads gradually from thence toward both poles4 ,upon the whole1 though the sun alone carries the nodes of the moon through an intire revolution in about #ai years1 the united force of both luminaries on the prominent parts of the earth will hardly carry round the equinox in a less space of time than #bfjjj years4 #ej4 ,,to this motion of the equinox we must add another consequence of this action of the sun and moon upon the elevated parts of the earth1 that this annular part of the earth about the equator1 and consequently the earth's axis1 will twice a year and twice a month change its inclination to the plane of the earth's motion1 and be again restored1 just as the inclination of the moon's orbit by the action of the sun is annually twice diminished1 and as often recovers its original magnitude4 ,but this change is very insensible4 #ea4 ,i ,,shall now finish the present chapter with our great author's inquiry into the figure of the secondary planets1 particularly of our moon1 upon the figure of which its fluid parts will have an influence4 ,the moon turns always the same side towards the earth1 and consequently revolves but once round its axis in the space of an entire month2 for a spectator placed without the circle1 in which the moon moves1 would in that time observe all the parts of the moon successively to pass once before his view and no more1 that is1 that the whole globe of the moon has turned once round4 ,now the great slowness of this motion will render the centrifugal force of the parts of the waters very weak1 so that the figure of the moon cannot1 as in the earth1 be much affected by this revolution upon its axis3 but the figure of those waters are made different from spherical by another cause1 viz4 the action of the earth upon them2 by which they will be reduced to an oblong oval form1 whose axis prolonged would pass through the earth2 for the same reason1 as we have above observed1 that the waters of the earth would take the like figure1 if they had moved so slowly1 as to keep pace with the moon4 ,and the solid part of the moon must correspond with this figure of the fluid part3 but this elevation of the parts of the moon is nothing near so great as is the protuberance of the earth at the equator1 for it will not exceed #ic english feet4 #eb4 ,the waters of the moon will have no tide1 except what will arise from the motion of the moon round the earth4 ,for the conversion of the moon about her axis is equable1 whereby the inequality in the motion round the earth discovers to us at some times small parts of the moon's surface towards the east or west1 which at other times lie hid2 and as the axis1 whereon the moon turns1 is oblique to her motion round the earth1 sometimes small parts of her surface toward the north1 and sometimes the like toward the south are visible1 which at other times are out of sight4 ,these appearances make what is called the libration of the moon1 discovered by ,,hevelius4 ,but now as the axis of the oval figure of the waters will he pointed towards the earth1 there must arise from hence some fluctuation in them2 and beside1 by the change of the moon's distance from the earth1 they will not always have the very same height4 ,illustration ,illustration ,,book ,,iii4 ,,chap ,i4 ,concerning the cause of ,,colours inherent in the ,,light4 ,,after this view which has been taken of ,sir ,,isaac ,,newton',s mathematical principles of philosophy1 and the use he has made of them1 in explaining the system of the world1 c4 the course of my design directs us to turn our eyes to that other philosophical work1 his treatise of ,optics1 in which we shall find our great author's inimitable genius discovering it self no less1 than in the former2 nay perhaps even more1 since this work gives as many instances of his singular force of reasoning1 and of his unbounded invention1 though unassisted in great measure by those rules and general precepts1 which facilitate the invention of mathematical theorems4 ,nor yet is this work inferior to the other in usefulness2 for as that has made known to us one great principle in nature1 by which the celestial motions are continued1 and by which the frame of each globe is preserved2 so does this point out to us another principle no less universal1 upon which depends all those operations in the smaller parts of matter1 for whose sake the greater frame of the universe is erected2 all those immense globes1 with which the whole heavens are filled1 being without doubt only design'd as so many convenient apartments for carrying on the more noble operations of nature in vegetation and animal life4 ,which single consideration gives abundant proof of the excellency of our author's choice1 in applying himself carefully to examine the action between light and bodies1 so necessary in all the varieties of these productions1 that none of them can be successfully promoted without the concurrence of heat in a greater or less degree4 #b4 ',,tis true1 our author has not made so full a discovery of the principle1 by which this mutual action between light and bodies is caused2 as he has in relation to the power1 by which the planets are kept in their courses3 yet he has led us to the very entrance upon it1 and pointed out the path so plainly which must be followed to reach it2 that one may be bold to say1 whenever mankind shall be blessed with this improvement of their knowledge1 it will be derived so directly from the principles laid down by our author in this book1 that the greatest share of the praise due to the discovery will belong to him4 #c4 ,,in speaking of the progress our author has made1 ,i shall distinctly pursue three things1 the two first relating to the colours of natural bodies3 for in the first head shall be shewn1 how those colours are derived from the properties of the light itself2 and in the second upon what properties of the bodies they depend3 but the third head of my discourse shall treat of the action of bodies upon light in refracting1 reflecting1 and inflecting it4 #d4 ,,the first of these1 which shall be the business of the present chapter1 is contained in this one proposition3 that the sun's direct light is not uniform in respect of colour1 not being disposed in every part of it to excite the idea of whiteness1 which the whole raises2 but on the contrary is a composition of different kinds of rays1 one sort of which if alone would give the sense of red1 another of orange1 a third of yellow1 a fourth of green1 a fifth of light blue1 a sixth of indigo1 and a seventh of a violet purple2 that all these rays together by the mixture of their sensations impress upon the organ of sight the sense of whiteness1 though each ray always imprints there its own colour2 and all the difference between the colours of bodies when viewed in open day light arises from this1 that coloured bodies do not reflect all the sorts of rays falling upon them in equal plenty1 but some sorts much more copiously than others2 the body appearing of that colour1 of which the light coming from it is most composed4 #e4 ,,that the light of the sun is compounded1 as has been said1 is proved by refracting it with a prism4 ,by a prism ,i here mean a glass or other body of a triangular form1 such as is represented in fig4 #aba4 ,but before we proceed to the illustration of the proposition we have just now laid down1 it will be necessary to spend a few words in explaining what is meant by the refraction of light2 as the design of our present labour is to give some notion of the subject1 we are engaged in1 to such as are not versed in the mathematics4 #f4 ,,it is well known1 that when a ray of light passing through the air falls obliquely upon the surface of any transparent body1 suppose water or glass1 and enters it1 the ray will not pass on in that body in the same line it described through the air1 but be turned off from the surface1 so as to be less inclined to it after passing it1 than before4 ,let ,a ,b ,c ,d 7in fig4 #abb47 represent a portion of water1 or glass1 ,a ,b the surface of it1 upon which the ray of light ,e ,f falls obliquely2 this ray shall not go right on in the course delineated by the line ,f ,g1 but be turned off from the surface ,a ,b into the line ,f ,h1 less inclined to the surface ,a ,b than the line ,e ,f is1 in which the ray is incident upon that surface4 #g4 ,,on the other hand1 when the light passes out of any such body into the air1 it is inflected the contrary way1 being after its emergence rendred more oblique to the surface it passes through1 than before4 ,thus the ray ,f ,h1 when it goes out of the surface ,c ,d1 will be turned up towards that surface1 going out into the air in the line ,h ,i4 #h4 ,,this turning of the light out of its way1 as it passes from one transparent body into another is called its refraction4 ,both these cases may be tried by an easy experiment with a bason and water4 ,for the first case set an empty bason in the sunshine or near a candle1 making a mark upon the bottom at the extremity of the shadow cast by the brim of the bason1 then by pouring water into the bason you will observe the shadow to shrink1 and leave the bottom of the bason enlightned to a good distance from the mark4 ,let ,a ,b ,c 7in fig4 #abc47 denote the empty bason1 ,e ,a ,d the light shining over the brim of it1 so that all the part ,a ,b ,d be shaded4 ,then a mark being made at ,d1 if water be poured into the bason 7as in fig4 #abd47 to ,f ,g1 you shall observe the light1 which before went on to ,d1 now to come much short of the mark ,d1 falling on the bottom in the point ,h1 and leaving the mark ,d a good way within the enlightened part2 which shews that the ray ,e ,a1 when it enters the water at ,i1 goes no longer straight forwards1 but is at that place incurvated1 and made to go nearer the perpendicular4 ,the other case may be tryed by putting any small body into an empty bason1 placed lower than your eye1 and then receding from the bason1 till you can but just see the body over the brim4 ,after which1 if the bason be filled with water1 you shall presently observe the body to be visible1 though you go farther off from the bason4 ,let ,a ,b ,c 7in fig4 #abe47 denote the bason as before1 ,d the body in it1 ,e the place of your eye1 when the body is seen just over the edge ,a1 while the bason is empty4 ,if it be then filled with water1 you will observe the body still to be visible1 though you take your eye farther off4 ,suppose you see the body in this case just over the brim ,a1 when your eye is at ,f1 it is plain that the rays of light1 which come from the body to your eye have not come straight on1 but are bent at ,a1 being turned downwards1 and more inclined to the surface of the water1 between ,a and your eye at ,f1 than they are between ,a and the body ,d4 #i4 ,,this we hope is sufficient to make all our readers apprehend1 what the writers of optics mean1 when they mention the refraction of the light1 or speak of the rays of light being refracted4 ,we shall therefore now go on to prove the assertion advanced in the forementioned proposition1 in relation to the different kinds of colours1 that the direct light of the sun exhibits to our sense3 which may be done in the following manner4 #aj4 ,,if a room be darkened1 and the sun permitted to shine into it through a small hole in the window shutter1 and be made immediately to fall upon a glass prism1 the beam of light shall in passing through such a prism be parted into rays1 which exhibit all the forementioned colours4 ,in this manner if ,a ,b 7in fig4 #abf7 represent the window shutter2 ,c the hole in it2 ,d ,e ,f the prism2 ,z ,y a beam of light coming from the sun1 which passes through the hole1 and falls upon the prism at ,y1 and if the prism were removed would go on to ,x1 but in entring the surface ,b ,f of the glass it shall be turned off1 as has been explained1 into the course ,y ,w falling upon the second surface of the prism ,d ,f in ,w1 going out of which into the air it shall be again farther inflected4 ,let the light now1 after it has passed the prism1 be received upon a sheet of paper held at a proper distance1 and it shall paint upon the paper the picture1 image1 or spectrum ,l ,m of an oblong figure1 whose length shall much exceed its breadth2 though the figure shall not be oval1 the ends ,l and ,m being semicircular and the sides straight4 ,but now this figure will be variegated with colours in this manner4 ,from the extremity ,m to some length1 suppose to the line n o1 it shall be of an intense red2 from n o to p q it shall be an orange2 from p q to r s it shall be yellow2 from thence to t u it shall be green2 from thence to w x blue2 from thence to y z indigo2 and from thence to the end violet4 #aa4 ,,thus it appears that the sun's white light by its passage through the prism1 is so changed as now to be divided into rays1 which exhibit all these several colours4 ,the question is1 whether the rays while in the sun's beam before this refraction possessed these properties distinctly2 so that some part of that beam would without the rest have given a red colour1 and another part alone have given an44orange1 c4 ,that this is possible to be the case1 appears from hence2 that if a convex glass be placed between the paper and the prism1 which may collect all the rays proceeding out of the prism into its focus1 as a burning glass does the sun's direct rays2 and if that focus fall upon the paper1 the spot formed by such a glass upon the paper shall appear white1 just like the sun's direct light4 ,illustration ,the rest remaining as before1 let ,p ,q4 7in fig4 #abg47 be the convex glass1 causing the rays to meet upon the paper ,h ,g ,i ,k in the point ,n1 ,i say that point or rather spot of light shall appear white1 without the least tincture of any colour4 ,but it is evident that into this spot are now gathered all those rays1 which before when separate gave all those different colours2 which shews that whiteness may be made by mixing those colours3 especially if we consider1 it can be proved that the glass ,p ,q does not alter the colour of the rays which pass through it4 ,which is done thus3 if the paper be made to approach the glass ,p ,q1 the colours will manifest themselves as far as the magnitude of the spectrum1 which the paper receives1 will permit4 ,suppose it in the situation h g i k1 and that it then receive the spectrum l m1 this spectrum shall be much smaller1 than if the glass ,p ,q were removed1 and therefore the colours cannot be so much separated2 but yet the extremity m shall manifestly appear red1 and the other extremity l shall be blue2 and these colours as well as the intermediate ones shall discover themselves more perfectly1 the farther the paper is removed from ,n1 that is1 the larger the spectrum is3 the same thing happens1 if the paper be removed farther off from ,p ,q than ,n4 ,suppose into the position 1 the spectrum painted upon it shall again discover its colours1 and that more distinctly1 the farther the paper is removed1 but only in an inverted order3 for as before1 when the paper was nearer the convex glass1 than at ,n1 the upper part of the image was blue1 and the under red2 now the upper part shall be red1 and the under blue3 because the rays cross at ,n4 #ab4 ,,nay farther that the whiteness at the focus ,n1 is made by the union of the colours may be proved without removing the paper out of the focus1 by intercepting with any opake body part of the light near the glass2 for if the under part1 that is the red1 or more properly the red-making rays1 as they are styled by our author1 are intercepted1 the spot shall take a bluish hue2 and if more of the inferior rays are cut off1 so that neither the red-making nor orange-making rays1 and if you please the yellow-making rays likewise1 shall fall upon the spot2 then shall the spot incline more and more to the remaining colours4 ,in like manner if you cut off the upper part of the rays1 that is the violet coloured or indigo-making rays1 the spot shall turn reddish1 and become1 more so1 the more of those opposite colours are intercepted4 #ac4 ,,this ,i think abundantly proves that whiteness may be produced by a mixture of all the colours of the spectrum4 ,at least there is but one way of evading the present arguments1 which is1 by asserting that the rays of light after passing the prism have no different properties to exhibit this or the other colour1 but are in that respect perfectly homogeneal1 so that the rays which pass to the under and red part of the image do not differ in any properties whatever from those1 which go to the upper and violet part of it2 but that the colours of the spectrum are produced only by some new modifications of the rays1 made at their incidence upon the paper by the different terminations of light and shadow3 if indeed this assertion can be allowed any place1 after what has been said2 for it seems to be sufficiently obviated by the latter part of the preceding experiment1 that by intercepting the inferior part of the light1 which comes from the prism1 the white spot shall receive a bluish cast1 and by stopping the upper part the spot shall turn red1 and in both cases recover its colour1 when the intercepted light is permitted to pass again2 though in all these trials there is the like termination of light and shadow4 ,however our author has contrived some experiments expresly to shew the absurdity of this supposition2 all which he has explained and enlarged upon in so distinct and expressive a manner1 that it would be wholly unnecessary to repeat them in this place#bhj4 ,i shall only mention that of them1 which may be tried in the experiment before us4 ,if you draw upon the paper ,h ,g ,i ,k1 and through the spot ,n1 the straight line w x parallel to the horizon1 and then if the paper be much inclined into the situation r s v t the line w x still remaining parallel to the horizon1 the spot ,n shall lose its whiteness and receive a blue tincture2 but if it be inclined as much the contrary way1 the same spot shall exchange its white colour for a reddish dye4 ,all which can never be accounted for by any difference in the termination of the light and shadow1 which here is none at all2 but are easily explained by supposing the upper part of the rays1 whenever they enter the eye1 disposed to give the sensation of the dark colours blue1 indigo and violet2 and that the under part is fitted to produce the bright colours yellow1 orange and red3 for when the paper is in the situation r s t u1 it is plain that the upper part of the light falls more directly upon it1 than the under part1 and therefore those rays will be most plentifully reflected from it2 and by their abounding in the reflected light will cause it to incline to their colour4 ,just so when the paper is inclined the contrary way1 it will receive the inferior rays most directly1 and therefore ting the light it reflects with their colour4 #ad4 ,,it is now to be proved that these dispositions of the rays of light to produce some one colour and some another1 which manifest themselves after their being refracted1 are not wrought by any action of the prism upon them1 but are originally inherent in those rays2 and that the prism only affords each species an occasion of shewing its distinct quality by separating them one from another1 which before1 while they were blended together in the direct beam of the sun's light1 lay conceal'd4 ,but that this is so1 will be proved1 if it can be shewn that no prism has any power upon the rays1 which after their passage through one prism are rendered uncompounded and contain in them but one colour1 either to divide that colour into several1 as the sun's light is divided1 or so much as to change it into any other colour4 ,this will be proved by the following experiment#bha4 ,the same thing remaining1 as in the first experiment1 let another prism ,n ,o 7in fig4 #abh47 be placed either immediately1 or at some distance after the first1 in a perpendicular posture1 so that it shall refract the rays issuing from the first sideways4 ,now if this prism could divide the light falling upon it into coloured rays1 as the first has done1 it would divide the spectrum breadthwise into colours1 as before it was divided lengthwise2 but no such thing is observed4 ,if ,l ,m were the spectrum1 which the first prism ,d ,e ,f would paint upon the paper ,h ,g ,i ,k2 ,p ,q lying in an oblique posture shall be the spectrum projected by the second1 and shall be divided lengthwise into colours corresponding to the colours of the spectrum ,l ,m1 and occasioned like them by the refraction of the first prism1 but its breadth shall receive no such division2 on the contrary each colour shall be uniform from side to side1 as much as in the spectrum ,l ,m1 which proves the whole assertion4 #ae4 ,,the same is yet much farther confirmed by another experiment4 ,our author teaches that the colours of the spectrum ,l ,m in the first experiment are yet compounded1 though not so much as in the sun's direct light4 ,he shews therefore how1 by placing the prism at a distance from the hole1 and by the use of a convex glass1 to separate the colours of the spectrum1 and make them uncompounded to any degree of exactness#bhb4 ,and he shews when this is done sufficiently1 if you make a small hole in the paper whereon the spectrum is received1 through which any one sort of rays may pass1 and then let that coloured ray fall so upon a prism1 as to be refracted by it1 it shall in no case whatever change its colour2 but shall always retain it perfectly as at first1 however it be refracted#bhc4 #af4 ,,nor yet will these colours after this full separation of them suffer any change by reflection from bodies of different colours2 on the other hand they make all bodies placed in these colours appear of the colour which falls upon them#bhd3 for minium in red light will appear as in open day light2 but in yellow light will appear yellow2 and which is more extraordinary1 in green light will appear green1 in blue1 blue2 and in the violet-purple coloured light will appear of a purple colour2 in like manner verdigrease1 or blue bise1 will put on the appearance of that colour1 in which it is placed2 so that neither bise placed in the red light shall be able to give that light the least blue tincture1 or any other different from red2 nor shall minium in the indigo or violet light exhibit the least appearance of red1 or any other colour distinct from that it is placed in4 ,the only difference is1 that each of these bodies appears most luminous and bright in the colour1 which corresponds with that it exhibits in the day light1 and dimmed in the colours most remote from that2 that is1 though minium and bise placed in blue light shall both appear blue1 yet the bise shall appear of a bright blue1 and the minium of a dusky and obscure blue3 but if minium and bise be compared together in red light1 the minium shall afford a brisk red1 the bise a duller colour1 though of the same species4 #ag4 ,,and this not only proves the immutability of all these simple and uncompounded colours2 but likewise unfolds the whole mystery1 why bodies appear in open day-light of such different colours1 it consisting in nothing more than this1 that whereas the white light of the day is composed of all sorts of colours1 some bodies reflect the rays of one sort in greater abundance than the rays of any other#bhe4 ,though it appears by the fore-cited experiment1 that almost all these bodies reflect some portion of the rays of every colour1 and give the sense of particular colours only by the predominancy of some sorts of rays above the rest4 ,and what has before been explained of composing white by mingling all the colours of the spectrum together shews clearly1 that nothing more is required to make bodies look white1 than a power to reflect indifferently rays of every colour4 ,but this will more fully appear by the following method3 if near the coloured spectrum in our first experiment a piece of white paper be so held1 as to be illuminated equally by all the parts of that spectrum1 it shall appear white2 whereas if it be held nearer to the red end of the image1 than to the other1 it shall turn reddish2 if nearer the blue end1 it shall seem bluish#bhf4 #ah4 ,,our indefatigable and circumspect author farther examined his theory by mixing the powders which painters use of several colours1 in order if possible to produce a white powder by such a composition#bhg4 ,but in this he found some difficulties for the following reasons4 ,each of these coloured powders reflects but part of the light1 which is cast upon them2 the red powders reflecting little green or blue1 and the blue powders reflecting very little red or yellow1 nor the green powders reflecting near so much of the red or indigo and purple1 as of the other colours3 and besides1 when any of these are examined in homogeneal light1 as our author calls the colours of the prism1 when well separated1 though each appears more bright and luminous in its own day-light colour1 than in any other2 yet white bodies1 suppose white paper for instance1 in those very colours exceed these coloured bodies themselves in brightness2 so that white bodies reflect not only more of the whole light than coloured bodies do in the day-light1 but even more of that very colour which they reflect most copiously4 ,all which considerations make it manifest that a mixture of these will not reflect so great a quantity of light1 as a white body of the same size2 and therefore will compose such a colour as would result from a mixture of white and black1 such as are all grey and dun colours1 rather than a strong white4 ,now such a colour he compounded of certain ingredients1 which he particularly sets down1 in so much that when the composition was strongly illuminated by the sun's direct beams1 it would appear much whiter than even white paper1 if considerably shaded4 ,nay he found by trials how to proportion the degree of illumination of the mixture and paper1 so that to a spectator at a proper distance it could not well be determined which was the more perfect colour2 as he experienced not only by himself1 but by the concurrent opinion of a friend1 who chanced to visit him while he was trying this experiment4 ,i must not here omit another method of trying the whiteness of such a mixture1 proposed in one of our author's letters on this subject#bhh3 which is to enlighten the composition by a beam of the sun let into a darkened room1 and then to receive the light reflected from it upon a piece of white paper1 observing whether the paper appears white by that reflection2 for if it does1 it gives proof of the composition's being white2 because when the paper receives the reflection from any coloured body1 it looks of that colour4 ,agreeable to this is the trial he made upon water impregnated with soap1 and agitated into a froth#bhi3 for when this froth after some short time exhibited upon the little bubbles1 which composed it1 a great variety of colours1 though these colours to a spectator at a small distance discover'd themselves distinctly2 yet when the eye was so far removed1 that each little bubble could no longer be distinguished1 the whole froth by the mixture of all these colours appeared intensly white4 #ai4 ,,our author having fully satisfied himself by these and many other experiments1 what the result is of mixing together all the prismatic colours2 he proceeds in the next place to examine1 whether this appearance of whiteness be raised by the rays of these different kinds acting so1 when they meet1 upon one another1 as to cause each of them to impress the sense of whiteness upon the optic nerve2 or whether each ray does not make upon the organ of sight the same impression1 as when separate and alone2 so that the idea of whiteness is not excited by the impression from any one part of the rays1 but results from the mixture of all those different sensations4 ,and that the latter sentiment is the true one1 he evinces by undeniable experiments4 #bj4 ,,in particular the foregoing experiment#bij1 wherein the convex glass was used1 furnishes proofs of this3 in that when the paper is brought into the situation 1 beyond1 beyond ,n the colours1 that at ,n disappeared1 begin to emerge again2 which shews that by mingling at ,n they did not lose their colorific qualities1 though for some reason they lay concealed4 ,this farther appears by that part of the experiment1 when the paper1 while in the focus1 was directed to be enclined different ways2 for when the paper was in such a situation1 that it must of necessity reflect the rays1 which before their arrival at the point ,n would have given a blue colour1 those rays in this very point itself by abounding in the reflected light tinged it with the same colour2 so when the paper reflects most copiously the rays1 which before they come to the point ,n exhibit redness1 those same rays tincture the light reflected by the paper from that very point with their own proper colour4 #ba4 ,,there is a certain condition relating to sight1 which affords an opportunity of examining this still more fully3 it is this1 that the impressions of light remain some short space upon the eye2 as when a burning coal is whirl'd about in a circle1 if the motion be very quick1 the eye shall not be able to distinguish the coal1 but shall see an entire circle of fire4 ,the reason of which appearance is1 that the impression made by the coal upon the eye in any one situation is not worn out1 before the coal returns again to the same place1 and renews the sensation4 ,this gives our author the hint to try1 whether these colours might not be transmitted successively to the eye so quick1 that no one of the colours should be distinctly perceived1 but the mixture of the sensations should produce a uniform whiteness2 when the rays could not act upon each other1 because they never should meet1 but come to the eye one after another4 ,and this thought he executed by the following expedient#bia4 ,he made an instrument in shape like a comb1 which he applied near the convex glass1 so that by moving it up and down slowly the teeth of it might intercept sometimes one and sometimes another colour2 and accordingly the light reflected from the paper1 placed at ,n1 should change colour continually4 ,but now when the comb-like instrument was moved very quick1 the eye lost all preception of the distinct colours1 which came to it from time to time1 a perfect whiteness resulting from the mixture of all those distinct impressions in the sensorium4 ,now in this case there can be no suspicion of the several coloured rays acting upon one another1 and making any change in each other's manner of affecting the eye1 seeing they do not so much as meet together there4 #bb4 ,,our author farther teaches us how to view the spectrum of colours produced in the first experiment with another prism1 so that it shall appear to the eye under the shape of a round spot and perfectly white#bib4 ,and in this case if the comb be used to intercept alternately some of the colours1 which compose the spectrum1 the round spot shall change its colour according to the colours intercepted2 but if the comb be moved too swiftly for those changes to be distinctly perceived1 the spot shall seem always white1 as before#bic4 #bc4 ,,besides this whiteness1 which results from an universal composition of all sorts of colours1 our author particularly explains the effects of other less compounded mixtures2 some of which compound other colours like some of the simple ones1 but others produce colours different from any of them4 ,for instance1 a mixture of red and yellow compound a colour like in appearance to the orange1 which in the spectrum lies between them2 as a composition of yellow and blue is made use of in all dyes to make a green4 ,but red and violet purple compounded make purples unlike to any of the prismatic colours1 and these joined with yellow or blue make yet new colours4 ,besides one rule is here to be observed1 that when many different colours are mixed1 the colour which arises from the mixture grows languid and degenerates into whiteness4 ,so when yellow green and blue are mixed together1 the compound will be green2 but if to this you add red and purple1 the colour shall first grow dull and less vivid1 and at length by adding more of these colours it shall turn to whiteness1 or some other colour#bid4 #bd4 ,,only here is one thing remarkable of those compounded colours1 which are like in appearance to the simple ones2 that the simple ones when viewed through a prism shall still retain their colour1 but the compounded colours seen through such a glass shall be parted into the simple ones of which they are the aggregate4 ,and for this reason any body illuminated by the simple light shall appear through a prism distinctly1 and have its minutest parts observable1 as may easily be tried with flies1 or other such little bodies1 which have very small parts2 but the same viewed in this manner when enlighten'd with compounded colours shall appear confused1 their smallest parts not being distinguishable4 ,how the prism separates these compounded colours1 as likewise how it divides the light of the sun into its colours1 has not yet been explained2 but is reserved for our third chapter4 #be4 ,,in the mean time what has been said1 ,i hope1 will suffice to give a taste of our author's way of arguing1 and in some measure to illustrate the proposition laid down in this chapter4 #bf4 ,,there are methods of separating the heterogeneous rays of the sun's light by reflection1 which perfectly conspire with and confirm this reasoning4 ,one of which ways may be this4 ,let ,a ,b 7in fig4 #abi7 represent the window shutter of a darkened room2 ,c a hole to let in the sun's rays2 ,d ,e ,f1 ,g ,h ,i two prisms so applied together1 that the sides ,e ,f and ,g ,i be contiguous1 and the sides ,d ,f1 ,g ,h parallel2 by this means the light will pass through them without any separation into colours3 but if it be afterwards received by a third prism ,i ,k ,l1 it shall be divided so as to form upon any white body ,p ,q the usual colours1 violet at m1 blue at n1 green at o1 yellow at r1 and red at s4 ,but because it never happens that the two adjacent surfaces ,e ,f and ,g ,i perfectly touch1 part only of the light incident upon the surface ,e ,f shall be transmitted1 and part shall be reflected4 ,let now the reflected part be received by a fourth prism 1 and passing through it paint upon a white body the colours of the prism1 red at t1 yellow at u1 green at w1 blue at x1 violet at y4 ,if the prisms ,d ,e ,f1 ,g ,h ,i be slowly turned about while they remain contiguous1 the colours upon the body ,p ,q shall not sensibly change their situation1 till such time as the rays become pretty oblique to the surface ,e ,f2 but then the light incident upon the surface ,e ,f shall begin to be wholly reflected4 ,and first of all the violet light shall be wholly reflected1 and thereupon will disappear at m1 appearing instead thereof at y1 and increasing the violet light falling there1 the other colours remaining as before4 ,if the prisms ,d ,e ,f1 ,g ,h ,i be turned a little farther about1 that the incident rays become yet more inclined to the surface ,e ,f1 the blue shall be totally reflected1 and shall disappear in n1 but appear at x by making the colour there more intense4 ,and the same may be continued1 till all the colours are successively removed from the surface ,p ,q to 4 ,but in any case1 suppose when the violet and the blue have forsaken the surface ,p ,q1 and appear upon the surface 1 1 the green1 yellow1 and red only remaining upon the surface ,p ,q2 if the light be received upon a paper held any where in its whole passage between the light's coming out of the prisms ,d ,e ,f1 ,g ,i ,h and its incidence upon the prism ,i ,k ,l1 it shall appear of the colour compounded of all the colours seen upon ,p ,q2 and the reflected ray1 received upon a piece of white paper held any where between the prisms ,d ,e ,f and shall exhibit the colour compounded of those the surface ,p ,q is deprived of mixed with the sun's light3 whereas before any of the light was reflected from the surface ,e ,f1 the rays between the prisms ,g ,h ,i and ,i ,k ,l would appear white2 as will likewise the reflected ray both before and after the total reflection1 provided the difference of refraction by the surfaces ,d ,f and ,d ,e be inconsiderable4 ,i call here the sun's light white1 as ,i have all along done2 but it is more exact to ascribe to it something of a yellowish tincture1 occasioned by the brighter colours abounding in it2 which caution is necessary in examining the colours of the reflected beam1 when all the violet and blue are in it3 for this yellowish turn of the sun's light causes the blue not to be quite so visible in it1 as it should be1 were the light perfectly white2 but makes the beam of light incline rather towards a pale white4 ,,chap4 ,,ii4 ,of the properties of ,,bodies1 upon which their ,,colours depend4 ,,after having shewn in the last chapter1 that the difference between the colours of bodies viewed in open day-light is only this1 that some bodies are disposed to reflect rays of one colour in the greatest plenty1 and other bodies rays of some other colour2 order now requires us to examine more particularly into the property of bodies1 which gives them this difference4 ,but this our author shews to be nothing more1 than the different magnitude of the particles1 which compose each body3 this ,i question not will appear no small paradox4 ,and indeed this whole chapter will contain scarce any assertions1 but what will be almost incredible1 though the arguments for them are so strong and convincing1 that they force our assent4 ,in the former chapter have been explained properties of light1 not in the least thought of before our author's discovery of them2 yet are they not difficult to admit1 as soon as experiments are known to give proof of their reality2 but some of the propositions to be stated here will1 ,i fear1 be accounted almost past belief2 notwithstanding that the arguments1 by which they are established are unanswerable4 ,for it is proved by our author1 that bodies are rendered transparent by the minuteness of their pores1 and become opake by having them large2 and more1 that the most transparent body by being reduced to a great thinness will become less pervious to the light4 #b4 ,,but whereas it had been the received opinion1 and yet remains so among all who have not studied this philosophy1 that light is reflected from bodies by its impinging against their solid parts1 rebounding from them1 as a tennis ball or other elastic substance would do1 when struck against any hard and resisting surface2 it will be proper to begin with declaring our author's sentiment concerning this1 who shews by many arguments that reflection cannot be caused by any such means#bie3 some few of his proofs ,i shall set down1 referring the reader to our author himself for the rest4 #c4 ,,it is well known1 that when light falls upon any transparent body1 glass for instance1 part of it is reflected and part transmitted2 for which it is ready to account1 by saying that part of the light enters the pores of the glass1 and part impinges upon its solid parts4 ,but when the transmitted light arrives at the farther surface of the glass1 in passing out of glass into air there is as strong a reflection caused1 or rather something stronger4 ,now it is not to be conceived1 how the light should find as many solid parts in the air to strike against as in the glass1 or even a greater number of them4 ,and to augment the difficulty1 if water be placed behind the glass1 the reflection becomes much weaker4 ,can we therefore say1 that water has fewer solid parts for the light to strike against1 than the air8 ,and if we should1 what reason can be given for the reflection's being stronger1 when the air by the air-pump is removed from behind the glass1 than when the air receives the rays of light4 ,besides the light may be so inclined to the hinder surface of the glass1 that it shall wholly be reflected1 which happens when the angle which the ray makes with the surface does not exceed about #di degrees2 but if the inclination be a very little increased1 great part of the light will be transmitted2 and how the light in one case should meet with nothing but the solid parts of the air1 and by so small a change of its inclination find pores in great plenty1 is wholly inconceivable4 ,it cannot be said1 that the light is reflected by striking against the solid parts of the surface of the glass2 because without making any change in that surface1 only by placing water contiguous to it instead of air1 great part of that light shall be transmitted1 which could find no passage through the air4 ,moreover in the last experiment recited in the preceding chapter1 when by turning the prisms ,d ,e ,f1 ,g ,h ,i1 the blue light became wholly reflected1 while the rest was mostly transmitted1 no possible reason can be assigned1 why the blue-making rays should meet with nothing but the solid parts of the air between the prisms1 and the rest of the light in the very same obliquity find pores in abundance4 ,nay farther1 when two glasses touch each other1 no reflection at all is made2 though it does not in the least appear1 how the rays should avoid the solid parts of glass1 when contiguous to other glass1 any more than when contiguous to air4 ,but in the last place upon this supposition it is not to be comprehended1 how the most polished substances could reflect the light in that regular manner we find they do2 for when a polished looking glass is covered over with quicksilver1 we cannot suppose the particles of light so much larger than those of the quicksilver that they should not be scattered as much in reflection1 as a parcel of marbles thrown down upon a rugged pavement4 ,the only cause of so uniform and regular a reflection must be some more secret cause1 uniformly spread over the whole surface of the glass4 #d4 ,,but now1 since the reflection of light from bodies does not depend upon its impinging against their solid parts1 some other reason must be sought for4 ,and first it is past doubt that the least parts of almost all bodies are transparent1 even the microscope shewing as much#bif2 besides that it may be experienced by this method4 ,take any thin plate of the opakest body1 and apply it to a small hole designed for the admission of light into a darkened room2 however opake that body may seem in open day-light1 it shall under these circumstances sufficiently discover its transparency1 provided only the body be very thin4 ,white metals indeed do not easily shew themselves transparent in these trials1 they reflecting almost all the light incident upon them at their first superficies2 the cause of which will appear in what follows#big4 ,but yet these substances1 when reduced into parts of extraordinary minuteness by being dissolved in aqua fortis or the like corroding liquors do also become transparent4 #e4 ,,since therefore the light finds free passage through the least parts of bodies1 let us consider the largeness of their pores1 and we shall find1 that whenever a ray of light has passed through any particle of a body1 and is come to its farther surface1 if it finds there another particle contiguous1 it will without interruption pass into that particle2 just as light will pass through one piece of glass into another piece in contact with it without any impediment1 or any part being reflected3 but as the light in passing out of glass1 or any other transparent body1 shall part of it be reflected back1 if it enter into air or other transparent body of a different density from that it passes out of2 the same thing will happen in the light's passage through any particle of a body1 whenever at its exit out of that particle it meets no other particle contiguous1 but must enter into a pore1 for in this case it shall not all pass through1 but part of it be reflected back4 ,thus will the light1 every time it enters a pore1 be in part reflected2 so that nothing more seems necessary to opacity1 than that the particles1 which compose any body1 touch but in very few places1 and that the pores of it are numerous and large1 so that the light may in part be reflected from it1 and the other part1 which enters too deep to be returned out of the body1 by numerous reflections may be stifled and lost#bih2 which in all probability happens1 as often as it impinges against the solid part of the body1 all the light which does so not being reflected back1 but stopt1 and deprived of any farther motion#bii4 #f4 ,,this notion of opacity is greatly confirmed by the observation1 that opake bodies become transparent by filling up the pores with any substance of near the same density with their parts4 ,as when paper is wet with water or oyl2 when linnen cloth is either dipt in water1 oyled1 or varnished2 or the oculus mundi stone steeped in water#cjj4 ,all which experiments confirm both the first assertion1 that light is not reflected by striking upon the solid parts of bodies2 and also the second1 that its passage is obstructed by the reflections it undergoes in the pores2 since we find it in these trials to pass in greater abundance through bodies1 when the number of their solid parts is increased1 only by taking away in great measure those reflections2 which filling the pores with a substance of near the same density with the parts of the body will do4 ,besides as filling the pores of a dark body makes it transparent2 so on the other hand evacuating the pores of a body transparent1 or separating the parts of such a body1 renders it opake4 ,as salts or wet paper by being dried1 glass by being reduced to powder or the surface made rough2 and it is well known that glass vessels discover cracks in them by their opacity4 ,just so water itself becomes impervious to the light by being formed into many small bubbles1 whether in froth1 or by being mixed and agitated with any quantity of a liquor with which it will not incorporate1 such as oyl of turpentine1 or oyl olive4 #g4 ,a ,,certain electrical experiment made by ,mr4 ,,hauksbee may not perhaps be useless to clear up the present speculation1 by shewing that something more is necessary besides mere porosity for transmitting freely other fine substances4 ,the experiment is this2 that a glass cane rubbed till it put forth its electric quality would agitate leaf brass inclosed under a glass vessel1 though not at so great a distance1 as if no body had intervened2 yet the same cane would lose all its influence on the leaf brass by the interposition of a piece of the finest muslin1 whose pores are immensely larger and more patent than those of glass4 #h4 ,,thus ,i have endeavoured to smooth my way1 as much as ,i could1 to the unfolding yet greater secrets in nature2 for ,i shall now proceed to shew the reason why bodies appear of different colours4 ,my reader no doubt will be sufficiently surprized1 when ,i inform him that the knowledge of this is deduced from that ludicrous experiment1 with which children divert themselves in blowing bubbles of water made tenacious by the solution of soap4 ,and that these bubbles1 as they gradually grow thinner and thinner till they break1 change successively their colours from the same principle1 as all natural bodies preserve theirs4 #i4 ,,our author after preparing water with soap1 so as to render it very tenacious1 blew it up into a bubble1 and placing it under a glass1 that it might not be irregularly agitated by the air1 observed as the water by subsiding changed the thickness of the bubble1 making it gradually less and less till the bubble broke2 there successively appeared colours at the top of the bubble1 which spread themselves into rings surrounding the top and descending more and more1 till they vanished at the bottom in the same order in which they appeared#cja4 ,the colours emerged in this order3 first red1 then blue2 to which succeeded red a second time1 and blue immediately followed2 after that red a third time1 succeeded by blue2 to which followed a fourth red1 but succeeded by green2 after this a more numerous order of colours1 first red1 then yellow1 next green1 and after that blue1 and at last purple2 then again red1 yellow1 green1 blue1 violet followed each other in order2 and in the last place red1 yellow1 white1 blue2 to which succeeded a dark spot1 which reflected scarce any light1 though our author found it did make some very obscure reflection1 for the image of the sun or a candle might be faintly discerned upon it2 and this last spot spread itself more and more1 till the bubble at last broke4 ,these colours were not simple and uncompounded colours1 like those which are exhibited by the prism1 when due care is taken to separate them2 but were made by a various mixture of those simple colours1 as will be shewn in the next chapter3 whence these colours1 to which ,i have given the name of blue1 green1 or red1 were not all alike1 but differed as follows4 ,the blue1 which appeared next the dark spot1 was a pure colour1 but very faint1 resembling the sky-colour2 the white next to it a very strong and intense white1 brighter much than the white1 which the bubble reflected1 before any of the colours appeared4 ,the yellow which preceded this was at first pretty good1 but soon grew dilute2 and the red which went before the yellow at first gave a tincture of scarlet inclining to violet1 but soon changed into a brighter colour2 the violet of the next series was deep with little or no redness in it2 the blue a brisk colour1 but came much short of the blue in the next order2 the green was but dilute and pale2 the yellow and red were very bright and full1 the best of all the yellows which appeared among any of the colours3 in the preceding orders the purple was reddish1 but the blue1 as was just now said1 the brightest of all2 the green pretty lively better than in the order which appeared before it1 though that was a good willow green2 the yellow but small in quantity1 though bright2 the red of this order not very pure3 those which appeared before yet more obscure1 being very dilute and dirty2 as were likewise the three first blues4 #aj4 ,,now it is evident1 that these colours arose at the top of the bubble1 as it grew by degrees thinner and thinner3 but what the express thickness of the bubble was1 where each of these colours appeared upon it1 could not be determined by these experiments2 but was found by another means1 viz4 by taking the object glass of a long telescope1 which is in a small degree convex1 and placing it upon a flat glass1 so as to touch it in one point1 and then water being put between them1 the same colours appeared as in the bubble1 in the form of circles or rings surrounding the point where the glasses touched1 which appeared black for want of any reflection from it1 like the top of the bubble when thinnest#cjb3 next to this spot lay a blue circle1 and next without that a white one2 and so on in the same order as before1 reckoning from the dark spot4 ,and henceforward ,i shall speak of each colour1 as being of the first1 second1 or any following order1 as it is the first1 second1 or any following one1 counting from the black spot in the center of these rings2 which is contrary to the order in which ,i must have mentioned them1 if ,i should have reputed them the first1 second1 or third1 c4 in order1 as they arise after one another upon the top of the bubble4 #aa4 ,but now by measuring the diameters of each of these rings1 and knowing the convexity of the telescope glass1 the thickness of the water at each of those rings may be determined with great exactness3 for instance the thickness of it1 where the white light of the first order is reflected1 is about #c such parts1 of which an inch contains #ajjjjjjcjc4 ,and this measure gives the thickness of the bubble1 where it appeared of this white colour1 as well as of the water between the glasses2 though the transparent body which surrounds the water in these two cases be very different3 for our author found1 that the condition of the ambient body would not alter the species of the colour at all1 though it might its strength and brightness2 for pieces of ,muscovy glass1 which were so thin as to appear coloured by being wet with water1 would have their colours faded and made less bright thereby2 but he could not observe their species at all to be changed4 ,so that the thickness of any transparent body determines its colour1 whatever body the light passes through in coming to it#cjd4 #ab4 ,,but it was found that different transparent bodies would not under the same thicknesses exhibit the same colours3 for if the forementioned glasses were laid upon each other without any water between their surfaces1 the air itself would afford the same colours as the water1 but more expanded1 insomuch that each ring had a larger diameter1 and all in the same proportion4 ,so that the thickness of the air proper to each colour was in the same proportion larger1 than the thickness of the water appropriated to the same#cje4 #ac4 ,,if we examine with care all the circumstances of these colours1 which will be enumerated in the next chapter1 we shall not be surprized1 that our author takes them to bear a great analogy to the colours of natural bodies#cjf4 ,for the regularity of those various and strange appearances relating to them1 which makes the most mysterious part of the action between light and bodies1 as the next chapter will shew1 is sufficient to convince us that the principle1 from which they flow1 is of the greatest importance in the frame of nature2 and therefore without question is designed for no less a purpose than to give bodies their various colours1 to which end it seems very fitly suited4 ,for if any such transparent substance of the thickness proper to produce any one colour should be cut into slender threads1 or broken into fragments1 it does not appear but these should retain the same colour2 and a heap of such fragments should frame a body of that colour4 ,so that this is without dispute the cause why bodies are of this or the other colour1 that the particles of which they are composed are of different sizes4 ,which is farther confirmed by the analogy between the colours of thin plates1 and the colours of many bodies4 ,for example1 these plates do not look of the same colour when viewed obliquely1 as when seen direct2 for if the rings and colours between a convex and plane glass are viewed first in a direct manner1 and then at different degrees of obliquity1 the rings will be observed to dilate themselves more and more as the obliquity is increased#cjg2 which shews that the transparent substance between the glasses does not exhibit the same colour at the same thickness in all situations of the eye3 just so the colours in the very same part of a peacock's tail change1 as the tail changes posture in respect of the sight4 ,also the colours of silks1 cloths1 and other substances1 which water or oyl can intimately penetrate1 become faint and dull by the bodies being wet with such fluids1 and recover their brightness again when dry2 just as it was before said that plates of ,muscovy glass grew faint and dim by wetting4 ,to this may be added1 that the colours which painters use will be a little changed by being ground very elaborately1 without question by the diminution of their parts4 ,all which particulars1 and many more that might be extracted from our author1 give abundant proof of the present point4 ,i shall only subjoin one more3 these transparent plates transmit through them all the light they do not reflect2 so that when looked through they exhibit those colours1 which result from the depriving white light of the colour reflected4 ,this may commodiously be tryed by the glasses so often mentioned2 which if looked through exhibit coloured rings as by reflected light1 but in a contrary order2 for the middle spot1 which in the other view appears black for want of reflected light1 now looks perfectly white1 opposite to the blue circle2 next without this spot the light appears tinged with a yellowish red2 where the white circle appeared before1 it now seems dark2 and so of the rest#cjh4 ,now in the same manner1 the light transmitted through foliated gold into a darkened room appears greenish by the loss of the yellow light1 which gold reflects4 #ad4 ,,hence it follows1 that the colours of bodies give a very probable ground for making conjecture concerning the magnitude of their constituent particles#cji4 ,my reason for calling it a conjecture is1 its being difficult to fix certainly the order of any colour4 ,the green of vegetables our author judges to be of the third order1 partly because of the intenseness of their colour2 and partly from the changes they suffer when they wither1 turning at first into a greenish or more perfect yellow1 and afterwards some of them to an orange or red2 which changes seem to be effected from their ringing particles growing denser by the exhalation of their moisture1 and perhaps augmented likewise by the accretion of the earthy and oily parts of that moisture4 ,how the mentioned colours should arise from increasing the bulk of those particles1 is evident2 seeing those colours lie without the ring of green between the glasses1 and are therefore formed where the transparent substance which reflects them is thicker4 ,and that the augmentation of the density of the colorific particles will conspire to the production of the same effect1 will be evident2 if we remember what was said of the different size of the rings1 when air was included between the glasses1 from their size when water was between them2 which shewed that a substance of a greater density than another gives the same colour at a less thickness4 ,now the changes likely to be wrought in the density or magnitude of the parts of vegetables by withering seem not greater1 than are sufficient to change their colour into those of the same order2 but the yellow and red of the fourth order are not full enough to agree with those1 into which these substances change1 nor is the green of the second sufficiently good to be the colour of vegetables2 so that their colour must of necessity be of the third order4 #ae4 ,,the blue colour of syrup of violets our author supposes to be of the third order2 for acids1 as vinegar1 with this syrup change it red1 and salt of tartar or other alcalies mixed therewith turn it green4 ,but if the blue colour of the syrup were of the second order1 the red colour1 which acids by attenuating its parts give it1 must be of the first order1 and the green given it by alcalies by incrassating its particles should be of the second2 whereas neither of those colours is perfect enough1 especially the green1 to answer those produced by these changes2 but the red may well enough be allowed to be of the second order1 and the green of the third2 in which case the blue must be likewise of the third order4 #af4 ,,the azure colour of the skies our author takes to be of the first order1 which requires the smallest particles of any colour1 and therefore most like to be exhibited by vapours1 before they have sufficiently coalesced to produce clouds of other colours4 #ag4 ,,the most intense and luminous white is of the first order1 if less strong it is a mixture of the colours of all the orders4 ,of the latter sort he takes the colour of linnen1 paper1 and such like substances to be2 but white metals to be of the former sort4 ,the arguments for it are these4 ,the opacity of all bodies has been shewn to arise from the number and strength of the reflections made within them2 but all experiments shew1 that the strongest reflection is made at those surfaces1 which intercede transparent bodies differing most in density4 ,among other instances of this1 the experiments before us afford one2 for when air only is included between the glasses1 the coloured rings are not only more dilated1 as has before been said1 than when water is between them2 but are likewise much more luminous and bright4 ,it follows therefore1 that whatever medium pervades the pores of bodies1 if so be there is any1 those substances must be most opake1 the density of whose parts differs most from the density of the medium1 which fills their pores4 ,but it has been sufficiently proved in the former part of this tract1 that there is no very dense medium lodging in1 at least pervading at liberty the pores of bodies4 ,and it is farther proved by the present experiments4 ,for when air is inclosed by the denser substance of glass1 the rings dilate themselves1 as has been said1 by being viewed obliquely2 this they do so very much1 that at different obliquities the same thickness of air will exhibit all sorts of colours4 ,the bubble of water1 though surrounded with the thinner substance of air1 does likewise change its colour by being viewed obliquely2 but not any thing near so much1 as in the other case2 for in that the same colour might be seen1 when the rings were viewed most obliquely1 at more than twelve times the thickness it appeared at under a direct view2 whereas in this other case the thickness was never found considerably above half as much again4 ,now the colours of bodies not depending only on the light1 that is incident upon them perpendicularly1 but likewise upon that1 which falls on them in all degrees of obliquity2 if the medium surrounding their particles were denser than those particles1 all sorts of colours must of necessity be reflected from them so copiously1 as would make the colours of all bodies white1 or grey1 or at best very dilute and imperfect4 ,but on the other hand1 if the medium in the pores of bodies be much rarer than their particles1 the colour reflected will be so little changed by the obliquity of the rays1 that the colour produced by the rays1 which fall near the perpendicular1 may so much abound in the reflected light1 as to give the body their colour with little allay4 ,to this may be added1 that when the difference of the contiguous transparent substances is the same1 a colour reflected from the denser substance reduced into a thin plate and surrounded by the rarer will be more brisk1 than the same colour will be1 when reflected from a thin plate formed of the rarer substance1 and surrounded by the denser2 as our author experienced by blowing glass very thin at a lamp furnace1 which exhibited in the open air more vivid colours1 than the air does between two glasses4 ,from these considerations it is manifest1 that if all other circumstances are alike1 the densest bodies will be most opake4 ,but it was observed before1 that these white metals can hardly be made so thin1 except by being dissolved in corroding liquors1 as to be rendred transparent2 though none of them are so dense as gold1 which proves their great opacity to have some other cause besides their density2 and none is more fit to produce this1 than such a size of their particles1 as qualifies them to reflect the white of the first order4 #ah4 ,,for producing black the particles ought to be smaller than for exhibiting any of the colours1 viz4 of a size answering to the thickness of the bubble1 where by reflecting little or no light it appears colourless2 but yet they must not be too small1 for that will make them transparent through deficiency of reflections in the inward parts of the body1 sufficient to stop the light from going through it2 but they must be of a size bordering upon that disposed to reflect the faint blue of the first order1 which affords an evident reason why blacks usually partake a little of that colour4 ,we see too1 why bodies dissolved by fire or putrefaction turn black3 and why in grinding glasses upon copper plates the dust of the glass1 copper1 and sand it is ground with1 become very black3 and in the last place why these black substances communicate so easily to others their hue2 which is1 that their particles by reason of the great minuteness of them easily overspread the grosser particles of others4 #ai4 ,i ,,shall now finish this chapter with one remark of the exceeding great porosity in bodies necessarily required in all that has here been said2 which1 when duly considered1 must appear very surprizing2 but perhaps it will be matter of greater surprize1 when ,i affirm that the sagacity of our author has discovered a method1 by which bodies may easily become so2 nay how any the least portion of matter may be wrought into a body of any assigned dimensions how great so ever1 and yet the pores of that body none of them greater1 than any the smallest magnitude proposed at pleasure2 notwithstanding which the parts of the body shall so touch1 that the body itself shall be hard and solid#caj4 ,the manner is this3 suppose the body be compounded of particles of such figures1 that when laid together the pores found between them may be equal in bigness to the particles2 how this may be effected1 and yet the body be hard and solid1 is not difficult to understand2 and the pores of such a body may be made of any proposed degree of smallness4 ,but the solid matter of a body so framed will take up only half the space occupied by the body2 and if each constituent particle be composed of other less particles according to the same rule1 the solid parts of such a body will be but a fourth part of its bulk2 if every one of these lesser particles again be compounded in the same manner1 the solid parts of the whole body shall be but one eighth of its bulk2 and thus by continuing the composition the solid parts of the body may be made to bear as small a proportion to the whole magnitude of the body1 as shall be desired1 notwithstanding the body will be by the contiguity of its parts capable of being in any degree hard4 ,which shews that this whole globe of earth1 nay all the known bodies in the universe together1 as far as we know1 may be compounded of no greater a portion of solid matter1 than might be reduced into a globe of one inch only in diameter1 or even less4 ,we see therefore how by this means bodies may easily be made rare enough to transmit light1 with all that freedom pellucid bodies are found to do4 ,though what is the real structure of bodies we yet know not4 ,,chap4 ,,iii4 ,of the ,,refraction1 ,,reflection1 and ,,inflection of ,,light4 ,,thus much of the colours of natural bodies2 our method now leads us to speculations yet greater1 no less than to lay open the causes of all that has hitherto been related4 ,for it must in this chapter be explained1 how the prism separates the colours of the sun's light1 as we found in the first chapter2 and why the thin transparent plates discoursed of in the last chapter1 and consequently the particles of coloured bodies1 reflect that diversity of colours only by being of different thicknesses4 #b4 ,,for the first it is proved by our author1 that the colours of the sun's light are manifested by the prism1 from the rays undergoing different degrees of refraction2 that the violet-making rays1 which go to the upper part of the coloured image in the first experiment of the first chapter1 are the most refracted2 that the indigo-making rays are refracted1 or turned out of their course by passing through the prism1 something less than the violet-making rays1 but more than the blue-making rays2 and the blue-making rays more than the green2 the green-making rays more than the yellow2 the yellow more than the orange2 and the orange-making rays more than the red-making1 which are least of all refracted4 ,the first proof of this1 that rays of different colours are refracted unequally is this4 ,if you take any body1 and paint one half of it red and the other half blue1 then upon viewing it through a prism those two parts shall appear separated from each other2 which can be caused no otherwise than by the prism's refracting the light of one half more than the light of the other half4 ,but the blue half will be most refracted2 for if the body be seen through the prism in such a situation1 that the body shall appear lifted upwards by the refraction1 as a body within a bason of water1 in the experiment mentioned in the first chapter1 appeared to be lifted up by the refraction of the water1 so as to be seen at a greater distance than when the bason is empty1 then shall the blue part appear higher than the red2 but if the refraction of the prism be the contrary way1 the blue part shall be depressed more than the other4 ,again1 after laying fine threads of black silk across each of the colours1 and the body well inlightened1 if the rays coming from it be received upon a convex glass1 so that it may by refracting the rays cast the image of the body upon a piece of white paper held beyond the glass2 then it will be seen that the black threads upon the red part of the image1 and those upon the blue part1 do not at the same time appear distinctly in the image of the body projected by the glass2 but if the paper be held so1 that the threads on the blue part may distinctly appear1 the threads cannot be seen distinct upon the red part2 but the paper must be drawn farther off from the convex glass to make the threads on this part visible2 and when the distance is great enough for the threads to be seen in this red part1 they become indistinct in the other4 ,whence it appears that the rays proceeding from each point of the blue part of the body are sooner united again by the convex glass than the rays which come from each point of the red parts#caa4 ,but both these experiments prove that the blue-making rays1 as well in the small refraction of the convex glass1 as in the greater refraction of the prism1 are more bent1 than the red-making rays4 #c4 ,,this seems already to explain the reason of the coloured spectrum made by refracting the sun's light with a prism1 though our author proceeds to examine that in particular1 and proves that the different coloured rays in that spectrum are in different degrees refracted2 by shewing how to place the prism in such a posture1 that if all the rays were refracted in the same manner1 the spectrum should of necessity be round3 whereas in that case if the angle made by the two surfaces of the prism1 through which the light passes1 that is the angle ,d ,f ,e in fig4 #abf1 be about #fc or #fd degrees1 the image instead of being round shall be near five times as long as broad2 a difference enough to shew a great inequality in the refractions of the rays1 which go to the opposite extremities of the image4 ,to leave no scruple unremoved1 our author is very particular in shewing by a great number of experiments1 that this inequality of refraction is not casual1 and that it does not depend upon any irregularities of the glass2 no nor that the rays are in their passage through the prism each split and divided2 but on the contrary that every ray of the sun has its own peculiar degree of refraction proper to it1 according to which it is more or less refracted in passing through pellucid substances always in the same manner#cab4 ,that the rays are not split and multiplied by the refraction of the prism1 the third of the experiments related in our first chapter shews very clearly2 for if they were1 and the length of the spectrum in the first refraction were thereby occasioned1 the breadth should be no less dilated by the cross refraction of the second prism2 whereas the breadth is not at all increased1 but the image is only thrown into an oblique posture by the upper part of the rays which were at first more refracted than the under part1 being again turned farthest out of their course4 ,but the experiment most expressly adapted to prove this regular diversity of refraction is this1 which follows#cac4 ,two boards ,a ,b1 ,c ,d 7in fig4 #acj47 being erected in a darkened room at a proper distance1 one of them ,a ,b being near the window-shutter ,e ,f1 a space only being left for the prism ,g ,h ,i to be placed between them2 so that the rays entring at the hole ,m of the window-shutter may after passing through the prism be trajected through a smaller hole ,k made in the board ,a ,b1 and passing on from thence go out at another hole ,l made in the board ,c ,d of the same size as the hole ,k1 and small enough to transmit the rays of one colour only at a time2 let another prism ,n ,o ,p be placed after the board ,c ,d to receive the rays passing through the holes ,k and ,l1 and after refraction by that prism let those rays fall upon the white surface ,q ,r4 ,suppose first the violet light to pass through the holes1 and to be refracted by the prism ,n ,o ,p to s1 which if the prism ,n ,o ,p were removed should have passed right onto ,w4 ,if the prism ,g ,h ,i be turned slowly about1 while the boards and prism ,n ,o ,p remain fixed1 in a little time another colour will fall upon the hole ,l1 which1 if the prism ,n ,o ,p were taken away1 would proceed like the former rays to the same point ,w2 but the refraction of the prism ,n ,o ,p shall not carry these rays to s1 but to some place less distant from ,w as to t4 ,suppose now the rays which go to t to be the indigo-making rays4 ,it is manifest that the boards ,a ,b1 ,c ,d1 and prism ,n ,o ,p remaining immoveable1 both the violet-making and indigo-making rays are incident alike upon the prism ,n ,o ,p1 for they are equally inclined to its surface ,o ,p1 and enter it in the same part of that surface2 which shews that the indigo-making rays are less diverted out of their course by the refraction of the prism1 than the violet-making rays under an exact parity of all circumstances4 ,farther1 if the prism ,g ,h ,i be more turned about1 'till the blue-making rays pass through the hole ,l1 these shall fall upon the surface ,q ,r below ,i1 as at v1 and therefore are subjected to a less refraction than the indigo-making rays4 ,and thus by proceeding it will be found that the green-making rays are less refracted than the blue-making rays1 and so of the rest1 according to the order in which they lie in the coloured spectrum4 #d4 ,,this disposition of the different coloured rays to be refracted some more than others our author calls their respective degrees of refrangibility4 ,and since this difference of refrangibility discovers it self to be so regular1 the next step is to find the rule it observes4 #e4 ,,it is a common principle in optics1 that the sine of the angle of incidence bears to the sine of the refracted angle a given proportion4 ,if ,a ,b 7in fig4 #aca1 #acb7 represent the surface of any refracting substance1 suppose of water or glass1 and ,c ,d a ray of light incident upon that face in the point ,d1 let ,d ,e be the ray1 after it has passed the surface ,a ,b2 if the ray pass out of the air into the substance whose surface is ,a ,b 7as in fig4 #aca7 it shall be turned from the surface1 and if it pass out of that substance into air it shall be bent towards it 7as in fig4 #acb7 ,but if ,f ,g be drawn through the point ,d perpendicular to the surface ,a ,b1 the angle under ,c ,d ,f made by the incident ray and this perpendicular is called the angle of incidence2 and the angle under ,e ,d ,g1 made by this perpendicular and the ray after refraction1 is called the refracted angle4 ,and if the circle ,h ,f ,i ,g be described with any interval cutting ,c ,d in ,h and ,d ,e in ,i1 then the perpendiculars ,h ,k1 ,i ,l being let fall upon ,f ,g1 ,h ,k is called the sine of the angle under ,c ,d ,f the angle of incidence1 and ,i ,l the sine of the angle under ,e ,d ,g the refracted angle4 ,the first of these sines is called the sine of the angle of incidence1 or more briefly the sine of incidence1 the latter is the sine of the refracted angle1 or the sine of refraction4 ,and it has been found by numerous experiments that whatever proportion the sine of incidence ,h ,k bears to the sine of refraction ,i ,l in any one case1 the same proportion shall hold in all cases2 that is1 the proportion between these sines will remain unalterably the same in the same refracting substance1 whatever be the magnitude of the angle under ,c ,d ,f4 #f4 ,,but now because optical writers did not observe that every beam of white light was divided by refraction1 as has been here explained1 this rule collected by them can only be understood in the gross of the whole beam after refraction1 and not so much of any particular part of it1 or at most only of the middle part of the beam4 ,it therefore was incumbent upon our author to find by what law the rays were parted from each other2 whether each ray apart obtained this property1 and that the separation was made by the proportion between the sines of incidence and refraction being in each species of rays different2 or whether the light was divided by some other rule4 ,but he proves by a certain experiment that each ray has its sine of incidence proportional to its sine of refraction2 and farther shews by mathematical reasoning1 that it must be so upon condition only that bodies refract the light by acting upon it1 in a direction perpendicular to the surface of the refracting body1 and upon the same sort of rays always in an equal degree at the same distances#cad4 #g4 ,,our great author teaches in the next place how from the refraction of the most refrangible and least refrangible rays to find the refraction of all the intermediate ones#cae4 ,the method is this3 if the sine of incidence be to the sine of refraction in the least refrangible rays as ,a to ,b ,c1 7in fig4 #acc7 and to the sine of refraction in the most refrangible as ,a to ,b ,d2 if ,c ,e be taken equal to ,c ,d1 and then ,e ,d be so divided in ,f1 ,g1 ,h1 ,i1 ,k1 ,l1 that ,e ,d1 ,e ,f1 ,e ,g1 ,e ,h1 ,e ,i1 ,e ,k1 ,e ,l1 ,e ,c1 shall be proportional to the eight lengths of musical chords1 which found the notes in an octave1 ,e ,d being the length of the key1 ,e ,f the length of the tone above that key1 ,e ,g the length of the lesser third1 ,e ,h of the fourth1 ,e ,i of the fifth1 ,e ,k of the greater sixth1 ,e ,l of the seventh1 and ,e ,c of the octave above that key2 that is if the lines ,e ,d1 ,e ,f1 ,e ,g1 ,e ,h1 ,e ,i1 ,e ,k1 ,e ,l1 and ,e ,c bear the same proportion as the numbers1 #a1 #i_/h1 #e_/f1 1 1 1 #i_/fa1 1 respectively then shall ,b ,d1 ,b ,f1 be the two limits of the sines of refraction of the violet-making rays1 that is the violet-making rays shall not all of them have precisely the same sine of refraction1 but none of them shall have a greater sine than ,b ,d1 nor a less than ,b ,f1 though there are violet-making rays which answer to any sine of refraction that can be taken between these two4 ,in the same manner ,b ,f and ,b ,g are the limits of the sines of refraction of the indigo-making rays2 ,b ,g1 ,b ,h are the limits belonging to the blue-making rays2 ,b ,h1 ,b ,i the limits pertaining to the green-making rays1 ,b ,i1 ,b ,k the limits for the yellow-making rays2 ,b ,k1 ,b ,l the limits for the orange-making rays2 and lastly1 ,b ,l and ,b ,c the extreme limits of the sines of refraction belonging to the red-making rays4 ,these are the proportions by which the heterogeneous rays of light are separated from each other in refraction4 #h4 ,,when light passes out of glass into air1 our author found ,a to ,b ,c as #ej to #gg1 and the same ,a to ,b ,d as #ej to #gh4 ,and when it goes out of any other refracting substance into air1 the excess of the sine of refraction of any one species of rays above its sine of incidence bears a constant proportion1 which holds the same in each species1 to the excess of the sine of refraction of the same sort of rays above the sine of incidence into the air out of glass2 provided the sines of incidence both in glass and the other substance are equal4 ,this our author verified by transmitting the light through prisms of glass included within a prismatic vessel of water2 and draws from those experiments the following observations3 that whenever the light in passing through so many surfaces parting diverse transparent substances is by contrary refractions made to emerge into the air in a direction parallel to that of its incidence1 it will appear afterwards white at any distance from the prisms1 where you shall please to examine it2 but if the direction of its emergence be oblique to its incidence1 in receding from the place of emergence its edges shall appear tinged with colours3 which proves that in the first case there is no inequality in the refractions of each species of rays1 but that when any one species is so refracted as to emerge parallel to the incident rays1 every sort of rays after refraction shall likewise be parallel to the same incident rays1 and to each other2 whereas on the contrary1 if the rays of any one sort are oblique to the incident light1 the several species shall be oblique to each other1 and be gradually separated by that obliquity4 ,from hence he deduces both the forementioned theorem1 and also this other2 that in each sort of rays the proportion of the sine of incidence to the sine of refraction1 in the passage of the ray out of any refracting substance into another1 is compounded of the proportion to which the sine of incidence would have to the sine of refraction in the passage of that ray out of the first substance into any third1 and of the proportion which the sine of incidence would have to the sine of refraction in the passage of the ray out of that third substance into the second4 ,from so simple and plain an experiment has our most judicious author deduced these important theorems1 by which we may learn how very exact and circumspect he has been in this whole work of his optics2 that notwithstanding his great particularity in explaining his doctrine1 and the numerous collection of experiments he has made to clear up every doubt which could arise1 yet at the same time he has used the greatest caution to make out every thing by the simplest and easiest means possible4 #i4 ,,our author adds but one remark more upon refraction1 which is1 that if refraction be performed in the manner he has supposed from the light's being pressed by the refracting power perpendicularly toward the surface of the refracting body1 and consequently be made to move swifter in the body than before its incidence2 whether this power act equally at all distances or otherwise1 provided only its power in the same body at the same distances remain without variation the same in one inclination of the incident rays as well as another2 he observes that the refracting powers in different bodies will be in the duplicate proportion of the tangents of the lead angles1 which the refracted light can make with the surfaces of the refracting bodies#caf4 ,this observation may be explained thus4 ,when the light passes into any refracting substance1 it has been shewn above that the sine of incidence bears a constant proportion to the sine of refraction4 ,suppose the light to pass to the refracting body ,a ,b ,c ,d 7in fig4 #acd7 in the line ,e ,f1 and to fall upon it at the point ,f1 and then to proceed within the body in the line ,f ,g4 ,let ,h ,i be drawn through ,f perpendicular to the surface ,a ,b1 and any circle ,k ,l ,m ,n be described to the center ,f4 ,then from the points ,o and ,p where this circle cuts the incident and refracted ray1 the perpendiculars ,o ,q1 ,p ,r being drawn1 the proportion of ,o ,q to ,p ,r will remain the same in all the different obliquities1 in which the same ray of light can fall on the surface ,a ,b4 ,now ,o ,q is less than ,f ,l the semidiameter of the circle ,k ,l ,m ,n1 but the more the ray ,e ,f is inclined down toward the surface ,a ,b1 the greater will ,o ,q be1 and will approach nearer to the magnitude of ,f ,l4 ,but the proportion of ,o ,q to ,p ,r remaining always the same1 when ,o ,q1 is largest1 ,p ,r will also be greatest2 so that the more the incident ray ,e ,f is inclined toward the surface ,a ,b1 the more the ray ,f ,g after refraction will be inclined toward the same4 ,now if the line ,f ,s ,t be so drawn1 that ,s ,v being perpendicular to ,f ,i shall be to ,f ,l the semidiameter of the circle in the constant proportion of ,p ,r to ,o ,q2 then the angle under ,n ,f ,t is that which ,i meant by the least of all that can be made by the refracted ray with this surface1 for the ray after refraction would proceed in this line1 if it were to come to the point ,f lying on the very surface ,a ,b2 for if the incident ray came to the point ,f in any line between ,a ,f and ,f ,h1 the ray after refraction would proceed forward in some line between ,f ,t and ,f ,i4 ,here if ,n ,w be drawn perpendicular to ,f ,n1 this line ,n ,w in the circle ,k ,l ,m ,n is called the tangent of the angle under ,n ,f ,s4 ,thus much being premised1 the sense of the forementioned proposition is this4 ,let there be two refracting substances 7in fig4 #ace7 ,a ,b ,c ,d1 and ,e ,f ,g ,h4 ,take a point1 as ,i1 in the surface ,a ,b1 and to the center ,i with any semidiameter describe the circle ,k ,l ,m4 ,in like manner on the surface ,e ,f take some point ,n1 as a center1 and describe with the same semidiameter the circle ,o ,p ,q4 ,let the angle under ,b ,i ,r be the least which the refracted light can make with the surface ,a ,b1 and the angle under ,f ,n ,s the least which the refracted light can make with the surface ,e ,f4 ,then if ,l ,t be drawn perpendicular to ,a ,b1 and ,p ,v perpendicular to ,e ,f2 the whole power1 wherewith the substance ,a ,b ,c ,d acts on the light1 will bear to the whole power wherewith the substance ,e ,f ,g ,h acts on1 the light1 a proportion1 which is duplicate of the proportion1 which ,l ,t bears to ,p ,v4 #aj4 ,,upon comparing according to this rule the refractive powers of a great many bodies it is found1 that unctuous bodies which abound most with sulphureous parts refract the light two or three times more in proportion to their density than others3 but that those bodies1 which seem to receive in their composition like proportions of sulphureous parts1 have their refractive powers proportional to their densities2 as appears beyond contradiction by comparing the refractive power of so rare a substance as the air with that of common glass or rock crystal1 though these substances are #bjjj times denser than air2 nay the same proportion is found to hold without sensible difference in comparing air with pseudo-topar and glass of antimony1 though the pseudo-topar be #cejj times denser than air1 and glass of antimony no less than #ddjj times denser4 ,this power in other substances1 as salts1 common water1 spirit of wine1 c4 seems to bear a greater proportion to their densities than these last named1 according as they abound with sulphurs more than these2 which makes our author conclude it probable1 that bodies act upon the light chiefly1 if not altogether1 by means of the sulphurs in them2 which kind of substances it is likely enters in some degree the composition of all bodies4 ,of all the substances examined by our author1 none has so great a refractive power1 in respect of its density1 as a diamond4 #aa4 ,,our author finishes these remarks1 and all he offers relating to refraction1 with observing1 that the action between light and bodies is mutual1 since sulphureous bodies1 which are most readily set on fire by the sun's light1 when collected upon them with a burning glass1 act more upon light in refracting it1 than other bodies of the same density do4 ,and farther1 that the densest bodies1 which have been now shewn to act most upon light1 contract the greatest heat by being exposed to the summer sun4 #ab4 ,,having thus dispatched what relates to refraction1 we must address ourselves to discourse of the other operation of bodies upon light in reflecting it4 ,when light passes through a surface1 which divides two transparent bodies differing in density1 part of it only is transmitted1 another part being reflected4 ,and if the light pass out of the denser body into the rarer1 by being much inclined to the foresaid surface at length no part of it shall pass through1 but be totally reflected4 ,now that part of the light1 which suffers the greatest refraction1 shall be wholly reflected with a less obliquity of the rays1 than the parts of the light which undergo a less degree of refraction2 as is evident from the last experiment recited in the first chapter2 where1 as the prisms ,d ,e ,f1 ,g ,h ,i1 7in fig4 #abi47 were turned about1 the violet light was first totally reflected1 and then the blue1 next to that the green1 and so of the rest4 ,in consequence of which our author lays down this proportion2 that the sun's light differs in reflexibility1 those rays being most reflexible1 which are most refrangible4 ,and collects from this1 in conjunction with other arguments1 that the refraction and reflection1 of light are produced by the same cause1 compassing those different effects only by the difference of circumstances with which it is attended4 ,another proof of this being taken by our author from what he has discovered of the passage of light through thin transparent plates1 viz4 that any particular species of light1 suppose1 for instance1 the red-making rays1 will enter and pass out of such a plate1 if that plate be of some certain thicknesses2 but if it be of other thicknesses1 it will not break through it1 but be reflected back3 in which is seen1 that the thickness of the plate determines whether the power1 by which that plate acts upon the light1 shall reflect it1 or suffer it to pass through4 #ac4 ,,but this last mentioned surprising property of the action between light and bodies affords the reason of all that has been said in the preceding chapter concerning the colours of natural bodies2 and must therefore more particularly be illustrated and explained1 as being what will principally unfold the nature of the action of bodies upon light4 #ad4 ,,to begin3 ,the object glass of a long telescope being laid upon a plane glass1 as proposed in the foregoing chapter1 in open day-light there will be exhibited rings of various colours1 as was there related2 but if in a darkened room the coloured spectrum be formed by the prism1 as in the first experiment of the first chapter1 and the glasses be illuminated by a reflection from the spectrum1 the rings shall not in this case exhibit the diversity of colours before described1 but appear all of the colour of the light which falls upon the glasses1 having dark rings between4 ,which shews that the thin plate of air between the glasses at some thicknesses reflects the incident light1 at other places does not reflect it1 but is found in those places to give the light passage2 for by holding the glasses in the light as it passes from the prism to the spectrum1 suppose at such a distance from the prism that the several sorts of light must be sufficiently separated from each other1 when any particular sort of light falls on the glasses1 you will find by holding a piece of white paper at a small distance beyond the glasses1 that at those intervals1 where the dark lines appeared upon the glasses1 the light is so transmitted1 as to paint upon the paper rings of light having that colour which falls upon the glasses4 ,this experiment therefore opens to us this very strange property of reflection1 that in these thin plates it should bear such a relation to the thickness of the plate1 as is here shewn4 ,farther1 by carefully measuring the diameters of each ring it is found1 that whereas the glasses touch where the dark spot appears in the center of the rings made by reflexion1 where the air is of twice the thickness at which the light of the first ring is reflected1 there the light by being again transmitted makes the first dark ring2 where the plate has three times that thickness which exhibits the first lucid ring1 it again reflects the light forming the second lucid ring2 when the thickness is four times the first1 the light is again transmitted so as to make the second dark ring2 where the air is five times the first thickness1 the third lucid ring is made2 where it has six times the thickness1 the third dark ring appears1 and so on3 in so much that the thicknesses1 at which the light is reflected1 are in proportion to the numbers #a1 #c1 #e1 #g1 #i1 c4 and the thicknesses1 where the light is transmitted1 are in the proportion of the numbers #j1 #b1 #d1 #f1 #h1 c4 ,and these proportions between the thicknesses which reflect and transmit the light remain the same in all situations of the eye1 as well when the rings are viewed obliquely1 as when looked on perpendicularly4 ,we must farther here observe1 that the light1 when it is reflected1 as well as when it is transmitted1 enters the thin plate1 and is reflected from its farther surface2 because1 as was before remarked1 the altering the transparent body behind the farther surface alters the degree of reflection as when a thin piece of ,muscovy glass has its farther surface wet with water1 and the colour of the glass made dimmer by being so wet2 which shews that the light reaches to the water1 otherwise its reflection could not be influenced by it4 ,but yet this reflection depends upon some power propagated from the first surface to the second2 for though made at the second surface it depends also upon the first1 because it depends upon the distance between the surfaces2 and besides1 the body through which the light passes to the first surface influences the reflection3 for in a plate of ,muscovy glass1 wetting the surface1 which first receives the light1 diminishes the reflection1 though not quite so much as wetting the farther surface will do4 ,since therefore the light in passing through these thin plates at some thicknesses is reflected1 but at others transmitted without reflection1 it is evident1 that this reflection is caused by some power propagated from the first surface1 which intermits and returns successively4 ,thus is every ray apart disposed to alternate reflections and transmissions at equal intervals2 the successive returns of which disposition our author calls the fits of easy reflection1 and of easy transmission4 ,but these fits1 which observe the same law of returning at equal intervals1 whether the plates are viewed perpendicularly or obliquely1 in different situations of the eye change their magnitude4 ,for what was observed before in respect of those rings1 which appear in open day-light1 holds likewise in these rings exhibited by simple lights2 namely1 that these two alter in bigness according to the different angle under which they are seen3 and our author lays down a rule whereby to determine the thicknesses of the plate of air1 which shall exhibit the same colour under different oblique views#cag4 ,and the thickness of the aereal plate1 which in different inclinations of the rays will exhibit to the eye in open day-light the same colour1 is also varied by the same rule#cah4 ,he contrived farther a method of comparing in the bubble of water the proportion between the thickness of its coat1 which exhibited any colour when seen perpendicularly1 to the thickness of it1 where the same colour appeared by an oblique view2 and he found the same rule to obtain here likewise#cai4 ,but farther1 if the glasses be enlightened successively by all the several species of light1 the rings will appear of different magnitudes2 in the red light they will be larger than in the orange colour1 in that larger than in the yellow1 in the yellow larger than in the green1 less in the blue1 less yet in the indigo1 and least of all in the violet3 which shew that the same thickness of the aereal plate is not fitted to reflect all colours1 but that one colour is reflected where another would have been transmitted2 and as the rays which are most strongly refracted form the least rings1 a rule is laid down by our author for determining the relation1 which the degree of refraction of each species of colour has to the thicknesses of the plate where it is reflected4 #ae4 ,,from these observations our author shews the reason of that great variety of colours1 which appears in these thin plates in the open white light of the day4 ,for when this white light falls on the plate1 each part of the light forms rings of its own colour2 and the rings of the different colours not being of the same bigness are variously intermixed1 and form a great variety of tints#cbj4 #af4 ,,in certain experiments1 which our author made with thick glasses1 he found1 that these fits of easy reflection and transmission returned for some thousands of times1 and thereby farther confirmed his reasoning concerning them#cba4 #ag4 ,,upon the whole1 our great author concludes from some of the experiments made by him1 that the reason why all transparent bodies refract part of the light incident upon them1 and reflect another part1 is1 because some of the light1 when it comes to the surface of the body1 is in a fit of easy transmission1 and some part of it in a fit of easy reflection2 and from the durableness of these fits he thinks it probable1 that the light is put into these fits from their first emission out of the luminous body2 and that these fits continue to return at equal intervals without end1 unless those intervals be changed by the light's entring into some refracting substance#cbb4 ,he likewise has taught how to determine the change which is made of the intervals of the fits of easy transmission and reflection1 when the light passes out of one transparent space or substance into another4 ,his rule is1 that when the light passes perpendicularly to the surface1 which parts any two transparent substances1 these intervals in the substance1 out of which the light passes1 bear to the intervals in the substance1 whereinto the light enters1 the same proportion1 as the sine of incidence bears to the sine of refraction#cbc4 ,it is farther to be observed1 that though the fits of easy reflection return at constant intervals1 yet the reflecting power never operates1 but at or near a surface where the light would suffer refraction2 and if the thickness of any transparent body shall be less than the intervals of the fits1 those intervals shall scarce be disturbed by such a body1 but the light shall pass through without any reflection#cbd4 #ah4 ,,what the power in nature is1 whereby this action between light and bodies is caused1 our author has not discovered4 ,but the effects1 which he has discovered1 of this power are very surprising1 and altogether wide from any conjectures that had ever been framed concerning it2 and from these discoveries of his no doubt this power is to be deduced1 if we ever can come to the knowledge of it4 ,sir ,,isaac ,,newton has in general hinted at his opinion concerning it2 that probably it is owing to some very subtle and elastic substance diffused through the universe1 in which such vibrations may be excited by the rays of light1 as they pass through it1 that shall occasion it to operate so differently upon the light in different places as to give rise to these alternate fits of reflection and transmission1 of which we have now been speaking#cbe4 ,he is of opinion1 that such a substance may produce this and other effects also in nature1 though it be so rare as not to give any sensible resistance to bodies in motion#cbf2 and therefore not inconsistent with what has been said above1 that the planets move in spaces free from resistance#cbg4 #ai4 ,,in order for the more full discovery of this action between light and bodies1 our author began another set of experiments1 wherein he found the light to be acted on as it passes near the edges of solid bodies2 in particular all small bodies1 such as the hairs of a man's head or the like1 held in a very small beam of the sun's light1 cast extremely broad shadows4 ,and in one of these experiments the shadow was #ce times the breadth of the body#cbh4 ,these shadows are also observed to be bordered with colours#cbi4 ,this our author calls the inflection of light2 but as he informs us1 that he was interrupted from prosecuting these experiments to any length1 ,i need not detain my readers with a more particular account of them4 ,,chap4 ,,iv4 ,of ,,optic ,,glasses4 ,,sir ,,isaac ,,newton having deduced from his doctrine of light and colours a surprising improvement of telescopes1 of which ,i intend here to give an account1 ,i shall first premise something in general concerning those instruments4 #b4 ,,it will be understood from what has been said above1 that when light falls upon the surface of glass obliquely1 after its entrance into the glass it is more inclined to the line drawn through the point of incidence perpendicular to that surface1 than before4 ,suppose a ray of light issuing from the point ,a 7in fig4 #acf7 falls on a piece of glass ,b ,c ,d ,e1 whose surface ,b ,c1 whereon the ray falls1 is of a spherical or globular figure1 the center whereof is ,f4 ,let the ray proceed in the line ,a ,g falling on the surface ,b ,c in the point ,g1 and draw ,f ,g ,h4 ,here the ray after its entrance into the glass will pass on in some line1 as ,g ,i1 more inclined toward the line ,f ,g ,h that the line ,a ,g is inclined thereto2 for the line ,f ,g ,h is perpendicular to the surface ,b ,c in the point ,g4 ,by this means1 if a number of rays proceeding from any one point fall on a convex spherical surface of glass1 they shall be inflected 7as is represented in fig4 #acg17 so as to be gathered pretty close together about the line drawn through the center of the glass from the point1 whence the rays proceed2 which line henceforward we shall call the axis of the glass3 or the point from whence the rays proceed may be so near the glass1 that the rays shall after entring the glass still go on to spread themselves1 but not so much as before2 so that if the rays were to be continued backward 7as in fig4 #ach17 they should gather together about the axis at a place more remote from the glass1 than the point is1 whence they actually proceed4 ,in these and the following figures ,a denotes the point to which the rays are related before refraction1 ,b the point to which they are directed afterwards1 and ,c the center of the refracting surface4 ,here we may observe1 that it is possible to form the glass of such a figure1 that all the rays which proceed from one point shall after refraction be reduced again exactly into one point on the axis of the glass4 ,but in glasses of a spherical form though this does not happen2 yet the rays1 which fall within a moderate distance from the axis1 will unite extremely near together4 ,if the light fall on a concave spherical surface1 after refraction it shall spread quicker than before 7as in fig4 #aci17 unless the rays proceed from a point between the center and the surface of the glass4 ,if we suppose the rays of light1 which fall upon the glass1 not to proceed from any point1 but to move so as to tend all to some point in the axis of the glass beyond the surface2 if the glass have a convex surface1 the rays shall unite about the axis sooner1 than otherwise they would do 7as in fig4 #adj17 unless the point to which they tended was between the surface and the center of that surface4 ,but if the surface be concave1 they shall not meet so soon3 nay perhaps converge4 7,see fig4 #ada and #adb47 #e4 ,,farther1 because the light in passing out of glass into the air is turned by the refraction farther off from the line drawn through the point of incidence perpendicular to the refracting surface1 than it was before2 the light which spreads from a point shall by parting through a convex surface of glass into the air be made either to spread less than before 7as in fig4 #adc17 or to gather about the axis beyond the glass 7as in fig4 #add47 ,but if the rays of light were proceeding to a point in the axis of the glass1 they should by the refraction be made to unite sooner about that axis 7as in fig4 #ade47 ,if the surface of the glass be concave1 rays which proceed from a point shall be made to spread faster 7as in fig #adf17 but rays which are tending to a point in the axis of the glass1 shall be made to gather about the axis farther from the glass 7as in fig4 #adg7 or even to diverge 7as in fig4 #adh17 unless the point1 to which the rays are directed1 lies between the surface of the glass and its center4 #d4 ,,the rays1 which spread themselves from a point1 are called diverging2 and such as move toward a point1 are called converging rays4 ,and the point in the axis of the glass1 about which the rays gather after refraction1 is called the focus of those rays4 #e4 ,,if a glass be formed of two convex spherical surfaces 7as in fig4 #adi17 where the glass ,,ab is formed of the surfaces ,a ,c ,b and ,a ,d ,b1 the line drawn through the centers of the two surfaces1 as the line ,e ,f1 is called the axis of the glass2 and rays1 which diverge from any point of this axis1 by the refraction of the glass will be caused to converge toward some part of the axis1 or at least to diverge as from a point more remote from the glass1 than that from whence they proceeded2 for the two surfaces both conspire to produce this effect upon the rays4 ,but converging rays will be caused by such a glass as this to converge sooner4 ,if a glass be formed of two concave surfaces1 as the glass ,a ,b 7in fig4 #aej17 the line ,c ,d drawn through the centers1 to which the two surfaces are formed1 is called the axis of the glass4 ,such a glass shall cause diverging rays1 which proceed from any point in the axis of the glass1 to diverge much more1 as if they came from some place in the axis of the glass nearer to it than the point1 whence the rays actually proceed4 ,but converging rays will be made either to converge less1 or even to diverge4 ,illustration #f4 ,,in these glasses rays1 which proceed from any point near the axis1 will be affected as it were in the same manner1 as if they proceeded from the very axis it self1 and such as converge toward a point at a small distance from the axis will suffer much the same effects from the glass1 as if they converged to some point in the very axis4 ,by this means any luminous body exposed to a convex glass may have an image formed upon any white body held beyond the glass4 ,this may be easily tried with a common spectacle-glass4 ,for if such a glass be held between a candle and a piece of white paper1 if the distances of the candle1 glass1 and paper be properly adjusted1 the image of the candle will appear very distinctly upon the paper1 but be seen inverted2 the reason whereof is this4 ,let ,a ,b 7in fig4 #aea7 be the glass1 ,c ,d an object placed cross the axis of the glass4 ,let the rays of light1 which issue from the point ,e1 where the axis of the glass crosses the object1 be so refracted by the glass1 as to meet again about the point ,f4 ,the rays1 which diverge from the point ,c of the object1 shall meet again almost at the same distance from the glass1 but on the other side of the axis1 as at ,g2 for the rays at the glass cross the axis4 ,in like manner the rays1 which proceed from the point ,d1 will meet about ,h on the other side of the axis4 ,none of these rays1 neither those which proceed from the point ,e in the axis1 nor those which issue from ,c or ,d1 will meet again exactly in one point2 but yet in one place1 as is here supposed at ,f1 ,g1 and ,h1 they will be crouded so close together1 as to make a distinct image of the object upon any body proper to reflect it1 which shall be held there4 #g4 ,,if the object be too near the glass for the rays to converge after the refraction1 the rays shall issue out of the glass1 as if they diverged from a point more distant from the glass1 than that from whence they really proceed 7as in fig4 #aeb17 where the rays coming from the point ,e of the object1 which lies on the axis of the glass ,a ,b1 issue out of the glass1 as if they came from the point ,f more remote from the glass than ,e2 and the rays proceeding from the point ,c issue out of the glass1 as if they proceeded from the point ,g2 likewise the rays which issue from the point ,d emerge out of the glass1 as if they came from the point ,h4 ,here the point ,g is on the same side of the axis1 as the point ,c2 and the point ,h on the same side1 as the point ,d4 ,in this case to an eye placed beyond the glass the object should appear1 as if it were in the situation ,g ,f ,h4 #h4 ,,if the glass ,a ,b had been concave 7as in1 fig4 #aec17 to an eye beyond the glass the object ,c ,d would appear in the situation ,g ,h1 nearer to the glass than really it is4 ,here also the object will not be inverted2 but the point ,g is on the same side the axe with the point ,c1 and ,h on the same side as ,d4 #i4 ,,hence may be understood1 why spectacles made with convex glasses help the sight in old age3 for the eye in that age becomes unfit to see objects distinctly1 except such as are remov'd to a very great distance2 whence all men1 when they first stand in need of spectacles1 are observed to read at arm's length1 and to hold the object at a greater distance1 than they used to do before4 ,but when an object is removed at too great a distance from the sight1 it cannot be seen clearly1 by reason that a less quantity of light from the object will enter the eye1 and the whole object will also appear smaller4 ,now by help of a convex glass an object may be held near1 and yet the rays of light issuing from it will enter the eye1 as if the object were farther removed4 #aj4 ,,after the same manner concave glasses assist such1 as are short sighted4 ,for these require the object to be brought inconveniently near to the eye1 in order to their seeing it distinctly2 but by such a glass the object may be removed to a proper distance1 and yet the rays of light enter the eye1 as if they came from a place much nearer4 #aa4 ,,whence these defects of the sight arise1 that in old age objects cannot be seen distinct within a moderate distance1 and in short-sightedness not without being brought too near1 will be easily understood1 when the manner of vision in general shall be explain'd2 which ,i shall now endeavour to do1 in order to be better understood in what follows4 ,the eye is form'd1 as is represented in fig4 #aed4 ,it is of a globular figure1 the fore part whereof scarce more protuberant than the rest is transparent4 ,underneath this transparent part is a small collection of an humour in appearance like water1 and it has also the same refractive power as common water2 this is called the aqueous humour1 and fills the space ,a ,b ,c ,d in the figure4 ,next beyond lies the body ,d ,e ,f ,g2 this is solid but transparent1 it is composed with two convex surfaces1 the hinder surface ,e ,f ,g being more convex1 than the anterior ,e ,d ,g4 ,between the outer membrane ,a ,b ,c1 and this body ,e ,d ,g ,f is placed that membrane1 which exhibits the colours1 that are seen round the sight of the eye2 and the black spot1 which is called the sight or pupil1 is a hole in this membrane1 through which the light enters1 whereby we see4 ,this membrane is fixed only by its outward circuit1 and has a muscular power1 whereby it dilates the pupil in a weak light1 and contracts it in a strong one4 ,the body ,d ,e ,f ,g is called the crystalline humour1 and has a greater refracting power than water4 ,behind this the bulk of the eye is filled up with what is called the vitreous humor1 this has much the same refractive power with water4 ,at the bottom of the eye toward the inner side next the nose the optic glass enters1 as at ,h1 and spreads it self all over the inside of the eye1 till within a small diftance from ,a and ,c4 ,now any object1 as ,i ,k1 being placed before the eye1 the rays of light issuing from each point of this object are so refracted by the convex surface of the aqueous humour1 as to be caused to converge2 after this being received by the convex surface ,e ,d ,g of the crystalline humour1 which has a greater refractive power than the aqueous1 the rays1 when they are entered into this surface1 still more converge1 and at going out of the surface ,e ,f ,g into a humour of a less refractive power than the crystalline they are made to converge yet farther4 ,by all these successive refractions they are brought to converge at the bottom of the eye1 so that a distinct image of the object as ,l ,m is impress'd on the nerve4 ,and by this means the object is seen4 #aa4 ,,it has been made a difficulty1 that the image of the object impressed on the nerve is inverted1 so that the upper part of the image is impressed on the lower part of the eye4 ,but this difficulty1 ,i think1 can no longer remain1 if we only consider1 that upper and lower are terms merely relative to the ordinary position of our bodies3 and our bodies1 when view'd by the eye1 have their image as much inverted as other objects2 so that the image of our own bodies1 and of other objects1 are impressed on the eye in the same relation to one another1 as they really have4 #ab4 ,,the eye can see objects equally distinct at very different distances1 but in one distance only at the same time4 ,that the eye may accomodate itself to different distances1 some change in its humours is requir'd4 ,it is my opinion1 that this change is made in the figure of the crystalline humour1 as ,i have indeavoured to prove in another place4 #ac4 ,,if any of the humours of the eye are too flat1 they will refract the light too little2 which is the case in old age4 ,if they are too convex1 they refract too much2 as in those who are short-sighted4 #ad4 ,,the manner of direct vision being thus explained1 ,i proceed to give some account of telescopes1 by which we view more distinctly remote objects2 and also of microscopes1 whereby we magnify the appearance of small objects4 ,in the first place1 the most simple sort of telescope is composed of two glasses1 either both convex1 or one convex1 and the other concave4 7,the first sort of these is represented in fig4 #aee1 the latter in fig4 #aef47 #ae4 ,,in fig4 #aee let ,a ,b represent the convex glass next the object1 ,c ,d the other glass more convex near the eye4 ,suppose the object-glass ,a ,b to form the image of the object at ,e ,f2 so that if a sheet of white paper were to be held in this place1 the object would appear4 ,now suppose the rays1 which pass the glass ,a ,b1 and are united about ,f1 to proceed to the eye glass ,c ,d1 and be there refracted4 ,three only of these rays are drawn in the figure1 those which pass by the extremities of the glass ,a ,b1 and that which passes its middle4 ,if the glass ,c ,d be placed at such a distance from the image ,e ,f1 that the rays1 which pass by the point ,f1 after having proceeded through the glass diverge so much1 as the rays do that come from an object1 which is at such a distance from the eye as to be seen distinctly1 these being received by the eye will make on the bottom of the eye a distinct representation of the point ,f4 ,in like manner the rays1 which pass through the object glass ,a ,b to the point ,e after proceeding through the eye-glass ,c ,d will on the bottom of the eye make a distinct representation of the point ,e4 ,but if the eye be placed where these rays1 which proceed from ,e1 cross those1 which proceed from ,f1 the eye will receive the distinct impression of both these points at the same time2 and consequently will also receive a distinct impression from all the intermediate parts of the image ,e ,f1 that is1 the eye will see the object1 to which the telescope is directed1 distinctly4 ,the place of the eye is about the point ,g1 where the rays ,h ,e1 ,h ,f cross1 which pass through the middle of the object-glass ,a ,b to the points ,e and ,f2 or at the place where the focus would be formed by rays coming from the point ,h1 and refracted by the glass ,c ,d4 ,to judge how much this instrument magnifies any object1 we must first observe1 that the angle under ,e ,h ,f1 in which the eye at the point ,h would see the image ,e ,f1 is nearly the same as the angle1 under which the object appears by direct vision2 but when the eye is in ,g1 and views the object through the telescope1 it sees the same under a greater angle2 for the rays1 which coming from ,e and ,f cross in ,g1 make a greater angle than the rays1 which proceed from the point ,h to these points ,e and ,f4 ,the angle at ,g is greater than that at ,h in the proportion1 as the distance between the glasses ,a ,b and ,c ,d is greater than the distance of the point ,g from the glass ,c ,d4 #af4 ,,this telescope inverts the object2 for the rays1 which came from the right-hand side of the object1 go to the point ,e the left side of the image2 and the rays1 which come from the left side of the object1 go to ,f the right side of the image4 ,these rays cross again in ,g1 so that the rays1 which come from the right side of the object1 go to the right side of the eye2 and the rays from the left side of the object go to the left side of the eye4 ,therefore in this telescope the image in the eye has the same situation as the object2 and seeing that in direct vision the image in the eye has an inverted situation1 here1 where the situation is not inverted1 the object must appear so4 ,this is no inconvenience to astronomers in celestial observations2 but for objects here on the earth it is usual to add two other convex glasses1 which may turn the object again 7as is represented in fig4 #aeg17 or else to use the other kind of telescope with a concave eye-glass4 #ag4 ,,in this other kind of telescope the effect is founded on the same principles1 as in the former4 ,the distinctness of the appearance is procured in the same manner4 ,but here the eye-glass ,c ,d 7in fig4 #aef7 is placed between the image ,e ,f1 and the object glass ,a ,b4 ,by this means the rays1 which come from the right-hand side of the object1 and proceed toward ,e the left side of the image1 being intercepted by the eye-glass are carried to the left side of the eye2 and the rays1 which come from the left side of the object1 go to the right side of the eye2 so that the impression in the eye being inverted the object appears in the same situation1 as when view'd by the naked eye4 ,the eye must here be placed close to the glass4 ,the degree of magnifying in this instrument is thus to be found4 ,let the rays1 which pass through the glass ,a ,b at ,h1 after the refraction of the eye-glass ,c ,d diverge1 as if they came from the point ,g2 then the rays1 which come from the extremities of the object1 enter the eye under the angle at ,g2 so that here also the object will be magnified in the proportion of the distance between the glasses1 to the distance of ,g from the eye-glass4 #ah4 ,,the space1 that can be taken in at one view in this telescope1 depends on the breadth of the pupil of the eye2 for as the rays1 which go to the points ,e1 ,f of the image1 are something distant from each other1 when they come out of the glass ,c ,d1 if they are wider asunder than the pupil1 it is evident1 that they cannot both enter the eye at once4 ,in the other telescope the eye is placed in the point ,g1 where the rays that come from the points ,e or ,f cross each other1 and therefore must enter the eye together4 ,on this account the telescope with convex glasses takes in a larger view1 than those with concave4 ,but in these also the extent of the view is limited1 because the eye-glass does not by the refraction towards its edges form so distinct a representation of the object1 as near the middle4 #ah4 ,,microscopes are of two sorts4 ,one kind is only a very convex glass1 by the means of which the object may be brought very near the eye1 and yet be seen distinctly4 ,this microscope magnifies in proportion1 as the object by being brought near the eye will form a broader impression on the optic nerve4 ,the other kind made with convex glasses produces its effects in the same manner as the telescope4 ,let the object ,a ,b 7in fig4 #aeh7 be placed under the glass ,c ,d1 and by this glass let an image be formed of this object4 ,above this image let the glass ,g ,h be placed4 ,by this glass let the rays1 which proceed from the points ,a and ,b1 be refracted1 as is expressed in the figure4 ,in particular1 let the rays1 which from each of these points pass through the middle of the glass ,c ,d1 cross in ,i1 and there let the eye be placed4 ,here the object will appear larger1 when seen through the microscope1 than if that instrument were removed1 in proportion as the angle1 in which these rays cross in ,i1 is greater than the angle1 which the lines would make1 that should be drawn from ,i to ,a and ,b2 that is1 in the proportion made up of the proportion of the distance of the object ,a ,b from ,i1 to the distance of ,i from the glass ,g ,h2 and of the proportion of the distance between the glasses1 to the distance of the object ,a ,b from the glass ,c ,d4 #ai4 ,i ,,shall now proceed to explain the imperfection in these instruments1 occasioned by the different refrangibility of the light which comes from every object4 ,this prevents the image of the object from being formed in the focus of the object glass with perfect distinctness2 so that if the eye-glass magnify the image overmuch1 the imperfections of it must be visible1 and make the whole appear confused4 ,our author more fully to satisfy himself1 that the different refrangibility of the several sorts of rays is sufficient to produce this irregularity1 underwent the labour of a very nice and difficult experiment1 whose process he has at large set down1 to prove1 that the rays of light are refracted as differently in the small refraction of telescope glasses as in the larger of the prism2 so exceeding careful has he been in searching out the true cause of this effect4 ,and he used1 ,i suppose1 the greater caution1 because another reason had before been generally assigned for it4 ,it was the opinion of all mathematicians1 that this defect in telescopes arose from the figure1 in which the glasses were formed2 a spherical refracting surface not collecting into an exact point all the rays which come from any one point of an object1 as has before been said#ccj4 ,but after our author has proved1 that in these small refractions1 as well as in greater1 the sine of incidence into air out of glass1 to the sine of refraction in the red-making rays1 is as #ej to #gg1 and in the blue-making rays #ej to #gh2 he proceeds to compare the inequalities of refraction arising from this different refrangibility of the rays1 with the inequalities1 which would follow from the figure of the glass1 were light uniformly refracted4 ,for this purpose he observes1 that if rays issuing from a point so remote from the object glass of a telescope1 as to be esteemed parallel1 which is the case of the rays1 which come from the heavenly bodies2 then the distance from the glass of the point1 in which the least refrangible rays are united1 will be to the distance1 at which the most refrangible rays unite1 as #bh to #bg2 and therefore that the least space1 into which all the rays can be collected1 will not be less than the #eeth part of the breadth of the glass4 ,for if ,a ,b 7in fig4 #aei7 be the glass1 ,c ,d its axis1 ,e ,a1 ,f ,b two rays of the light parallel to that axis entring the glass near its edges2 after refraction let the least refrangible part of these rays meet in ,g1 the most refrangible in ,h2 then1 as has been said1 ,g ,i will be to ,i ,h1 as #bh to #bg2 that is1 ,g ,h will be the #bhth part of ,g ,i1 and the #bgth part of ,h ,i2 whence if ,k ,l be drawn through ,g1 and ,m ,n through ,h1 perpendicular to ,c ,d1 ,m ,n will be the a #bhth part of ,a ,b1 the breadth of the glass1 and ,k ,l the #bgth part of the same2 so that ,o ,p the least space1 into which the rays are gathered1 will be about half the mean between these two1 that is the #eeth part of ,a ,b4 #bj4 ,,this is the error arising from the different refrangibility of the rays of light1 which our author finds vastly to exceed the other1 consequent upon the figure of the glass4 ,in particular1 if the telescope glass be flat on one side1 and convex on the other2 when the flat side is turned towards the object1 by a theorem1 which he has laid down1 the error from the figure comes out above #ejjj times less than the other4 ,this other inequality is so great1 that telescopes could not perform so well as they do1 were it not that the light does not equally fill all the space ,o ,p1 over which it is scattered1 but is much more dense toward the middle of that space than at the extremities4 ,and besides1 all the kinds of rays affect not the sense equally strong1 the yellow and orange being the strongest1 the red and green next to them1 the blue indigo and violet being much darker and fainter colours2 and it is shewn that all the yellow and orange1 and three fifths of the brighter half of the red next the orange1 and as great a share of the brighter half of the green next the yellow1 will be collected into a space whose breadth is not above the #bejth part of the breadth of the glass4 ,illustration ,and the remaining colours1 which fall without this space1 as they are much more dull and obscure than these1 so will they be likewise much more diffused2 and therefore call hardly affect the sense in comparison of the other4 ,and agreeable to this is the observation of astronomers1 that telescopes between twenty and sixty feet in length represent the fixed stars1 as being about #e or #f1 at most about #h or #aj seconds in diameter4 ,whereas other arguments shew us1 that they do not really appear to us of any sensible magnitude any otherwise than as their light is dilated by refraction4 ,one proof that the fixed stars do not appear to us under any sensible angle is1 that when the moon passes over any of them1 their light does not1 like the planets on the same occasion1 disappear by degrees1 but vanishes at once4 #ba4 ,,our author being thus convinced1 that telescopes were not capable of being brought to much greater perfection than at present by refractions1 contrived one by reflection1 in which there is no separation made of the different coloured light2 for in every kind of light the rays after reflection have the same degree of inclination to the surface1 from whence they are reflected1 as they have at their incidence1 so that those rays which come to the surface in one line1 will go off also in one line without any parting from one another4 ,accordingly in the attempt he succeeded so well1 that a short one1 not much exceeding six inches in length1 equalled an ordinary telescope whose length was four feet4 ,instruments of this kind to greater lengths1 have of late been made1 which fully answer expectation#cca4 ,,chap4 ,v4 ,of the ,,rainbow4 ,i ,,shall now explain the rainbow4 ,the manner of its production was understood1 in the general1 before ,sir ,,isaac ,,newton had discovered his theory of colours2 but what caused the diversity of colours in it could not then be known1 which obliges him to explain this appearance particularly2 whom we shall imitate as follows4 ,the first person1 who expressly shewed the rainbow to be formed by the reflection of the sun-beams from drops of falling rain1 was ,,antonio ,,de ,,dominis4 ,but this was afterwards more fully and distinctly explained by ,,descartes4 #b4 ,,there appears most frequently two rainbows2 both of which are caused by the foresaid reflection of the sun-beams from the drops of falling rain1 but are not produced by all the light which falls upon and are reflected from the drops4 ,the inner bow is produced by those rays only which enter the drop1 and at their entrance are so refracted as to unite into a point1 as it were1 upon the farther surface of the drop1 as is represented in fig4 #afj2 where the contiguous rays a b1 c d1 e f1 coming from the sun1 and therefore to sense parallel1 upon their entrance into the drop in the points b1 d1 f1 are so refracted as to meet together in the point g1 upon the farther surface of the drop4 ,now these rays being reflected nearly from the same point of the surface1 the angle of incidence of each ray upon the point g being equal to the angle of reflection1 the rays will return in the lines g h1 g k1 g l1 in the same manner inclined to each other1 as they were before their incidence upon the point g1 and will make the same angles with the surface of the drop at the points b1 k1 l1 as at the points b1 d1 f1 after their entrance2 and therefore after their emergence out of the drop each ray will be inclined to the surface in the same angle1 as when it first entered it2 whence the lines b m1 k n1 l o1 in which the rays emerge1 must be parallel to each other1 as well as the lines a b1 c d1 e f1 in which they were incident4 ,but these emerging rays being parallel will not spread nor diverge from each other in their passage from the drop1 and therefore will enter the eye conveniently situated in sufficient plenty to cause a sensation4 ,whereas all the other rays1 whether those nearer the center of the drop1 as p q1 r s1 or those farther off1 as t u1 w x1 will be reflected from other points in the hinder surface of the drop2 namely1 the ray p q from the point y1 r s from z1 t v from 1 and w x from 4 ,and for this reason by their reflection and succeeding refraction they will be scattered after their emergence from the forementioned rays and from each other1 and therefore cannot enter the eye placed to receive them copious enough to excite any distinct sensation4 #c4 ,,the external rainbow is formed by two reflections made between the incidence and emergence of the rays2 for it is to be noted1 that the rays g h1 g k1 g l1 at the points h1 k1 l1 do not wholly pass out of the drop1 but are in part reflected back2 though the second reflection of these particular rays does not form the outer bow4 ,for this bow is made by those rays1 which after their entrance into the drop are by the refraction of it united1 before they arrive at the farther surface1 at such a distance from it1 that when they fall upon that surface1 they may be reflected in parallel lines1 as is represented in fig4 #afa2 where the rays a b1 c d1 e f1 are collected by the refraction of the drop into the point g1 and passing on from thence strike upon the surface of the drop in the points h1 k1 l1 and are thence reflected to m1 n1 o1 passing from h to m1 from k to n1 and from l to o in parallel lines4 ,for these rays after reflection at m1 n1 o1 will meet again in the point p1 at the same distance from these points of reflection m1 n1 o1 as the point g is from the former points of reflection h1 k1 l4 ,therefore these rays in passing from p to the surface of the drop will fall upon that surface in the points q1 r1 s in the same angles1 as these rays made with the surface in b1 d1 f1 after refraction4 ,consequently1 when these rays emerge out of the drop into the air1 each ray will make with the surface of the drop the same angle1 as it made at its first incidence2 so that the lines q t1 r v1 s w1 in which they come from the drop1 will be parallel to each other1 as well as the lines a b1 c d1 e f1 in which they came to the drop4 ,by this means these rays to a spectator commodiously situated will become visible4 ,but all the other rays1 as well those nearer the center of the drop x y1 z 1 as those more remote from it 1 1 will be reflected in lines not parallel to the lines h m1 k n1 l o2 namely1 the ray x y1 in the line 1 the ray in the line 1 the ray in the line 1 and the ray in the line 4 ,whence these rays after their next reflection and subsequent refraction will be scattered from the forementioned rays1 and from one another1 and by that means become invisible4 #d4 ,,it is farther to be remarked1 that if in the first case the incident rays a b1 c d1 e f1 and their correspondent emergent rays h m1 k n1 l o1 are produced till they meet1 they will make with each other a greater angle1 than any other incident ray will make with its corresponding emergent ray4 ,and in the latter case1 on the contrary1 the emergent rays q t1 r v1 s w make with the incident rays an acuter angle1 than is made by any other of the emergent rays4 #e4 ,,our author delivers a method of finding each of these extream angles from the degree of refraction being given2 by which method it appears1 that the first of these angles is the less1 and the latter the greater1 by how much the refractive power of the drop1 or the refrangibility of the rays is greater4 ,and this last consideration fully compleats the doctrine of the rainbow1 and shews1 why the colours of each bow are ranged in the order wherein they are seen4 #f4 ,,suppose ,a 7in fig4 #afb47 to be the eye1 ,b1 ,c1 ,d1 ,e1 ,f1 drops of rain1 ,m n1 ,o p1 ,q r1 ,s t1 ,v w parcels of rays of the sun1 which entring the drops ,b1 ,c1 ,d1 ,e1 ,f after one reflection pass out to the eye in ,a4 ,now let ,m n be produced to till it meets with the emergent ray likewise produced1 let ,o p produced meet its emergent ray produced in 1 let ,q r meet its emergent ray in 1 let ,s t meet its emergent ray in 1 and let ,v w meet its emergent ray produced in 4 ,if the angle under ,m ,a be that1 which is derived from the refraction of the violet-making rays by the method we have here spoken of1 it follows that the violet light will only enter the eye from the drop ,b1 all the other coloured rays passing below it1 that is1 all those rays which are not scattered1 but go out parallel so as to cause a sensation4 ,for the angle1 which these parallel emergent rays makes with the incident in the most refrangible or violet-making rays1 being less than this angle in any other sort of rays1 none of the rays which emerge parallel1 except the violet-making1 will enter the eye under the angle ,m ,a1 but the rest making with the incident ray ,m a greater angle than this will pass below the eye4 ,in like manner if the angle under ,o ,a agrees to the blue-making rays1 the blue rays only shall enter the eye from the drop ,c1 and all the other coloured rays will pass by the eye1 the violet-coloured rays passing above1 the other colours below4 ,farther1 the angle ,q ,a corresponding to the green-making rays1 those only shall enter the eye from the drop ,d1 the violet and blue-making rays passing above1 and the other colours1 that is the yellow and red1 below4 ,and if the angle ,s ,a answers to the refraction of the yellow-making rays1 they only shall come to the eye from the drop ,e4 ,and in the last place1 if the angle ,v ,a belongs to the red-making and least refrangible rays1 they only shall enter the eye from the drop ,f1 all the other coloured rays passing above4 #g4 ,,but now it is evident1 that all the drops of water found in any of the lines ,a 1 ,a 1 ,a 1 ,a 1 whether farther from the eye1 or nearer than the drops ,b1 ,c1 ,d1 ,e1 ,f1 will give the same colours as these do1 all the drops upon each line giving the same colour2 so that the light reflected from a number of these drops will become copious enough to be visible2 whereas the reflection from one minute drop alone could not be perceived4 ,but besides1 it is farther manifest1 that if the line ,a be drawn from the sun through the eye1 that is1 parallel to the lines ,m n1 ,o p1 ,q r1 ,s t1 ,v w1 and if drops of water are placed all round this line1 the same colour will be exhibited by all the drops at the same distance from this line4 ,hence it follows1 that when the sun is moderately elevated above the horizon1 if it rains opposite to it1 and the sun shines upon the drops as they fall1 a spectator with his back turned to the sun must observe a coloured circular arch reaching to the horizon1 being red without1 next to that yellow1 then green1 blue1 and on the inner edge violet2 only this last colour appears faint by being diluted with the white light of the clouds1 and from another cause to be mentioned hereafter#ccb4 #h4 ,,thus is caused the interior or primary bow4 ,the drops of rain at some distance without this bow will cause the exterior or secondary bow by two reflections of the sun's light4 ,let these drops be ,g1 ,h1 ,i1 ,k1 ,l2 ,x y1 ,z 1 1 1 denoting parcels of rays which enter each drop4 ,now it has been remarked1 that these rays make with the visible refracted rays the greatest angle in those rays1 which are most refrangible4 ,suppose therefore the visible refracted rays1 which pass out from each drop after two reflections1 and enter the eye in ,a1 to intersect the incident rays in 1 1 1 1 respectively4 ,it is manifest1 that the angle under ,a is the greatest of all1 next to that the angle under ,a1 the next in bigness will be the angle under ,a1 the next to this the angle under ,z ,a1 and the least of all the angle under ,x ,a4 ,from the drop ,l therefore will come to the eye the violet-making1 or most refrangible rays1 from ,k the blue1 from ,i the green1 from ,h the yellow1 and from ,g the red-making rays2 and the like will happen to all the drops in the lines ,a 1 ,a 1 ,a 1 ,a 1 and also to all the drops at the same distances from the line ,a all round that line4 ,whence appears the reason of the secondary bow1 which is seen without the other1 having its colours in a contrary order1 violet without and red within2 though the colours are fainter than in the other bow1 as being made by two reflections1 and two refractions2 whereas the other bow is made by two refractions1 and one reflection only4 #i4 ,,there is a farther appearance in the rainbow particularly described about five years ago#ccc1 which is1 that under the upper part or the inner bow there appears often two or three orders of very faint colours1 making alternate arches of green1 and a reddish purple4 ,at the time this appearance was taken notice of1 ,i gave my thoughts concerning the cause of it#ccd1 which ,i shall here repeat4 ,sir ,,isaac ,,newton has observed1 that in glass1 which is polished and quick-silvered1 there is an irregular refraction made1 whereby some small quantity of light is scattered from the principal reflected beam#cce4 ,if we allow the same thing to happen in the reflection whereby the rainbow is caused1 it seems sufficient to produce the appearance now mentioned4 #aj4 ,,let ,a ,b 7in fig4 #afb47 represent a globule of water1 ,b the point from whence the rays of any determinate species being reflected to ,c1 and afterwards emerging in the line ,c ,d1 would proceed to the eye1 and cause the appearance of that colour in the rainbow1 which appertains to this species4 ,here suppose1 that besides what is reflected regularly1 some small part of the light is irregularly scattered every way2 so that from the point ,b1 besides the rays that are regularly reflected from ,b to ,c1 some scattered rays will return in other lines1 as in ,b ,e1 ,b ,f1 ,b ,g1 ,b ,h1 on each side the line ,b ,c4 ,now it has been observed above#ccf1 that the rays of light in their passage from one superficies of a refracting body to the other undergo alternate fits of easy transmission and reflection1 succeeding each other at equal intervals2 insomuch that if they reach the farther superficies in one sort of those fits1 they shall be transmitted2 if in the other kind of them1 they shall rather be reflected back4 ,whence the rays that proceed from ,b to ,c1 and emerge in the line ,c ,d1 being in a fit of easy transmission1 the scattered rays1 that fall at a small distance without these on either side 7suppose the rays that pass in the lines ,b ,e1 ,b ,g7 shall fall on the surface in a fit of easy reflection1 and shall not emerge2 but the scattered rays1 that pass at some distance without these last1 shall arrive at the surface of the globule in a fit of easy transmission1 and break through that surface4 ,suppose these rays to pass in the lines ,b ,f1 ,b ,h2 the former of which rays shall have had one fit more of easy transmission1 and the latter one fit less1 than the rays that pass from ,b to ,c4 ,now both these rays1 when they go out of the globule1 will proceed by the refraction of the water ,in the lines ,f ,i1 ,h ,k1 that will be inclined almost equally to the rays incident on the globule1 which come from the sun2 but the angles of their inclination will be less than the angle1 in which the rays emerging in the line ,c ,d are inclined to those incident rays4 ,and after the same manner rays scattered from the point ,b at a certain distance without these will emerge out of the globule1 while the intermediate rays are intercepted2 and these emergent rays will be inclined to the rays incident on the globule in angles still less than the angles1 in which the rays ,f ,i and ,h ,k are inclined to them2 and without these rays will emerge other rays1 that shall be inclined to the incident rays in angles yet less4 ,illustration ,now by this means may be formed of every kind of rays1 besides the principal arch1 which goes to the formation of the rainbow1 other arches within every one of the principal of the same colour1 though much more faint2 and this for divers successions1 as long as these weak lights1 which in every arch grow more and more obscure1 shall continue visible4 ,now as the arches produced by each colour will be variously mixed together1 the diversity of colours observ'd in these secondary arches may very possibly arise from them4 #aa4 ,,in the darker colours these arches may reach below the bow1 and be seen distinct4 ,in the brighter colours these arches are lost in the inferior part of the principal light of the rainbow2 but in all probability they contribute to the red tincture1 which the purple of the rainbow usually has1 and is most remarkable when these secondary colours appear strongest4 ,however these secondary arches in the brightest colours may possibly extend with a very faint light below the bow1 and tinge the purple of these secondary arches with a reddish hue4 #ab4 ,,the precise distances between the principal arch and these fainter arches depend on the magnitude of the drops1 wherein they are formed4 ,to make them any degree separate it is necessary the drop be exceeding small4 ,it is most likely1 that they are formed in the vapour of the cloud1 which the air being put in motion by the fall of the rain may carry down along with the larger drops2 and this may be the reason1 why these colours appear under the upper part of the bow only1 this vapour not descending very low4 ,as a farther confirmation of this1 these colours are seen strongest1 when the rain falls from very black clouds1 which cause the fiercest rains1 by the fall whereof the air will be most agitated4 #ac4 ,,to the like alternate return of the fits of easy transmission and reflection in the passage of light through the globules of water1 which compose the clouds1 ,sir ,,isaac ,,newton ascribes some of those coloured circles1 which at times appear about the sun and moon#ccg4 ,illustration ,illustration ,,conclusion4 ,,sir ,,isaac ,,newton having concluded each of his philosophical treatises with some general reflections1 ,i shall now take leave of my readers with a short account of what he has there delivered4 ,at the end of his mathematical principles of natural philosophy he has given us his thoughts concerning the ,deity4 ,wherein he first observes1 that the similitude found in all parts of the universe makes it undoubted1 that the whole is governed by one supreme being1 to whom the original is owing of the frame of nature1 which evidently is the effect of choice and design4 ,he then proceeds briefly to state the best metaphysical notions concerning ,god4 ,in short1 we cannot conceive either of space or time otherwise than as necessarily existing2 this ,being therefore1 on whom all others depend1 must certainly exist by the same necessity of nature4 ,consequently wherever space and time is found1 there ,god must also be4 ,and as it appears impossible to us1 that space should be limited1 or that time should have had a beginning1 the ,deity must be both immense and eternal4 #b4 ,,at the end of his treatise of optics he has proposed some thoughts concerning other parts of nature1 which he had not distinctly searched into4 ,he begins with some farther reflections concerning light1 which he had not fully examined4 ,in particular he declares his sentiments at large concerning the power1 whereby bodies and light act on each other4 ,in some parts of his book he had given short hints at his opinion concerning this#cch1 but here he expressly declares his conjecture1 which we have already mentioned#cci1 that this power is lodged in a very subtle spirit of a great elastic force diffused thro' the universe1 producing not only this1 but many other natural operations4 ,he thinks it not impossible1 that the power of gravity itself should be owing to it4 ,on this occasion he enumerates many natural appearances1 the chief of which are produced by chymical experiments4 ,from numerous observations of this kind he makes no doubt1 that the smallest parts of matter1 when near contact1 act strongly on each other1 sometimes being mutually attracted1 at other times repelled4 #c4 ,,the attractive power is more manifest than the other1 for the parts of all bodies adhere by this principle4 ,and the name of attraction1 which our author has given to it1 has been very freely made use of by many writers1 and as much objected to by others4 ,he has often complained to me of having been misunderstood in this matter4 ,what he lays upon this head was not intended by him as a philosophical explanation of any appearances1 but only to point out a power in nature not hitherto distinctly observed1 the cause of which1 and the manner of its acting1 he thought was worthy of a diligent enquiry4 ,to acquiesce in the explanation of any appearance by asserting it to be a general power of attraction1 is not to improve our knowledge in philosophy1 but rather to put a stop to our farther search4 ,,finis4 ,illustration ,,footnotes3 #a ,philosoph4 ,nat4 princ4 math4 ,l4 iii4 introduct4 #b ,nov4 ,org4 ,scient4 ,l4 i4 ,aphorism4 #i4 #c ,nov4 ,org4 ,l4 i4 ,aph4 #ai4 #d ,ibid4 ,aph4 #be4 #e ,aph4 #cj4 ,errores radicales in prima digestione mentis ab excellentia functionum remediorum sequentium non curantur4 #f ,aph4 #ch4 #g ,ibid4 #h ,aph4 #ci4 #i ,aph4 #da4 #aj ,aph4 #aj1 #bd4 #aa ,aph4 #de4 #ab ,de ,cartes ,princ4 ,phil4 ,part4 #c4 4 #eb4 #ac ,fermat1 in ,oper4 pag4 #aef1 c4 #ad ,nov4 ,org4 ,aph4 #df4 #ae ,aph4 #ej4 #af ,ibid4 #ag ,aph #ec4 #ah ,aph4 #ed4 #ai ,aph4 #ef4 #bj ,aph4 #ee4 #ba ,locke1 ,on human understanding1 ,b4 iii4 #bb ,nov4 ,org4 ,aph4 #ei4 #bc ,in the conclusion4 #bd ,nov4 ,org4 ,l4 i4 ,aph4 #ei4 #be ,ibid4 ,aph4 #fj4 #bf ,ibid4 ,aph4 #fb4 #bg ,aph4 #fc4 #bh ,aph4 #fd4 #bi ,aph4 #fe4 #cj ,see above1 #d1 #e4 #ca ,nov4 ,org4 ,l4 i4 ,aph4 #fi4 #cb ,ibid4 #cc ,ibid4 ,aph4 #aji4 #cd ,book ,,iii4 ,chap4 iv4 #ce ,book ,i4 ,chap4 #b4 #ad4 #cf ,ibid4 #he1 c4 #cg ,see ,book ,,ii4 ,ch4 #c4 #c1 #d4 of this treatise4 #ch ,see ,book ,,ii4 ,ch4 #c4 of this treatise4 #ci ,see ,chap4 #d4 #dj ,at the end of his ,optics4 in ,qu4 #ba4 #da ,see the same treatise1 in ,advertisement #b4 #db ,nov4 ,org4 ,lib4 i4 ,ax4 #aje4 #dc ,princip4 philos4 pag4 #ac1 #ad4 #dd ,princ4 ,philos4 ,l4 ,,ii4 prop4 #bd4 corol4 #g4 ,see also ,b4 ,,ii4 ,ch4 #e4 #c4 of this treatise4 #de ,how this degree of elasticity is to be found by experiment1 will be shewn below in #gd4 #df ,in oper4 posthum de ,motu corpor4 ex percussion4 prop4 #i4 #dg ,in the above-cited place4 #dh ,in the place above-cited4 #di ,these experiments are described in #gc4 #ej ,book ,,ii4 ,chap4 #e4 #ea ,chap4 #a4 #be1 #bf1 #bg1 compared with #ae1 c4 #eb ,book ,,ii4 ,chap4 #e4 #c4 #ec ,see ,euclid's ,elements1 ,book ,,xii4 prop4 #ac4 #ed ,archimed4 de quipond4 prop4 #aa4 #ee ,ibid4 prop4 #ab4 #ef ,lucas ,valerius ,de centr4 gravit4 solid4 ,l4 ,i4 prop4 #b4 #eg ,idem ,l4 ,,ii4 prop4 #b4 #eh #be4 #ei #bg4 #fj ,pag4 #fe1 #fh4 #fa #bc4 #fb #bj #fc #ag4 #fd #bg4 #fe ,hugen4 ,horolog4 oscillat4 pag4 #ada1 #adb4 #ff ,see ,hugen4 ,horolog4 ,oscillat4 p4 #adb4 #fg ,princip4 ,philos4 pag4 #bb4 #fh ,chap4 #a4 #bi4 #fi ,princip4 ,philos4 pag4 #be4 #gj #ga4 #ga ,see ,method4 ,increment4 prop4 #be4 #gb ,lib4 ,,xi4 ,def4 #gc ,chap4 #b4 #ag4 #gd ,see above ,ch4 #b4 #ag4 #ge ,from ,b ,,ii4 ,ch4 #c4 #gf ,prin4 ,philos4 pag4 #g1 c4 #gg ,see ,newton1 princip4 philos4 pag4 #i4 lin4 #cj4 #gh ,princip4 ,philos4 pag4 #aj4 #gi ,renat4 ,des ,cart4 ,princ4 ,philos4 ,part4 ,,ii4 #be4 #hj ,ibid4 #cj4 #ha #he1 c4 #hb ,princip4 ,philos4 ,lib4 ,i4 prop4 #i4 #hc #ib4 #hd ,ch4 ,,ii4 #bb4 #he ,viz4 ,l4 ,i4 prop4 #cj1 #bi1 #bf4 #hf ,ch4 ,,ii4 #ba1 #bb4 #hg viz4 ,his doctrine of prime and ultimate ratios4 #hh #eg #hi #c4 #ij ,ch4 #b4 #bb4 #ia #ab4 #ib ,ch4 #a4 sect4 #ba1 #bb4 #ic ,elem4 ,book ,i4 p4 #cg4 #id #ab4 #ie ,ch #a #bd4 #if ,ch #b select4 #ag4 #ig ,newt4 ,princ4 ,l4 ,,ii4 prop4 #b2 #e1 #f1 #g2 #aa1 #ab4 #ih ,prop4 #c2 #h1 #i2 #ac1 #ad4 #ii ,prop4 #d4 #ajj ,prlect4 ,geometr4 pag4 #abc4 #aja ,newton4 ,princ4 ,lib4 ,,ii4 prop4 #aj4 #ajb ,newton4 ,princ4 ,lib ,,ii4 prop #aj4 in schol4 #ajc ,torricelli de motu gravium4 #ajd ,ch4 #b #he1 c4 #aje ,newt4 ,princ ,l4 ,,ii4 sect #f4 #ajf ,l4 ,,ii4 sect4 #d4 #ajg ,see ,b4 ,,ii4 ,ch #f4 #g4 of this treatise4 #ajh ,lib4 ,i4 sect4 #aj4 #aji ,de la ,pesanteur1 pag4 #afi1 and the following4 #aaj ,newton4 ,princ4 ,l4 ,,ii4 prop #d4 schol4 #aaa ,see his ,tract on the admirable rarifaction of the air4 #aab ,book ,,ii4 ,ch4 #f4 #aac ,princ4 philos4 ,lib4 ,,ii4 prop4 #bc4 #aad ,book ,i4 ,ch4 #b4 #cj4 #aae ,princ4 philos4 ,lib4 ,,ii4 prop4 #bc1 in schol4 #aaf ,princ4 philos4 ,lib4 ,,ii4 prop4 #cc4 coroll4 #aag ,lib4 ,,ii4 ,ch4 #e4 #aah ,ibid4 ,prop4 #ce4 coroll4 #b4 #aai ,ibid4 coroll4 #c4 #abj ,vid4 ibid4 coroll4 #f4 #aba ,in #b4 #abb ,princ4 philos4 ,lib4 ,,ii4 ,prop4 #ce4 #abc ,ibid4 #abd ,id4 #abe h4 #a4 #bi4 #abf ,princ4 philos4 ,lib4 ,,ii4 ,prop4 #ch1 compared with coroll4 #a of prop4 #ce4 #abg ,l4 ,,ii4 ,lem4 #g4 schol4 pag4 #cda4 #abh ,lib4 ,,ii4 ,prop4 #cd4 #abi ,lib4 ,,ii4 ,lem4 #g4 p4 #cda4 #acj ,schol4 to ,lem4 #g4 #aca ,prop4 #cd4 schol4 #acb ,ibid4 #acc ,ibid4 #acd ,book ,,ii4 ,ch4 ,i4 #f4 #ace ,vid4 ,newt4 princ4 in schol4 to ,lem4 #g1 of ,lib4 ,,ii4 pag4 #cda4 #acf ,sect4 #ag4 of this chapter4 #acg ,see ,princ4 philos4 ,lib4 ,,ii4 prop4 #cd4 #ach ,vid4 ,princ4 philos4 ,lib4 ,,ii4 ,lem4 #e4 p4 #cad4 #aci ,lemm4 #f4 #adj ,ibid4 #g4 #ada ,newt4 ,princ4 ,lib4 ,,ii4 prop4 #dj1 in schol4 #adb ,lib4 ,,ii4 in schol4 post prop4 #ca4 #adc ,book ,i4 ch4 #b #hb4 #add ,book ,i4 ,ch4 #c #bi4 #ade ,ch4 #c4 of this present book4 #adf ,ch4 #d4 #adg ,in ,princ4 philos4 part4 #c4 #adh ,philos4 princ4 mathem4 ,lib4 ,,ii4 prop4 #b4 schol4 #adi ,ibid4 prop #ec4 #aej ,philos4 princ4 prop4 #eb4 coroll4 #d4 #aea ,ibid4 #aeb ,coroll4 #aa4 #aec ,see ibid4 schol4 post prop4 #ec4 #aed ,princ4 philos4 pag4 #caf1 #cag4 #aee ,ch4 ,i4 #g4 #aef ,book ,i4 ,ch4 #c4 #aeg ,book ,i4 ,ch4 #c4 #bi4 #aeh ,ibid4 ,ch4 #b4 #cj1 #ag4 #aei ,book ,i4 ,ch4 #c4 #afj ,ch4 #a4 #g4 #afa ,chap4 #e4 #h4 #afb ,princ4 pag4 #fj4 #afc ,street1 in ,astron4 ,carolin4 #afd ,see ,chap4 #e4 #i1 c4 #afe ,in the foregoing page4 #aff ,see ,newton4 ,princ4 ,lib4 ,,iii4 prop4 #ac4 #afg ,chap4 #e4 #aj4 #afh ,princ4 ,lib4 ,i4 prop4 #fj4 #afi ,book ,i1 ,chap4 #b4 #hj4 #agj ,princ4 philos4 ,lib4 ,i4 prop4 #eh4 coroll4 #c4 #aga ,newt4 ,optics4 pag4 #cgh4 #agb ,newton4 ,princ4 ,lib4 ,,iii4 prop4 #a4 #agc ,newton1 ,princ4 ,lib4 ,,iii4 pag4 #cij1cia4 compared with pag4 #cic4 #agd ,book ,i4 ,ch4 #c4 #bi4 #age ,princ4 philos4 ,lib4 ,i4 prop4 #d4 #agf ,ibid4 coroll4 #agg ,newt4 ,princ4 philos4 ,lib4 ,,iii4 pag4 #cij4 #agh ,newt4 ,princ4 philos4 ,lib4 ,,iii4 pag4 #cia1 #cib4 #agi ,book ,,iii4 ,ch4 #d4 #ahj ,newt4 ,princ4 philos4 ,lib4 ,,iii4 pag4 #cia4 #aha ,ibid4 pag4 #cib4 #ahb ,see ,book ,i4 ,ch4 #b4 #fj1 #fd4 #ahc ,book ,i4 ,ch4 #b4 #ag4 #ahd ,see ,ch4 ,,ii4 #f4 #ahe ,the second of the laws of motion laid down in ,book ,i4 ,ch4 #a4 #ahf ,newton4 ,princ4 philos4 ,lib4 ,,iii4 prop4 #f4 pag4 #dja4 #ahg ,newton's ,princ4 philos4 ,lib4 ,,iii4 prop4 #bb1 #bc4 #ahh ,newton4 ,princ4 ,lib4 ,i4 prop4 #ff4 coroll4 #g4 #ahi ,menelai ,sphaeric4 ,lib4 ,i4 prop4 #aj4 #aij ,vid4 ,newt4 ,princ4 ,lib4 ,i4 prop4 #ff4 coroll4 #aj4 #aia ,vid4 ,newt4 ,princ4 ,lib4 ,,iii prop4 #cj4 p4 #ddj4 #aib ,ibid4 ,lib4 ,i4 prop4 #ff4 coroll4 #aj4 #aic ,what this proportion is1 may be known from ,coroll4 #b prop4 #dd4 ,lib4 ,i4 ,princ4 philos4 ,newton4 #aid ,princ4 ,phil4 ,newt4 ,lib4 ,i4 prop4 #de4 ,coroll4 #a4 #aie ,pr4 ,phil4 ,newt4 ,lib4 ,i4 prop4 #ff4 ,coroll4 #g4 #aif ,see #ai of this chapter4 #aig ,phil4 ,nat4 ,pr4 ,math ,lib4 ,i4 prop4 #ff4 cor4 #h4 #aih ,ibid4 ,coroll4 #h4 #aii ,ibid4 #bjj ,ibid4 #bja ,newt4 ,princ4 ,lib4 ,,iii4 prop4 #bi4 #bjb ,ibid4 prop4 #bh4 #bjc ,ibid4 prop4 #ca4 #bjd ,newt4 ,princ4 pag4 #dei4 #bje ,in ,princ4 philos4 part4 #c4 #da4 #bjf ,chap4 #a4 #aa4 #bjg ,newton4 ,princ4 philos4 ,lib4 ,,iii4 ,lemm4 #d4 pag4 #dgh4 #bjh ,princ4 philos4 ,lib4 ,,iii4 prop4 #dj4 #bji ,book ,i4 chap4 #b4 #hb4 #baj ,princ4 philos4 ,lib4 ,,iii4 pag4 #dii1 #ejj4 #baa ,ibid4 pag4 #ejj1 and #ebj1 c4 #bab ,princ4 ,philos4 ,lib4 ,,iii4 prop4 #dj4 #bac ,ibid4 prop4 #da4 #bad ,ibid4 pag4 #ebb4 #bae ,ibid4 prop4 #db4 #baf ,newt4 ,princ4 philos4 edit4 #b4 p4 #dfd1 #dfe4 #bag ,ibid4 edit4 #c4 p #eja1 #ejb4 #bah ,ibid4 pag4 #eai4 #bai ,ibid4 pag4 #ebd4 #bbj ,newt4 ,princ4 philos4 p4 #ebe4 #bba ,ibid4 #bbb ,ibid4 pag4 #ejh4 #bbc ,ibid4 #bbd ,ibid4 pag4 #dhd4 #bbe ,ibid4 pag4 #dhb1 #dhc4 #bbf ,ibid4 pag4 #dha4 #bbg ,ibid4 pag4 #eji4 #bbh ,see the fore-cited place4 #bbi ,ibid4 and ,cartes4 ,princ4 ,phil4 part4 #c4 #acd1 c4 #bcj ,vid4 ,phil4 ,nat4 princ4 ,math4 p4 #eaa4 #bca ,book ,i4 ,ch4 #d4 #aa4 #bcb ,ch4 #e4 #bcc ,all these arguments are laid down in ,philos4 ,nat4 ,princ4 ,lib4 ,,iii4 from p4 #eji1 to #eag4 #bcd ,philos4 ,nat4 ,princ4 ,lib4 ,,iii4 p4 #eae4 #bce ,ch4 #e4 #bcf ,see ,ch4 #a4 #aa4 #bcg ,newt4 ,princ4 ,philos4 pag4 #ebe1 #ebf4 ,an account of all the stars of both these kinds1 which have appeared within the last #aej years may be seen in the ,philosophical transactions1 vol4 #bi4 numb4 #cdf4 #bch ,newt4 ,princ4 ,philos4 ,nat4 ,lib4 ,,iii4 prop4 #f4 #bci ,ch4 #c4 #f4 #bdj ,book ,i4 ,ch4 #b4 #bd4 #bda ,newt4 ,princ4 ,lib4 ,,iii4 prop4 #f4 #bdb ,ch4 #c4 #f4 #bdc ,newt4 ,princ4 philos4 ,lib4 ,,iii4 prop4 #g4 cor4 #a4 #bdd ,see ,book ,i4 ,ch4 #a4 #ae4 #bde ,ibid4 #e1 #f4 #bdf ,chap4 #b4 #h4 #bdg ,newt4 ,princ4 ,lib4 ,i4 prop4 #fc4 #bdh #h4 #bdi ,see ,introd4 #bc4 #bej #d1 #e4 #bea ,newt4 ,princ4 philos4 ,lib4 ,i4 prop4 #gd4 #beb ,ibid4 coroll4 #c4 #bec ,lib4 ,i4 ,prop4 #ge4 and ,lib4 ,,iii4 prop4 #h4 #bed ,lib4 ,i4 ,prop4 #gf4 #bee ,ibid4 cor4 #e4 #bef ,vid4 ,lib4 ,,iii4 ,prop4 #g4 coroll4 #a #beg ,newt4 ,princ4 ,lib4 ,,iii4 prop4 #h4 coroll4 #a4 #beh ,ibid4 coroll4 #b4 #bei ,book ,i4 ,ch4 #d4 #b4 #bfj ,newt4 ,princ4 ,lib4 ,,iii4 prop4 #h4 coroll4 #c4 #bfa ,ibid4 coroll4 #d4 #bfb ,book ,i4 ,ch4 #d4 #bfc ,lib4 ,,ii4 prop4 #bj4 cor4 #b4 #bfd ,chap4 #d4 #ag4 #bfe ,ibid4 #bff ,vid4 ,newt4 ,princ4 ,lib4 ,,ii4 prop4 #df4 #bfg ,princ4 philos4 ,lib4 ,,ii4 prop4 #di4 #bfh ,chap4 #c4 #ah4 #bfi ,newt4 ,princ4 philos4 ,lib4 ,i4 prop4 #ff4 coroll4 #ah4 #bgj #h4 #bga ,ch4 #c4 #e4 #bgb ,ch4 #c #ag4 #bgc ,ibid4 #bgd ,see below #dd4 #bge ,newton ,princ4 ,lib4 ,,iii4 prop4 #ai4 #bgf ,lib4 ,,iii4 prop4 #ai4 #bgg ,lib4 ,i4 prop4 #gc4 #bgh ,lib4 ,,iii4 prop4 #bj4 #bgi ,ibid4 #bhj ,opt4 ,b4 ,i4 part4 #b4 prop4 #a4 #bha ,newt4 ,opt4 ,b4 #a4 part #a4 experim4 #e4 #bhb ,ibid4 prop4 #d4 #bhc ,newt4 ,opt4 ,b4 #a4 part #b4 exper4 #e4 #bhd ,ibid exper4 #f4 #bhe ,newton ,opt4 ,b4 ,i4 prop4 #aj4 #bhf ,ibid exp4 #i4 #bhg ,newt4 ,opt4 ,b4 ,i4 part #a4 exp #ae4 #bhh ,philos4 ,transact4 ,n4 #hh1 p4 #ejii4 #bhi ,opt ,b4 ,i4 par4 #b4 exp4 #ad4 #bij ,ibid4 exp4 #aj4 #bia ,opt4 pag4 #abb4 #bib ,opt4 ,b4 ,i4 part #b4 exp4 #aa4 #bic ,ibid prop4 #d1 #f4 #bid ,opt4 pag4 #ea4 #bie ,opt4 ,book ,,ii4 prop4 #h4 #bif ,opt4 ,book ,,ii4 par4 #c4 prop4 #b4 #big #ag4 #bih ,opt4 ,book ,,ii4 par4 #c4 prop4 #d4 #bii ,opt4 ,book ,,ii4 pag4 #bda4 #cjj ,ibid4 pag4 #bbd4 #cja ,ibid4 ,obs4 #ag4 c4 #cjb ,ibid4 ,obs4 #aj4 #cjc ,ibid4 pag4 #bjf4 #cjd ,obser4 #ba4 #cje ,observ4 #e4 compared with ,observ4 #aj #cjf ,ibid4 prop4 #e4 #cjg ,observ4 #g4 #cjh ,observ4 #i4 #cji ,ibid prop4 #g4 #caj ,opt4 pag4 #bdc4 #caa ,newt4 ,opt4 ,b4 ,i4 part4 #a4 prop4 ,i4 #cab ,opt4 ,b4 ,i4 part4 #a4 prop4 #b4 #cac ,opt4 ,b4 ,i4 part #a4 ,expec4 #f4 #cad ,opt4 pag4 #fg1 #fh1 c4 #cae ,ibid4 ,b4 #a4 par4 #b4 prop4 #c4 #caf ,opt4 ,b4 ,,ii4 par4 #c4 prop4 #aj4 #cag ,opt4 ,b4 ,,ii4 par4 #c4 prop4 #ae4 #cah ,ibid4 par4 #a4 observ4 #g4 #cai ,ibid4 ,observ4 #ai4 #cbj ,opt4 ,b4 ,,ii4 par4 #b4 pag4 #aii4 c4 #cba ,ibid4 par4 #d #cbb ,ibid4 part4 #c4 prop4 #ac4 #cbc ,ibid4 prop4 #ag4 #cbd ,ibid4 prop4 #ac4 #cbe ,opt4 ,qu4 #ah1 c4 #cbf ,see ,concl4 ,s4 #b4 #cbg ,b4 ,,ii4 ,ch4 #a4 #cbh ,opt4 ,b4 ,,iii4 ,obs4 #a4 #cbi ,ibid4 ,obs4 #b4 #ccj #b4 #cca ,philos4 ,trans4 ,no4 #cgh4 #ccb #aa4 #ccc ,philos4 ,transact ,no4 #cge4 #ccd ,ibid4 #cce ,opt4 ,b4 ,,ii4 part #d4 #ccf ,ch4 #c4 #ad4 #ccg ,opt4 ,b4 ,,ii4 part #d4 obs4 #ac4 #cch ,opt4 pag4 #bee4 #cci ,ch4 #c4 #ah4