,transcriber's ,note3 ,this text is intended for users whose text readers cannot use the real 7,unicode_/,,utf-#h7 version of the file4 ,letters with overline 7representing following n or m7 have been written out1 without further marking4 ,the ae ligature is shown as two letters4 ,the pilcrow is given as ,sec44,greek--mainly in the introduction--has been transliterated and shown between marks1 with eta and omega rendered as e3 and o34 ,unless otherwise noted1 spelling1 punctuation and capitalization are unchanged4 ,details are given at the end of the e-text4 ,superscripts are shown with a caret 1 adding braces where necessary for clarity4 ,terminology3 right line straight line gemow 7line7 parallel gemew twin square quadrilateral 7also spelled squire1 squyre7 also angle square as described under hexagons 7siseangles7 likeiamme parallelogram iam7me7 jamb limb1 side longsquare rectangle touch line tangent cantle segment of a circle cantle slice 99 99 99 99 99 99 99 99 99 99 99 99 99 99 ,the pathway to ,,knowledg1 ,,contai- ,,ning ,,the ,,first ,,prin- ciples of ,geometrie1 as they may moste aptly be applied vn- to practice1 bothe for vse of instrumentes ,geome- tricall1 and astrono- micall and also for proiection of plattes in euerye kinde1 and therfore much ne- cessary for all sortes of men4 ,geometries verdicte ,all fresshe fine wittes by me are filed1 ,all grosse dull wittes wishe me exiled3 ,thoughe no mannes witte reiect will ,i1 ,yet as they be1 ,i wyll them trye4 ,the argumentes of the foure bookes ,the first booke declareth the definitions of the termes and names vsed in ,geometry1 with certaine of the chiefe grounds whereon the arte is founded4 ,and then teacheth those conclusions1 which may serue diuersely in al workes ,geometricall4 ,the second booke doth sette forth the ,theoremes1 7whiche maye be called approued truthes7 seruinge for the due knowledge and sure proofe of all conclusions and workes in ,geometrye4 ,the third booke intreateth of diuers formes1 and sondry protractions thereto belonging1 with the vse of certain conclusions4 ,the fourth booke teacheth the right order of measuringe all platte formes1 and bodies also1 by reson ,geometricall4 ,,to ,,the ,,gentle ,,reader4 ,excvse me1 gentle reder if oughte be amisse1 straung paths ar not troden al truly at the first3 the way muste needes be comberous1 wher none hathe gone before4 ,where no man hathe geuen light1 lighte is it to offend1 but when the light is shewed ones1 light is it to amende4 ,if my light may so light some other1 to espie and marke my faultes1 ,i wish it may so lighten them1 that they may voide offence4 ,of staggeringe and stomblinge1 and vnconstaunt turmoilinge3 often offending1 and seldome amending1 such vices to eschewe1 and their fine wittes to shew that they may winne the praise1 and ,i to hold the candle1 whilest they their glorious works with eloquence sette forth1 so cunningly inuented1 so finely indited1 that my bokes maie seme worthie to occupie no roome4 ,for neither is mi wit so finelie filed1 nother mi learning so largly lettred1 nother yet mi laiser so quiet and vncombered1 that ,i maie perform iustlie so learned a laboure or accordinglie to accomplishe so haulte an enforcement1 yet maie ,i thinke thus3 ,this candle did ,i light3 this light haue ,i kindeled3 that learned men maie se1 to practise their pennes1 their eloquence to aduaunce1 to register their names in the booke of memorie ,i drew the platte rudelie1 whereon thei maie builde1 whom god hath indued with learning and liuelihod4 ,for liuing by laboure doth learning so hinder1 that learning serueth liuinge1 whiche is a peruers trade4 ,yet as carefull familie shall cease hir cruell callinge1 and suffre anie laiser to learninge to repaire1 ,i will not cease from trauaile the pathe so to trade1 that finer wittes maie fashion them selues with such glimsinge dull light1 a more complete woorke at laiser to finisshe1 with inuencion agreable1 and aptnes of eloquence4 ,and this gentle reader ,i hartelie protest where erroure hathe happened ,i wisshe it redrest4 ,,to ,,the ,,most ,,no- ble and puissaunt prince ,edwarde the sixte by the grace of ,god1 of ,en- gland ,fraunce and ,ireland kynge1 de- fendour of the faithe1 and of the ,churche of ,england and ,ire- lande in earth the su- preme head4 ,it is not vnknowen to youre maiestie1 moste soueraigne lorde1 what great disceptacion hath been amongest the wyttie men of all nacions1 for the exacte knoweledge of true felicitie1 bothe what it is1 and wherein it consisteth3 touchynge whiche thyng1 their opinions almoste were as many in numbre1 as were the persons of them1 that either disputed or wrote thereof4 ,but and if the diuersitie of opinions in the vulgar sort for placyng of their felicitie shall be considered also1 the varietie shall be found so great1 and the opinions so dissonant1 yea plainly monsterouse1 that no honest witte would vouchesafe to lose time in hearyng them1 or rather 7as ,i may saie7 no witte is of so exact remembrance1 that can consider together the monsterouse multitude of them all4 ,and yet not withstandyng this repugnant diuersitie1 in two thynges do they all agree4 ,first all do agre1 that felicitie is and ought to be the stop and end of all their doynges1 so that he that hath a full desire to any thyng how so euer it be estemed of other men1 yet he estemeth him self happie1 if he maie obtain it3 and contrary waies vnhappie if he can not attaine it4 ,and therfore do all men put their whole studie to gette that thyng1 wherin they haue perswaded them self that felicitie doth consist4 ,wherfore some whiche put their felicitie in fedyng their bellies1 thinke no pain to be hard1 nor no dede to be vnhonest1 that may be a meanes to fill that foule panche4 ,other which put their felicitie in play and ydle pastimes1 iudge no time euill spent1 that is employed thereabout3 nor no fraude vnlawfull that may further their winning4 ,if ,i should particularly ouerrunne but the common sortes of men1 which put their felicitie in their desires1 it wold make a great boke of it self4 ,therfore wyl ,i let them al go1 and conclude as ,i began1 ,that all men employ their whole endeuour to that thing1 wherin thei thinke felicitie to stand4 whiche thyng who so listeth to mark exactly1 shall be able to espie and iudge the natures of al men1 whose conuersacion he doth know1 though thei vse great dissimulacion to colour their desires1 especially when they perceiue other men to mislyke that which thei so much desire3 ,for no man wold gladly haue his appetite improued4 ,and herof commeth that seconnde thing wherin al agree1 that euery man would most gladly win all other men to his sect1 and to make them of his opinion1 and as far as he dare1 will dispraise all other mens iudgementes1 and praise his own waies only1 onles it be when he dissimuleth1 and that for the furtherance of his own purpose4 ,and this propertie also doth geue great light to the full knowledge of mens natures1 which as all men ought to obserue1 so princes aboue other haue most cause to mark for sundrie occasions which may lye them on1 wherof ,i shall not nede to speke any farther1 consideryng not only the greatnes of wit1 and exactnes of iudgement whiche god hath lent vnto your highnes person1 but also ye most graue wisdom and profound knowledge of your maiesties most honorable councel1 by whom your highnes may so sufficiently vnderstand all thinges conuenient1 that lesse shal it nede to vnderstand by priuate readying1 but yet not vtterly to refuse to read as often as occasion may serue1 for bokes dare speake1 when men feare to displease4 ,but to returne agayne to my firste matter1 if none other good thing maie be lerned at their maners1 which so wrongfully place their felicity1 in so miserable a condition 7that while they thinke them selfes happy1 their felicitie must nedes seme vnluckie1 to be by them so euill placed7 yet this may men learn at them1 by those 4ij4 spectacles to espye the secrete natures and dispositions of others1 whiche thyng vnto a wise man is muche auailable4 ,and thus will ,i omit this great tablement of vnhappie hap1 and wil come to 4iij4 other sortes of a better degre1 wherof the one putteth felicitie to consist in power and royaltie4 ,the second sorte vnto power annexeth worldly wisdome1 thinkyng him full happie1 that could attayn those two1 wherby he might not onely haue knowledge in all thynges1 but also power to bryng his desires to ende4 ,the thyrd sort estemeth true felicitie to consist in wysdom annexed with vertuouse maners1 thinkyng that they can take harme of nothyng1 if they can with their wysedome ouercome all vyces4 ,of the firste of those three sortes there hath been a great numbre in all ages1 yea many mightie kinges and great gouernoures which cared not greately howe they myght atchieue their pourpose1 so that they dyd preuayle3 nor did not take any greatter care for gouernance1 then to kepe the people in onely feare of them1 ,whose common sentence was alwaies this3 ,oderint dum metuant4 ,and what good successe suche menne had1 all histories doe report4 ,yet haue they not wanted excuses3 yea ,iulius ,caesar 7whiche in dede was of the second sorte7 maketh a kynde of excuse by his common sentence1 for them of that fyrste sorte1 for he was euer woonte to saie3 eiper gar adikeim chre31 turannidos peri kallistom adikeim1 t' alla d' eusebeim chreo3m4 ,whiche sentence ,i wysshe had neuer been learned out of ,grecia4 ,but now to speake of the second sort1 of whiche there hathe been verye many also1 yet for this present time amongest them all1 ,i wyll take the exaumples of kynge ,phylippe of ,macedonie1 and of ,alexander his sonne1 that valiaunt conquerour4 ,first of kinge ,phylip it appeareth by his letter sente vnto ,aristotle that famous philosopher1 that he more delited in the birthe of his sonne1 for the hope of learning and good education1 that might happen to him by the said ,aristotle1 then he didde reioyse in the continuaunce of his succession1 for these were his wordes and his whole epistle1 worthye to bee remembred and registred euery where4 ,philippos ,aristotelei chaireim4 esthe moi gegonota huom4 polle3m oum tois theois charim echo31 ouch houto3s epi te3 genne3sei tou paidos1 ho3s epi to3 kata te3m se3m he3likiam autom gegonenai elpizo3 gar autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimo3m kai te3s to3m pragmato3m diadoche3s4 ,that is thus in sense1 ,philip vnto ,aristotle sendeth gretyng4 ,you shall vnderstande1 that ,i haue a sonne borne1 for whiche cause ,i yelde vnto ,god moste hartie thankes1 not so muche for the byrthe of the childe1 as that it was his chaunce to be borne in your tyme4 ,for my trust is1 that he shall be so brought vp and instructed by you1 that he shall become worthie not only to be named our sonne1 but also to be the successour of our affayres4 ,and his good desire was not all vayne1 for it appered that ,alexander was neuer so busied with warres 7yet was he neuer out of moste terrible battaile7 but that in the middes thereof he had in remembraunce his studies1 and caused in all countreies as he went1 all strange beastes1 fowles and fisshes1 to be taken and kept for the ayd of that knowledg1 which he learned of ,aristotle3 ,and also to be had with him alwayes a greate numbre of learned men4 ,and in the moste busye tyme of all his warres against ,darius kinge of ,persia1 when he harde that ,aristotle had putte forthe certaine bookes of suche knowledge wherein he hadde before studied1 hee was offended with ,aristotle1 and wrote to hym this letter4 ,alezandros ,arisotelei eu pratteim4 ,ouk ortho3s epoie3sas ekdous tous akroamatikous to3m logo3m1 tini gar dioisome3n he3meis to3m allo3m1 ei kath' hous epaideuthe3men logous1 houtoi panto3m esontai koinoi1 ego3 de bouloi me3m am tais peri ta arista empeiriais1 e3 tais dunamesi diapherim4 erro3so4 that is ,alexander vnto ,aristotle sendeth greeting4 ,you haue not doone well1 to put forthe those bookes of secrete phylosophy intituled1 akroamatikoi4 ,for wherin shall we excell other1 yf that knowledge that wee haue studied1 shall be made commen to all other men1 namely sithe our desire is to excelle other men in experience and knowledge1 rather then in power and strength4 ,farewell4 ,by whyche lettre it appeareth that hee estemed learninge and knowledge aboue power of men4 ,and the like iudgement did he vtter1 when he beheld the state of ,diogenes ,linicus1 adiudginge it the beste state next to his owne1 so that he said3 ,if ,i were not ,alexander1 ,i wolde wishe to be ,diogenes4 ,whereby apeareth1 how he esteemed learning1 and what felicity he putte therin1 reputing al the worlde saue him selfe to be inferiour to ,diogenes4 ,and bi al coniecturs1 ,alexander did esteme ,diogenes one of them whiche contemned the vaine estimation of the disceitfull world1 and put his whole felicity in knowledg of vertue1 and practise of the same1 though some reporte that he knew more vertue then he folowed3 ,but whatso euer he was1 it appeareth that ,socrates and ,plato and many other did forsake their liuings and sel away their patrimony1 to the intent to seeke and trauaile for learning1 which examples ,i shall not need to repete to your ,maiesty1 partly for that your highnes doth often reade them and other lyke1 and partly sith your maiesty hath at hand such learned schoolemaysters1 which can much better then ,i1 declare them vnto your highnes1 and that more largely also then the shortenes of thys epistle will permit4 ,but thys may ,i yet adde1 that ,king ,salomon whose renoume spred so farre abroad1 was very greatlye estemed for his wonderfull power and exceading treasure1 but yet much more was he estemed for his wisdom4 ,and him selfe doth bear witnes1 that wisedom is better then pretious stones 4 yea all thinges that can be desired ar not to be compared to it4 ,but what needeth to alledge one sentence of him1 whose bookes altogither do none other thing1 then set forth the praise of wisedom knowledg8 ,and his father king ,dauid ioyneth uertuous conuersacion and knowledg togither1 as the summe of perfection and chief felicity4 ,wherfore ,i maye iustelye conclude1 that true felicity doth consist in wisdome and vertu4 ,then if wisdome be as ,cicero defineth it1 ,diuinarum atq2 humanarum rerum scientia1 then ought all men to trauail for knowledg in matters both of religion and humaine docrine1 if he shall be counted wyse1 and able to attaine true felicitie3 ,but as the study of religious matters is most principall1 so ,i leue it for this time to them that better can write of it then ,i can4 ,and for humaine knowledge thys wil ,i boldly say1 that who soeuer wyll attain true iudgment therein1 must not only trauail in ye knowledg of the tungs1 but must also before al other arts1 taste of the mathematical sciences1 specially ,arithmetike and ,geometry1 without which it is not possible to attayn full knowledg in any art4 ,which may sufficiently by gathered by ,aristotle not only in his bookes of demonstration 7whiche can not be vnderstand without ,geometry7 but also in all his other workes4 ,and before him ,plato his maister wrote this sentence on his schole house dore4 ,ageometre3tos oudeis eisito34 ,let no man entre here 7saith he7 without knowledg in ,geometry4 ,wherfore moste mighty prince1 as your most excellent ,maiesty appeareth to be borne vnto most perfect felicity1 not only by reason that ,god moued with the long prayers of this realme1 did send your highnes as moste comfortable inheritour to the same1 but also in that your ,maiesty was borne in the time of such skilful schoolmaisters learned techers1 as your highnes doth not a little reioyse in1 and profite by them in all kind of vertu knowledg4 ,amongst which is that heauenly knowledg most worthely to be praised1 wherbi the blindnes of errour superstition is exiled1 good hope conceiued that al the sedes fruts therof1 with all kindes of vice iniquite1 wherby vertu is hindered1 iustice defaced1 shal be clean extrirped and rooted out of this realm1 which hope shal increase more and more1 if it may appear that learning be estemed florish within this realm4 ,and al be it the chief learning be the diuine scriptures1 which instruct the mind principally1 nexte therto the lawes politike1 which most specially defend the right of goodes1 yet is it not possible that those two can long be wel vsed1 if that ayde want that gouerneth health and expelleth sicknes1 which thing is done by ,physik1 these require the help of the vij4 liberall sciences1 but of none more then of ,arithmetik and ,geometry1 by which not only great thinges ar wrought touching accomptes in al kinds1 in suruaiyng measuring of landes1 but also al arts depend partly of them1 building which is most necessary can not be wtout them1 which thing considering1 moued me to help to serue your maiesty in this point as wel as other wais1 to do what mai be in me1 yt not only thei which studi principalli for lerning1 mai haue furderance bi mi poore help1 but also those whiche haue no tyme to trauaile for exacter knowlege1 may haue some helpe to vnderstand in those ,mathematicall artes1 in whiche as ,i haue all readye set forth sumwhat of ,arithmetike1 so god willing ,i intend shortly to setforth a more exacter worke therof4 ,and in the meane ceason for a taste of ,geometry1 ,i haue sette forthe this small introduction1 desiring your grace not so muche to beholde the simplenes of the woorke1 in comparison to your ,maiesties excellencye1 as to fauour the edition thereof1 for the ayde of your humble subiectes1 which shal thinke them selues more and more dayly bounden to your highnes1 if when they shall perceaue your graces desyre to haue theym profited in all knowledge and vertue4 ,and ,i for my poore ability considering your ,maiesties studye for the increase of learning generally through all your highenes dominions1 and namely in the vniuersities of ,oxforde and ,camebridge1 as ,i haue an earnest good will as far as my simple seruice and small knowledg will suffice1 to helpe toward the satisfiyng of your graces desire1 so if ,i shall perceaue that my seruice may be to your maiesties contentacion1 ,i wil not only put forth the other two books1 whiche shoulde haue beene sette forth with these two1 yf misfortune had not hindered it1 but also ,i wil set forth other bookes of more exacter arte1 bothe in the ,latine tongue and also in the ,englyshe1 whereof parte bee all readye written1 and newe instrumentes to theym deuised1 and the residue shall bee eanded with all possible speede4 ,i was boldened to dedicate this booke of ,geometrye vnto your ,maiestye1 not so muche bycause it is the firste that euer was sette forthe in ,englishe1 and therefore for the noueltye a straunge presente1 but for that ,i was perswaded1 that suche a wyse prince doothe desire to haue a wise sorte of subiectes4 ,for it is a kynges chiefe reioysinge and glory1 if his subiectes be riche in substaunce1 and wytty in knowledge3 and contrarye waies nothyng can bee more greuouse to a noble kyng1 then that his realme should be other beggerly or full of ignoraunce3 ,but as god hath geuen your grace a realme bothe riche in commodities and also full of wyttie men1 so ,i truste by the readyng of wyttie artes 7whiche be as the whette stones of witte7 they muste needes increase more and more in wysedome1 and peraduenture fynde some thynge towarde the ayde of their substaunce1 whereby your grace shall haue newe occasion to reioyce1 seyng your subiectes to increase in substance or wisdom1 or in both4 ,and thei again shal haue new and new causes to pray for your maiestie1 perceiuyng so graciouse a mind towarde their benefite4 ,and ,i truste 7as ,i desire7 that a great numbre of gentlemen1 especially about the courte1 whiche vnderstand not the latin tong1 or els for the hardnesse of the mater could not away with other mens writyng1 will fall in trade with this easie forme of teachyng in their vulgar tong1 and so employe some of their tyme in honest studie1 whiche were wont to bestowe most part of their time in triflyng pastime3 ,for vndoubtedly if they mean other your maiesties seruice1 other their own wisdome1 they will be content to employ some tyme aboute this honest and wittie exercise4 ,for whose encouragement to the intent they maie perceiue what shall be the vse of this science1 ,i haue not onely written somewhat of the vse of ,geometrie1 but also ,i haue annexed to this boke the names and brefe argumentes of those other bokes whiche ,i will set forth hereafter1 and that as shortly as it shall appeare vnto your maiestie by coniecture of their diligent vsyng of this first boke1 that they wyll vse well the other bokes also4 ,in the meane ceason1 and at all times ,i wil be a continuall petitioner1 that god may work in all english hartes an ernest mynde to all honest exercises1 wherby thei may serue the better your maiestie and the realm4 ,and for your highnes ,i besech the most mercifull god1 as he hath most fauourably sent you vnto vs1 as our chefe comforter in earthe1 so that he will increase your maiestie daiely in all vertue and honor with moste prosperouse successe1 and augment in vs your most humble subiectes1 true loue to godward1 and iust obedience toward your highnes with all reuerence and subiection4 ,at ,london the 4xxviij4 daie of ,ianuarie4 ,m4 ,d4 ,l ,i4 ,your maiesties moste humble seruant and obedient subiect1 ,robert ,recorde4 ,,the ,,preface1 declaring briefely the commodi- tes of ,geometrye1 and the necessitye thereof4 ,geometrye may thinke it selfe to sustaine great iniury1 if it shall be inforced other to show her manifold commodities1 or els not to prease into the sight of men1 and therefore might this wayes answere briefely3 ,other ,i am able to do you much good1 or els but litle4 ,if ,i bee able to doo you much good1 then be you not your owne friendes1 but greatlye your owne enemies to make so little of me1 which maye profite you so muche4 ,for if ,i were as vncurteous as you vnkind1 ,i shuld vtterly refuse to do them any good1 which will so curiously put me to the trial and profe of my commodities1 or els to suffre exile1 and namely sithe ,i shal only yeld benefites to other1 and receaue none againe4 ,but and if you could saye truely1 that my benefites be nother many nor yet greate1 yet if they bee anye1 ,i doo yelde more to you1 then ,i doo receaue againe of you1 and therefore ,i oughte not to bee repelled of them that loue them selfe1 althoughe they loue me not all for my selfe4 ,but as ,i am in nature a liberall science1 so canne ,i not againste nature contende with your inhumanitye1 but muste shewe my selfe liberall euen to myne enemies4 ,yet this is my comforte againe1 that ,i haue none enemies but them that knowe me not1 and therefore may hurte themselues1 but can not noye me4 ,yf they dispraise the thinge that they know not1 all wise men will blame them and not credite them1 and yf they thinke they knowe me1 lette theym shewe one vntruthe and erroure in me1 and ,i wyll geue the victorye4 ,yet can no humayne science saie thus1 but ,i onely1 that there is no sparke of vntruthe in me3 but all my doctrine and workes are without any blemishe of errour that mans reason can discerne4 ,and nexte vnto me in certaintie are my three systers1 ,arithmetike1 ,musike1 and ,astronomie1 whiche are also so nere knitte in amitee1 that he that loueth the one1 can not despise the other1 and in especiall ,geometrie1 of whiche not only these thre1 but all other artes do borow great ayde1 as partly hereafter shall be shewed4 ,but first will ,i beginne with the vnlearned sorte1 that you maie perceiue how that no arte can stand without me4 ,for if ,i should declare how many wayes my helpe is vsed1 in measuryng of ground1 for medow1 corne1 and wodde3 in hedgyng1 in dichyng1 and in stackes makyng1 ,i thinke the poore ,husband man would be more thankefull vnto me1 then he is nowe1 whyles he thinketh that he hath small benefite by me4 ,yet this maie he coniecture certainly1 that if he kepe not the rules of ,geometrie1 he can not measure any ground truely4 ,and in dichyng1 if he kepe not a proportion of bredth in the mouthe1 to the bredthe of the bottome1 and iuste slopenesse in the sides agreable to them bothe1 the diche shall be faultie many waies4 ,when he doth make stackes for corne1 or for heye1 he practiseth good ,geometrie1 els would thei not long stand3 ,so that in some stakes1 whiche stand on foure pillers1 and yet made round1 doe increase greatter and greatter a good height1 and then againe turne smaller and smaller vnto the toppe3 you maie see so good ,geometrie1 that it were very difficult to counterfaite the lyke in any kynde of buildyng4 ,as for other infinite waies that he vseth my benefite1 ,i ouerpasse for shortnesse4 ,carpenters1 ,karuers1 ,ioyners1 and ,masons1 doe willingly acknowledge that they can worke nothyng without reason of ,geometrie1 in so muche that they chalenge me as a peculiare science for them4 ,but in that they should do wrong to all other men1 seyng euerie kynde of men haue som benefit by me1 not only in buildyng1 whiche is but other mennes costes1 and the arte of ,carpenters1 ,masons1 and the other aforesayd1 but in their owne priuate profession1 whereof to auoide tediousnes ,i make this rehersall4 ,sith ,merchauntes by shippes great riches do winne1 ,i may with good righte at their seate beginne4 ,the ,shippes on the sea with ,saile and with ,ore1 were firste founde1 and styll made1 by ,geometries lore4 ,their ,compas1 their ,carde1 their ,pulleis1 their ,ankers1 were founde by the skill of witty ,geometers4 ,to sette forth the ,capstocke1 and eche other parte1 wold make a greate showe of ,geometries arte4 ,carpenters1 ,caruers1 ,ioiners and ,masons1 ,painters and ,limners with suche occupations1 ,broderers1 ,goldesmithes1 if they be cunning1 ,must yelde to ,geometrye thankes for their learning4 ,the ,carte and the ,plowe1 who doth them well marke1 ,are made by good ,geometrye4 ,and so in the warke ,of ,tailers and ,shoomakers1 in all shapes and fashion1 ,the woorke is not praised1 if it wante proportion4 ,so weauers by ,geometrye hade their foundacion1 ,their ,loome is a frame of straunge imaginacion4 ,the wheele that doth spinne1 the stone that doth grind1 ,the ,myll that is driuen by water or winde1 ,are workes of ,geometrye straunge in their trade1 ,fewe could them deuise1 if they were vnmade4 ,and all that is wrought by waight or by measure1 without proofe of ,geometry can neuer be sure4 ,clockes that be made the times to deuide1 ,the wittiest inuencion that euer was spied1 ,nowe that they are common they are not regarded1 ,the artes man contemned1 the woorke vnrewarded4 ,but if they were scarse1 and one for a shewe1 ,made by ,geometrye1 then shoulde men know1 ,that neuer was arte so wonderfull witty1 ,so needefull to man1 as is good ,geometry4 ,the firste findinge out of euery good arte1 ,seemed then vnto men so godly a parte1 ,that no recompence might satisfye the finder1 ,but to make him a god1 and honoure him for euer4 ,so ,ceres and ,pallas1 and ,mercury also1 ,eolus and ,neptune1 and many other mo1 ,were honoured as goddes1 bicause they did teache1 ,firste tillage and weuinge and eloquent speache1 ,or windes to obserue1 the seas to saile ouer1 ,they were called goddes for their good indeuour4 ,then were men more thankefull in that golden age3 ,this yron wolde nowe vngratefull in rage1 ,wyll yelde the thy reward for trauaile and paine1 ,with sclaunderous reproch1 and spitefull disdaine4 ,yet thoughe other men vnthankfull will be1 ,suruayers haue cause to make muche of me4 ,and so haue all ,lordes1 that landes do possesse3 ,but ,tennaunted ,i feare will like me the lesse4 ,yet do ,i not wrong but measure all truely1 ,all yelde the full right of euerye man iustely4 ,proportion ,geometricall hath no man opprest1 ,yf anye bee wronged1 ,i wishe it redrest4 ,but now to procede with learned professions1 in ,logike and ,rhetorike and all partes of phylosophy1 there neadeth none other proofe then ,aristotle his testimony1 whiche without ,geometry proueth almost nothinge4 ,in ,logike all his good syllogismes and demonstrations1 hee declareth by the principles of ,geometrye4 ,in philosophye1 nether motion1 nor time1 nor ayrye impressions could hee aptely declare1 but by the helpe of ,geometrye as his woorkes do witnes4 ,yea the faculties of the minde dothe hee expresse by similitude to figures of ,geometrye4 ,and in morall phylosophy he thought that iustice coulde not wel be taught1 nor yet well executed without proportion geometricall4 ,and this estimacion of ,geometry he maye seeme to haue learned of his maister ,plato1 which without ,geometrye wolde teache nothinge1 nother wold admitte any to heare him1 except he were experte in ,geometry4 ,and what merualle if he so muche estemed geometrye1 seinge his opinion was1 that ,godde was alwaies workinge by ,geometrie8 ,whiche sentence ,plutarche declareth at large4 ,and although ,platto do vse the helpe of ,geometrye in all the most waighte matter of a common wealth1 yet it is so generall in vse1 that no small thinges almost can be wel done without it4 ,and therfore saith he3 that ,geometrye is to be learned1 if it were for none other cause1 but that all other artes are bothe soner and more surely vnderstand by helpe of it4 ,what greate help it dothe in physike1 ,galene doth so often and so copiousely declare1 that no man whiche hath redde any booke almoste of his1 can be ignorant thereof1 in so much that he coulde neuer cure well a rounde vlcere1 tyll reason geometricall dydde teache it hym4 ,hippocrates is earnest in admonyshynge that study of geometrie must prepare the way to physike1 as well as to all other artes4 ,i shoulde seeme somewhat to tedious1 if ,i shoulde recken vp1 howe the diuines also in all their mysteries of scripture doo vse healpe of geometrie3 and also that lawyers can neuer vnderstande the hole lawe1 no nor yet the firste title therof exactly without ,geometrie4 ,for if lawes can not well be established1 nor iustice duelie executed without geometricall proportion1 as bothe ,plato in his ,politike bokes1 and ,aristotle in his ,moralles doo largely declare4 ,yea sithe ,lycurgus that cheefe lawmaker amongest the ,lacedemonians1 is moste praised for that he didde chaunge the state of their common wealthe frome the proportion ,arithmeticall to a proportion geometricall1 whiche without knowledg of bothe he coulde not dooe1 than is it easye to perceaue howe necessarie ,geometrie is for the lawe and studentes thereof4 ,and if ,i shall saie preciselie and freelie as ,i thinke1 he is vtterlie destitute of all abilitee to iudge in anie arte1 that is not sommewhat experte in the ,theoremes of ,geometrie4 ,and that caused ,galene to say of hym selfe1 that he coulde neuer perceaue what a demonstration was1 no not so muche1 as whether there were any or none1 tyll he had by geometrie gotten abilitee to vnderstande it1 although he heard the beste teachers that were in his tyme4 ,it shuld be to longe and nedelesse also to declare what helpe all other artes ,mathematicall haue by geometrie1 sith it is the grounde of all theyr certeintie1 and no man studious in them is so doubtful therof1 that he shall nede any persuasion to procure credite thereto4 ,for he can not reade 4ij4 lines almoste in any mathematicall science1 but he shall espie the nedefulnes of geometrie4 ,but to auoyde tediousnesse ,i will make an ende hereof with that famous sentence of auncient ,pythagoras1 ,that who so will trauayle by learnyng to attayne wysedome1 shall neuer approche to any excellencie without the artes mathematicall1 and especially ,arithmetike and ,geometrie4 ,and yf ,i shall somewhat speake of noble men1 and gouernours of realmes1 howe needefull ,geometrye maye bee vnto them1 then must ,i repete all that ,i haue sayde before1 sithe in them ought all knowledge to abounde1 namely that maye appertaine either to good gouernaunce in time of peace1 eyther wittye pollicies in time of warre4 ,for ministration of good lawes in time of peace ,lycurgus example with the testimonies of ,plato and ,aristotle may suffise4 ,and as for warres1 ,i might thinke it sufficient that ,vegetius hath written1 and after him ,valturius in commendation of ,geometry1 for vse of warres1 but all their woordes seeme to saye nothinge1 in comparison to the example of ,archimedes worthy woorkes make by geometrie1 for the defence of his countrey1 to reade the wonderfull praise of his wittie deuises1 set foorthe by the most famous hystories of ,liuius1 ,plutarche1 and ,plinie1 and all other hystoriographiers1 whyche wryte of the stronge siege of ,syracusae made by that valiant capitayne1 and noble warriour ,marcellus1 whose power was so great1 that all men meruayled how that one citee coulde withstande his wonderfull force so longe4 ,but much more woulde they meruaile1 if they vnderstode that one man onely dyd withstand all ,marcellus strength1 and with counter engines destroied his engines to the vtter astonyshment of ,marcellus1 and all that were with hym4 ,he had inuented suche balastelas that dyd shoote out a hundred dartes at one shotte1 to the great destruction of ,marcellus souldiours1 wherby a fonde tale was spredde abrode1 how that in ,syracusae there was a wonderfull gyant1 whiche had a hundred handes1 and coulde shoote a hundred dartes at ones4 ,and as this fable was spredde of ,archimedes1 so many other haue been fayned to bee gyantes and monsters1 bycause they dyd suche thynges1 whiche farre passed the witte of the common people4 ,so dyd they feyne ,argus to haue a hundred eies1 bicause they herde of his wonderfull circumspection1 and thoughte that as it was aboue their capacitee1 so it could not be1 onlesse he had a hundred eies4 ,so imagined they ,ianus to haue two faces1 one lokyng forwarde1 and an other backwarde1 bycause he coulde so wittily compare thynges paste with thynges that were to come1 and so duely pondre them1 as yf they were all present4 ,of like reason did they feyn ,lynceus to haue such sharp syght1 that he could see through walles and hylles1 bycause peraduenture he dyd by naturall iudgement declare what commoditees myght be digged out of the grounde4 ,and an infinite noumbre lyke fables are there1 whiche sprange all of lyke reason4 ,for what other thyng meaneth the fable of the great gyant ,atlas1 whiche was ymagined to beare vp heauen on his shulders8 but that he was a man of so high a witte1 that it reached vnto the skye1 and was so skylfull in ,astronomie1 and coulde tell before hande of ,eclipses1 and other like thynges as truely as though he dyd rule the sterres1 and gouerne the planettes4 ,so was ,eolus accompted god of the wyndes1 and to haue theim all in a caue at his pleasure1 by reason that he was a wittie man in naturall knowlege1 and obserued well the change of wethers1 aud was the fyrst that taught the obseruation of the wyndes4 ,and lyke reson is to be geuen of al the old fables4 ,but to retourne agayne to ,archimedes1 he dyd also by arte perspectiue 7whiche is a parte of geometrie7 deuise such glasses within the towne of ,syracusae1 that dyd bourne their ennemies shyppes a great way from the towne1 whyche was a meruaylous politike thynge4 ,and if ,i shulde repete the varietees of suche straunge inuentions1 as ,archimedes and others haue wrought by geometrie1 ,i should not onely excede the order of a preface1 but ,i should also speake of suche thynges as can not well be vnderstande in talke1 without somme knowledge in the principles of geometrie4 ,but this will ,i promyse1 that if ,i may perceaue my paynes to be thankfully taken1 ,i wyll not onely write of suche pleasant inuentions1 declaryng what they were1 but also wil teache howe a great numbre of them were wroughte1 that they may be practised in this tyme also4 ,wherby shallbe plainly perceaued1 that many thynges seme impossible to be done1 whiche by arte may very well be wrought4 ,and whan they be wrought1 and the reason therof not vnderstande1 than say the vulgare people1 that those thynges are done by negromancy4 ,and hereof came it that fryer ,bakon was accompted so greate a negromancier1 whiche neuer vsed that arte 7by any coniecture that ,i can fynde7 but was in geometrie and other mathematicall sciences so experte1 that he coulde dooe by theim suche thynges as were wonderfull in the syght of most people4 ,great talke there is of a glasse that he made in ,oxforde1 in whiche men myght see thynges that were doon in other places1 and that was iudged to be done by power of euyll spirites4 ,but ,i knowe the reason of it to bee good and naturall1 and to be wrought by geometrie 7sythe perspectiue is a parte of it7 and to stande as well with reason as to see your face in common glasse4 ,but this conclusion and other dyuers of lyke sorte1 are more mete for princes1 for sundry causes1 than for other men1 and ought not to bee taught commonly4 ,yet to repete it1 ,i thought good for this cause1 that the worthynes of geometry myght the better be knowen1 partly vnderstanding geuen1 what wonderfull thynges may be wrought by it1 and so consequently how pleasant it is1 and how necessary also4 ,and thus for this tyme ,i make an end4 ,the reason of som thynges done in this boke1 or omitted in the same1 you shall fynde in the preface before the ,theoremes4 ,the definitions of the principles of ,,geometry4 ,geometry teacheth the drawyng1 ,measuring and proporcion of figures4 but in as muche as no figure can bee drawen1 but it muste haue certayne boundes and inclosures of lines3 and euery lyne also is begon and ended at some certaine prycke1 fyrst it shal be meete to know these smaller partes of euery figure1 that therby the whole figures may the better bee iudged1 and distincte in sonder4 ,sidenote3 ,a poincte4 ,a ,poynt or a ,prycke1 is named of ,geometricians that small and vnsensible shape1 whiche hath in it no partes1 that is to say3 nother length1 breadth nor depth4 ,but as their exactnes of definition is more meeter for onlye ,theorike speculacion1 then for practise and outwarde worke 7consideringe that myne intent is to applye all these whole principles to woorke7 ,i thynke meeter for this purpose1 to call a poynt or prycke1 that small printe of penne1 pencyle1 or other instrumente1 whiche is not moued1 nor drawen from his fyrst touche1 and therfore hath no notable length nor bredthe3 as this example doeth declare4 ,illustration3 therefore symbol ,where ,i haue set 4iij4 prickes1 eche of them hauyng both length and bredth1 thogh it be but smal1 and thefore not notable4 ,nowe of a great numbre of these prickes1 is made a ,lyne1 as you may perceiue by this forme ensuyng4 444444444444444444444444 where as ,i haue set a numbre of prickes1 so if you with your pen will set in more other prickes betweene euerye two of these1 ,sidenote3 ,a lyne4 then wil it be a lyne1 as here you may see -------- and this lyne1 is called of ,geometricians1 ,lengthe withoute breadth4 ,but as they in theyr theorikes 7which ar only mind workes7 do precisely vnderstand these definitions1 so it shal be sufficient for those men1 whiche seke the vse of the same thinges1 as sense may duely iudge them1 and applye to handy workes if they vnderstand them so to be true1 that outwarde sense canne fynde none erroure therein4 ,of lynes there bee two principall kyndes4 ,the one is called a right or straight lyne1 and the other a croked lyne4 ,sidenote3 ,a streghte lyne4 ,a ,straight lyne1 is the shortest that maye be drawenne between two prickes4 ,sidenote3 ,a crokyd lyne4 ,and all other lines1 that go not right forth from prick to prick1 but boweth any waye1 such are called ,croked lynes as in these examples folowyng ye may se1 where ,i haue set but one forme of a straight lyne1 for more formes there be not1 but of crooked lynes there bee innumerable diuersities1 whereof for examples sum ,i haue sette here4 ,illustration3 ,a right lyne4 ,illustration3 ,croked lynes4 ,illustration3 ,croked lines4 ,so now you must vnderstand1 that euery lyne is drawen betwene twoo prickes1 wherof the one is at the beginning1 and the other at the ende4 ,illustration ,therefore when soeuer you do see any formes of lynes to touche at one notable pricke1 as in this example1 then shall you not call it one croked lyne1 but rather twoo lynes3 ,sidenote3 an ,angle4 in as muche as there is a notable and sensible angle by 4,a4 whiche euermore is made by the meetyng of two seuerall lynes4 ,and likewayes shall you iudge of this figure1 whiche is made of two lines1 and not of one onely4 ,illustration ,so that whan so euer any suche meetyng of lines doth happen1 the place of their metyng is called an ,angle or corner4 ,of angles there be three generall kindes3 a sharpe angle1 a square angle1 and a blunte angle4 ,sidenote3 ,a righte angle4 ,the square angle1 whiche is commonly named a right corner1 is made of twoo lynes meetyng together in fourme of a squire1 whiche two lines1 if they be drawen forth in length1 will crosse one an other3 as in the examples folowyng you maie see4 ,sidenote3 ,a sharpe corner4 ,a sharpe angle is so called1 because it is lesser than is a square angle1 and the lines that make it1 do not open so wide in their departynge as in a square corner1 and if thei be drawen crosse1 all fower corners will not be equall4 ,sidenote3 ,a blunte angle4 ,a blunt or brode corner1 is greater then is a square angle1 and his lines do parte more in sonder then in a right angle1 of whiche all take these examples4 ,illustration3 ,right angles4 ,illustration3 ,sharpe angles4 ,illustration3 ,blunte or brode angles4 ,and these angles 7as you see7 are made partly of streght lynes1 partly of croken lines1 and partly of both together4 ,howbeit in right angles ,i haue put none example of croked lines1 because it would muche trouble a lerner to iudge them3 for their true iudgment doth appertaine to arte perspectiue1 and as ,i may say1 rather to reason then to sense4 ,but now as of many prickes there is made one line1 so of diuerse lines are there made sundry formes1 figures1 and shapes1 whiche all yet be called by one propre name1 ,sidenote3 ,a platte forme4 ,platte formes1 and thei haue bothe length and bredth1 but yet no depenesse4 ,and the boundes of euerie platte forme are lines3 as by the examples you maie perceiue4 ,of platte formes some be plain1 and some be croked1 and some parly plaine1 and partlie croked4 ,sidenote3 ,a plaine platte4 ,a plaine platte is that1 whiche is made al equall in height1 so that the middle partes nother bulke vp1 nother shrink down more then the bothe endes4 ,sidenote3 ,a crooked platte4 ,for whan the one parte is higher then the other1 then is it named a ,croked platte4 ,and if it be partlie plaine1 and partlie crooked1 then is it called a ,myxte platte1 of all whiche1 these are exaumples4 ,illustration3 ,a plaine platte4 ,illustration3 ,a croked platte4 ,illustration3 ,a myxte platte4 ,and as of many prickes is made a line1 and of diuerse lines one platte forme1 ,sidenote3 ,a bodie4 so of manie plattes is made a bodie1 whiche conteigneth ,lengthe1 bredth1 and depenesse4 ,sidenote3 ,depenesse4 ,by ,depenesse ,i vnderstand1 not as the common sort doth1 the holownesse of any thing1 as of a well1 a diche1 a potte1 and suche like1 but ,i meane the massie thicknesse of any bodie1 as in exaumple of a potte3 the depenesse is after the common name1 the space from his brimme to his bottome4 ,but as ,i take it here1 the depenesse of his bodie is his thicknesse in the sides1 whiche is an other thyng cleane different from the depenesse of his holownes1 that the common people meaneth4 ,now all bodies haue platte formes for their boundes1 ,sidenote3 ,cubike4 so in a dye 7whiche is called a cubike bodie7 by geomatricians1 ,sidenote3 ,asheler4 and an ashler of masons1 there are 4vi4 sides1 whiche are 4vi4 platte formes1 and are the boundes of the dye4 ,sidenote3 ,a globe4 ,but in a ,globe1 7whiche is a bodie rounde as a bowle7 there is but one platte forme1 and one bounde1 and these are the exaumples of them bothe4 ,illustration3 ,a dye or ashler4 ,illustration3 ,a globe4 ,but because you shall not muse what ,i dooe call a bound1 ,sidenote3 ,a bounde4 ,i mean therby a generall name1 betokening the beginning1 end and side1 of any forme4 ,sidenote3 ,forme1 ,fygure4 ,a forme1 figure1 or shape1 is that thyng that is inclosed within one bond or manie bondes1 so that you vnderstand that shape1 that the eye doth discerne1 and not the substance of the bodie4 ,of figures there be manie sortes1 for either thei be made of prickes1 lines1 or platte formes4 ,not withstandyng to speake properlie1 a figure is euer made by platte formes1 and not of bare lines vnclosed1 neither yet of prickes4 ,yet for the lighter forme of teachyng1 it shall not be vnsemely to call all suche shapes1 formes and figures1 whiche ye eye maie discerne distinctly4 ,and first to begin with prickes1 there maie be made diuerse formes of them1 as partely here doeth folowe4 ,illustration3 ,a lynearic numbre4 ,trianguler numbres ,longsquare numbre4 ,iust square numbres a threcornered spire4 ,a square spire4 ,and so maie there be infinite formes more1 whiche ,i omitte for this time1 considering that their knowledg appertaineth more to ,arithmetike figurall1 than to ,geometrie4 ,but yet one name of a pricke1 whiche he taketh rather of his place then of his fourme1 maie ,i not ouerpasse4 ,and that is1 when a pricke standeth in the middell of a circle 7as no circle can be made by compasse without it7 then is it called a centre4 ,sidenote3 ,a centre ,and thereof doe masons1 and other worke menne call that patron1 a centre1 whereby thei drawe the lines1 for iust hewyng of stones for arches1 vaultes1 and chimneies1 because the chefe vse of that patron is wrought by findyng that pricke or centre1 from whiche all the lynes are drawen1 as in the thirde booke it doeth appere4 ,lynes make diuerse figures also1 though properly thei maie not be called figures1 as ,i said before 7vnles the lines do close7 but onely for easie maner of teachyng1 all shall be called figures1 that the eye can discerne1 of whiche this is one1 when one line lyeth flatte 7whiche is named ,sidenote3 ,a ground line4 the ground line7 and an other commeth downe on it1 and is called ,sidenote3 ,a perpendicular4 ,sidenote3 ,a plume lyne4 a perpendiculer or plumme lyne1 as in this example you may see4 where 4,a4,b4 is the grounde line1 and ,c4,d4 the plumbe line4 ,illustration ,and like waies in this figure there are three lines1 the grounde lyne whiche is ,a4,b4 the plumme line that is ,a4,c4 and the bias line1 whiche goeth from the one of them to the other1 and lieth against the right corner in such a figure whiche is here 4,c4,b4 ,illustration ,but consideryng that ,i shall haue occasion to declare sundry figures anon1 ,i will first shew some certaine varietees of lines that close no figures1 but are bare lynes1 and of the other lines will ,i make mencion in the description of the figures4 ,illustration3 tortuouse paralleles4 ,sidenote3 ,parallelys ,sidenote3 ,gemowe lynes4 ,paralleles1 or gemowe lynes be suche lines as be drawen foorth still in one distaunce1 and are no nerer in one place then in an other1 for and if they be nerer at one ende then at the other1 then are they no paralleles1 but maie bee called bought lynes1 and loe here exaumples of them bothe4 ,illustration3 parallelis4 ,illustration3 bought lines ,illustration3 parallelis3 circular4 ,concentrikes4 ,i haue added also paralleles tortuouse1 whiche bowe contrarie waies with their two endes3 and paralleles circular1 whiche be lyke vnperfecte compasses3 for if they bee whole circles1 ,sidenote3 ,concentrikes then are they called concentrikes1 that is to saie1 circles drawen on one centre4 ,here might ,i note the error of good ,albert ,durer1 which affirmeth that no perpendicular lines can be paralleles4 which errour doeth spring partlie of ouersight of the difference of a streight line1 and partlie of mistakyng certain principles geometrical1 which al ,i wil let passe vntil an other tyme1 and wil not blame him1 which hath deserued worthyly infinite praise4 ,and to returne to my matter4 ,sidenote3 ,a twine line4 an other fashioned line is there1 which is named a twine or twist line1 and it goeth as a wreyth about some other bodie4 ,sidenote3 ,a spirall line4 ,and an other sorte of lines is there1 that is called a spirall line1 ,sidenote3 ,a worme line4 or a worm line1 whiche representeth an apparant forme of many circles1 where there is not one in dede3 of these 4ii4 kindes of lines1 these be examples4 ,illustration3 ,a twiste lyne4 ,illustration3 ,a spirail lyne ,illustration3 ,a touche lyne4 ,sidenote3 ,a tuch line4 ,a touche lyne1 is a line that runneth a long by the edge of a circle1 onely touching it1 but doth not crosse the circumference of it1 as in this exaumple you maie see4 ,sidenote3 ,a corde1 ,and when that a line doth crosse the edg of the circle1 then is it called a cord1 as you shall see anon in the speakynge of circles4 ,sidenote3 ,matche corners ,in the meane season must ,i not omit to declare what angles bee called matche corners1 that is to saie1 suche as stande directly one against the other1 when twoo lines be drawen a crosse1 as here appereth4 ,illustration3 ,matche corner4 ,matche corner4 ,where ,a4 and ,b4 are matche corners1 so are ,c4 and ,d4 but not ,a4 and ,c4 nother ,d4 and ,a4 ,nowe will ,i beginne to speak of figures1 that be properly so called1 of whiche all be made of diuerse lines1 except onely a circle1 an egge forme1 and a tunne forme1 which 4iij4 haue no angle and haue but one line for their bounde1 and an eye fourme whiche is made of one lyne1 and hath an angle onely4 ,sidenote3 ,a circle4 ,a circle is a figure made and enclosed with one line1 and hath in the middell of it a pricke or centre1 from whiche all the lines that be drawen to the circumference are equall all in length1 as here you see4 ,illustration ,sidenote3 ,circumference4 ,and the line that encloseth the whole compasse1 is called the circumference4 ,sidenote3 ,a diameter4 ,and all the lines that bee drawen crosse the circle1 and goe by the centre1 are named diameters1 whose halfe1 ,i meane from the center to the circumference any waie1 ,sidenote3 ,semidiameter4 is called the semidiameter1 or halfe diameter4 ,illustration ,but and if the line goe crosse the circle1 and passe beside the centre1 ,sidenote3 ,a cord1 or a stringlyne4 then is it called a corde1 or a stryng line1 as ,i said before1 and as this exaumple sheweth3 where ,a4 is the corde4 ,and the compassed line that aunswereth to it1 ,sidenote3 ,an archline is called an arche lyne1 ,sidenote3 ,a bowline4 or a bowe lyne1 whiche here marked with ,b4 and the diameter with ,c4 ,illustration ,but and if that part be separate from the rest of the circle 7as in this example you see7 then ar both partes called cantelles1 ,sidenote3 ,a cantle the one the greatter cantle as ,e4 and the other the lesser cantle1 as ,d4 ,and if it be parted iuste by the centre 7as you see in ,f47 ,sidenote3 ,a semyecircle then is it called a semicircle1 or halfe compasse4 ,illustration ,sometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference 7as ,g4 is7 ,sidenote3 ,a nooke cantle and then maie it be called a nooke cantle1 and if it be not parted from the reste of the circle 7as you see in ,h47 ,sidenote3 ,a nooke4 then is it called a nooke plainely without any addicion4 ,and the compassed lyne in it is called an arche lyne1 as the exaumple here doeth shewe4 ,illustration3 ,an arche4 ,nowe haue you heard as touchyng circles1 meetely sufficient instruction1 so that it should seme nedeles to speake any more of figures in that kynde1 saue that there doeth yet remaine ij4 formes of an imperfecte circle1 for it is lyke a circle that were brused1 and thereby did runne out endelong one waie1 whiche forme ,geometricians dooe call an ,sidenote3 ,an egge fourme4 egge forme1 because it doeth represent the figure and shape of an egge duely proportioned 7as this figure sheweth7 hauyng the one ende greate then the other4 ,illustration3 ,an egge forme ,illustration3 ,a tunne forme4 ,sidenote3 ,a tunne or barrel form ,for if it be lyke the figure of a circle pressed in length1 and bothe endes lyke bygge1 then is it called a tunne forme1 or barrell forme1 the right makyng of whiche figures1 ,i wyll declare hereafter in the thirde booke4 ,an other forme there is1 whiche you maie call a nutte forme1 and is made of one lyne muche lyke an egge forme1 saue that it hath a sharpe angle4 ,and it chaunceth sometyme that there is a right line drawen crosse these figures1 ,sidenote3 ,an axtre or axe lyne4 and that is called an axelyne1 or axtre4 ,howe be it properly that line that is called an axtre1 whiche gooeth throughe the myddell of a ,globe1 for as a diameter is in a circle1 so is an axe lyne or axtre in a ,globe1 that lyne that goeth from side to syde1 and passeth by the middell of it4 ,and the two poyntes that suche a lyne maketh in the vtter bounde or platte of the globe1 are named polis1 wch you may call aptly in englysh1 tourne pointes3 of whiche ,i do more largely intreate1 in the booke that ,i haue written of the vse of the globe4 ,illustration ,but to returne to the diuersityes of figures that remayne vndeclared1 the most simple of them ar such ones as be made but of two lynes1 as are the cantle of a circle1 and the halfe circle1 of which ,i haue spoken allready4 ,likewyse the halfe of an egge forme1 the cantle of an egge forme1 the halfe of a tunne fourme1 and the cantle of a tunne fourme1 and besyde these a figure moche like to a tunne fourne1 saue that it is sharp couered at both the endes1 and therfore doth consist of twoo lynes1 where a tunne forme is made of one lyne1 ,sidenote3 ,an yey fourme and that figure is named an yey fourme4 ,illustration ,sidenote3 ,a triangle ,the nexte kynd of figures are those that be made of 4iij4 lynes other be all right lynes1 all crooked lynes1 other some right and some crooked4 ,but what fourme so euer they be of1 they are named generally triangles4 for a triangle is nothinge els to say1 but a figure of three corners4 ,and thys is a generall rule1 looke how many lynes any figure hath1 so mannye corners it hath also1 yf it bee a platte forme1 and not a bodye4 ,for a bodye hath dyuers lynes metyng sometime in one corner4 ,illustration3 ,a ,now to geue you example of triangles1 there is one whiche is all of croked lynes1 and may be taken fur a portion of a globe as the figur marked wt ,a4 ,illustration3 ,b ,an other hath two compassed lines and one right lyne1 and is as the portion of halfe a globe1 example of ,b4 ,illustration3 ,c ,an other hath but one compassed lyne1 and is the quarter of a circle1 named a quadrate1 and the ryght lynes make a right corner1 as you se in ,c4 ,otherlesse then it as you se ,d1 whose right lines make a sharpe corner1 or greater then a quadrate1 as is ,f1 and then the right lynes of it do make a blunt corner4 ,illustration3 ,d ,also some triangles haue all righte lynes and they be distincted in sonder by their angles1 or corners4 for other their corners bee all sharpe1 as you see in the figure1 ,e4 other ij4 sharpe and one blunt1 as is the figure ,g4 other ij4 sharp and one blunt as in the figure ,h4 ,illustration3 ,e ,illustration3 ,f ,there is also an other distinction of the names of triangles1 according to their sides1 whiche other be all equal as in the figure ,e1 and that the ,greekes doo call ,isopleuron1 ,sidenote3 isopleurom4 and ,latine men aeequilaterum3 and in english it may be called a threlike triangle1 other els two sydes bee equall and the thyrd vnequall1 which the ,greekes call ,isosceles1 ,sidenote3 isoskeles4 the ,latine men aequicurio1 and in english tweyleke may they be called1 as in ,g1 ,h1 and ,k4 ,for1 they may be of iij4 kinds that is to say1 with one square angle1 as is ,g1 or with a blunte corner as ,h1 or with all in sharpe korners1 as you see in ,k4 ,illustration3 ,g ,illustration3 ,h ,illustration3 ,k ,further more it may be yt they haue neuer a one syde equall to an other1 and they be in iij kyndes also distinct lyke the twilekes1 as you maye perceaue by these examples 4,m4 ,n1 and ,o4 where ,m4 hath a right angle1 ,n1 a blunte angle1 and ,o1 all sharpe angles ,sidenote3 skalenom4 these the ,greekes and latine men do cal scalena and in englishe theye may be called nouelekes1 for thei haue no side equall1 or like long1 to ani other in the same figur4 ,here it is to be noted1 that in a triangle al the angles bee called innerangles except ani side bee drawenne forth in lengthe1 for then is that fourthe corner caled an vtter corner1 as in this example because ,a4,b1 is drawen in length1 therfore the angle ,c1 is called an vtter angle4 ,illustration3 ,m ,illustration3 ,n ,illustration3 ,o ,illustration ,illustration3 ,q ,sidenote3 ,quadrangle ,and thus haue ,i done with trianguled figures1 and nowe foloweth quadrangles1 which are figures of iiij4 corners and of iiij4 lines also1 of whiche there be diuers kindes1 but chiefely v4 that is to say1 ,sidenote3 ,a square quadrate4 a square quadrate1 whose sides bee all equall1 and al the angles square1 as you se here in this figure ,q4 ,sidenote3 ,a longe square4 ,the second kind is called a long square1 whose foure corners be all square1 but the sides are not equall eche to other1 yet is euery side equall to that other that is against it1 as you maye perceaue in this figure4 ,r4 ,illustration3 ,r ,sidenote3 ,a losenge ,the thyrd kind is called losenges ,sidenote3 ,a diamond4 or diamondes1 whose sides bee all equall1 but it hath neuer a square corner1 for two of them be sharpe1 and the other two be blunt1 as appeareth in 4,s4 ,illustration3 ,s ,the iiij4 sorte are like vnto losenges1 saue that they are longer one waye1 and their sides be not equal1 yet ther corners are like the corners of a losing1 and therfore ar they named ,sidenote3 ,a losenge lyke4 losengelike or diamondlike1 whose figur is noted with ,t4 ,here shal you marke that al those squares which haue their sides al equal1 may be called also for easy vnderstandinge1 likesides1 as ,q4 and ,s4 and those that haue only the contrary sydes equal1 as ,r4 and ,t4 haue1 those wyll ,i call likeiammys1 for a difference4 ,illustration3 ,t ,illustration ,the fift sorte doth containe all other fashions of foure cornered figurs1 and ar called of the ,grekes trapezia1 of ,latin men mensulae and of ,arabitians1 helmuariphe1 they may be called in englishe borde formes1 ,sidenote3 ,borde formes4 they haue no syde equall to an other as these examples shew1 neither keepe they any rate in their corners1 and therfore are they counted vnruled formes1 and the other foure kindes onely are counted ruled formes1 in the kynde of quadrangles4 ,of these vnruled formes ther is no numbre1 they are so mannye and so dyuers1 yet by arte they may be changed into other kindes of figures1 and therby be brought to measure and proportion1 as in the thirtene conclusion is partly taught1 but more plainly in my booke of measuring you may see it4 ,and nowe to make an eande of the dyuers kyndes of figures1 there dothe folowe now figures of 4v4 sydes1 other 4v4 corners1 which we may call cink-angles1 whose sydes partlye are all equall as in ,a1 and those are counted ruled cinkeangles1 and partlye vnequall1 as in ,b1 and they are called vnruled4 ,illustration3 ,a ,illustration3 ,b ,likewyse shall you iudge of siseangles1 which haue sixe corners1 septangles1 whiche haue seuen angles1 and so forth1 for as mannye numbres as there maye be of sydes and angles1 so manye diuers kindes be there of figures1 vnto which yow shall geue names according to the numbre of their sides and angles1 of whiche for this tyme ,i wyll make an ende1 ,sidenote3 ,a squyre4 and wyll sette forthe on example of a syseangle1 whiche ,i had almost forgotten1 and that is it1 whose vse commeth often in ,geometry1 and is called a squire1 is made of two long squares ioyned togither1 as this example sheweth4 ,illustration ,and thus ,i make an eand to speake of platte formes1 and will briefelye saye somwhat touching the figures of bodeis which partly haue one platte forme for their bound1 and yt iust round as a globe hath1 or ended long as in an egge1 and a tunne fourme1 whose pictures are these4 ,illustration3 ,the globe as is before4 ,howe be it you must marke that ,i meane not the very figure of a tunne1 when ,i saye tunne form1 but a figure like a tunne1 for a tune fourme1 hath but one plat forme1 and therfore must needs be round at the endes1 where as a tunne hath thre platte formes1 and is flatte at eche end1 as partly these pictures do shewe4 ,bodies of two plattes1 are other cantles or halues of those other bodies1 that haue but one platte forme1 or els they are lyke in foorme to two such cantles ioyned togither as this ,a4 doth partly eppresse3 or els it is called a rounde spire1 or stiple fourme1 as in this figure is some what expressed4 ,sidenote3 ,a rounde spier4 ,nowe of three plattes there are made certain figures of bodyes1 as the cantels and halues of all bodyes that haue but ij4 plattys1 and also the halues of halfe globys and canteles of a globe4 ,lykewyse a rounde piller1 and a spyre made of a rounde spyre1 slytte in ij4 partes long ways4 ,but as these formes be harde to be iudged by their pycturs1 so ,i doe entende to passe them ouer with a great number of other formes of bodyes1 which afterwarde shall be set forth in the boke of ,perspectiue1 bicause that without perspectiue knowledge1 it is not easy to iudge truly the formes of them in flatte protacture4 ,and thus ,i made an ende for this tyme1 of the definitions ,geometricall1 appertayning to this parte of practise1 and the rest wil ,i prosecute as cause shall serue4 ,,the ,,practike ,,workinge ,,of sondry conclusions geometrical4 ,,the ,,fyrst ,,conclvsion4 ,to make a threlike triangle on any lyne measurable4 ,take the iuste length of the lyne with your compasse1 and stay the one foot of the compas in one of the endes of that line1 turning the other vp or doun at your will1 drawyng the arche of a circle against the midle of the line1 and doo like wise with the same compasse vnaltered1 at the other end of the line1 and wher these ij4 croked lynes doth crosse1 frome thence drawe a lyne to ech end of your first line1 and there shall appear a threlike triangle drawen on that line4 ,illustration ,example4 ,a4,b4 is the first line1 on which ,i wold make the threlike triangle1 therfore ,i open the compasse as wyde as that line is long1 and draw two arch lines that mete in ,c1 then from ,c1 ,i draw ij other lines one to ,a1 another to ,b1 and than ,i haue my purpose4 ,,the 4,,ii4 ,,conclvsion ,if you wil make a twileke or a nouelike triangle on ani certaine line4 ,illustration ,consider fyrst the length that yow will haue the other sides to containe1 and to that length open your compasse1 and then worke as you did in the threleke triangle1 remembryng this1 that in a nouelike triangle you must take ij4 lengthes besyde the fyrste lyne1 and draw an arche lyne with one of them at the one ende1 and with the other at the other end1 the example is as in the other before4 ,illustration ,,the ,,iii4 ,,concl4 ,to diuide an angle of right lines into ij4 equal partes4 ,first open your compasse as largely as you can1 so that it do not excede the length of the shortest line yt incloseth the angle4 ,then set one foote of the compasse in the verye point of the angle1 and with the other fote draw a compassed arch from the one lyne of the angle to the other1 that arch shall you deuide in halfe1 and then draw a line from the angle to ye middle of ye arch1 and so ye angle is diuided into ij4 equall partes4 ,illustration ,example4 ,let the triangle be ,a4,b4,c1 then set ,i one foot of ye compasse in ,b1 and with the other ,i draw ye arch ,d4,e1 which ,i part into ij4 equall parts in ,f1 and then draw a line from ,b1 to ,f1 so ,i haue mine intent4 ,,the ,,iiii4 ,,concl4 ,to deuide any measurable line into ij4 equall partes4 ,illustration ,open your compasse to the iust length of ye line4 ,and then set one foote steddely at the one ende of the line1 wt the other fote draw an arch of a circle against ye midle of the line1 both ouer it1 and also vnder it1 then doo lykewaise at the other ende of the line4 ,and marke where those arche lines do meet crosse waies1 and betwene those ij4 pricks draw a line1 and it shall cut the first line in two equall portions4 ,example4 ,the lyne is ,a4,b4 accordyng to which ,i open the compasse and make 4iiij4 arche lines1 whiche meete in ,c4 and ,d1 then drawe ,i a lyne from ,c1 so haue ,i my purpose4 ,this conlusion serueth for makyng of quadrates and squires1 beside many other commodities1 howebeit it maye bee don more readylye by this conclusion that foloweth nexte4 ,,the ,,fift ,,conclvsion4 ,to make a plumme line or any pricke that you will in any right lyne appointed4 ,open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line1 but rather shorter4 ,then sette the one foote of the compasse in the first pricke appointed1 and with the other fote marke ij4 other prickes1 one of eche syde of that fyrste1 afterwarde open your compasse to the wydenes of those ij4 new prickes1 and draw from them ij4 arch lynes1 as you did in the fyrst conclusion1 for making of a threlyke triangle4 then if you do mark their crossing1 and from it drawe a line to your fyrste pricke1 it shall bee a iust plum lyne on that place4 ,illustration ,example4 ,the lyne is ,a4,b4 the prick on whiche ,i shoulde make the plumme lyne1 is ,c4 then open ,i the compasse as wyde as ,a4,c1 and sette one foot in ,c4 and with the other doo ,i marke out ,c4,a4 and ,c4,b1 then open ,i the compasse as wide as ,a4,b1 and make ij4 arch lines which do crosse in ,d1 and so haue ,i doone4 ,howe bee it1 it happeneth so sommetymes1 that the pricke on whiche you would make the perpendicular or plum line1 is so nere the eand of your line1 that you can not extende any notable length from it to thone end of the line1 and if so be it then that you maie not drawe your line lenger from that end1 then doth this conclusion require a newe ayde1 for the last deuise will not serue4 ,in suche case therfore shall you dooe thus3 ,if your line be of any notable length1 deuide it into fiue partes4 ,and if it be not so long that it maie yelde fiue notable partes1 then make an other line at will1 and parte it into fiue equall portions3 so that thre of those partes maie be found in your line4 ,then open your compas as wide as thre of these fiue measures be1 and sette the one foote of the compas in the pricke1 where you would haue the plumme line to lighte 7whiche ,i call the first pricke17 and with the other foote drawe an arche line righte ouer the pricke1 as you can ayme it3 then open youre compas as wide as all fiue measures be1 and set the one foote in the fourth pricke1 and with the other foote draw an other arch line crosse the first1 and where thei two do crosse1 thense draw a line to the poinct where you woulde haue the perpendicular line to light1 and you haue doone4 ,example4 ,illustration ,the line is ,a4,b4 and ,a4 is the prick1 on whiche the perpendicular line must light4 ,therfore ,i deuide ,a4,b4 into fiue partes equall1 then do ,i open the compas to the widenesse of three partes 7that is ,a4,d47 and let one foote staie in ,a4 and with the other ,i make an arche line in ,c4 ,afterwarde ,i open the compas as wide as ,a4,b4 7that is as wide as all fiue partes7 and set one foote in the 4iiij4 pricke1 which is ,e1 drawyng an arch line with the other foote in ,c4 also4 ,then do ,i draw thence a line vnto ,a1 and so haue ,i doone4 ,but and if the line be to shorte to be parted into fiue partes1 ,i shall deuide it into iij4 partes only1 as you see the liue ,f4,g1 and then make ,d4 an other line 7as is ,k4,l47 whiche ,i deuide into 4v4 suche diuisions1 as ,f4,g4 containeth 4iij1 then open ,i the compass as wide as 4iiij4 partes 7whiche is ,k4,m47 and so set ,i one foote of the compas in ,f1 and with the other ,i drawe an arch lyne toward ,h1 then open ,i the compas as wide as ,k4,l4 7that is all 4v4 partes7 and set one foote in ,g1 7that is the iij4 pricke7 and with the other ,i draw an arch line toward ,h4 also3 and where those 4ij4 arch lines do crosse 7whiche is by ,h47 thence draw ,i a line vnto ,f1 and that maketh a very plumbe line to ,f4,g1 as my desire was4 ,the maner of workyng of this conclusion1 is like to the second conlusion1 but the reason of it doth depend of the 4xlvi4 proposicion of ye first boke of ,euclide4 ,an other waie yet4 set one foote of the compas in the prick1 on whiche you would haue the plumbe line to light1 and stretche forth thother foote toward the longest end of the line1 as wide as you can for the length of the line1 and so draw a quarter of a compas or more1 then without stirryng of the compas1 set one foote of it in the same line1 where as the circular line did begin1 and extend thother in the circular line1 settyng a marke where it doth light1 then take half that quantitie more there vnto1 and by that prick that endeth the last part1 draw a line to the pricke assigned1 and it shall be a perpendicular4 ,illustration ,example4 ,a4,b4 is the line appointed1 to whiche ,i must make a perpendicular line to light in the pricke assigned1 which is ,a4 ,therfore doo ,i set one foote of the compas in ,a1 and extend the other vnto ,d4 makyng a part of a circle1 more then a quarter1 that is ,d4,e4 ,then do ,i set one foote of the compas vnaltered in ,d1 and stretch the other in the circular line1 and it doth light in ,f1 this space betwene ,d4 and ,f4 ,i deuide into halfe in the pricke ,g1 whiche halfe ,i take with the compas1 and set it beyond ,f4 vnto ,h1 and thefore is ,h4 the point1 by whiche the perpendicular line must be drawn1 so say ,i that the line ,h4,a1 is a plumbe line to ,a4,b1 as the conclusion would4 ,,the 4,,vi4 ,,conclvsion4 ,to drawe a streight line from any pricke that is not in a line1 and to make it perpendicular to an other line4 ,open your compas as so wide that it may extend somewhat farther1 then from the prick to the line1 then sette the one foote of the compas in the pricke1 and with the other shall you draw a compassed line1 that shall crosse that other first line in 4ij4 places4 ,now if you deuide that arch line into 4ij4 equall partes1 and from the middell pricke therof vnto the prick without the line you drawe a streight line1 it shalbe a plumbe line to that firste lyne1 accordyng to the conclusion4 ,illustration ,example4 ,c4 is the appointed pricke1 from whiche vnto the line ,a4,b4 ,i must draw a perpendicular4 ,thefore ,i open the compas so wide1 that it may haue one foote in ,c1 and thother to reach ouer the line1 and with yt foote ,i draw an arch line as you see1 betwene ,a4 and ,b1 which arch line ,i deuide in the middell in the point ,d4 ,then drawe ,i a line from ,c4 to ,d1 and it is perpendicular to the line ,a4,b1 accordyng as my desire was4 ,,the 4,,vii4 ,,conclvsion4 ,to make a plumbe lyne on any porcion of a circle1 and that on the vtter or inner bughte4 ,mark first the prick where ye plumbe line shal lyght3 and prick out of ech side of it 4ij4 other poinctes equally distant from that first pricke4 ,then set the one foote of the compas in one of those side prickes1 and the other foote in the other side pricke1 and first moue one of the feete and drawe an arche line ouer the middell pricke1 then set the compas steddie with the one foote in the other side pricke1 and with the other foote drawe an other arche line1 that shall cut that first arche1 and from the very poincte of their meetyng1 drawe a right line vnto the firste pricke1 where you do minde that the plumbe line shall lyghte4 ,and so haue you performed thintent of this conclusion4 ,example4 ,illustration ,the arche of the circle on whiche ,i would erect a plumbe line1 is ,a4,b4,c4 and ,b4 is the pricke where ,i would haue the plumbe line to light4 ,therfore ,i meate out two equall distaunces on eche side of that pricke ,b4 and they are ,a4,c4 ,then open ,i the compas as wide as ,a4,c4 and settyng one of the feete in ,a4 with the other ,i drawe an arche line which goeth by ,g4 ,like waies ,i set one foote of the compas steddily in ,c4 and with the other ,i drawe an arche line1 goyng by ,g4 also4 ,now consideryng that ,g4 is the pricke of their meetyng1 it shall be also the poinct fro whiche ,i must drawe the plumbe line4 ,then draw ,i a right line from ,g4 to ,b4 and so haue mine intent4 ,now as ,a4,b4,c4 hath a plumbe line erected on his vtter bought1 so may ,i erect a plumbe line on the inner bught of ,d4,e4,f1 doynge with it as ,i did with the other1 that is to saye1 fyrste settyng forthe the pricke where the plumbe line shall light1 which is ,e1 and then markyng one other on eche syde1 as are ,d4 and ,f4 ,and then proceding as ,i dyd in the example before4 ,,the ,,viii4 ,,conclvsyon4 ,how to deuide the arche of a circle into two equall partes1 without measuring the arche4 ,deuide the corde of that line info ij4 equall portions1 and then from the middle prycke erecte a plumbe line1 and it shal parte that arche in the middle4 ,illustration ,example4 ,the arch to be diuided ys ,a4,d4,c1 the corde is ,a4,b4,c1 this corde is diuided in the middle with ,b1 from which prick if ,i erect a plum line as ,b4,d1 then will it diuide the arch in the middle1 that is to say1 in ,d4 ,,the ,,ix4 ,,conclvsion4 ,to do the same thynge other wise4 ,and for shortenes of worke1 if you wyl make a plumbe line without much labour1 you may do it with your squyre1 so that it be iustly made1 for yf you applye the edge of the squyre to the line in which the prick is1 and foresee the very corner of the squyre doo touche the pricke4 ,and than frome that corner if you drawe a lyne by the other edge of the squyre1 yt will be perpendicular to the former line4 ,illustration ,example4 ,a4,b4 is the line1 on which ,i wold make the plumme line1 or perpendicular4 ,and therefore ,i marke the prick1 from which the plumbe lyne muste rise1 which here is ,c4 ,then do ,i sette one edg of my squyre 7that is ,b4,c47 to the line ,a4,b1 so at the corner of the squyre do touche ,c4 iustly4 ,and from ,c4 ,i drawe a line by the other edge of the squire1 7which is ,c4,d47 ,and so haue ,i made the plumme line ,d4,c1 which ,i sought for4 ,,the ,x4 ,,conclvsion4 ,how to do the same thinge an other way yet ,if so be it that you haue an arche of suche greatnes1 that your squyre wyll not suffice therto1 as the arche of a brydge or of a house or window1 then may you do this4 ,mete vnderneth the arch where ye midle of his cord wyl be1 and ther set a mark4 ,then take a long line with a plummet1 and holde the line in suche a place of the arch1 that the plummet do hang iustely ouer the middle of the corde1 that you didde diuide before1 and then the line doth shewe you the middle of the arche4 ,illustration ,example4 ,the arch is ,a4,d4,b1 of which ,i trye the midle thus4 ,i draw a corde from one syde to the other 7as here is ,a4,b17 which ,i diuide in the middle in ,c4 ,then take ,i a line with a plummet 7that is ,d4,e17 and so hold ,i the line that the plummet ,e1 dooth hange ouer ,c1 ,and then ,i say that ,d4 is the middle of the arche4 ,and to thentent that my plummet shall point the more iustely1 ,i doo make it sharpe at the nether ende1 and so may ,i trust this woorke for certaine4 ,,the ,,xi4 ,,conclvsion4 ,when any line is appointed and without it a pricke1 whereby a parallel must be drawen howe you shall doo it4 ,take the iuste measure beetwene the line and the pricke1 accordinge to which you shal open your compasse4 ,then pitch one foote of your compasse at the one ende of the line1 and with the other foote draw a bowe line right ouer the pytche of the compasse1 lyke-wise doo at the other ende of the lyne1 then draw a line that shall touche the vttermoste edge of bothe those bowe lines1 and it will bee a true parallele to the fyrste lyne appointed4 ,example4 ,illustration ,a4,b1 is the line vnto which ,i must draw an other gemow line1 which muste passe by the prick ,c1 first ,i meate with my compasse the smallest distance that is from ,c4 to the line1 and that is ,c4,f1 wherfore staying the compasse at that distaunce1 ,i seete the one foote in ,a1 and with the other foot ,i make a bowe lyne1 which is ,d1 then like wise set ,i the one foote of the compasse in ,b1 and with the other ,i make the second bow line1 which is ,e4 ,and then draw ,i a line1 so that it toucheth the vttermost edge of bothe these bowe lines1 and that lyne passeth by the pricke ,c1 end is a gemowe line to ,a4,b1 as my sekyng was4 ,,the 4,,xii4 ,,conclvsion4 ,to make a triangle of any 4iij4 lines1 so that the lines be suche1 that any 4ij4 of them be longer then the thirde4 ,for this rule is generall1 that any two sides of euerie triangle taken together1 are longer then the other side that remaineth4 ,if you do remember the first and seconde conclusions1 then is there no difficultie in this1 for it is in maner the same woorke4 ,first consider the 4iij4 lines that you must take1 and set one of them for the ground line1 then worke with the other 4ij4 lines as you did in the first and second conclusions4 ,example4 ,illustration ,i haue 4iij4 ,a4,b4 and ,c4,d4 and ,e4,f4 of whiche ,i put 4,c4,d4 for my ground line1 then with my compas ,i take the length of 4,a4,b4 and set the one foote of my compas in ,c1 and draw an arch line with the other foote4 ,likewaies ,i take the length of ,e4,f1 and set one foote in ,d1 and with the other foote ,i make an arch line crosse the other arche1 and the pricke of their metyng 7whiche is ,g47 shall be the thirde corner of the triangle1 for in all suche kyndes of woorkynge to make a tryangle1 if you haue one line drawen1 there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee1 for two of them must needes be at the two eandes of the lyne that is drawen4 ,,the ,,xiii4 ,,conclvsion4 ,if you haue a line appointed1 and a pointe in it limited1 howe you maye make on it a righte lined angle1 equall to an other right lined angle1 all ready assigned4 ,fyrste draw a line against the corner assigned1 and so is it a triangle1 then take heede to the line and the pointe in it assigned1 and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned1 and if it bee longe enoughe1 then prick out there the length of one of the lines1 and then woorke with the other two lines1 accordinge to the laste conlusion1 makynge a triangle of thre like lynes to that assigned triangle4 ,if it bee not longe inoughe1 thenn lengthen it fyrste1 and afterwarde doo as ,i haue sayde beefore4 ,illustration ,example4 ,lette the angle appoynted bee ,a4,b4,c1 and the corner assigned1 ,b4 ,farthermore let the lymited line bee ,d4,g1 and the pricke assigned ,d4 ,fyrste therefore by drawinge the line ,a4,c1 ,i make the triangle ,a4,b4,c4 ,illustration ,then consideringe that ,d4,g1 is longer thanne ,a4,b1 you shall cut out a line from ,d4 toward ,g1 equal to ,a4,b1 as for example ,d4,f4 ,then measure oute the other ij4 lines and worke with them according as the conclusion with the fyrste also and the second teacheth yow1 and then haue you done4 ,,the ,,xiiii4 ,,conclvsion4 ,to make a square quadrate of any righte lyne appoincted4 ,first make a plumbe line vnto your line appointed1 whiche shall light at one of the endes of it1 accordyng to the fifth conclusion1 and let it be of like length as your first line is1 then open your compasse to the iuste length of one of them1 and sette one foote of the compasse in the ende of the one line1 and with the other foote draw an arche line1 there as you thinke that the fowerth corner shall be1 after that set the one foote of the same compasse vnsturred1 in the eande of the other line1 and drawe an other arche line crosse the first archeline1 and the poincte that they do crosse in1 is the pricke of the fourth corner of the square quadrate which you seke for1 therfore draw a line from that pricke to the eande of eche line1 and you shall therby haue made a square quadrate4 ,example4 ,illustration ,a4,b4 is the line proposed1 of whiche ,i shall make a square quadrate1 therefore firste ,i make a plumbe line vnto it1 whiche shall lighte in ,a1 and that plumb line is ,a4,c1 then open ,i my compasse as wide as the length of ,a4,b1 or ,a4,c1 7for they must be bothe equall7 and ,i set the one foote of thend in ,c1 and with the other ,i make an arche line nigh vnto ,d1 afterward ,i set the compas again with one foote in ,b1 and with the other foote ,i make an arche line crosse the first arche line in ,d1 and from the prick of their crossyng ,i draw 4ij4 lines1 one to ,b1 and an other to ,c1 and so haue ,i made the square quadrate that ,i entended4 ,,the 4,,xv4 ,,conclvsion4 ,to make a likeiamme equall to a triangle appointed1 and that in a right lined angle limited4 ,first from one of the angles of the triangle1 you shall drawe a gemowe line1 whiche shall be a parallele to that syde of the triangle1 on whiche you will make that likeiamme4 ,then on one end of the side of the triangle1 whiche lieth against the gemowe lyne1 you shall draw forth a line vnto the gemow line1 so that one angle that commeth of those 4ij4 lines be like to the angle which is limited vnto you4 ,then shall you deuide into ij4 equall partes that side of the triangle whiche beareth that line1 and from the pricke of that deuision1 you shall raise an other line parallele to that former line1 and continewe it vnto the first gemowe line1 and then of those 4ij4 last gemowe lynes1 and the first gemowe line1 with the halfe side of the triangle1 is made a lykeiamme equall to the triangle appointed1 and hath an angle lyke to an angle limited1 accordyng to the conclusion4 ,illustration ,example4 ,b4,c4,g1 is the triangle appoincted vnto1 whiche ,i muste make an equall likeiamme4 ,and ,d1 is the angle that the likeiamme must haue4 ,therfore first entendyng to erecte the likeiamme on the one side1 that the ground line of the triangle 7whiche is ,b4,g47 ,i do draw a gemow line by ,c1 and make it parallele to the ground line ,b4,g1 and that new gemow line is ,a4,h4 ,then do ,i raise a line from ,b4 vnto the gemowe line1 7whiche line is ,a4,b7 and make an angle equall to ,d1 that is the appointed angle 7accordyng as the 4viij4 conclusion teacheth7 and that angle is ,b4,a4,e4 ,then to procede1 ,i doo parte in ye middle the said ground line ,b4,g1 in the prick ,f1 from which prick ,i draw to the first gemowe line 7,a4,h47 an other line that is parallele to ,a4,b1 and that line is ,e4,f4 ,now saie ,i that the likeiamme ,b4,a4,e4,f1 is equall to the triangle ,b4,c4,g4 ,and also that it hath one angle 7that is ,b4,a4,e47 like to ,d4 the angle that was limitted4 ,and so haue ,i mine intent4 ,the profe of the equalnes of those two figures doeth depend of the 4xli4 proposition of ,euclides first boke1 and is the 4xxxi4 proposition of this second boke of ,theoremis1 whiche saieth1 that whan a tryangle and a likeiamme be made betwene 4ij4 selfe same gemow lines1 and haue their ground line of one length1 then is the likeiamme double to the triangle1 wherof it foloweth1 that if 4ij4 suche figures so drawen differ in their ground line onely1 so that the ground line of the likeiamme be but halfe the ground line of the triangle1 then be those 4ij4 figures equall1 as you shall more at large perceiue by the boke of ,theoremis1 in ye 4xxxi4 theoreme4 ,,the 4,,xvi4 ,,conclvsion4 ,to make a likeiamme equall to a triangle appoincted1 accordyng to an angle limitted1 and on a line also assigned4 ,in the last conclusion the sides of your likeiamme wer left to your libertie1 though you had an angle appoincted4 ,nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted1 which must be the side of the likeiamme4 ,therfore thus shall you procede4 ,firste accordyng to the laste conclusion1 make a likeiamme in the angle appoincted1 equall to the triangle that is assigned4 ,then with your compasse take the length of your line appointed1 and set out two lines of the same length in the second gemowe lines1 beginnyng at the one side of the likeiamme1 and by those two prickes shall you draw an other gemowe line1 whiche shall be parallele to two sides of the likeiamme4 ,afterward shall you draw 4ij4 lines more for the accomplishement of your worke1 which better shall be perceaued by a shorte exaumple1 then by a greate numbre of wordes1 only without example1 therefore ,i wyl by example sette forth the whole worke4 ,example4 ,illustration ,fyrst1 according to the last conclusion1 ,i make the likeiamme ,e4,f4,c4,g1 equal to the triangle ,d1 in the appoynted angle whiche is ,e4 ,then take ,i the lengthe of the assigned line 7which is ,a4,b17 and with my compas ,i sette forthe the same length in the ij4 gemow lines ,n4,f4 and ,h4,g1 setting one foot in ,e1 and the other in ,n1 and againe settyng one foote in ,c1 and the other in ,h4 ,afterward ,i draw a line from ,n4 to ,h1 whiche is a gemow lyne1 to ij4 sydes of the likeiamme4 thenne drawe ,i a line also from ,n4 vnto ,c4 and extend it vntyll it crosse the lines1 ,e4,l4 and ,f4,g1 which both must be drawen forth longer then the sides of the likeiamme4 and where that lyne doeth crosse ,f4,g1 there ,i sette ,m4 ,nowe to make an ende1 ,i make an other gemowe line1 whiche is parallel to ,n4,f4 and ,h4,g1 and that gemowe line doth passe by the pricke ,m1 and then haue ,i done4 ,now say ,i that ,h4,c4,k4,l1 is a likeiamme equall to the triangle appointed1 whiche was ,d1 and is made of a line assigned that is ,a4,b1 for ,h4,c1 is equall vnto ,a4,b1 and so is ,k4,l4 ,the profe of ye equalnes of this likeiam vnto the triangle1 dependeth of the thirty and two ,theoreme3 as in the boke of ,theoremes doth appear1 where it is declared1 that in al likeiammes1 when there are more then one made about one bias line1 the filsquares of euery of them muste needes be equall4 ,,the ,,xvii4 ,,conclvsion4 ,to make a likeiamme equal to any right lined figure1 and that on an angle appointed4 ,the readiest waye to worke this conclusion1 is to tourn that rightlined figure into triangles1 and then for euery triangle together an equal likeiamme1 according vnto the eleuen conclusion1 and then to ioine al those likeiammes into one1 if their sides happen to be equal1 which thing is euer certain1 when al the triangles happen iustly betwene one pair of gemow lines4 but and if they will not frame so1 then after that you haue for the firste triangle made his likeiamme1 you shall take the length of one of his sides1 and set that as a line assigned1 on whiche you shal make the other likeiams1 according to the twelft conclusion1 and so shall you haue al your likeiammes with ij4 sides equal1 and ij4 like angles1 so yt you mai easily ioyne them into one figure4 ,illustration ,example4 ,if the right lined figure be like vnto ,a1 then may it be turned into triangles that wil stand betwene ij4 parallels anye ways1 as you mai se by ,c4 and ,d1 for ij4 sides of both the trianngles ar parallels4 ,also if the right lined figure be like vnto ,e1 then wil it be turned into triangles1 liyng betwene two parallels also1 as ye other did before1 as in the example of ,f4,g4 ,but and if ye right lined figure be like vnto ,h1 and so turned into triangles as you se in ,k4,l4,m1 wher it is parted into iij triangles1 then wil not all those triangles lye betwen one pair of parallels or gemow lines1 but must haue many1 for euery triangle must haue one paire of parallels seuerall1 yet it maye happen that when there bee three or fower triangles1 ij4 of theym maye happen to agre to one pair of parallels1 whiche thinge ,i remit to euery honest witte to serche1 for the manner of their draught wil declare1 how many paires of parallels they shall neede1 of which varietee bicause the examples ar infinite1 ,i haue set forth these few1 that by them you may coniecture duly of all other like4 ,illustration ,further explicacion you shal not greatly neede1 if you remembre what hath ben taught before1 and then diligently behold how these sundry figures be turned into triangles4 ,in the fyrst you se ,i haue made v4 triangles1 and four paralleles4 in the seconde vij4 triangles and foure paralleles4 in the thirde thre triangles1 and fiue parallels1 in the iiij4 you se fiue triangles four parallels4 in the fift1 iiij4 triangles and 4iiij4 parallels1 in ye sixt ther ar fiue triangles iiij4 paralels4 ,howbeit a man maye at liberty alter them into diuers formes of triangles therefore ,i leue it to the discretion of the woorkmaister1 to do in al suche cases as he shal thinke best1 for by these examples 7if they bee well marked7 may all other like conclusions be wrought4 ,,the ,,xviii4 ,,conclvsion4 ,to parte a line assigned after suche a sorte1 that the square that is made of the whole line and one of his parts1 shal be equal to the squar that cometh of the other parte alone4 ,first deuide your lyne into ij4 equal parts1 and of the length of one part make a perpendicular to light at one end of your line assigned4 then adde a bias line1 and make thereof a triangle1 this done if you take from this bias line the halfe lengthe of your line appointed1 which is the iuste length of your perpendicular1 that part of the bias line whiche dothe remayne1 is the greater portion of the deuision that you seke for1 therefore if you cut your line according to the lengthe of it1 then will the square of that greater portion be equall to the square that is made of the whole line and his lesser portion4 ,and contrary wise1 the square of the whole line and his lesser parte1 wyll be equall to the square of the greater parte4 ,illustration ,example4 ,a4,b1 is the lyne assigned4 ,e4 is the middle pricke of ,a4,b1 ,b4,c4 is the plumb line or perpendicular1 made of the halfe of ,a4,b1 equall to ,a4,e1 other ,b4,e1 the byas line is ,c4,a1 from whiche ,i cut a peece1 that is ,c4,d1 equall to ,c4,b1 and accordyng to the lengthe lo the peece that remaineth 7whiche is ,d4,a17 ,i doo deuide the line ,a4,b1 at whiche diuision ,i set ,f4 ,now say ,i1 that this line ,a4,b1 7wch was assigned vnto me7 is so diuided in this point ,f1 yt ye square of ye hole line ,a4,b1 of the one portion 7yt is ,f4,b1 the lesser part7 is equall to the square of the other parte1 whiche is ,f4,a1 and is the greater part of the first line4 ,the profe of this equalitie shall you learne by the 4xl4 ,theoreme4 ,transcriber's ,note3 ,there are two ways to make this ,example work3 --transpose ,e and ,f in the illustration1 and change one occurrence of ,e to ,f in the text 7at whiche diuision ,i set44471 or3 --keep the illustration as printed1 and transpose all other occurrences of ,e and ,f in the text4 ,,the 4,,xix4 ,,conclvsion4 ,to make a square quadrate equall to any right lined figure appoincted4 ,first make a likeiamme equall to that right lined figure1 with a right angle1 accordyng to the 4xi4 conclusion1 then consider the likeiamme1 whether it haue all his sides equall1 or not3 for yf they be all equall1 then haue you doone your conclusion4 but and if the sides be not all equall1 then shall you make one right line iuste as long as two of those vnequall sides1 that line shall you deuide in the middle1 and on that pricke drawe half a circle1 then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme1 and from that pointe of diuision shall you erecte a perpendicular1 which shall touche the edge of the circle4 ,and that perpendicular shall be the iuste side of the square quadrate1 equall both to the lykeiamme1 and also to the right lined figure appointed1 as the conclusion willed4 ,example4 ,illustration ,k1 is the right lined figure appointed1 and ,b4,c4,d4,e1 is the likeiamme1 with right angles equall vnto ,k1 but because that this likeiamme is not a square quadrate1 ,i must turne it into such one after this sort1 ,i shall make one right line1 as long as 4ij4 vnequall sides of the likeiamme1 that line here is ,f4,g1 whiche is equall to ,b4,c1 and ,c4,e4 ,then part ,i that line in the middle in the pricke ,m1 and on that pricke ,i make halfe a circle1 accordyng to the length of the diameter ,f4,g4 ,afterward ,i cut awaie a peece from ,f4,g1 equall to ,c4,e1 markyng that point with ,h4 ,and on that pricke ,i erecte a perpendicular ,h4,k1 whiche is the iust side to the square quadrate that ,i seke for1 therfore accordyng to the doctrine of the 4x4 conclusion1 of the lyne ,i doe make a square quadrate1 and so haue ,i attained the practise of this conclusion4 ,,the 4,,xx4 ,,conclvsion4 ,when any 4ij4 square quadrates are set forth1 how you maie make one equall to them bothe4 ,first drawe a right line equall to the side of one of the quadrates3 and on the ende of it make a perpendicular1 equall in length to the side of the other quadrate1 then drawe a byas line betwene those 4ij4 other lines1 makyng thereof a right angeled triangle4 ,and that byas lyne wyll make a square quadrate1 equall to the other 4ij4 quadrates appointed4 ,illustration ,example4 ,a4,b4 and ,c4,d1 are the two square quadrates appointed1 vnto which ,i must make one equall square quadrate4 ,first therfore ,i dooe make a righte line ,e4,f1 equall to one of the sides of the square quadrate ,a4,b4 ,and on the one end of it ,i make a plumbe line ,e4,g1 equall to the side of the other quadrate ,d4,c4 ,then drawe ,i a byas line ,g4,f1 which beyng made the side of a quadrate 7accordyng to the tenth conclusion7 will accomplishe the worke of this practise3 for the quadrate ,h4 is muche iust as the other two4 ,i meane ,a4,b4 and ,d4,c4 ,,the 4,,xxi4 ,,conclvsion4 ,when any two quadrates be set forth1 howe to make a squire about the one quadrate1 whiche shall be equall to the other quadrate4 ,determine with your selfe about whiche quadrate you wil make the squire1 and drawe one side of that quadrate forth in lengte1 accordyng to the measure of the side of the other quadrate1 whiche line you maie call the grounde line1 and then haue you a right angle made on this line by an other side of the same quadrate3 ,therfore turne that into a right cornered triangle1 accordyng to the worke in the laste conclusion1 by makyng of a byas line1 and that byas lyne will performe the worke of your desire4 ,for if you take the length of that byas line with your compasse1 and then set one foote of the compas in the farthest angle of the first quadrate 7whiche is the one ende of the groundline7 and extend the other foote on the same line1 accordyng to the measure of the byas line1 and of that line make a quadrate1 enclosyng ye first quadrate1 then will there appere the forme of a squire about the first quadrate1 which squire is equall to the second quadrate4 ,illustration ,example4 ,the first square quadrate is ,a4,b4,c4,d1 and the seconde is ,e4 ,now would ,i make a squire about the quadrate ,a4,b4,c4,d1 whiche shall bee equall vnto the quadrate ,e4 ,therfore first ,i draw the line ,a4,d1 more in length1 accordyng to the measure of the side of ,e1 as you see1 from ,d4 vnto ,f1 and so the hole line of bothe these seuerall sides is ,a4,f1 then make ,i a byas line from ,c1 to ,f1 whiche byas line is the measure of this woorke4 wherefore ,i open my compas accordyng to the length of that byas line ,c4,f1 and set the one compas foote in ,a1 and extend thother foote of the compas toward ,f1 makyng this pricke ,g1 from whiche ,i erect a plumbeline ,g4,h1 and so make out the square quadrate ,a4,g4,h4,k1 whose sides are equall eche of them to ,a4,g4 ,and this square doth contain the first quadrate ,a4,b4,c4,d1 and also a squire ,g4,h4,k1 whiche is equall to the second quadrate ,e1 for as the last conclusion declareth1 the quadrate ,a4,g4,h4,k1 is equall to bothe the other quadrates proposed1 that is ,a4,b4,c4,d1 and ,e4 ,then muste the squire ,g4,h4,k1 needes be equall to ,e1 consideryng that all the rest of that great quadrate is nothyng els but the quadrate self1 ,a4,b4,c4,d1 and so haue ,i thintent of this conclusion4 ,,the 4,,xxii4 ,,conclvsion4 ,to find out the centre of any circle assigned4 ,draw a corde or stryngline crosse the circle1 then deuide into 4ij4 equall partes1 both that corde1 and also the bowe line1 or arche line1 that serueth to that corde1 and from the prickes of those diuisions1 if you drawe an other line crosse the circle1 it must nedes passe by the centre4 ,therfore deuide that line in the middle1 and that middle pricke is the centre of the circle proposed4 ,example4 ,illustration ,let the circle be ,a4,b4,c4,d1 whose centre ,i shall seke4 ,first therfore ,i draw a corde crosse the circle1 that is ,a4,c4 ,then do ,i deuide that corde in the middle1 in ,e1 and likewaies also do ,i deuide his arche line ,a4,b4,c1 in the middle1 in the pointe ,b4 ,afterward ,i drawe a line from ,b4 to ,e1 and so crosse the circle1 whiche line is ,b4,d1 in which line is the centre that ,i seeke for4 ,therefore if ,i parte that line ,b4,d1 in the middle in to two equall portions1 that middle pricke 7which here is ,f7 is the verye centre of the sayde circle that ,i seke4 ,this conclusion may other waies be wrought1 as the moste part of conclusions haue sondry formes of practise1 and that is1 by makinge thre prickes in the circumference of the circle1 at liberty where you wyll1 and then findinge the centre to those thre pricks1 ,which worke bicause it serueth for sondry vses1 ,i think meet to make it a seuerall conclusion by it selfe4 ,,the ,,xxiii4 ,,conclvsion4 ,to find the commen centre belongyng to anye three prickes appointed1 if they be not in an exacte right line4 ,it is to be noted1 that though euery small arche of a greate circle do seeme to be a right lyne1 yet in very dede it is not so1 for euery part of the circumference of al circles is compassed1 though in litle arches of great circles the eye cannot discerne the crokednes1 yet reason doeth alwais declare it1 therfore iij4 prickes in an exact right line can not bee brought into the circumference of a circle4 ,but and if they be not in a right line how so euer they stande1 thus shall you find their common centre4 ,open your compas so wide1 that it be somewhat more then the halfe distance of two of those prickes4 ,then sette the one foote of the compas in the one pricke1 and with the other foot draw an arche lyne toward the other pricke1 ,then againe putte the foot of your compas in the second pricke1 and with the other foot make an arche line1 that may crosse the firste arch line in ij4 places4 ,now as you haue done with those two pricks1 so do with the middle pricke1 and the thirde that remayneth4 ,then draw ij4 lines by the poyntes where those arche lines do crosse1 and where those two lines do meete1 there is the centre that you seeke for4 ,example ,illustration ,the iij4 prickes ,i haue set to be ,a4,b1 and ,c1 whiche ,i wold bring into the edg of one common circle1 by finding a centre commen to them all1 fyrst therefore ,i open my compas1 so that thei occupye more then ye halfe distance betwene ij4 pricks 7as are ,a4,b47 and so settinge one foote in ,a4 and extendinge the other toward ,b1 ,i make the arche line ,d4,e4 ,likewise setting one foot in ,b1 and turninge the other toward ,a1 ,i draw an other arche line that crosseth the first in ,d4 and ,e4 ,then from ,d4 to ,e1 ,i draw a right lyne ,d4,h4 ,after this ,i open my compasse to a new distance1 and make ij4 arche lines betwene ,b4 and ,c1 whiche crosse one the other in ,f4 and ,g1 by whiche two pointes ,i draw an other line1 that is ,f4,h4 ,and bycause that the lyne ,d4,h4 and the lyne ,f4,h4 doo meete in ,h1 ,i saye that ,h4 is the centre that serueth to those iij4 prickes4 ,now therfore if you set one foot of your compas in ,h1 and extend the other to any of the iij4 pricks1 you may draw a circle wch shal enclose those iij4 pricks in the edg of his circumference thus haue you attained ye vse of this conclusion4 ,,the ,,xxiiii4 ,,conclvsion4 ,to drawe a touche line onto a circle1 from any poincte assigned4 ,here must you vnderstand that the pricke must be without the circle1 els the conclusion is not possible4 ,but the pricke or poinct beyng without the circle1 thus shall you procede3 ,open your compas1 so that the one foote of it maie be set in the centre of the circle1 and the other foote on the pricke appoincted1 and so draw an other circle of that largenesse about the same centre3 and it shall gouerne you certainly in makyng the said touche line4 ,for if you draw a line from the pricke appointed vnto the centre of the circle1 and marke the place where it doeth crosse the lesser circle1 and from that poincte erect a plumbe line that shall touche the edge of the vtter circle1 and marke also the place where that plumbe line crosseth that vtter circle1 and from that place drawe an other line to the centre1 takyng heede where it crosseth the lesser circle1 if you drawe a plumbe line from that pricke vnto the edge of the greatter circle1 that line ,i say is a touche line1 drawen from the point assigned1 according to the meaning of this conclusion4 ,illustration ,example4 ,let the circle be called ,b4,c4,d1 and his centre ,e1 and ye prick assigned ,a1 open your compas now of such widenes1 yt the one foote may be set in ,e1 wch is ye centre of ye circle1 ye other in ,a1 wch is ye pointe assigned1 so make an other greter circle 7as here is ,a4,f4,g7 then draw a line from ,a4 vnto ,e1 and wher that line doth cross ye inner circle 7wch heere is in the prick ,b47 there erect a plumb line vnto the line4 ,a4,e4 and let that plumb line touch the vtter circle1 as it doth here in the point ,f1 so shall ,b4,f4 bee that plumbe lyne4 ,then from ,f4 vnto ,e4 drawe an other line whiche shal be ,f4,e1 and it will cutte the inner circle1 as it doth here in the point ,c1 from which pointe ,c4 if you erect a plumb line vnto ,a1 then is that line ,a4,c1 the touche line1 whiche you shoulde finde4 ,not withstandinge that this is a certaine waye to fynde any touche line1 and a demonstrable forme1 yet more easyly by many folde may you fynde and make any suche line with a true ruler1 layinge the edge of the ruler to the edge of the circle and to the pricke1 and so drawing a right line1 as this example sheweth1 where the circle is ,e1 the pricke assigned is ,a4 and the ruler ,c4,d4 by which the touch line is drawen1 and that is ,a4,b1 and as this way is light to doo1 so is it certaine inoughe for any kinde of workinge4 ,illustration ,,the ,,xxv4 ,,conclvsion4 ,when you haue any peece of the circumference of a circle assigned1 howe you may make oute the whole circle agreynge therevnto4 ,first seeke out of the centre of that arche1 according to the doctrine of the seuententh conclusion1 and then setting one foote of your compas in the centre1 and extending the other foot vnto the edge of the arche or peece of the circumference1 it is easy to drawe the whole circle4 ,example4 ,a peece of an olde pillar was found1 like in forme to thys figure ,a4,d4,b4 ,now to knowe howe muche the compasse of the hole piller was1 seing by this parte it appereth that it was round1 thus shal you do4 ,make in a table the like draught of yt circumference by the self patron1 vsing it as it wer a croked ruler4 ,then make 4iij4 prickes in that arche line1 as ,i haue made1 ,c4 ,d4 and ,e4 ,and then finde out the common centre to them all1 as the 4xvij4 conclusion teacheth4 ,and that centre is here ,f1 nowe settyng one foote of your compas in ,f1 and the other in ,c4 ,d1 other in ,e1 and so makyng a compasse1 you haue youre whole intent4 ,illustration ,,the ,,xxvi4 ,,conclvsion4 ,to finde the centre to any arche of a circle4 ,if so be it that you desire to find the centre by any other way then by those 4iij4 prickes1 consideryng that sometimes you can not haue so much space in the thyng where the arche is drawen1 as should serue to make those 4iiij4 bowe lines1 then shall you do thus3 ,parte that arche line into two partes1 equall other vnequall1 it maketh no force1 and vnto ech portion draw a corde1 other a stringline4 ,and then accordyng as you dyd in one arche in the 4xvi4 conclusion1 so doe in bothe those arches here1 that is to saie1 deuide the arche in the middle1 and also the corde1 and drawe then a line by those two deuisions1 so then are you sure that that line goeth by the centre4 ,afterward do lykewaies with the other arche and his corde1 and where those 4ij4 lines do crosse1 there is the centre1 that you seke for4 ,example4 ,illustration ,the arche of the circle is ,a4,b4,c1 vnto whiche ,i must seke a centre1 therfore firste ,i do deuide it into 4ij4 partes1 the one of them is ,a4,b1 and the other is ,b4,c4 ,then doe ,i cut euery arche in the middle1 so is ,e4 the middle of ,a4,b1 and ,g4 is the middle of ,b4,c4 ,likewaies1 ,i take the middle of their cordes1 whiche ,i mark with ,f4 and ,h1 settyng ,f4 by ,e1 and ,h4 by ,g4 ,then drawe ,i a line from ,e4 to ,f1 and from ,g4 to ,h1 and they do crosse in ,d1 wherefore saie ,i1 that ,d4 is the centre1 that ,i seke for4 ,,the ,,xxvii4 ,,conclvsion4 ,to drawe a circle within a triangle appoincted4 ,for this conclusion and all other lyke1 you muste vnderstande1 that when one figure is named to be within an other1 that it is not other waies to be vnderstande1 but that eyther euery syde of the inner figure dooeth touche euerie corner of the other1 other els euery corner of the one dooeth touche euerie side of the other4 ,so ,i call that triangle drawen in a circle1 whose corners do touche the circumference of the circle4 ,and that circle is contained in a triangle1 whose circumference doeth touche iustely euery side of the triangle1 and yet dooeth not crosse ouer any side of it4 ,and so that quadrate is called properly to be drawen in a circle1 when all his fower angles doeth touche the edge of the circle1 ,and that circle is drawen in a quadrate1 whose circumference doeth touche euery side of the quadrate1 and lykewaies of other figures4 ,examples are these4 ,a4,b4,c4,d4,e4,f4 ,illustration3 ,a4 is a circle in a triangle4 ,b4 a triangle in a circle4 ,c4 a quadrate in a circle4 ,d4 a circle in a quadrate4 ,in these 4ij4 last figures ,e4 and ,f1 the circle is not named to be drawen in a triangle1 because it doth not touche the sides of the triangle1 neither is the triangle counted to be drawen in the circle1 because one of his corners doth not touche the circumference of the circle1 yet 7as you see7 the circle is within the triangle1 and the triangle within the circle1 but nother of them is properly named to be in the other4 ,now to come to the conclusion4 ,if the triangle haue all 4iij4 sides lyke1 then shall you take the middle of euery side1 and from the contrary corner drawe a right line vnto that poynte1 and where those lines do crosse one an other1 there is the centre4 ,then set one foote of the compas in the centre and stretche out the other to the middle pricke of any of the sides1 and so drawe a compas1 whiche shall touche euery side of the triangle1 but shall not passe with out any of them4 ,example4 ,the triangle is ,a4,b4,c1 whose sides ,i do part into 4ij4 equall partes1 eche by it selfe in these pointes ,d4,e4,f1 puttyng ,f4 betwene ,a4,b1 and ,d4 betwene ,b4,c1 and ,e4 betwene ,a4,c4 ,then draw ,i a line from ,c4 to ,f1 and an other from ,a4 to ,d1 and the third from ,b4 to ,e4 ,illustration ,and where all those lines do mete 7that is to saie ,m4 ,g17 ,i set the one foote of my compasse1 because it is the common centre1 and so drawe a circle accordyng to the distaunce of any of the sides of the triangle4 ,and then find ,i that circle to agree iustely to all the sides of the triangle1 so that the circle is iustely made in the triangle1 as the conclusion did purporte4 ,and this is euer true1 when the triangle hath all thre sides equall1 other at the least 4ij4 sides lyke long4 ,but in the other kindes of triangles you must deuide euery angle in the middle1 as the third conclusion teaches you4 ,and so drawe lines from eche angle to their middle pricke4 ,and where those lines do crosse1 there is the common centre1 from which you shall draw a perpendicular to one of the sides4 ,then sette one foote of the compas in that centre1 and stretche the other foote accordyng to the length of the perpendicular1 and so drawe your circle4 ,illustration ,example4 ,the triangle is ,a4,b4,c1 whose corners ,i haue diuided in the middle with ,d4,e4,f1 and haue drawen the lines of diuision ,a4,d4 ,b4,e1 and ,c4,f1 which crosse in ,g1 therfore shall ,g4 be the common centre4 ,then make ,i one perpendicular from ,g4 vnto the side ,b4,c1 and that is ,g4,h4 ,now sette ,i one fote of the compas in ,g1 and extend the other foote vnto ,h4 and so drawe a compas1 whiche wyll iustly answere to that triangle according to the meaning of the conclusion4 ,,the ,,xxviii4 ,,conclvsion4 ,to drawe a circle about any triangle assigned4 ,fyrste deuide two sides of the triangle equally in half and from those ij4 prickes erect two perpendiculars1 which muste needes meet in crosse1 and that point of their meting is the centre of the circle that must be drawen1 therefore sette one foote of the compasse in that pointe1 and extend the other foote to one corner of the triangle1 and so make a circle1 and it shall touche all iij4 corners of the triangle4 ,example4 ,illustration ,a4,b4,c4 is the triangle1 whose two sides ,a4,c4 and ,b4,c4 are diuided into two equall partes in ,d4 and ,e1 settyng ,d4 betwene ,b4 and ,c1 and ,e4 betwene ,a4 and ,c4 ,and from eche of those two pointes is ther erected a perpendicular 7as you se ,d4,f1 and ,e4,f47 which mete1 and crosse in ,f1 and stretche forth the other foot of any corner of the triangle1 and so make a circle1 that circle shal touch euery corner of the triangle1 and shal enclose the whole triangle1 accordinge1 as the conclusion willeth4 ,an other way to do the same4 ,and yet an other waye may you doo it1 accordinge as you learned in the seuententh conclusion1 for if you call the three corners of the triangle iij4 prickes1 and then 7as you learned there7 yf you seeke out the centre to those three prickes1 and so make it a circle to include those thre prickes in his circumference1 you shall perceaue that the same circle shall iustelye include the triangle proposed4 ,example4 ,illustration ,a4,b4,c4 is the triangle1 whose iij4 corners ,i count to be iij4 pointes4 ,then 7as the seuentene conclusion doth teache7 ,i seeke a common centre1 on which ,i may make a circle1 that shall enclose those iij prickes4 that centre as you se is ,d1 for in ,d4 doth the right lines1 that passe by the angles of the arche lines1 meete and crosse4 ,and on that centre as you se1 haue ,i made a circle1 which doth inclose the iij4 angles of the triangle1 and consequentlye the triangle itselfe1 as the conclusion dydde intende4 ,,the ,,xxix4 ,,conclvsion4 ,to make a triangle in a circle appoynted whose corners shal be equall to the corners of any triangle assigned4 ,when ,i will draw a triangle in a circle appointed1 so that the corners of that triangle shall be equall to the corners of any triangle assigned1 then must ,i first draw a tuche lyne vnto that circle1 as the twenty conclusion doth teach1 and in the very poynte of the touche muste ,i make an angle1 equall to one angle of the triangle1 and that inwarde toward the circle3 likewise in the same pricke must ,i make an other angle wt the other halfe of the touche line1 equall to an other corner of the triangle appointed1 and then betwen those two corners will there resulte a third angle1 equall to the third corner of that triangle4 ,nowe where those two lines that entre into the circle1 doo touche the circumference 7beside the touche line7 there set ,i two prickes1 and betwene them ,i drawe a thyrde line4 ,and so haue ,i made a triangle in a circle appointed1 whose corners bee equall to the corners of the triangle assigned4 ,example4 ,illustration ,a4,b4,c1 is the triangle appointed1 and ,f4,g4,h4 is the circle1 in which ,i muste make an other triangle1 with lyke angles to the angles of ,a4,b4,c4 the triangle appointed4 ,therefore fyrst ,i make the touch lyne ,d4,f4,e4 ,and then make ,i an angle in ,f1 equall to ,a1 whiche is one of the angles of the triangle4 ,and the lyne that maketh that angle with the touche line1 is ,f4,h1 whiche ,i drawe in lengthe vntill it touche the edge of the circle4 ,then againe in the same point ,f1 ,i make an other corner equall to the angle ,c4 and the line that maketh that corner with the touche line1 is ,f4,g4 whiche also ,i drawe foorthe vntill it touche the edge of the circle4 ,and then haue ,i made three angles vpon that one touch line1 and in yt one point ,f1 and those iij4 angles be equall to the iij4 angles of the triangle assigned1 whiche thinge doth plainely appeare1 in so muche as they bee equall to ij4 right angles1 as you may gesse by the fixt theoreme4 ,and the thre angles of euerye triangle are equill also to ij4 righte angles1 as the two and twenty theoreme dothe show1 so that bicause they be equall to one thirde thinge1 they must needes be equal togither1 as the common sentence saith4 ,then do ,i draw a line frome ,g4 to ,h1 and that line maketh a triangle ,f4,g4,h1 whole angles be equall to the angles of the triangle appointed4 ,and this triangle is drawn in a circle1 as the conclusion didde wyll4 ,the proofe of this conclusion doth appeare in the seuenty and iiij4 ,theoreme4 ,,the ,,xxx4 ,,conclvsion4 ,to make a triangle about a circle assigned which shall haue corners1 equall to the corners of any triangle appointed4 ,first draw forth in length the one side of the triangle assigned so that therby you may haue ij4 vtter angles1 vnto which two vtter angles you shall make ij4 other equall on the centre of the circle proposed1 drawing thre halfe diameters frome the circumference1 whiche shal enclose those ij4 angles1 then draw iij4 touche lines which shall make ij4 right angles1 eche of them with one of those semidiameters4 ,those iij4 lines will make a triangle equally cornered to the triangle assigned1 and that triangle is drawen about a circle apointed1 as the conclusion did wil4 ,example4 ,a4,b4,c1 is the triangle assigned1 and ,g4,h4,k1 is the circle appointed1 about which ,i muste make a triangle hauing equall angles to the angles of that triangle ,a4,b4,c4 ,fyrst therefore ,i draw ,a4,c4 7which is one of the sides of the triangle7 in length that there may appeare two vtter angles in that triangle1 as you se ,b4,a4,d1 and ,b4,c4,e4 ,illustration ,then drawe ,i in the circle appointed a semidiameter1 which is here ,h4,f1 for ,f4 is the centre of the circle ,g4,h4,k4 ,then make ,i on that centre an angle equall to the vtter angle ,b4,a4,d1 and that angle is ,h4,f4,k4 ,like waies on the same centre by drawyng an other semidiameter1 ,i make an other angle ,h4,f4,g1 equall to the second vtter angle of the triangle1 whiche is ,b4,c4,e4 ,and thus haue ,i made 4iij4 semidiameters in the circle appointed4 ,then at the ende of eche semidiameter1 ,i draw a touche line1 whiche shall make righte angles with the semidiameter4 ,and those 4iij4 touch lines mete1 as you see1 and make the trianagle ,l4,m4,n1 whiche is the triangle that ,i should make1 for it is drawen about a circle assigned1 and hath corners equall to the corners of the triangle appointed1 for the corner ,m4 is equall to ,c4 ,likewaies ,l4 to ,a1 and ,n4 to ,b1 whiche thyng you shall better perceiue by the vi4 ,theoreme1 as ,i will declare in the booke of proofes4 ,,the ,,xxxi4 ,,conclvsion4 ,to make a portion of a circle on any right line assigned1 whiche shall conteine an angle equall to a right lined angle appointed4 ,the angle appointed1 maie be a sharpe angle1 a right angle1 other a blunte angle1 so that the worke must be diuersely handeled according to the diuersities of the angles1 but consideringe the hardenes of those seuerall woorkes1 ,i wyll omitte them for a more meter time1 and at this tyme wyll shewe you one light waye which serueth for all kindes of angles1 and that is this4 ,when the line is proposed1 and the angle assigned1 you shall ioyne that line proposed so to the other twoo lines contayninge the angle assigned1 that you shall make a triangle of theym1 for the easy dooinge whereof1 you may enlarge or shorten as you see cause1 anye of the two lynes contayninge the angle appointed4 ,and when you haue made a triangle of those iij4 lines1 then accordinge to the doctrine of the seuen and twenty coclusion1 make a circle about that triangle4 ,and so haue you wroughte the request of this conclusion4 ,whyche yet you maye woorke by the twenty and eight conclusion also1 so that of your line appointed1 you make one side of the triangle be equal to ye angle assigned as youre selfe mai easily gesse4 ,illustration ,example4 ,first for example of a sharpe angle let ,a4 stand ,b4,c shal be ye lyne assigned4 ,then do ,i make a triangle1 by adding ,b4,c1 as a thirde side to those other ij4 which doo include the angle assigned1 and that triangle is ,d4,e4,f1 so yt ,e4,f4 is the line appointed1 and ,d4 is the angle assigned4 ,then doo ,i drawe a portion of a circle about that triangle1 from the one ende of that line assigned vnto the other1 that is to saie1 from ,e4 a long by ,d4 vnto ,f1 whiche portion is euermore greatter then the halfe of the circle1 by reason that the angle is a sharpe angle4 ,but if the angle be right 7as in the second exaumple you see it7 then shall the portion of the circle that containeth that angle1 euer more be the iuste halfe of a circle4 ,and when the angle is a blunte angle1 as the thirde exaumple dooeth propounde1 then shall the portion of the circle euermore be lesse then the halfe circle4 ,so in the seconde example1 ,g4 is the right angle assigned1 and ,h4,k4 is the lyne appointed1 and ,l4,m4,n4 the portion of the circle aunsweryng thereto4 ,in the third exaumple1 ,o4 is the blunte corner assigned1 ,p4,q4 is the line1 and ,r4,s4,t4 is the portion of the circle1 that containeth that blunt corner1 and is drawen on ,r4,t4 the line appointed4 ,,the ,,xxxii4 ,,conclvsion4 ,to cutte of from a circle appointed1 a portion containyng an angle equall to a right lyned angle assigned4 ,when the angle and the circle are assigned1 first draw a touch line vnto that circle1 and then drawe an other line from the pricke of the touchyng to one side of the circle1 so that thereby those two lynes do make an angle equall to the angle assigned4 ,then saie ,i that the portion of the circle of the contrarie side to the angle drawen1 is the parte that you seke for4 ,example4 ,illustration ,a4 is the angle appointed1 and ,d4,e4,f4 is the circle assigned1 from which ,i must cut away a portion that doth contain an angle equall to this angle ,a4 ,therfore first ,i do draw a touche line to the circle assigned1 and that touch line is ,b4,c1 the very pricke of the touche is ,d1 from whiche ,d4 ,i drawe a lyne ,d4,e1 so that the angle made of those two lines be equall to the angle appointed4 ,then say ,i1 that the arch of the circle ,d4,f4,e1 is the arche that ,i seke after4 ,for if ,i doo deuide that arche in the middle 7as here is done in ,f47 and so draw thence two lines1 one to ,d1 and the other to ,e1 then will the angle ,f1 be equall to the angle assigned4 ,,the ,,xxxiii4 ,,conclvsion4 ,to make a square quadrate in a circle assigned4 ,draw 4ij4 diameters in the circle1 so that they runne a crosse1 and that they make 4iiij4 right angles4 ,then drawe 4iiij4 lines1 that may ioyne the 4iiij4 ends of those diameters1 one to an other1 and then haue you made a square quadrate in the circle appointed4 ,example4 ,illustration ,a4,b4,c4,d4 is the circle assigned1 and ,a4,c4 and ,b4,d4 are the two diameters which crosse in the centre ,e1 and make 4iiij4 right corners4 ,then do ,i make fowre other lines1 that is ,a4,b1 ,b4,c1 ,c4,d1 and ,d4,a1 which do ioyne together the fowre endes of the ij4 diameters4 ,and so is the square quadrate made in the circle assigned1 as the conclusion willeth4 ,,the ,,xxxiiii4 ,,conclvsion4 ,to make a square quadrate aboute annye circle assigned4 ,drawe two diameters in crosse waies1 so that they make foure righte angles in the centre4 ,then with your compasse take the length of the halfe diameter1 and set one foote of the compas in eche end of the compas1 so shall you haue viij4 archelines4 ,then yf you marke the prickes wherin those arch lines do crosse1 and draw betwene those iiij4 prickes iiij right lines1 then haue you made the square quadrate accordinge to the request of the conclusion4 ,example4 ,illustration ,a4,b4,c4 is the circle assigned in which first ,i draw two diameters1 in crosse waies1 making iiij4 righte angles1 and those ij4 diameters are ,a4,c4 and ,b4,d4 ,then sette ,i my compasse 7whiche is opened according to the semidiameter of the said circle7 fixing one foote in the end of euery semidiameter1 and drawe with the other foote twoo arche lines1 one on euery side4 ,as firste1 when ,i sette the one foote in ,a1 then with the other foote ,i doo make twoo arche lines1 one in ,e1 and an other in ,f4 ,then sette ,i the one foote of the compasse in ,b1 and drawe twoo arche lines ,f4 and ,g4 ,like wise setting the compasse foote in ,c1 ,i drawe twoo other arche lines1 ,g4 and ,h1 and on ,d4 ,i make twoo other1 ,h4 and ,e4 ,then frome the crossinges of those eighte arche lines ,i drawe iiij4 straighte lynes1 that is to saye1 ,e4,f1 and ,f4,g1 also ,g4,h1 and ,h4,e1 whiche iiij4 straighte lynes do make the square quadrate that ,i should draw about the circle assigned4 ,,the ,,xxxv4 ,,conclvsion4 ,to draw a circle in any square quadrate appointed4 ,fyrste deuide euery side of the quadrate into twoo equall partes1 and so drawe two lynes betwene eche two contrary poinctes1 and where those twoo lines doo crosse1 there is the centre of the circle4 ,then sette the foote of the compasse in that point1 and stretch forth the other foot1 according to the length of halfe one of those lines1 and so make a compas in the square quadrate assigned4 ,example4 ,illustration ,a4,b4,c4,d4 is the quadrate appointed1 in whiche ,i muste make a circle4 ,therefore first ,i do deuide euery side in ij4 equal partes1 and draw ij4 lines acrosse1 betwene eche ij4 contrary prickes1 as you se ,e4,g1 and ,f4,h1 whiche mete in ,k1 and therfore shal ,k1 be the centre of the circle4 ,then do ,i set one foote of the compas in ,k4 and open the other as wide as ,k4,e1 and so draw a circle1 which is made accordinge to the conclusion4 ,,the ,,xxxvi4 ,,conclvsion4 ,to draw a circle about a square quadrate4 ,draw ij4 lines betwene the iiij4 corners of the quadrate1 and where they mete in crosse1 ther is the centre of the circle that you seeke for4 ,then set one foot of the compas in that centre1 and extend the other foote vnto one corner of the quadrate1 and so may you draw a circle which shall iustely inclose the quadrate proposed4 ,example4 ,illustration ,a4,b4,c4,d4 is the square quadrate proposed1 about which ,i must make a circle4 ,therfore do ,i draw ij4 lines crosse the square quadrate from angle to angle1 as you se ,a4,c4 ,b4,d4 ,and where they ij4 do crosse 7that is to say in ,e47 there set ,i the one foote of the compas as in the centre1 and the other foote ,i do extend vnto one angle of the quadrate1 as for example to ,a1 and so make a compas1 whiche doth iustly inclose the quadrate1 according to the minde of the conclusion4 ,,the ,,xxxvii4 ,,conclvsion4 ,to make a twileke triangle1 whiche shall haue euery of the ij4 angles that lye about the ground line1 double to the other corner4 ,fyrste make a circle1 and deuide the circumference of it into fyue equall partes4 ,and thenne drawe frome one pricke 7which you will7 two lines to ij4 other prickes1 that is to say to the iij4 and iiij4 pricke1 counting that for the first1 wherhence you drewe both those lines1 ,then drawe the thyrde lyne to make a triangle with those other twoo1 and you haue doone according to the conclusion1 and haue made a twelike triangle1 whose ij4 corners about the grounde line1 are eche of theym double to the other corner4 ,example4 ,illustration ,a4,b4,c4 is the circle1 whiche ,i haue deuided into fiue equal portions4 ,and from one of the prickes 7which is ,a17 ,i haue drawen ij4 lines1 ,a4,b4 and ,b4,c1 whiche are drawen to the third and iiij4 prickes4 ,then draw ,i the third line ,c4,b1 which is the grounde line1 and maketh the triangle1 that ,i would haue1 for the angle ,c4 is double to the angle ,a1 and so is the angle ,b4 also4 ,,the ,,xxxviii4 ,,conclvsion4 ,to make a cinkangle of equall sides1 and equall corners in any circle appointed4 ,deuide the circle appointed into fiue equall partes1 as you didde in the laste conclusion1 and drawe ij4 lines from euery pricke to the other ij4 that are nexte vnto it4 ,and so shall you make a cinkangle after the meanynge of the conclusion4 ,example4 ,yow se here this circle ,a4,b4,c4,d4,e4 deuided into fiue equall portions4 ,and from eche pricke ij4 lines drawen to the other ij4 nexte prickes1 so from ,a4 are drawen ij4 lines1 one to ,b1 and the other to ,e1 and so from ,c4 one to ,b4 and an other to ,d1 and likewise of the reste4 ,so that you haue not only learned hereby how to make a sinkangle in anye circle1 but also how you shal make a like figure spedely1 whanne and where you will1 onlye drawinge the circle for the intente1 readylye to make the other figure 7,i meane the cinkangle7 thereby4 ,illustration ,,the ,,xxxix4 ,,conclvsion4 ,how to make a cinkangle of equall sides and equall angles about any circle appointed4 ,deuide firste the circle as you did in the last conclusion into fiue equall portions1 and draw fiue semidiameters in the circle4 ,then make fiue touche lines1 in suche sorte that euery touche line make two right angles with one of the semidiameters4 ,and those fiue touche lines will make a cinkangle of equall sides and equall angles4 ,illustration ,example4 ,a4,b4,c4,d4,e4 is the circle appointed1 which is deuided into fiue equal partes4 ,and vnto euery prycke is drawen a semidiameter1 as you see4 ,then doo ,i make a touche line in the pricke ,b1 whiche is ,f4,g1 making ij4 right angles with the semidiameter ,b1 and lyke waies on ,c4 is made ,g4,h1 on ,d4 standeth ,h4,k1 and on ,e1 is set ,k4,l1 so that of those 4v4 touche lynes are made the 4v4 sides of a cinkeangle1 accordyng to the conclusion4 ,an other waie4 ,another waie also maie you drawe a cinkeangle aboute a circle1 drawyng first a cinkeangle in the circle 7whiche is an easie thyng to doe1 by the doctrine of the 4xxxvij4 conclusion7 and then drawing 4v4 touche lines whiche shall be iuste paralleles to the 4v4 sides of the cinkeangle in the circle1 forseeyng that one of them do not crosse ouerthwarte an other and then haue you done4 ,the exaumple of this 7because it is easie7 ,i leaue to your owne exercise4 ,,the ,,xl4 ,,conclvsion4 ,to make a circle in any appointed cinkeangle of equall sides and equall corners4 ,drawe a plumbe line from any one corner of the cinkeangle1 vnto the middle of the side that lieth iuste against that angle4 ,and do likewaies in drawyng an other line from some other corner1 to the middle of the side that lieth against that corner also4 ,and those two lines wyll meete in crosse in the pricke of their crossyng1 shall you iudge the centre of the circle to be4 ,therfore set one foote of the compas in that pricke1 and extend the other to the end of the line that toucheth the middle of one side1 whiche you liste1 and so drawe a circle4 ,and it shall be iustly made in the cinkeangle1 according to the conclusion4 ,example4 ,the cinkeangle assigned is ,a4,b4,c4,d4,e1 in whiche ,i muste make a circle1 wherefore ,i draw a right line from the one angle 7as from ,b17 to the middle of the contrary side 7whiche is ,e4 ,d17 and that middle pricke is ,f4 ,then lykewaies from an other corner 7as from ,e7 ,i drawe a right line to the middle of the side that lieth against it 7whiche is ,b4,c47 and that pricke is ,g4 ,nowe because that these two lines do crosse in ,h1 ,i saie that ,h4 is the centre of the circle1 whiche ,i would make4 ,therfore ,i set one foote of the compasse in ,h1 and extend the other foote vnto ,g1 or ,f4 7whiche are the endes of the lynes that lighte in the middle of the side of that cinkeangle7 and so make ,i the circle in the cinkangle1 right as the conclusion meaneth4 ,illustration ,,the ,,xli4 ,,conclvsion ,to make a circle about any assigned cinkeangle of equall sides1 and equall corners4 ,drawe 4ij4 lines within the cinkeangle1 from 4ij4 corners to the middle on tbe 4ij4 contrary sides 7as the last conclusion teacheth7 and the pointe of their crossyng shall be the centre of the circle that ,i seke for4 ,then sette ,i one foote of the compas in that centre1 and the other foote ,i extend to one of the angles of the cinkangle1 and so draw ,i a circle about the cinkangle assigned4 ,example4 ,a4,b4,c4,d4,e1 is the cinkangle assigned1 about which ,i would make a circle4 ,therfore ,i drawe firste of all two lynes 7as you see7 one from ,e4 to ,g1 and the other from ,c4 to ,f1 and because thei do meete in ,h1 ,i saye that ,h4 is the centre of the circle that ,i woulde haue1 wherfore ,i sette one foote of the compasse in ,h4 and extende the other to one corner 7whiche happeneth fyrste1 for all are like distaunte from ,h47 and so make ,i a circle aboute the cinkeangle assigned4 ,illustration ,an other waye also4 ,another waye maye ,i do it1 thus presupposing any three corners of the cinkangle to be three prickes appointed1 vnto whiche ,i shoulde finde the centre1 and then drawinge a circle touchinge them all thre1 accordinge to the doctrine of the seuentene1 one and twenty1 and two and twenty conclusions4 ,and when ,i haue founde the centre1 then doo ,i drawe the circle as the same conclusions do teache1 and this forty conclusion also4 ,,the ,,xlii4 ,,conclvsion4 ,to make a siseangle of equall sides1 and equall angles1 in any circle assigned4 ,yf the centre of the circle be not knowen1 then seeke oute the centre according to the doctrine of the sixtenth conclusion4 ,and with your compas take the quantitee of the semidiameter iustly4 ,and then sette one foote in one pricke of the circumference of the circle1 and with the other make a marke in the circumference also towarde both sides4 ,then sette one foote of the compas stedily in eche of those new prickes1 and point out two other prickes4 ,and if you haue done well1 you shal perceaue that there will be but euen sixe such diuisions in the circumference4 ,whereby it dothe well appeare1 that the side of anye sisangle made in a circle1 is equalle to the semidiameter of the same circle4 ,example4 ,illustration ,the circle is ,b4,c4,d4,e4,f4,g1 whose centre ,i finde to bee ,a4 ,therefore ,i sette one foote of the compas in ,a1 and do extend the other foote to ,b1 thereby takinge the semidiameter4 ,then sette ,i one foote of the compas vnremoued in ,b1 and marke with the other foote on eche side ,c4 and ,g4 ,then from ,c4 ,i marke ,d1 and from ,d1 ,e3 from ,e4 marke ,i ,f4 ,and then haue ,i but one space iuste vnto ,g4 and so haue ,i made a iuste siseangle of equall sides and equall angles1 in a circle appointed4 ,,the ,,xliii4 ,,conclvsion4 ,to make a siseangle of equall sides1 and equall angles about any circle assigned4 ,,the ,,xliiii4 ,,conclvsion4 ,to make a circle in any siseangle appointed1 of equall sides and equal angles4 ,,the ,,xlv4 ,,conclvsion4 ,to make a circle about any sise angle limited of equall sides and equall angles4 ,bicause you maye easily coniecture the makinge of these figures by that that is saide before of cinkangles1 only consideringe that there is a difference in the numbre of sides1 ,i thought beste to leue these vnto your owne deuice1 that you should study in some thinges to exercise your witte withall and that you mighte haue the better occasion to perceaue what difference there is betwene eche twoo of those conclusions4 ,for thoughe it seeme one thing to make a siseangle in a circle1 and to make a circle about a siseangle1 yet shall you perceaue1 that is not one thinge1 nother are those twoo conclusions wrought one way4 ,likewaise shall you thinke of those other two conclusions4 ,to make a siseangle about a circle1 and to make a circle in a siseangle1 thoughe the figures be one in fashion1 when they are made1 yet are they not one in working1 as you may well perceaue by the xxxvij4 xxxviij4 xxxix4 and xl4 conclusions1 in whiche the same workes are taught1 touching a circle and a cinkangle1 yet this muche wyll ,i saye1 for your helpe in working1 that when you shall seeke the centre in a siseangle 7whether it be to make a circle in it other about it7 you shall drawe the two crosselines1 from one angle to the other angle that lieth againste it1 and not to the middle of any side1 as you did in the cinkangle4 ,,the ,,xlvi4 ,,conclvsion4 ,to make a figure of fifteene equall sides and angles in any circle appointed4 ,this rule is generall1 that how many sides the figure shall haue1 that shall be drawen in any circle1 into so many partes iustely muste the circles bee deuided4 ,and therefore it is the more easier woorke commonly1 to drawe a figure in a circle1 then to make a circle in an other figure4 ,now therefore to end this conclusion1 deuide the circle firste into fiue partes1 and then eche of them into three partes againe3 ,or els first deuide it into three partes1 and then ech of them into fiue other partes1 as you list1 and canne most readilye4 ,then draw lines betwene euery two prickes that be nighest togither1 and ther wil appear rightly drawen the figure1 of fiftene sides1 and angles equall4 ,and so do with any other figure of what numbre of sides so euer it bee4 ,,finis4 ,,the ,,second ,,booke ,,of ,,the ,,principles of ,geometry1 containing certaine ,theoremes1 whiche may be cal- led ,approued truthes4 ,and be as it were the moste certaine groundes1 wheron the practike conclusions of ,geometry ar founded4 ,leaf ,whervnto are annexed certaine declarations by examples1 for the right vnderstanding of the same1 to the ende that the simple reader might not iustly complain of hardnes or obscuritee1 and for the same cause ar the demonstra- tions and iust profes omitted1 vntill a more conueni- ent time4 #aeea4 ,if truthe maie trie it selfe1 ,by ,reasons prudent skyll1 ,if reason maie preuayle by right1 ,and rule the rage of will1 ,i dare the triall byde1 ,for truthe that ,i pretende4 ,and though some lyst at me repine1 ,iuste truthe shall me defende4 ,,the ,,preface ,,vnto the ,theoremes4 ,i ,doubt not gentle reader1 but as my argument is straunge and vnacquainted with the vulgare toungue1 so shall ,i of many men be straungly talked of1 and as straungly iudged4 ,some men will saye peraduenture1 ,i mighte haue better imployed my tyme in some pleasaunte historye1 comprisinge matter of chiualrye4 ,some other wolde more haue preised my trauaile1 if ,i hadde spente the like time in some morall matter1 other in deciding some controuersy of religion4 ,and yet some men 7as ,i iudg7 will not mislike this kind of mater1 but then will they wishe that ,i had vsed a more certaine order in placinge bothe the ,propositions and ,theoremes1 and also a more exacter proofe of eche of theim bothe1 by demonstrations mathematicall4 ,some also will mislike my shortenes and simple plainesse1 as other of other affections diuersely shall espye somwhat that they shall thinke blame worthy1 and shal misse somewhat1 that thei wold with to haue bene here vsed1 so that euerie manne shall giue his verdicte of me according to his phantasie1 vnto whome ioinctly1 ,i make this my firste answere3 that as they ar many and in opinions verie diuers1 so were it scarse possible to please them all with anie one argumente1 of what kinde so euer it were4 ,and for my seconde aunswere1 ,i saye thus4 ,that if annye one argumente mighte please them all1 then should thei be thankfull vnto me for this kind of matter4 ,for nother is there anie matter more straunge in the englishe tungue1 then this whereof neuer booke was written before now1 in that tungue1 and therefore oughte to delite all them1 that desire to vnderstand strange matters1 as most men commonlie doo4 ,and againe the practise is so pleasaunt in vsinge1 and so profitable in appliynge1 that who so euer dothe delite in anie of bothe1 ought not of right to mislike this arte4 ,and if any manne shall like the arte welle for it selfe1 but shall mislyke the fourme that ,i haue vsed in teachyng of it1 to hym ,i shall saie1 ,firste1 that ,i dooe wishe with hym that some other man1 whiche coulde better haue doone it1 hadde shewed his good will1 and vsed his diligence in suche sorte1 that ,i myght haue bene therby occasioned iustely to haue left of my laboure1 or after my trauaile to haue suppressed my bookes4 ,but sithe no manne hath yet attempted the like1 as far as ,i canne learne1 ,i truste all suche as bee not exercised in the studie of ,geometrye1 shall finde greate ease and furtheraunce by this simple1 plaine1 and easie forme of writinge4 ,and shall perceaue the exacte woorkes of ,theon1 and others that write on ,euclide1 a great deale the soner1 by this blunte delineacion afore hande to them taughte4 ,for ,i dare presuppose of them1 that thing which ,i haue sette in my selfe1 and haue marked in others1 that is to saye1 that it is not easie for a man that shall trauaile in a straunge arte1 to vnderstand at the beginninge bothe the thing that is taught and also the iuste reason whie it is so4 ,and by experience of teachinge ,i haue tried it to bee true1 for whenne ,i haue taughte the proposition1 as it is imported in meaninge1 and annexed the demonstration with all1 ,i didde perceaue that it was a greate trouble and a painefull vexacion of mynde to the learner1 to comprehend bothe those thinges at ones4 ,and therfore did ,i proue firste to make them to vnderstande the sence of the propositions1 and then afterward did they conceaue the demonstrations muche soner1 when they hadde the sentence of the propositions first ingrafted in their mindes4 ,this thinge caused me in bothe these bookes to omitte the demonstrations1 and to vse onlye a plaine forme of declaration1 which might best serue for the firste introduction4 ,whiche example hath beene vsed by other learned menne before nowe1 for not only ,georgius ,ioachimus ,rheticus1 but also ,boetius that wittye clarke did set forth some whole books of ,euclide1 without any demonstration or any other declaration at al4 ,but if ,i shal hereafter perceaue that it maie be a thankefull trauaile to sette foorth the propositions of geometrie with demonstrations1 ,i will not refuse to dooe it1 and that with sundry varietees of demonstrations1 bothe pleasaunt and profitable also4 ,and then will ,i in like maner prepare to sette foorth the other bookes1 whiche now are lefte vnprinted1 by occasion not so muche of the charges in cuttyng of the figures1 as for other iuste hynderances1 whiche ,i truste hereafter shall bee remedied4 ,in the meane season if any man muse why ,i haue sette the ,conclusions beefore the ,teoremes1 seynge many of the ,theoremes seeme to include the cause of some of the conclusions1 and therfore oughte to haue gone before them1 as the cause goeth before the effecte4 ,here vnto ,i saie1 that although the cause doo go beefore the effect in order of nature1 yet in order of teachyng the effect must be fyrst declared1 and than the cause therof shewed1 for so that men best vnderstand things ,first to lerne that such thinges ar to be wrought1 and secondarily what thei ar1 and what thei do import1 and than thirdly what is the cause therof4 ,an other cause why yt the theoremes be put after the conclusions is this1 whan ,i wrote these first connclusions 7which was 4iiiij4 yeres passed7 ,i thought not then to haue added any theoremes1 but next vnto ye conclusions to haue taught the order how to haue applied them to work1 for drawing of plottes such like vses4 ,but afterward considering the great commoditie yt thei serue for1 and the light that thei do geue to all sortes of practise geometricall1 besyde other more notable benefites1 whiche shall be declared more specially in a place conuenient1 ,i thoughte beste to geue you some taste of theym1 and the pleasaunt contemplation of suche geometrical propositions1 which might serue diuerselye in other bookes for the demonstrations and proofes of all ,geometricall woorkes4 ,and in theim1 as well as in the propositions1 ,i haue drawen in the ,linearie examples many tymes more lynes1 than be spoken of in the explication of them1 whiche is doone to this intent1 that yf any manne lyst to learne the demonstrations by harte1 7as somme learned men haue iudged beste to doo7 those same men should find the ,linearye exaumples to serue for this purpose1 and to wante no thyng needefull to the iuste proofe1 whereby this booke may bee wel approued to be more complete then many men wolde suppose it4 ,and thus for this tyme ,i wyll make an ende without any larger declaration of the commoditiees of this arte1 or any farther answeryng to that may bee obiected agaynst my handelyng of it1 wyllyng them that myslike it1 not to medle with it3 and vnto those that will not disdaine the studie of it1 ,i promise all suche aide as ,i shall be able to shewe for their farther procedyng both in the same1 and in all other commoditees that thereof maie ensue4 ,and for their incouragement ,i haue here annexed the names and brefe argumentes of suche bookes1 as ,i intende 7,god willyng7 shortly to sette forth1 if ,i shall perceaue that my paynes maie profyte other1 as my desyre is4 ,the brefe argumentes of suche bokes as ar appoynted shortly to be set forth by the author herof4 ,,the seconde part of ,arithmetike1 teachyng the workyng by fractions1 with extraction of rootes both square and cubike3 ,and declaryng the rule of allegation1 with sundrye plesaunt exaumples in metalles and other thynges4 ,also the rule of false position1 with dyuers examples not onely vulgar1 but some appertaynyng to the rule of ,algeber1 applied vnto quantitees partly rationall1 and partly surde4 ,,the arte of ,measuryng by the quadrate geometricall1 and the disorders committed by vsyng the same1 not only reueled but reformed also 7as muche as to the instrument pertayneth7 by the deuise of a new quadrate newely inuented by the author hereof4 ,,the arte of measuryng by the astronomers staffe1 and by the astronomers ryng1 and the form of makyng them both4 ,,the arte of makyng of ,dials1 bothe for the daie and the nyght1 with certayn new formes of fixed dialles for the moon and other for the sterres1 whiche may bee sette in glasse windowes to serue by daie and by night4 ,and howe you may by those dialles knowe in what degree of the ,zodiake not only the sonne1 but also the moone is4 ,and how many howrs old she is4 ,and also by the same dial to know whether any eclipse shall be that moneth1 of the sonne or of the moone4 ,the makyng and vse of an instrument1 wherby you maye not onely measure the distance at ones of all places that you can see togyther1 howe muche eche one is from you1 and euery one from other1 but also therby to drawe the plotte of any countreie that you shall come in1 as iustely as maie be1 by mannes diligence and labour4 ,,the vse bothe of the ,globe and the ,sphere1 and therin also of the arte of ,nauigation1 and what instrumentes serue beste thervnto1 and of the trew latitude and longitude of regions and townes4 ,euclides woorkes in foore partes1 with diuers demonstrations ,arithmeticall and ,geometricall or ,linearie4 ,the fyrst parte of platte formes4 ,the second of numbres and quantitees surde or irrationall4 ,the third of bodies and solide formes4 ,the fourthe of perspectiue1 and other thynges thereto annexed4 ,,beside these ,i haue other sundrye woorkes partely ended1 and partely to bee ended1 ,of the peregrination of man1 and the originall of al nations1 ,the state of tymes1 and mutations of realmes1 ,the image of a perfect common welth1 with diuers other woorkes in naturall sciences1 ,of the wonderfull workes and effectes in beastes1 plantes1 and minerals1 of whiche at this tyme1 ,i will omitte the argumentes1 beecause thei doo appertaine littel to this arte1 and handle other matters in an other sorte4 ,to haue1 or leaue1 ,nowe maie you chuse1 ,no paine to please1 ,will ,i refuse4 ,the ,theoremes of ,geometry1 before ,,whiche ,,are ,,set ,,forthe certaine grauntable requestes which serue for demonstrations ,mathematicall4 ,sidenote3 ,i4 ,that from any pricke to one other1 there may be drawen a right line4 ,as for example ,a--------,b4 ,a4 being the one pricke1 and ,b4 the other1 you maye drawe betwene them from the one to the other1 that is to say1 frome ,a4 vnto ,b1 and from ,b4 to ,a4 ,sidenote3 ,,ii4 ,that any right line of measurable length may be drawen forth longer1 and straight4 ,illustration ,example of ,a4,b1 which as it is a line of measurable lengthe1 so may it be drawen forth farther1 as for example vnto ,c1 and that in true streightenes without crokinge4 ,sidenote3 ,,iii4 ,illustration ,that vpon any centre1 there may be made a circle of anye quantitee that a man wyll4 ,let the centre be set to be ,a1 what shal hinder a man to drawe a circle aboute it1 of what quantitee that he lusteth1 as you se the forme here3 other bygger or lesse1 as it shall lyke him to doo3 ,that all right angles be equall eche to other4 ,illustration ,set for an example ,a4 and ,b1 of which two though ,a4 seme the greatter angle to some men of small experience1 it happeneth only bicause that the lines aboute ,a1 are longer then the lines about ,b1 as you may proue by drawing them longer1 for so that ,b4 seme the greater angle yf you make his lines longer then the lines that make the angle ,a4 ,and to proue it by demonstration1 ,i say thus4 ,if any ij4 right corners be not equal1 then one right corner is greater then an other1 but that corner which is greatter then a right angle1 is a blunt corner 7by his definition7 so must one corner be both a right corner and a blunt corner also1 which is not possible3 ,and againe3 the lesser right corner must be a sharpe corner1 by his definition1 bicause it is lesse then a right angle4 which thing is impossible4 ,therefore ,i conclude that all right angles be equall4 ,yf one right line do crosse two other right lines1 and make ij4 inner corners of one side lesser then ij4 righte corners1 it is certaine1 that if those two lines be drawen forth right on that side that the sharpe inner corners be1 they wil at length mete togither1 and crosse on an other4 ,illustration ,the ij4 lines beinge as ,a4,b4 and ,c4,d1 and the third line crossing them as dooth heere ,e4,f1 making ij inner cornes 7as ar ,g4,h47 lesser then two right corners1 sith ech of them is lesse then a right corner1 as your eyes maye iudge1 then say ,i1 if those ij4 lines ,a4,b4 and ,c4,d4 be drawen in lengthe on that side that ,g4 and ,h4 are1 the will at length meet and crosse one an other4 ,two right lines make no platte forme4 ,illustration ,a platte forme1 as you harde before1 hath bothe length and bredthe1 and is inclosed with lines as with his boundes1 but ij4 right lines cannot inclose al the bondes of any platte forme4 ,take for an example firste these two right lines ,a4,b4 and ,a4,c4 whiche meete togither in ,a1 but yet cannot be called a platte forme1 bicause there is no bond from ,b4 to ,c1 but if you will drawe a line betwene them twoo1 that is frome ,b4 to ,c1 then will it be a platte forme1 that is to say1 a triangle1 but then are there iij4 lines1 and not only ij4 ,likewise may you say of ,d4,e4 and ,f4,g1 whiche doo make a platte forme1 nother yet can they make any without helpe of two lines more1 whereof the one must be drawen from ,d4 to ,f1 and the other frome ,e4 to ,g1 and then will it be a longe rquare4 ,so then of two right lines can bee made no platte forme4 ,but of ij4 croked lines be made a platte forme1 as you se in the eye form4 ,and also of one right line1 one croked line1 maye a platte fourme bee made1 as the semicircle ,f4 doothe sette forth4 ,certayn common sentences manifest to sence1 and acknowledged of all men4 ,the firste common sentence4 ,what so euer things be equal to one other thinge1 those same bee equall betwene them selues4 ,illustration ,examples therof you may take both in greatnes and also in numbre4 ,first 7though it pertaine not proprely to geometry1 but to helpe the vnderstandinge of the rules1 whiche may bee wrought by bothe artes7 thus may you perceaue4 ,if the summe of monnye in my purse1 and the mony in your purse be equall eche of them to the mony that any other man hathe1 then must needes your mony and mine be equall togyther4 ,likewise1 if anye ij4 quantities1 as ,a4 and ,b1 be equal to an other1 as vnto ,c1 then muste nedes ,a4 and ,b4 be equall eche to other1 as ,a4 equall to ,b1 and ,b4 equall to ,a1 whiche thinge the better to perceaue1 tourne these quantities into numbre1 so shall ,a4 and ,b4 make sixteene1 and ,c4 as many4 ,as you may perceaue by multipliyng the numbre of their sides togither4 ,the seconde common sentence4 ,and if you adde equall portions to thinges that be equall1 what so amounteth of them shall be equall4 ,example1 ,yf you and ,i haue like summes of mony1 and then receaue eche of vs like summes more1 then our summes wil be like styll4 ,also if ,a4 and ,b4 7as in the former example7 bee equall1 then by adding an equal portion to them both1 as to ech of them1 the quarter of ,a4 7that is foure7 they will be equall still4 ,the thirde common sentence4 ,and if you abate euen portions from things that are equal1 those partes that remain shall be equall also4 ,this you may perceaue by the last example4 ,for that that was added there1 is subtracted heere4 and so the one doothe approue the other4 ,the fourth common sentence4 ,if you abate equalle partes from vnequal thinges1 the remainers shall be vnequall4 ,as bicause that a hundreth and eight and forty be vnequal if ,i take tenne from them both1 there will remaine nynetye and eight and thirty1 which are also vnequall4 and likewise in quantities it is to be iudged4 ,the fifte common sentence4 ,when euen portions are added to vnequalle thinges1 those that amounte shalbe vnequall4 ,so if you adde twenty to fifty1 and lyke ways to nynty1 you shall make seuenty and a hundred and ten whiche are no lesse vnequall1 than were fifty and nynty4 ,the syxt common sentence4 ,if two thinges be double to any other1 those same two thinges are equal togither4 ,illustration ,bicause ,a4 and ,b4 are eche of them double to ,c1 therefore must ,a4 and ,b4 nedes be equall togither4 ,for as v4 times viij4 maketh xl4 which is double to iiij4 times v1 that is xx so iiij4 times x1 likewise is double to xx4 7for it maketh fortie7 and therefore muste neades be equall to forty4 ,the seuenth common sentence4 ,if any two thinges be the halfes of one other thing1 then are thei 4ij4 equall togither4 ,so are ,d4 and ,c4 in the laste example equal togyther1 bicause they are eche of them the halfe of ,a4 other of ,b1 as their numbre declareth4 ,the eyght common sentence4 ,if any one quantitee be laide on an other1 and thei agree1 so that the one excedeth not the other1 then are they equall togither4 ,illustration ,as if this figure ,a4,b4,c1 be layed on that other ,d4,e4,f1 so that ,a4 be layed to ,d1 ,b4 to ,e1 and ,c4 to ,f1 you shall see them agre in sides exactlye and the one not to excede the other1 for the line ,a4,b4 is equall to ,d4,e1 and the third lyne ,c4,a1 is equall to ,f4,d so that eueryside in the one is equall to some one side of the other4 ,wherfore it is playne1 that the two triangles are equall togither4 ,the nynth common sentence4 ,euery whole thing is greater than any of his partes4 ,this sentence nedeth none example4 ,for the thyng is more playner then any declaration1 yet considering that other common sentence that foloweth nexte that4 ,the tenthe common sentence4 ,euery whole thinge is equall to all his partes taken togither4 ,illustration ,illustration ,it shall be mete to expresse both wt one example1 for of thys last sentence many men at the first hearing do make a doubt4 ,therfore as in this example of the circle deuided into sundry partes it doeth appere that no parte can be so great as the whole circle1 7accordyng to the meanyng of the eight sentence7 so yet it is certain1 that all those eight partes together be equall vnto the whole circle4 ,and this is the meanyng of that common sentence 7whiche many vse1 and fewe do rightly vnderstand7 that is1 that ,all the partes of any thing are nothing els1 but the whole4 ,and contrary waies3 ,the whole is nothing els1 but all his partes taken togither4 whiche saiynges some haue vnderstand to meane thus3 that all the partes are of the same kind that the whole thyng is3 but that that meanyng is false1 it doth plainly appere by this figure ,a4,b1 whose partes ,a4 and ,b1 are triangles1 and the whole figure is a square1 and so are they not of one kind4 ,but and if they applie it to the matter or substance of thinges 7as some do7 then it is most false1 for euery compound thyng is made of partes of diuerse matter and substance4 ,take for example a man1 a house1 a boke1 and all other compound thinges4 ,some vnderstand it thus1 that the partes all together can make none other forme1 but that that the whole doth shewe1 whiche is also false1 for ,i maie make fiue hundred diuerse figures of the partes of some one figure1 as you shall better perceiue in the third boke4 ,and in the meane season take for an example this square figure following ,a4,b4,c4,d1 wch is deuided but in two parts1 and yet 7as you se7 ,i haue made fiue figures more beside the firste1 with onely diuerse ioynyng of those two partes4 ,but of this shall ,i speake more largely in an other place4 ,in the mean season content your self with these principles1 whiche are certain of the chiefe groundes wheron all demonstrations mathematical are fourmed1 of which though the moste parte seeme so plaine1 that no childe doth doubte of them1 thinke not therfore that the art vnto whiche they serue1 is simple1 other childishe1 but rather consider1 howe certayne the profes of that arte is1 yt hath for his groundes soche playne truthes1 as ,i may say1 suche vndowbtfull and sensible principles1 ,and this is the cause why all learned menne dooth approue the certenty of geometry1 and consequently of the other artes mathematical1 which haue the grounds 7as ,arithmeticke1 musike and astronomy7 aboue all other artes and sciences1 that be vsed amongest men4 ,thus muche haue ,i sayd of the first principles1 and now will ,i go on with the theoremes1 whiche ,i do only by examples declare1 minding to reserue the proofes to a peculiar boke which ,i will then set forth1 when ,i perceaue this to be thankfully taken of the readers of it4 ,illustration ,the theoremes of ,geometry brieflye declared by shorte examples4 ,the firste ,theoreme4 ,when 4ij4 triangles be so drawen1 that the one of them hath ij4 sides equal to ij sides of the other triangle1 and that the angles enclosed with those sides1 bee equal also in bothe triangles1 then is the thirde side likewise equall in them4 ,and the whole triangles be of one greatnes1 and euery angle in the one equall to his matche angle in the other1 ,i meane those angles that be inclosed with like sides4 ,example4 ,illustration ,this triangle ,a4,b4,c4 hath ij4 sides 7that is to say7 ,c4,a4 and ,c4,b1 equal to ij4 sides of the other triangle ,f4,g4,h1 for ,a4,c4 is equall to ,f4,g1 and ,b4,c4 is equall to ,g4,h4 ,and also the angle ,c4 contayned beetweene ,f4,g1 and ,g4,h1 for both of them answere to the eight parte of a circle4 ,therfore doth it remayne that ,a4,b4 whiche is the thirde lyne in the firste triangle1 doth agre in lengthe with ,f4,h1 wch is the third line in ye second triangle ye hole triangle4 ,a4,b4,c4 must nedes be equal to ye hole triangle ,f4,g4,h4 ,and euery corner equall to his match1 that is to say1 ,a4 equall to ,f1 ,b4 to ,h1 and ,c4 to ,g1 for those bee called match corners1 which are inclosed with like sides1 other els do lye against like sides4 ,the second ,theoreme4 ,in twileke triangles the ij4 corners that be about the ground line1 are equal togither4 ,and if the sides that be equal1 be drawen out in length then wil the corners that are vnder the ground line1 be equal also togither4 ,example ,illustration ,a4,b4,c4 is a twileke triangle1 for the one side ,a4,c1 is equal to the other side ,b4,c4 ,and therfore ,i saye that the inner corners ,a4 and ,b1 which are about the ground lines1 7that is ,a4,b47 be equall togither4 ,and farther if ,c4,a4 and ,c4,b4 bee drawen forthe vnto ,d4 and ,e4 as you se that ,i haue drawen them1 then saye ,i that the two vtter angles vnder ,a4 and ,b1 are equal also togither3 as the theorem said4 ,the profe wherof1 as of al the rest1 shal apeare in ,euclide1 whome ,i intende to set foorth in english with sondry new additions1 if ,i may perceaue that it wilbe thankfully taken4 ,the thirde ,theoreme4 ,if in annye triangle there bee twoo angles equall togither1 then shall the sides1 that lie against those angles1 be equal also4 ,illustration ,example4 ,this triangle ,a4,b4,c4 hath two corners equal eche to other1 that is ,a4 and ,b1 as ,i do by supposition limite1 wherfore it foloweth that the side ,a4,c1 is equal to that other side ,b4,c1 for the side ,a4,c1 lieth againste the angle ,b1 and the side ,b4,c1 lieth against the angle ,a4 ,the fourth ,theoreme4 ,when two lines are drawen from the endes of anie one line1 and meet in anie pointe1 it is not possible to draw two other lines of like lengthe ech to his match that shal begin at the same pointes1 and end in anie other pointe then the twoo first did4 ,example4 ,illustration ,the first line is ,a4,b1 on which ,i haue erected two other lines ,a4,c1 and ,b4,c1 that meete in the pricke ,c1 wherefore ,i say1 it is not possible to draw ij4 other lines from ,a4 and ,b4 which shal mete in one point 7as you se ,a4,d4 and ,b4,d4 mete in ,d47 but that the match lines shalbe vnequal1 ,i mean by match lines1 the two lines on one side1 that is the ij4 on the right hand1 or the ij4 on the lefte hand1 for as you se in this example ,a4,d4 is longer then ,a4,c1 and ,b4,c4 is longer then ,b4,d4 ,and it is not possible1 that ,a4,c4 and ,a4,d4 shall bee of one lengthe1 if ,b4,d4 and ,b4,c4 bee like longe4 ,for if one couple of matche lines be equall 7as the same example ,a4,e4 is equall to ,a4,c4 in length7 then must ,b4,e4 needes be vnequall to ,b4,c4 as you see1 it is here shorter4 ,the fifte ,theoreme4 ,if two triangles haue there ij4 sides equal one to an other1 and their ground lines equal also1 then shall their corners1 whiche are contained betwene like sides1 be equall one to the other4 ,example4 ,illustration ,because these two triangles ,a4,b4,c1 and ,d4,e4,f4 haue two sides equall one to an other4 ,for ,a4,c4 is equall to ,d4,f1 and ,b4,c4 is equall to ,e4,f1 and again their ground lines ,a4,b4 and ,d4,e4 are lyke in length1 therfore is eche angle of the one triangle equall to ech angle of the other1 comparyng together those angles that are contained within lyke sides1 so is ,a4 equall to ,d1 ,b4 to ,e1 and ,c4 to ,f1 for they are contayned within like sides1 as before is said4 ,the sixt ,theoreme4 ,when any right line standeth on an other1 the ij4 angles that thei make1 other are both right angles1 or els equall to 4ij4 righte angles4 ,example4 ,illustration ,a4,b4 is a right line1 and on it there doth light another right line1 drawen from ,c4 perpendicularly on it1 therefore saie ,i1 that the 4ij4 angles that thei do make1 are 4ij4 right angles as maie be iudged by the definition of a right angle4 ,but in the second part of the example1 where ,a4,b4 beyng still the right line1 on which ,d4 standeth in slope wayes1 the two angles that be made of them are not righte angles1 but yet they are equall to two righte angles1 for so muche as the one is to greate1 more then a righte angle1 so muche iuste is the other to little1 so that bothe togither are equall to two right angles1 as you maye perceiue4 ,the seuenth ,theoreme4 ,if 4ij4 lines be drawen to any one pricke in an other lyne1 and those 4ij4 lines do make with the fyrst lyne1 two right angles1 other suche as be equall to two right angles1 and that towarde one hande1 than those two lines doo make one streyght lyne4 ,example4 ,illustration ,a4,b4 is a streyght lyne1 on which there doth lyght two other lines one frome ,d1 and the other frome ,c1 but considerynge that they meete in one pricke ,e1 and that the angles on one hand be equal to two right corners 7as the laste theoreme dothe declare7 therfore maye ,d4,e4 and ,e4,c4 be counted for one ryght lyne4 ,the eight ,theoreme4 ,when two lines do cut one an other crosseways they do make their matche angles equall4 ,illustration ,example4 ,what matche angles are1 ,i haue tolde you in the definitions of the termes4 ,and here ,a1 and ,b4 are matche corners in this example1 as are also ,c4 and ,d1 so that the corner ,a1 is equall to ,b1 and the angle ,c1 is equall to ,d4 ,the nynth ,theoreme4 ,whan so euer in any triangle the line of one side is drawen forthe in lengthe1 that vtter angle is greater than any of the two inner corners1 that ioyne not with it4 ,example4 ,illustration ,the triangle ,a4,d4,c hathe hys grounde lyne ,a4,c4 drawen forthe in lengthe vnto ,b1 so that the vtter corner that it maketh at ,c1 is greater then any of the two inner corners that lye againste it1 and ioyne not wyth it1 whyche are ,a4 and ,d1 for they both are lesser then a ryght angle1 and be sharpe angles1 but ,c4 is a blonte angle1 and therfore greater then a ryght angle4 ,the tenth ,theoreme4 ,in euery triangle any 4ij4 corners1 how so euer you take them1 ar lesse then ij4 right corners4 ,example4 ,illustration ,in the firste triangle ,e1 whiche is a threlyke1 and therfore hath all his angles sharpe1 take anie twoo corners that you will1 and you shall perceiue that they be lesser then ij4 right corners1 for in euery triangle that hath all sharpe corners 7as you see it to be in this example7 euery corner is lesse then a right corner4 ,and therfore also euery two corners must nedes be lesse then two right corners4 ,furthermore in that other triangle marked with ,m1 whiche hath 4ij4 sharpe corners and one right1 any 4ij4 of them also are lesse then two right angles4 ,for though you take the right corner for one1 yet the other whiche is a sharpe corner1 is lesse then a right corner4 ,and so it is true in all kindes of triangles1 as you maie perceiue more plainly by the 4xxij4 ,theoreme4 ,the 4xi4 ,theoreme4 ,in euery triangle1 the greattest side lieth against the greattest angle4 ,example4 ,illustration ,as in this triangle ,a4,b4,c1 the greattest angle is ,c4 ,and ,a4,b4 7whiche is the side that lieth against it7 is the greatest and longest side4 ,and contrary waies1 as ,a4,c4 is the shortest side1 so ,b4 7whiche is the angle liyng against it7 is the smallest and sharpest angle1 for this doth folow also1 that is the longest side lyeth against the greatest angle1 so it that foloweth ,the twelft ,theoreme4 ,in euery triangle the greattest angle lieth against the longest side4 ,for these ij4 theoremes are one in truthe4 ,the thirtenth theoreme4 ,in euerie triangle anie ij4 sides togither how so euer you take them1 are longer then the thirde4 ,illustration ,for example you shal take this triangle ,a4,b4,c4 which hath a very blunt corner1 and therfore one of his sides greater a good deale then any of the other1 and yet the ij4 lesser sides togither ar greater then it4 ,and if it bee so in a blunte angeled triangle1 it must nedes be true in all other1 for there is no other kinde of triangles that hathe the one side so greate aboue the other sids1 as thei yt haue blunt corners4 ,the fourtenth theoreme4 ,if there be drawen from the endes of anie side of a triangle 4ij4 lines metinge within the triangle1 those two lines shall be lesse then the other twoo sides of the triangle1 but yet the corner that thei make1 shall bee greater then that corner of the triangle1 whiche standeth ouer it4 ,example4 ,illustration ,a4,b4,c4 is a triangle4 on whose ground line ,a4,b4 there is drawen ij4 lines1 from the ij4 endes of it1 ,i say from ,a4 and ,b1 and they meete within the triangle in the pointe ,d1 wherfore ,i say1 that as those two lynes ,a4,d4 and ,b4,d1 are lesser then ,a4,c4 and ,b4,c1 so the angle ,d1 is greatter then the angle ,c1 which is the angle against it4 ,the fiftenth ,theoreme4 ,if a triangle haue two sides equall to the two sides of an other triangle1 but yet the angle that is contained betwene those sides1 greater then the like angle in the other triangle1 then is his grounde line greater then the grounde line of the other triangle4 ,illustration ,example4 ,a4,b4,c4 is a triangle1 whose sides ,a4,c4 and ,b4,c1 are equall to ,e4,d4 and ,d4,f1 the two sides of the triangle ,d4,e4,f1 but bicause the angle in ,d1 is greatter then the angle ,c4 7whiche are the ij4 angles contayned betwene the equal lynes7 therfore muste the ground line ,e4,f4 nedes bee greatter thenne the grounde line ,a4,b1 as you se plainely4 ,illustration ,the xvi4 ,theoreme4 ,if a triangle haue twoo sides equalle to the two sides of an other triangle1 but yet hathe a longer ground line then that other triangle1 then is his angle that lieth betwene the equall sides1 greater then the like corner in the other triangle4 ,example4 ,this ,theoreme is nothing els1 but the sentence of the last ,theoreme turned backward1 and therfore nedeth none other profe nother declaration1 then the other example4 ,the seuententh ,theoreme4 ,if two triangles be such sort1 that two angles of the one be equal to ij4 angles of the other1 and that one side of the one be equal to on side of the other1 whether that side do adioyne to one of the equall corners1 or els lye againste one of them1 then shall the other twoo sides of those triangles bee equalle togither1 and the thirde corner also shall be equall in those two triangles4 ,example4 ,illustration ,bicause that ,a4,b4,c1 the one triangle hath two corners ,a4 and ,b1 equal to ,d4,e1 that are twoo corners of the other triangle4 ,d4,e4,f4 and that they haue one side in theym bothe equall1 that is ,a4,b1 which is equall to ,d4,e1 therefore shall both the other ij4 sides be equall one to an other1 as ,a4,c4 and ,b4,c4 equall to ,d4,f4 and ,e4,f1 and also the thirde angle in them both shal be equall1 that is1 the angle ,c4 shal be equall to the angle ,f4 ,the eightenth ,theoreme4 ,when on ij4 right lines ther is drawen a third right line crosse waies1 and maketh 4ij4 matche corners of the one line equall to the like twoo matche corners of the other line1 then ar those two lines gemmow lines1 or paralleles4 ,example4 ,illustration ,the 4ij4 fyrst lynes are ,a4,b4 and ,c4,d1 the thyrd lyne that crosseth them is ,e4,f4 ,and bycause that ,e4,f4 maketh ij4 matche angles with ,a4,b1 equall to 4ij4 other lyke matche angles on ,c4,d1 7that is to say ,e4,g1 equall to ,k4,f1 and ,m4,n4 equall also to ,h4,l47 therfore are those ij4 lynes ,a4,b4 and ,c4,d4 gemow lynes1 vnderstand here by lyke matche corners1 those that go one way as doth ,e4,g1 and ,k4,f1 lyke ways ,n4,m1 and ,h4,l1 for as ,e4,g4 and ,h4,l1 other ,n4,m4 and ,k4,f4 go not one waie1 so be not they lyke match corners4 ,the nyntenth ,theoreme4 ,when on two right lines there is drawen a thirde right line crossewaies1 and maketh the ij4 ouer corners towarde one hande equall togither1 then ar those 4ij4 lines paralleles4 ,and in like maner if two inner corners toward one hande1 be equall to 4ii4 right angles4 ,example4 ,as the ,theoreme dothe speake of 4ij4 ouer angles1 so muste you vnderstande also of 4ij4 nether angles1 for the iudgement is lyke in bothe4 ,take for example the figure of the last theoreme1 where ,a4,b1 and ,c4,d1 be called paralleles also1 bicause ,e4 and ,k1 7whiche are 4ij4 ouer corners7 are equall1 and lykewaies ,l4 and ,m4 ,and so are in lyke maner the nether corners ,n4 and ,h1 and ,g4 and ,f4 ,nowe to the seconde parte of the theoreme1 those 4ij4 lynes ,a4,b4 and ,c4,d1 shall be called paralleles1 because the ij4 inner corners4 ,as for example those two that bee toward the right hande 7that is ,g4 and ,l47 are equall 7by the fyrst parte of this nyntenth theoreme7 therfore muste ,g4 and ,l4 be equall to two ryght angles4 ,the xx4 ,theoreme4 ,when a right line is drawen crosse ouer 4ij4 right gemow lines1 it maketh 4ij4 matche corners of the one line1 equall to two matche corners of the other line1 and also bothe ouer corners of one hande equall togither1 and bothe nether corners like waies1 and more ouer two inner corners1 and two vtter corners also towarde one hande1 equall to two right angles4 ,example4 ,bycause ,a4,b4 and ,c4,d1 7in the laste figure7 are paralleles1 therefore the two matche corners of the one lyne1 as ,e4,g4 be equall vnto the 4ij4 matche corners of the other line1 that is ,k4,f1 and lykewaies ,m4,n1 equall to ,h4,l4 ,and also ,e4 and ,k4 bothe ouer corners of the lefte hande equall togyther1 and so are ,m4 and ,l1 the two ouer corners on the ryghte hande1 in lyke maner ,n4 and ,h1 the two nether corners on the lefte hande1 equall eche to other1 and ,g4 and ,f4 the two nether angles on the right hande equall togither4 ,sec4 ,farthermore yet ,g4 and ,l4 the 4ij4 inner angles on the right hande bee equall to two right angles1 and so are ,m4 and ,f4 the 4ij4 vtter angles on the same hande1 in lyke manner shall you say of ,n4 and ,k4 the two inner corners on the left hand4 and of ,e4 and ,h4 the two vtter corners on the same hande4 ,and thus you see the agreable sentence of these 4iii4 theoremes to tende to this purpose1 to declare by the angles how to iudge paralleles1 and contrary waies howe you may by paralleles iudge the proportion of the angles4 ,the xxi4 ,theoreme4 ,what so euer lines be paralleles to any other line1 those same be paralleles togither4 ,example4 ,illustration ,a4,b4 is a gemow line1 or a parallele vnto ,c4,d4 ,and ,e4,f1 lykewaies is a parallele vnto ,c4,d4 ,wherfore it foloweth1 that ,a4,b4 must nedes bee a parallele vnto ,e4,f4 ,the 4xxij4 theoreme4 ,in euery triangle1 when any side is drawen forth in length1 the vtter angle is equall to the ij4 inner angles that lie againste it4 ,and all iij4 inner angles of any triangle are equall to ij4 right angles4 ,illustration ,example4 ,the triangle beeyng ,a4,d4,e4 and the syde ,a4,e4 drawen foorthe vnto ,b1 there is made an vtter corner1 whiche is ,c1 and this vtter corner ,c1 is equall to bother the inner corners that lye agaynst it1 whyche are ,a4 and ,d4 ,and all thre inner corners1 that is to say1 ,a4,d4 and ,e1 are equall to two ryght corners1 whereof it foloweth1 that all the three corners of any one triangle are equall to all the three corners of euerye other triangle4 ,for what so euer thynges are equalle to anny one thyrde thynge1 those same are equalle togitther1 by the fyrste common sentence1 so that bycause all the 4iij4 angles of euery triangle are equall to two ryghte angles1 and all ryghte angles bee equall togyther 7by the fourth request7 therfore must it nedes folow1 that all the thre corners of euery triangle 7accomptyng them togyther7 are equall to iij4 corners of any triangle1 taken all togyther4 ,the 4xxiii4 theoreme4 ,when any ij4 right lines doth touche and couple 4ij4 other righte lines1 whiche are equall in length and paralleles1 and if those 4ij4 lines bee drawen towarde one hande1 then are thei also equall together1 and paralleles4 ,example4 ,illustration ,a4,b4 and ,c4,d4 are ij4 ryght lynes and paralleles and equall in length1 and they ar touched and ioyned togither by ij4 other lynes ,a4,c4 and ,b4,d1 this beyng so1 and ,a4,c4 and ,b4,d4 beyng drawen towarde one syde 7that is to saye1 bothe towarde the lefte hande7 therefore are ,a4,c4 and ,b4,d4 bothe equall and also paralleles4 ,the 4xxiiij4 theoreme4 ,in any likeiamme the two contrary sides ar equall togither1 and so are eche 4ij4 contrary angles1 and the bias line that is drawen in it1 dothe diuide it into two equall portions4 ,example4 ,illustration ,here ar two likeiammes ioyned togither1 the one is a longe square ,a4,b4,e1 and the other is a losengelike ,d4,c4,e4,f4 which ij4 likeiammes ar proued equall togither1 bycause they haue one ground line1 that is1 ,f4,e1 ,and are made betwene one payre of gemow lines1 ,i meane ,a4,d4 and ,e4,h4 ,by this ,theoreme may you know the arte of the righte measuringe of likeiammes1 as in my booke of measuring ,i wil more plainly declare4 ,the xxvi4 ,theoreme4 ,all likeiammes that haue equal grounde lines and are drawen betwene one paire of paralleles1 are equal togither4 ,example4 ,fyrste you muste marke the difference betwene this ,theoreme and the laste1 for the laste ,theoreme presupposed to the diuers likeiammes one ground line common to them1 but this theoreme doth presuppose a diuers ground line for euery likeiamme1 only meaning them to be equal in length1 though they be diuers in numbre4 ,as for example4 ,in the last figure ther are two parallels1 ,a4,d4 and ,e4,h1 and betwene them are drawen thre likeiammes1 the firste is1 ,a4,b4,e4,f1 the second is ,e4,c4,d4,f1 and the thirde is ,c4,g4,h4,d4 ,the firste and the seconde haue one ground line1 7that is ,e4,f47 and therfore in so muche as they are betwene one paire of paralleles1 they are equall accordinge to the fiue and twentye ,theoreme1 but the thirde likeiamme that is ,c4,g4,h4,d4 hathe his grounde line ,g4,h1 seuerall frome the other1 but yet equall vnto it4 wherefore the third likeiam is equall to the other two firste likeiammes4 ,and for a proofe that ,g4,h4 being the ground or ground line of the third likeiamme1 is equal to ,e4,f1 whiche is the ground line to both the other likeiams1 that may be thus declared1 ,g4,h4 is equall to ,c4,d1 seynge they are the contrary sides of one likeiamme 7by the foure and twenty theoreme7 and so are ,c4,d4 and ,e4,f4 by the same theoreme4 ,therfore seynge both those ground lines ,e4,f4 and ,g4,h1 are equall to one thirde line 7that is ,c4,d47 they must nedes bee equall togyther by the firste common sentence4 ,the xxvii4 ,theoreme4 ,all triangles hauinge one grounde lyne1 and standing betwene one paire of parallels1 ar equall togither4 ,example4 ,illustration ,a4,b4 and ,c4,f4 are twoo gemowe lines1 betweene which there be made two triangles1 ,a4,d4,e4 and ,d4,e4,b1 so that ,d4,e1 is the common ground line to them bothe4 wherfore it doth folow1 that those two triangles ,a4,d4,e4 and ,d4,e4,b4 are equall eche to other4 ,the xxviij4 ,theoreme4 ,all triangles that haue like long ground lines1 and bee made betweene one paire of gemow lines1 are equall togither4 ,example4 ,example of this ,theoreme you may see in the last figure1 where as sixe triangles made betwene those two gemowe lines ,a4,b4 and ,c4,f1 the first triangle is ,a4,c4,d1 the seconde is ,a4,d4,e1 the thirde is ,a4,d4,b1 the fourth is ,a4,b4,e1 the fifte is ,d4,e4,b1 and the sixte is ,b4,e4,f1 of which sixe triangles1 ,a4,d4,e4 and ,d4,e4,b4 are equall1 bicause they haue one common grounde line4 ,and so likewise ,a4,b4,e4 and ,a4,b4,d1 whose commen grounde line is ,a4,b1 but ,a4,c4,d4 is equal to ,b4,e4,f1 being both betwene one couple of parallels1 not bicause thei haue one ground line1 but bicause they haue their ground lines equall1 for ,c4,d4 is equall to ,e4,f1 as you may declare thus4 ,c4,d1 is equall to ,a4,b4 7by the foure and twenty ,theoreme7 for thei are two contrary sides of one lykeiamme4 ,a4,c4,d4,b1 and ,e4,f by the same theoreme1 is equall to ,a4,b1 for thei ar the two ye contrary sides of the likeiamme1 ,a4,e4,f4,b1 wherfore ,c4,d4 must needes be equall to ,e4,f4 like wise the triangle ,a4,c4,d1 is equal to ,a4,b4,e1 bicause they ar made betwene one paire of parallels and haue their groundlines like1 ,i meane ,c4,d4 and ,a4,b4 ,againe ,a4,d4,e1 is equal to eche of them both1 for his ground line ,d4,e1 is equall to ,a4,b1 inso muche as they are the contrary sides of one likeiamme1 that is the long square ,a4,b4,d4,e4 ,and thus may you proue the equalnes of all the reste4 ,the xxix4 ,theoreme4 ,al equal triangles that are made on one grounde line1 and rise one waye1 must needes be betwene one paire of parallels4 ,example4 ,take for example ,a4,d4,e1 and ,d4,e4,b1 which 7as the xxvij4 conclusion dooth proue7 are equall togither1 and as you see1 they haue one ground line ,d4,e4 ,and againe they rise towarde one side1 that is to say1 vpwarde toward the line ,a4,b1 wherfore they must needes be inclosed betweene one paire of parallels1 which are heere in this example ,a4,b4 and ,d4,e4 ,the thirty ,theoreme4 ,equal triangles that haue their ground lines equal1 and be drawen toward one side1 ar made betwene one paire of paralleles4 ,example4 ,the example that declared the last theoreme1 maye well serue to the declaracion of this also4 ,for those ij4 theoremes do diffre but in this one pointe1 that the laste theoreme meaneth of triangles1 that haue one ground line common to them both1 and this theoreme dothe presuppose the grounde lines to bee diuers1 but yet of one length1 as ,a4,c4,d1 and ,b4,e4,f1 as they are ij4 equall triangles approued1 by the eighte and twentye ,theorem1 so in the same ,theorem it is declared1 yt their ground lines are equall togither1 that is ,c4,d1 and ,e4,f1 now this beeynge true1 and considering that they are made towarde one side1 it foloweth1 that they are made betwene one paire of parallels when ,i saye1 drawen towarde one side1 ,i meane that the triangles must be drawen other both vpward frome one parallel1 other els both downward1 for if the one be drawen vpward and the other downward1 then are they drawen betwene two paire of parallels1 presupposinge one to bee drawen by their ground line1 and then do they ryse toward contrary sides4 ,the xxxi4 theoreme4 ,if a likeiamme haue one ground line with a triangle1 and be drawen betwene one paire of paralleles1 then shall the likeiamme be double to the triangle4 ,example4 ,illustration ,a4,h4 and ,b4,g4 are 4ij4 gemow lines1 betwene which there is made a triangle ,b4,c4,g1 and a lykeiamme1 ,a4,b4,g4,c1 whiche haue a grounde lyne1 that is to saye1 ,b4,g4 ,therfore doth it folow that the lyke iamme ,a4,b4,g4,c4 is double to the triangle ,b4,c4,g4 ,for euery halfe of that lykeiamme is equall to the triangle1 ,i meane ,a4,b4,f4,e4 other ,f4,e4,c4,g4 as you may coniecture by the 4xi4 conclusion geometrical4 ,and as this ,theoreme dothe speake of a triangle and likeiamme that haue one groundelyne1 so is it true also1 yf theyr groundelynes bee equall1 though they bee dyuers1 so that thei be made betwene one payre of paralleles4 ,and hereof may you perceaue the reason1 why in measuryng the platte of a triangle1 you must multiply the perpendicular lyne by halfe the grounde lyne1 or els the hole grounde lyne by halfe the perpendicular1 for by any of these bothe waies is there made a lykeiamme equall to halfe suche a one as shulde be made on the same hole grounde lyne with the triangle1 and betweene one payre of paralleles4 ,therfore as that lykeiamme is double to the triangle1 so the halfe of it1 must needes be equall to the triangle4 ,compare the 4xi4 conclusion with this theoreme4 ,the 4xxxij4 ,theoreme4 ,in all likeiammes where there are more than one made aboute one bias line1 the fill squares of euery of them must nedes be equall4 ,illustration ,example4 ,fyrst before ,i declare the examples1 it shal be mete to shew the true vnderstandyng of this theorem4 ,sidenote3 ,bias lyne4 ,therfore by the ,bias line1 ,i meane that lyne1 whiche in any square figure dooth runne from corner to corner4 ,and euery square which is diuided by that bias line into equall halues from corner to corner 7that is to say1 into 4ij4 equall triangles7 those be counted to stande aboute one bias line1 and the other squares1 whiche touche that bias line1 with one of their corners onely1 those doo ,i call ,fyll squares1 ,sidenote3 ,fyll squares4 accordyng to the greke name1 which is anapleromata1 ,sidenote3 anaple3ro3mata and called in latin supplementa1 bycause that they make one generall square1 includyng and enclosyng the other diuers squares1 as in this example ,h4,c4,e4,n4 is one square likeiamme1 and ,l4,m4,g4,c4 is an other1 whiche bothe are made aboute one bias line1 that is ,n4,m1 than ,k4,l4,h4,c4 and ,c4,e4,f4,g4 are 4ij4 fyll squares1 for they doo fyll vp the sydes of the 4ij4 fyrste square lykeiammes1 in suche sorte1 that all them foure is made one greate generall square ,k4,m4,f4,n4 ,nowe to the sentence of the theoreme1 ,i say1 that the 4ij4 fill squares1 ,h4,k4,l4,c4 and ,c4,e4,f4,g4 are both equall togither1 7as it shall bee declared in the booke of proofes7 bicause they are the fill squares of two likeiammes made aboute one bias line1 as the exaumple sheweth4 ,conferre the twelfthe conclusion with this theoreme4 ,the xxxiij4 ,theoreme4 ,in all right anguled triangles1 the square of that side whiche lieth against the right angle1 is equall to the 4ij4 squares of both the other sides4 ,example4 ,illustration ,a4,b4,c4 is a triangle1 hauing a ryght angle in ,b4 ,wherfore it foloweth1 that the square of ,a4,c1 7whiche is the side that lyeth agaynst the right angle7 shall be as muche as the two squares of ,a4,b4 and ,b4,c4 which are the other 4ij4 sides4 ,sec4 ,by the square of any lyne1 you muste vnderstande a figure made iuste square1 hauyng all his iiij4 sydes equall to that line1 whereof it is the square1 so is ,a4,c4,f1 the square of ,a4,c4 ,lykewais ,a4,b4,d4 is the square of ,a4,b4 ,and ,b4,c4,e4 is the square of ,b4,c4 ,now by the numbre of the diuisions in eche of these squares1 may you perceaue not onely what the square of any line is called1 but also that the theoreme is true1 and expressed playnly bothe by lines and numbre4 ,for as you see1 the greatter square 7that is ,a4,c4,f47 hath fiue diuisions on eche syde1 all equall togyther1 and those in the whole square are twenty and fiue4 ,nowe in the left square1 whiche is ,a4,b4,d4 there are but 4iij4 of those diuisions in one syde1 and that yeldeth nyne in the whole4 ,so lykeways you see in the meane square ,a4,c4,e4 in euery syde 4iiij4 partes1 whiche in the whole amount vnto sixtene4 ,nowe adde togyther all the partes of the two lesser squares1 that is to saye1 sixtene and nyne1 and you perceyue that they make twenty and fiue1 whyche is an equall numbre to the summe of the greatter square4 ,by this theoreme you may vnderstand a redy way to know the syde of any ryght anguled triangle that is vnknowen1 so that you knowe the lengthe of any two sydes of it4 ,for by tournynge the two sydes certayne into theyr squares1 and so addynge them togyther1 other subtractynge the one from the other 7accordyng as in the vse of these theoremes ,i haue sette foorthe7 and then fyndynge the roote of the square that remayneth1 which roote 7,i meane the syde of the square7 is the iuste length of the vnknowen syde1 whyche is sought for4 ,but this appertaineth to the thyrde booke1 and therefore ,i wyll speake no more of it at this tyme4 ,the xxxiiij4 ,theoreme4 ,if so be it1 that in any triangle1 the square of the one syde be equall to the 4ij4 squares of the other 4ij4 sides1 than must nedes that corner be a right corner1 which is conteined betwene those two lesser sydes4 ,example4 ,as in the figure of the laste ,theoreme1 bicause ,a4,c1 made in square1 is asmuch as the square of ,a4,b1 and also as the square of ,b4,c4 ioyned bothe togyther1 therefore the angle that is inclosed betwene those 4ij4 lesser lynes1 ,a4,b4 and ,b4,c4 7that is to say7 the angle ,b4 whiche lieth against the line ,a4,c1 must nedes be a ryght angle4 ,this theoreme dothe so depende of the truthe of the laste1 that whan you perceaue the truthe of the one1 you can not iustly doubt of the others truthe1 for they conteine one sentence1 contrary waies pronounced4 ,the 4xxxv4 theoreme4 ,if there be set forth 4ij4 right lines1 and one of them parted into sundry partes1 how many or few so euer they be1 the square that is made of those ij4 right lines proposed1 is equal to all the squares1 that are made of the vndiuided line1 and euery parte of the diuided line4 ,illustration ,example4 ,the ij4 lines proposed ar ,a4,b4 and ,c4,d1 and the lyne ,a4,b4 is deuided into thre partes by ,e4 and ,f4 ,now saith this theoreme1 that the square that is made of those two whole lines ,a4,b4 and ,c4,d1 so that the line ,a4,b4 standeth for the length of the square1 and the other line ,c4,d4 for the bredth of the same4 ,that square 7,i say7 wil be equall to all the squares that be made1 of the vndiueded lyne 7which is ,c4,d47 and euery portion of the diuided line4 ,and to declare that particularly1 ,fyrst ,i make an other line ,g4,k1 equall to the line 4,c4,d1 and the line ,g4,h4 to be equal to the line ,a4,b1 and to bee diuided into iij4 like partes1 so that ,g4,m4 is equall to ,a4,e1 and ,m4,n4 equal to ,e4,f1 and then muste ,n4,h4 nedes remaine equall to ,f4,b4 ,then of those ij4 lines ,g4,k1 vndeuided1 and ,g4,h4 which is deuided1 ,i make a square1 that is ,g4,h4,k4,l1 ,in which square if ,i drawe crosse lines frome one side to the other1 according to the diuisions of the line ,g4,h1 then will it appear plaine1 that the theoreme doth affirme4 ,for the first square ,g4,m4,o4,k1 must needes be equal to the square of the line ,c4,d1 and the first portion of the diuided line1 which is ,a4,e1 for bicause their sides are equall4 ,and so the seconde square that is ,m4,n4,p4,o1 shall be equall to the square of ,c4,d1 and the second part of ,a4,b1 that is ,e4,f4 ,also the third square which is ,n4,h4,l4,p1 must of necessitee be equal to the square of ,c4,d1 and ,f4,b1 bicause those lines be so coupeled that euery couple are equall in the seuerall figures4 ,and so shal you not only in this example1 but in all other finde it true1 that if one line be deuided into sondry partes1 and an other line whole and vndeuided1 matched with him in a square1 that square which is made of these two whole lines1 is as muche iuste and equally1 as all the seuerall squares1 whiche bee made of the whole line vndiuided1 and euery part seuerally of the diuided line4 ,the xxxvi4 ,theoreme4 ,if a right line be parted into ij4 partes1 as chaunce may happe1 the square that is made of the whole line1 is equall to bothe the squares that are made of the same line1 and the twoo partes of it seuerally4 ,example4 ,illustration ,the line propounded beyng ,a4,b4 and deuided1 as chaunce happeneth1 in ,c4 into ij4 vnequall partes1 ,i say that the square made of the hole line ,a4,b1 is equal to the two squares made of the same line with the twoo partes of itselfe1 as with ,a4,c1 and with ,c4,b1 for the square ,d4,e4,f4,g4 is equal to the two other partial squares of ,d4,h4,k4,g and ,h4,e4,f4,k1 but that the greater square is equall to the square of the whole line ,a4,b1 and the partiall squares equall to the squares of the second partes of the same line ioyned with the whole line1 your eye may iudg without muche declaracion1 so that ,i shall not neede to make more exposition therof1 but that you may examine it1 as you did in the laste ,theoreme4 ,the xxxvij4 ,theoreme4 ,if a right line be deuided by chaunce1 as it maye happen1 the square that is made of the whole line1 and one of the partes of it which soeuer it be1 shal be equall to that square that is made of the ij4 partes ioyned togither1 and to an other square made of that part1 which was before ioyned with the whole line4 ,example4 ,illustration ,the line ,a4,b4 is deuided in ,c4 into twoo partes1 though not equally1 of which two partes for an example ,i take the first1 that is ,a4,c1 and of it ,i make one side of a square1 as for example ,d4,g4 accomptinge those two lines to be equall1 the other side of the square is ,d4,e1 whiche is equall to the whole line ,a4,b4 ,now may it appeare1 to your eye1 that the great square made of the whole line ,a4,b1 and of one of his partes that is ,a4,c1 7which is equall with ,d4,g47 is equal to two partiall squares1 whereof the one is made of the saide greatter portion ,a4,c1 in as muche as not only ,d4,g1 beynge one of his sides1 but also ,d4,h4 beinge the other side1 are eche of them equall to ,a4,c4 ,the second square is ,h4,e4,f4,k1 in which the one side ,h4,e1 is equal to ,c4,b1 being the lesser parte of the line1 ,a4,b1 and ,e4,f4 is equall to ,a4,c4 which is the greater parte of the same line4 ,so that those two squares ,d4,h4,k4,g and ,h4,e4,f4,k1 bee bothe of them no more then the greate square ,d4,e4,f4,g1 accordinge to the wordes of the ,theoreme afore saide4 ,the xxxviij4 ,theoreme4 ,if a righte line be deuided by chaunce1 into partes1 the square that is made of that whole line1 is equall to both the squares that ar made of eche parte of the line1 and moreouer to two squares made of the one portion of the diuided line ioyned with the other in square4 ,illustration ,example4 ,lette the diuided line bee ,a4,b1 and parted in ,c1 into twoo partes3 ,nowe saithe the ,theoreme1 that the square of the whole lyne ,a4,b1 is as mouche iuste as the square of ,a4,c1 and the square of ,c4,b1 eche by it selfe1 and more ouer by as muche twise1 as ,a4,c4 and ,c4,b4 ioyned in one square will make4 ,for as you se1 the great square ,d4,e4,f4,g1 conteyneth in hym foure lesser squares1 of whiche the first and the greatest is ,n4,m4,f4,k1 and is equall to the square of the lyne ,a4,c4 ,the second square is the lest of them all1 that is ,d4,h4,l4,n1 and it is equall to the square of the line ,c4,b4 ,then are there two other longe squares both of one bygnes1 that is ,h4,e4,n4,m4 and ,l4,n4,g4,k1 eche of them both hauyng 4ij4 sides equall to ,a4,c1 the longer parte of the diuided line1 and there other two sides equall to ,c4,b1 beeyng he shorter parte of the said line ,a4,b4 ,so is that greatest square1 beeyng made of the hole lyne ,a4,b1 equal to the ij4 squares of eche of his partes seuerally1 and more by as muche iust as 4ij4 longe squares1 made of the longer portion of the diuided lyne ioyned in square with the shorter parte of the same diuided line1 as the theoreme wold4 ,and as here ,i haue put an example of a lyne diuided into 4ij4 partes1 so the theoreme is true of all diuided lines1 of what number so euer the partes be1 foure1 fyue1 or syxe4 etc4 ,this theoreme hath great vse1 not only in geometrie1 but also in arithmetike1 as herafter ,i will declare in conuenient place4 ,the 4xxxix4 theoreme4 ,if a right line be deuided into two equall partes1 and one of these 4ij4 partes diuided agayn into two other partes1 as happeneth the longe square that is made of the thyrd or later part of that diuided line1 with the residue of the same line1 and the square of the mydlemoste parte1 are bothe togither equall to the square of halfe the firste line4 ,example4 ,illustration ,the line ,a4,b4 is diuided into ij4 equal partes in ,c1 and that parte ,c4,b4 is diuided agayne as hapneth in ,d4 ,wherfore saith the ,theorem that the long square made of ,d4,b4 and ,a4,d1 with the square of ,c4,d4 7which is the mydle portion7 shall bothe be equall to the square of half the lyne ,a4,b1 that is to saye1 to the square of ,a4,c1 or els of ,c4,d1 which make all one4 ,the long square ,f4,g4,n4,o4 whiche is the longe square that the theoreme speaketh of1 is made of 4ij4 long squares1 wherof the fyrst is ,f4,g4,m4,k1 and the seconde is ,k4,n4,o4,m4 ,the square of the myddle portion is ,l4,m4,o4,p4 and the square of the halfe of the fyrste lyne is ,e4,k4,q4,l4 ,nowe by the theoreme1 that longe square ,f4,g4,m4,o1 with the iuste square ,l4,m4,o4,p1 muste bee equall to the greate square ,e4,k4,q4,l1 whyche thynge bycause it seemeth somewhat difficult to vnderstande1 althoughe ,i intende not here to make demonstrations of the ,theoremes1 bycause it is appoynted to be done in the newe edition of ,euclide1 yet ,i wyll shew you brefely how the equalitee of the partes doth stande4 ,and fyrst ,i say1 that where the comparyson of equalitee is made betweene the greate square 7whiche is made of halfe the line ,a4,b47 and two other1 where of the fyrst is the longe square ,f4,g4,n4,o1 and the second is the full square ,l4,m4,o4,p1 which is one portion of the great square all redye1 and so is that longe square ,k4,n4,m4,o1 beynge a parcell also of the longe square ,f4,g4,n4,o1 ,wherfore as those two partes are common to bothe partes compared in equalitee1 and therfore beynge bothe abated from eche parte1 if the reste of bothe the other partes bee equall1 than were those whole partes equall before3 ,nowe the reste of the great square1 those two lesser squares beyng taken away1 is that longe square ,e4,n4,p4,q1 whyche is equall to the long square ,f4,g4,k4,m1 beyng the rest of the other parte4 ,and that they two be equall1 theyr sydes doo declare4 ,for the longest lynes that is ,f4,k and ,e4,q are equall1 and so are the shorter lynes1 ,f4,g1 and ,e4,n1 and so appereth the truthe of the ,theoreme4 ,the 4xl4 theoreme4 ,if a right line be diuided into 4ij4 euen partes1 and an other right line annexed to one ende of that line1 so that it make one righte line with the firste4 ,the longe square that is made of this whole line so augmented1 and the portion that is added1 with the square of halfe the right line1 shall be equall to the square of that line1 whiche is compounded of halfe the firste line1 and the parte newly added4 ,example4 ,illustration ,the fyrst lyne propounded is ,a4,b1 and it is diuided into ij4 equall partes in ,c1 and an other ryght lyne1 ,i meane ,b4,d annexed to one ende of the fyrste lyne4 ,nowe say ,i1 that the long square ,a4,d4,m4,k1 is made of the whole lyne so augmented1 that is ,a4,d1 and the portion annexed1 yt is ,d4,m1 for ,d4,m is equall to ,b4,d1 wherfore yt long square ,a4,d4,m4,k1 with the square of halfe the first line1 that is ,e4,g4,h4,l1 is equall to the great square ,e4,f4,d4,c4 whiche square is made of the line ,c4,d4 that is to saie1 of a line compounded of halfe the first line1 beyng ,c4,b1 and the portion annexed1 that is ,b4,d4 ,and it is easyly perceaued1 if you consyder that the longe square ,a4,c4,l4,k4 7whiche onely is lefte out of the great square7 hath another longe square equall to hym1 and to supply his steede in the great square1 and that is ,g4,f4,m4,h4 ,for their sydes be of lyke lines in length4 ,the xli4 ,theoreme4 ,if a right line bee diuided by chaunce1 the square of the same whole line1 and the square of one of his partes are iuste equall to the long square of the whole line1 and the sayde parte twise taken1 and more ouer to the square of the other parte of the sayd line4 ,example4 ,illustration ,a4,b4 is the line diuided in ,c4 ,and ,d4,e4,f4,g1 is the square of the whole line1 ,d4,h4,k4,m4 is the square of the lesser portion 7whyche ,i take for an example7 and therfore must bee twise reckened4 ,nowe ,i saye that those ij4 squares are equall to two longe squares of the whole line ,a4,b1 and his sayd portion ,a4,c1 and also to the square of the other portion of the sayd first line1 whiche portion is ,c4,b1 and his square ,k4,n4,f4,l4 ,in this theoreme there is no difficultie1 if you consyder that the litle square ,d4,h4,k4,m4 is 4iiij4 tymes reckened1 that is to say1 fyrst of all as a parte of the greatest square1 whiche is ,d4,e4,f4,g4 ,secondly he is rekned by him selfe4 ,thirdely he is accompted as parcell of the long square ,d4,e4,n4,m1 ,and fourthly he is taken as a part of the other long square ,d4,h4,l4,g1 so that in as muche as he is twise reckened in one part of the comparison of equalitee1 and twise also in the second parte1 there can rise none occasion of errour or doubtfulnes therby4 ,the xlij4 ,theoreme4 ,if a right line be deuided as chance happeneth the iiij4 long squares1 that may be made of that whole line and one of his partes with the square of the other part1 shall be equall to the square that is made of the whole line and the saide first portion ioyned to him in lengthe as one whole line4 ,example4 ,illustration ,the firste line is ,a4,b1 and is deuided by ,c4 into two vnequall partes as happeneth4 ,the long square of yt1 and his lesser portion ,a4,c1 is foure times drawen1 the first is ,e4,g4,m4,k1 the seconde is ,k4,m4,q4,o1 the third is ,h4,k4,r4,s1 and the fourthe is ,k4,l4,s4,t4 ,and where as it appeareth that one of the little squares 7,i meane ,k4,l4,p4,o7 is reckened twise1 ones as parcell of the second long square and agayne as parte of the thirde long square1 to auoide ambiguite1 you may place one insteede of it1 an other square of equalitee1 with it4 that is to saye1 ,d4,e4,k4,h1 which was at no tyme accompting as parcell of any one of them1 and then haue you iiij4 long squares distinctly made of the whole line ,a4,b1 and his lesser portion ,a4,c4 ,and within them is there a greate full square ,p4,q4,t4,v4 whiche is the iust square of ,b4,c1 beynge the greatter portion of the line ,a4,b4 ,and that those fiue squares doo make iuste as muche as the whole square of that longer line ,d4,g1 7whiche is as longe as ,a4,b1 and ,a4,c4 ioyned togither7 it may be iudged easyly by the eye1 sith that one greate square doth comprehend in it all the other fiue squares1 that is to say1 foure long squares 7as is before mencioned7 and one full square4 which is the intent of the ,theoreme4 ,the xliij4 ,theoreme4 ,if a right line be deuided into ij4 equal partes first1 and one of those parts again into other ij4 parts1 as chaunce hapeneth1 the square that is made of the last part of the line so diuided1 and the square of the residue of that whole line1 are double to the square of halfe that line1 and to the square of the middle portion of the same line4 ,example4 ,illustration ,the line to be deuided is ,a4,b1 and is parted in ,c4 into two equall partes1 and then ,c4,b1 is deuided againe into two partes in ,d1 so that the meaninge of the ,theoreme1 is that the square of ,d4,b4 which is the latter parte of the line1 and the square of ,a4,d1 which is the residue of the whole line4 ,those two squares1 ,i say1 ar double to the square of one halfe of the line1 and to the square of ,c4,d1 which is the middle portion of those thre diuisions4 ,which thing that you maye more easilye perceaue1 ,i haue drawen foure squares1 whereof the greatest being marked with ,e4 is the square of ,a4,d4 ,the next1 which is marked with ,g1 is the square of halfe the line1 that is1 of ,a4,c1 ,and the other two little squares marked with ,f4 and ,h1 be both of one bignes1 by reason that ,i did diuide ,c4,b4 into two equall partes1 so that you amy take the square ,f1 for the square of ,d4,b1 and the square ,h1 for the square of ,c4,d4 ,now ,i thinke you doubt not1 but that the square ,e4 and the square ,f1 ar double so much as the square ,g4 and the square ,h1 which thing the easyer is to be vnderstande1 bicause that the greate square hath in his side iij4 quarters of the firste line1 which multiplied by itselfe maketh nyne quarters1 and the square ,f4 containeth but one quarter1 so that bothe doo make tenne quarters4 ,then ,g4 contayneth iiij4 quarters1 seynge his side containeth twoo1 and ,h4 containeth but one quarter1 whiche both make but fiue quarters1 and that is but halfe of tenne4 ,whereby you may easylye coniecture1 that the meanynge of the theoreme is verified in the figures of this example4 ,the xliiij4 ,theoreme4 ,if a right line be deuided into ij4 partes equally1 and an other portion of a righte lyne annexed to that firste line1 the square of this whole line so compounded1 and the square of the portion that is annexed1 ar doule as much as the square of the halfe of the firste line1 and the square of the other halfe ioyned in one with the annexed portion1 as one whole line4 ,example4 ,illustration ,the line is ,a4,b1 and is diuided firste into twoo equal partes in ,c1 and then is there annexed to it an other portion whiche is ,b4,d4 ,now saith the ,theoreme1 that the square of ,a4,d1 and the square of ,b4,d1 ar double to the square of ,a4,c1 and to the square of ,c4,d4 ,the line ,a4,b4 containing four partes1 then must needes his halfe containe ij4 partes of such partes ,i suppose ,b4,d4 7which is the annexed line7 to containe thre1 so shal the hole line comprehend vij4 parts1 and his square xlix4 parts1 where vnto if you ad ye square of the annexed lyne1 whiche maketh nyne1 than those bothe doo yelde1 lviij4 whyche must be double to the square of the halfe lyne with the annexed portion4 ,the halfe lyne by it selfe conteyneth but 4ij4 partes1 and therfore his square dooth make foure4 ,the halfe lyne with the annexed portion conteyneth fiue1 and the square of it is 4xxv1 now put foure to 4xxv1 and it maketh iust 4xxix1 the euen halfe of fifty and eight1 wherby appereth the truthe of the theoreme4 ,the 4xlv4 theoreme4 ,in all triangles that haue a blunt angle1 the square of the side that lieth against the blunt angle1 is greater than the two squares of the other twoo sydes1 by twise as muche as is comprehended of the one of those 4ij4 sides 7inclosyng the blunt corner7 and the portion of the same line1 beyng drawen foorth in lengthe1 which lieth betwene the said blunt corner and a perpendicular line lightyng on it1 and drawen from one of the sharpe angles of the foresayd triangle4 ,example4 ,for the declaration of this theoreme and the next also1 whose vse are wonderfull in the practise of ,geometrie1 and in measuryng especially1 it shall be nedefull to declare that euery triangle that hath no ryght angle as those whyche are called 7as in the boke of practise is declared7 sharp cornered triangles1 and blunt cornered triangles1 yet may they be brought to haue a ryght angle1 eyther by partyng them into two lesser triangles1 or els by addyng an other triangle vnto them1 whiche may be a great helpe for the ayde of measuryng1 as more largely shall be sette foorthe in the boke of measuryng4 ,but for this present place1 this forme wyll ,i vse1 7whiche ,theon also vseth7 to adde one triangle vnto an other1 to bryng the blunt cornered triangle into a ryght angled triangle1 whereby the proportion of the squares of the sides in suche a blunt cornered triangle may the better bee knowen4 ,illustration ,fyrst therfore ,i sette foorth the triangle ,a4,b4,c1 whose corner by ,c4 is a blunt corner as you maye well iudge1 than to make an other triangle of yt with a ryght angle1 ,i must drawe forth the side ,b4,c4 vnto ,d1 and from the sharp corner by ,a4 ,i brynge a plumbe lyne or perpendicular on ,d4 ,and so is there nowe a newe triangle ,a4,b4,d4 whose angle by ,d4 is a right angle4 ,nowe accordyng to the meanyng of the ,theoreme1 ,i saie1 that in the first triangle ,a4,b4,c1 because it hath a blunt corner at ,c1 the square of the line ,a4,b4 whiche lieth against the said blunte corner1 is more then the square of the line ,a4,c1 and also of the lyne ,b4,c1 7whiche inclose the blunte corner7 by as muche as will amount twise of the line ,b4,c1 and that portion ,d4,c4 whiche lieth betwene the blunt angle by ,c1 and the perpendicular line ,a4,d4 ,the square of the line ,a4,b1 is the great square marked with ,e4 ,the square of ,a4,c1 is the meane square marked with ,f4 ,the square of ,b4,c1 is the least square marked with ,g4 ,and the long square marked with ,k1 is sette in steede of two squares made of ,b4,c1 and ,c4,d4 ,for as the shorter side is the iuste lengthe of ,c4,d1 so the other longer side is iust twise so longe as ,b4,c1 ,wherfore ,i saie now accordyng to the ,theoreme1 that the greatte square ,e1 is more then the other two squares ,f4 and ,g1 by the quantitee of the longe square ,k1 wherof ,i reserue the profe to a more conuenient place1 where ,i will also teache the reason howe to fynde the lengthe of all suche perpendicular lynes1 and also of the line that is drawen betweene the blunte angle and the perpendicular line1 with sundrie other very pleasant conclusions4 ,the 4xlvi4 ,theoreme4 ,in sharpe cornered triangles1 the square of anie side that lieth against a sharpe corner1 is lesser then the two squares of the other two sides1 by as muche as is comprised twise in the long square of that side1 on whiche the perpendicular line falleth1 and the portion of that same line1 liyng betweene the perpendicular1 and the foresaid sharpe corner4 ,example4 ,illustration ,fyrst ,i sette foorth the triangle ,a4,b4,c1 and in yt ,i draw a plumbe line from the angle ,c4 vnto the line ,a4,b1 and it lighteth in ,d4 ,nowe by the theoreme the square of ,b4,c4 is not so muche as the square of the other two sydes1 that of ,b4,a4 and of ,a4,c4 by as muche as is twise conteyned in the long square made of ,a4,b1 and ,a4,d1 ,a4,b4 beyng the line or syde on which the perpendicular line falleth1 and ,a4,d4 beeyng that portion of the same line whiche doth lye betwene the perpendicular line1 and the sayd sharpe angle limitted1 whiche angle is by ,a4 ,for declaration of the figures1 the square marked with ,e4 is the square of ,b4,c1 whiche is the syde that lieth agaynst the sharpe angle1 the square marked with ,g4 is the square of ,a4,b1 and the square marked with ,f4 is the square of ,a4,c1 and the two longe squares marked with ,h4,k1 are made of the hole line ,a4,b1 and one of his portions ,a4,d4 ,and truthe it is that the square ,e4 is lesser than the other two squares ,c4 and ,f4 by the quantitee of those two long squares ,h4 and ,k4 ,wherby you may consyder agayn1 an other proportion of equalitee1 that is to saye1 that the square ,e4 with the twoo longsquares ,h4,k1 are iuste equall to the other twoo squares ,c4 and ,f4 ,and so maye you make1 as it were an other theoreme4 ,that in al sharpe cornered triangles1 where a perpendicular line is drawen frome one angle to the side that lyeth againste it1 the square of anye one side1 with the ij4 longesquares made at that hole line1 whereon the perpendicular line doth lighte1 and of that portion of it1 which ioyneth to that side whose square is all ready taken1 those thre figures1 ,i say1 are equall to the ij4 squares1 of the other ij4 sides of the triangle4 ,in whiche you muste vnderstand1 that the side on which the perpendiculare falleth1 is thrise vsed1 yet is his square but ones mencioned1 for twise he is taken for one side of the two long squares4 ,and as ,i haue thus made as it were an other theoreme out of this fourty and sixe theoreme1 so mighte ,i out of it1 and the other that goeth nexte before1 make as manny as woulde suffice for a whole booke1 so that when they shall bee applyed to practise1 and consequently to expresse their benefite1 no manne that hathe not well wayde their wonderfull commoditee1 would credite the possibilitie of their wonderfull vse1 and large ayde in knowledge4 ,but all this wyll ,i remitte to a place conuenient4 ,the xlvij4 ,theoreme4 ,if ij4 points be marked in the circumference of a circle1 and a right line drawen frome the one to the other1 that line must needes fal within the circle4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 the ij4 poinctes are ,a4,b1 the righte line that is drawenne frome the one to the other1 is the line ,a4,b1 which as you see1 must needes lyghte within the circle4 ,so if you putte the pointes to be ,a4,d1 or ,d4,c1 or ,a4,c1 other ,b4,c1 or ,b4,d1 in any of these cases you see1 that the line that is drawen from the one pricke to the other dothe euermore run within the edge of the circle1 els canne it be no right line4 ,how be it1 that a croked line1 especially being more croked then the portion of the circumference1 maye bee drawen from pointe to pointe withoute the circle4 ,but the theoreme speaketh only of right lines1 and not of croked lines4 ,the xlviij4 ,theoreme4 ,if a righte line passinge by the centre of a circle1 doo crosse an other right line within the same circle1 passinge beside the centre1 if he deuide the saide line into twoo equal partes1 then doo they make all their angles righte4 ,and contrarie waies1 if they make all their angles righte1 then doth the longer line cutte the shorter in twoo partes4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 the line that passeth by the centre1 is ,a4,e4,c1 the line that goeth beside the centre is ,d4,b4 ,nowe saye ,i1 that the line ,a4,e4,c1 dothe cutte that other line ,d4,b4 into twoo iuste partes1 and therefore all their four angles ar righte angles4 ,and contrarye wayes1 bicause all their angles are righte angles1 therfore it muste be true1 that the greater cutteth the lesser into two equal partes1 accordinge as the ,theoreme would4 ,the xlix4 ,theoreme4 ,if twoo right lines drawen in a circle doo crosse one an other1 and doo not passe by the centre1 euery of them dothe not deuide the other into equall partions4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 and the centre is ,e1 the one line ,a4,c1 and the other is ,b4,d1 which two lines crosse one an other1 but yet they go not by the centre1 wherefore accordinge to the woordes of the theoreme1 eche of theim doth cuytte the other into equall portions4 ,for as you may easily iudge1 ,a4,c4 hath one portion longer and an other shorter1 and so like wise ,b4,d4 ,howbeit1 it is not so to be vnderstand1 but one of them may be deuided into ij4 euen parts1 but bothe to bee cutte equally in the middle1 is not possible1 onles both passe through the centre1 therfore much rather when bothe go beside the centre1 it can not be that eche of theym shoulde be iustely parted into ij4 euen partes4 ,the ,l4 ,theoreme4 ,if two circles crosse and cut one an other1 then haue not they both one centre4 ,example4 ,illustration ,this theoreme seemeth of it selfe so manifest1 that it neadeth nother demonstration nother declaracion4 ,yet for the plaine vnderstanding of it1 ,i haue sette forthe a figure here1 where ij4 circles be drawen1 so that one of them doth crosse the other 7as you see7 in the pointes ,b4 and ,g1 and their centres appear at the firste sighte to bee diuers4 ,for the centre of the one is ,f1 and the centre of the other is ,e1 which diffre as farre asondre as the edges of the circles1 where they bee most distaunte in sonder4 ,the ,li4 ,theoreme4 ,if two circles be so drawen1 that one of them do touche the other1 then haue they not one centre4 ,example4 ,illustration ,there are two circles made1 as you see1 the one is ,a4,b4,c1 and hath his centre by ,g1 the other is ,b4,d4,e1 and his centre is by ,f1 so that it is easy enough to perceaue that their centres doe dyffer as muche a sonder1 as the halfe diameter of the greater circle is longer then the half diameter of the lesser circle4 ,and so must it needes be thought and said of all other circles in lyke kinde4 ,the 4lij4 theoreme4 ,if a certaine pointe be assigned in the diameter of a circle1 distant from the centre of the said circle1 and from that pointe diuerse lynes drawen to the edge and circumference of the same circle1 the longest line is that whiche passeth by the centre1 and the shortest is the residew of the same line4 ,and of al the other lines that is euer the greatest1 that is nighest to the line1 which passeth by the centre4 ,and contrary waies1 that is the shortest1 that is farthest from it4 ,and amongest them all there can be but onely 4ij4 equall together1 and they must nedes be so placed1 that the shortest line shall be in the iust middle betwixte them4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d4,e4,h1 and his centre is ,f1 the diameter is ,a4,e1 in whiche diameter ,i haue taken a certain point distaunt from the centre1 and that pointe is ,g1 from which ,i haue drawen 4iiij4 lines to the circumference1 beside the two partes of the diameter1 whiche maketh vp vi4 lynes in all4 ,nowe for the diuersitee in quantitie of these lynes1 ,i saie accordyng to the ,theoreme1 that the line whiche goeth by the centre is the longest line1 that is to saie1 ,a4,g1 and the residewe of the same diameter beeyng ,g4,e1 is the shortest lyne4 ,and of all the other that lyne is longest1 that is neerest vnto that parte of the diameter whiche gooeth by the centre1 and that is shortest1 that is farthest distant from it1 wherefore ,i saie1 that ,g4,b1 is longer then ,g4,c1 and therfore muche more longer then ,g4,d1 sith ,g4,c1 also is longer then ,g4,d1 and by this maie you soone perceiue1 that it is not possible to drawe 4ij4 lynes on any one side of the diameter1 whiche might be equall in lengthe together1 but on the one side of the diameter maie you easylie make one lyne equall to an other1 on the other side of the same diameter1 as you see in this example ,g4,h1 to bee equall to ,g4,d1 betweene whiche the lyne ,g4,e1 7as the shortest in all the circle7 doothe stande euen distaunte from eche of them1 and it is the precise knoweledge of their equalitee1 if they be equally distaunt from one halfe of the diameter4 ,where as contrary waies if the one be neerer to any one halfe of the diameter then the other is1 it is not possible that they two may be equall in lengthe1 namely if they dooe ende bothe in the circumference of the circle1 and be bothe drawen from one poynte in the diameter1 so that the saide poynte be 7as the ,theoreme doeth suppose7 somewhat distaunt from the centre of the said circle4 ,for if they be drawen from the centre1 then must they of necessitee be all equall1 howe many so euer they bee1 as the definition of a circle dooeth importe1 withoute any regarde how neere so euer they be to the diameter1 or how distante from it4 ,and here is to be noted1 that in this ,theoreme1 by neerenesse and distaunce is vnderstand the nereness and distaunce of the extreeme partes of those lynes where they touche the circumference4 ,for at the other end they do all meete and touche4 ,the 4liij4 ,theoreme4 ,if a pointe bee marked without a circle1 and from it diuerse lines drawen crosse the circle1 to the circumference on the other side1 so that one of them passe by the centre1 then that line whiche passeth by the centre shall be the loongest of them all that crosse the circle4 ,and of the other lines those are longest1 that be nexte vnto it that passeth by the centre4 ,and those ar shortest1 that be farthest distant from it4 ,but among those partes of those lines1 whiche ende in the outewarde circumference1 that is most shortest1 whiche is parte of the line that passeth by the centre1 and amongeste the othere eche1 of them1 the nerer they are vnto it1 the shorter they are1 and the farther from it1 the longer they be4 ,and amongest them all there can not be more then 4ij4 of any one length1 and they two muste be on the two contrarie sides of the shortest line4 ,example4 ,illustration ,take the circle to be ,a4,b4,c1 and the point assigned without it to be ,d4 ,now say ,i1 that if there be drawen sundrie lines from ,d1 and crosse the circle1 endyng in the circumference on the contrary side1 as here you see1 ,d4,a1 ,d4,e1 ,d4,f1 and ,d4,b1 then of all these lines the longest must needes be ,d4,a1 which goeth by the centre of the circle1 and the nexte vnto it1 that is ,d4,e1 is the longest amongest the rest4 ,and contrarie waies1 ,d4,b1 is the shorteste1 because it is farthest distaunt from ,d4,a4 ,and so maie you iudge of ,d4,f1 because it is nerer vnto ,d4,a1 then is ,d4,b1 therefore is it longer then ,d4,b4 ,and likewaies because it is farther of from ,d4,a1 then is ,d4,e1 therfore is it shorter then ,d4,e4 ,now for those partes of the lines whiche bee withoute the circle 7as you see7 ,d4,c1 is the shortest4 because it is the parte of that line which passeth by the centre1 ,and ,d4,k1 is next to it in distance1 and therefore also in shortnes1 so ,d4,g1 is farthest from it in distance1 and therfore is the longest of them4 ,now ,d4,h1 beyng nerer then ,d4,g1 is also shorter then it1 and beynge farther of1 then ,d4,k1 is longer then it4 ,so that for this parte of the theoreme 7as ,i think7 you do plainly perceaue the truthe thereof1 so the residue hathe no difficulte4 ,for seing that the nearer any line is to ,d4,c1 7which ioyneth with the diameter7 the shorter it is and the farther of from it1 the longer it is4 ,and seyng two lynes can not be of like distaunce beinge bothe on one side1 therefore if they shal be of one lengthe1 and consequently of one distaunce1 they must needes bee on contrary sides of the saide line ,d4,c4 ,and so appeareth the meaning of the whole ,theoreme4 ,and of this ,theoreme dothe there folowe an other lyke4 whiche you maye calle other a theoreme by it selfe1 or else a ,corollary vnto this laste theoreme1 ,i passe not so muche for the name4 ,but his sentence is this3 when so euer any lynes be drawen frome any pointe1 withoute a circle1 whether they crosse the circle1 or eande in the utter edge of his circumference1 those two lines that bee equally distaunt from the least line are equal togither1 and contrary waies1 if they be equall togither1 they ar also equally distant from that least line4 ,for the declaracion of this proposition1 it shall not need to vse any other example1 then that which is brought for the explication of this laste theoreme1 by whiche you may without any teachinge easyly perceaue both the meanyng and also the truth of this proposition4 ,the ,liiij4 ,theoreme4 ,if a point be set forthe in a circle1 and from that pointe vnto the circumference many lines drawen1 of which more then two are equal togither1 then is that point the centre of that circle4 ,example4 ,illustration ,the circle is ,a4,b4,c1 and within it ,i haue sette fourth for an example three prickes1 which are ,d4,e4 and ,f1 from euery one of them ,i haue drawen 7at the leaste7 iiij4 lines vnto the circumference of the circle but frome ,d1 ,i haue drawen more1 yet maye it appear readily vnto your eye1 that of all the lines whiche be drawen from ,e4 and ,f1 vnto the circumference1 there are but twoo equall1 and more can not bee1 for ,g4,e4 nor ,e4,h4 hath none other equal to theim1 nor canne not haue any beinge drawen from the same point ,e4 ,no more can ,l4,f1 or ,f4,k1 haue anye line equall to either of theim1 beinge drawen from the same pointe ,f4 ,and yet from either of those two poinctes are there drawen twoo lines equall togither1 as ,a4,e1 is equall to ,e4,b1 and ,b4,f1 is equall to ,f4,c1 but there can no third line be drawen equall to either of these two couples1 and that is by reason that they be drawen from a pointe distaunte from the centre of the circle4 ,but from ,d1 althoughe there be seuen lines drawen1 to the circumference1 yet all bee equall1 bicause it is the centre of the circle4 ,and therefore if you drawe neuer so mannye more from it vnto the circumference1 all shall be equal1 so that this is the priuilege 7as it were of the centre7 and therfore no other point can haue aboue two equal lines drawen from it vnto the circumference4 ,and from all pointes you maye drawe ij4 equall lines to the circumference of the circle1 whether that pointe be within the circle or without it4 ,the lv4 ,theoreme4 ,no circle canne cut an other circle in more pointes then two4 ,example4 ,illustration ,the first circle is ,a4,b4,f4,e1 the second circle is ,b4,c4,d4,e1 and they crosse one an other in ,b4 and in ,e1 and in no more pointes4 ,nother is it possible that they should1 but other figures ther be1 which maye cutte a circle in foure partes1 as you se in this example4 ,where ,i haue set forthe one tunne forme1 and one eye forme1 and eche of them cutteth euery of their two circles into foure partes4 ,but as they be irregulare formes1 that is to saye1 suche formes as haue no precise measure nother proportion in their draughte1 so can there scarcely be made any certaine theorem of them4 ,but circles are regulare formes1 that is to say1 such formes as haue in their protracture a iuste and certaine proportion1 so that certain and determinate truths may be affirmed of them1 sith they ar vniforme and vnchaungable4 ,the lvi4 ,theoreme4 ,if two circles be so drawen1 that the one be within the other1 and that they touche one an other3 ,if a line bee drawen by bothe their centres1 and so forthe in lengthe1 that line shall runne to that pointe1 where the circles do touche4 ,example4 ,illustration ,the one circle1 which is the greattest and vttermost is ,a4,b4,c1 the other circle that is ye lesser1 and is drawen within the firste1 is ,a4,d4,e4 ,the centre of the greater circle is ,f1 and the centre of the lesser circle is ,g1 the pointe where they touche is ,a4 ,and now you may see the truthe of the theoreme so plainely1 that it needeth no farther declaracion4 ,for you maye see1 that drawinge a line from ,f4 to ,g1 and so forth in lengthe1 vntill it come to the circumference1 it wyll lighte in the very poincte ,a1 where the circles touche one an other4 ,the ,lvij4 ,theoreme4 ,if two circles bee drawen so one withoute an other1 that their edges doo touche and a right line bee drawnenne frome the centre of the one to the centre of the other1 that line shall passe by the place of their touching4 ,example4 ,illustration ,the firste circle is ,a4,b4,e1 and his centre is ,k1 ,the second circle is ,d4,b4,c1 and his centre is ,h1 the point wher they do touch is ,b4 ,nowe doo you se that the line ,k4,h1 whiche is drawen from ,k1 that is centre of the firste circle1 vnto ,h1 beyng centre of the second circle1 doth passe 7as it must nedes by the pointe ,b17 whiche is the verye poynte wher they do to touche together4 ,the 4lviij4 theoreme4 ,one circle can not touche an other in more pointes then one1 whether they touche within or without4 ,example4 ,illustration ,for the declaration of this ,theoreme1 ,i haue drawen iiij4 circles1 the first is ,a4,b4,c1 and his centre ,h4 the second is ,a4,d4,g1 and his centre ,f4 the third is ,l4,m1 and his centre ,k4 the 4iiij4 is ,d4,g4,l4,m1 and his centre ,e4 ,nowe as you perceiue the second circle ,a4,d4,g1 toucheth the first in the inner side1 in so much as it is drawen within the other1 and yet it toucheth him but in one point1 that is to say in ,a1 so lykewaies the third circle ,l4,m1 is drawen without the firste circle and toucheth hym1 as you maie see1 but in one place4 ,and now as for the 4iiij4 circle1 it is drawen to declare the diuersitie betwene touchyng and cuttyng1 or crossyng4 ,for one circle maie crosse and cutte a great many other circles1 yet can be not cutte any one in more places then two1 as the fiue and fiftie ,theoreme affirmeth4 ,the 4lix4 ,theoreme4 ,in euerie circle those lines are to be counted equall1 whiche are in lyke distaunce from the centre1 ,and contrarie waies they are in lyke distance from the centre1 whiche be equall4 ,example4 ,illustration ,in this figure you see firste the circle drawen1 whiche is ,a4,b4,c4,d1 and his centre is ,e4 ,in this circle also there are drawen two lines equally distaunt from the centre1 for the line ,a4,b1 and the line ,d4,c1 are iuste of one distaunce from the centre1 whiche is ,e1 and therfore are they of one length4 ,again thei are of one lengthe 7as shall be proued in the boke of profes7 and therefore their distaunce from the centre is all one4 ,the lx4 ,theoreme4 ,in euerie circle the longest line is the diameter1 and of all the other lines1 thei are still longest that be nexte vnto the centre1 and they be the shortest1 that be farthest distaunt from it4 ,example4 ,illustration ,in this circle ,a4,b4,c4,d1 ,i haue drawen first the diameter1 whiche is ,a4,d1 whiche passeth 7as it must7 by the centre ,e1 ,then haue ,i drawen ij4 other lines as ,m4,n1 whiche is neerer the centre1 and ,f4,g1 that is farther from the centre4 ,the fourth line also on the other side of the diameter1 that is ,b4,c1 is neerer to the centre then the line ,f4,g1 for it is of lyke distance as is the lyne ,m4,n4 ,nowe saie ,i1 that ,a4,d1 beyng the diameter1 is the longest of all those lynes1 and also of any other that maie be drawen within that circle1 ,and the other line ,m4,n1 is longer then ,f4,g4 ,also the line ,f4,g1 is shorter then the line ,b4,c1 for because it is farther from the centre then is the lyne ,b4,c4 ,and thus maie you iudge of al lines drawen in any circle1 how to know the proportion of their length1 by the proportion of their distance1 and contrary waies1 howe to discerne the proportion of their distance by their lengthes1 if you knowe the proportion of their length4 ,and to speake of it by the waie1 it is a maruaylouse thyng to consider1 that a man maie knowe an exacte proportion betwene two thynges1 and yet can not name nor attayne the precise quantitee of those two thynges1 ,as for exaumple1 ,if two squares be sette foorthe1 whereof the one containeth in it fiue square feete1 and the other contayneth fiue and fortie foote1 of like square feete1 ,i am not able to tell1 no nor yet anye manne liuyng1 what is the precyse measure of the sides of any of those 4ij4 squares1 and yet ,i can proue by vnfallible reason1 that their sides be in a triple proportion1 that is to saie1 that the side of the greater square 7whiche containeth 4xlv4 foote7 is three tymes so long iuste as the side of the lesser square1 that includeth but fiue foote4 ,but this seemeth to be spoken out of ceason in this place1 therfore ,i will omitte it now1 reseruyng the exacter declaration therof to a more conuenient place and time1 and will procede with the residew of the ,theoremes appointed for this boke4 ,the 4lxi4 ,theoreme4 ,if a right line be drawen at any end of a diameter in perpendicular forme1 and do make a right angle with the diameter1 that right line shall light without the circle1 and yet so iointly knitte to it1 that it is not possible to draw any other right line betwene that saide line and the circumference of the circle4 ,and the angle that is made in the semicircle is greater then any sharpe angle that may be made of right lines1 but the other angle without1 is lesser then any that can be made of right lines4 ,example4 ,illustration ,in this circle ,a4,b4,c1 the diameter is ,a4,c1 the perpendicular line1 which maketh a right angle with the diameter1 is ,c4,a1 whiche line falleth without the circle1 and yet ioyneth so exactly vnto it1 that it is not possible to draw an other right line betwene the circumference of the circle and it1 whiche thyng is so plainly seene of the eye1 that it needeth no farther declaracion4 ,for euery man wil easily consent1 that betwene the croked line ,a4,f1 7whiche is a parte of the circumference of the circle7 and ,a4,e 7which is the said perpendicular line7 there can none other line bee drawen in that place where they make the angle4 ,nowe for the residue of the theoreme4 ,the angle ,d4,a4,b1 which is made in the semicircle1 is greater then anye sharpe angle that may bee made of ryghte lines4 and yet is it a sharpe angle also1 in as much as it is lesser then a right angle1 which is the angle ,e4,a4,d1 and the residue of that right angle1 which lieth without the circle1 that is to saye1 ,e4,a4,b1 is lesser then any sharpe angle that can be made of right lines also4 ,for as it was before rehersed1 there canne no right line be drawen to the angle1 betwene the circumference and the right line ,e4,a4 ,then must it needes folow1 that there can be made no lesser angle of righte lines4 ,and againe1 if ther canne be no lesser then the one1 then doth it sone appear1 that there canne be no greater then the other1 for they twoo doo make the whole right angle1 so that if anye corner coulde be made greater then the one parte1 then shoulde the residue bee lesser then the other parte1 so that other bothe partes muste be false1 or els bothe graunted to be true4 ,the lxij4 ,theoreme4 ,if a right line doo touche a circle1 and an other right line drawen frome the centre of the circle to the pointe where they touche1 that line whiche is drawenne frome the centre1 shall be a perpendicular line to the touch line4 ,example4 ,illustration ,the circle is ,a4,b4,c1 and his centre is ,f4 ,the touche line is ,d4,e1 and the point wher they touch is ,c4 ,now by reason that a right line is drawen frome the centre ,f4 vnto ,c1 which is the point of the touche1 therefore saith the theoreme1 that the sayde line ,f4,c1 muste needes bee a perpendicular line vnto the touche line ,d4,e4 ,the lxiij4 ,theoreme4 ,if a righte line doo touche a circle1 and an other right line be drawen from the pointe of their touchinge1 so that it doo make righte corners with the touche line1 then shal the centre of the circle bee in that same line1 so drawen4 ,example4 ,illustration ,the circle is ,a4,b4,c1 and the centre of it is ,g4 ,the touche line is ,d4,c4,e1 and the pointe where it toucheth1 is ,c4 ,nowe it appeareth manifest1 that if a righte line be drawen from the pointe where the touch line doth ioine with the circle1 and that the said lyne doo make righte corners with the touche line1 then muste it needes go by the centre of the circle1 and then consequently it must haue the sayde centre in him4 ,for if the saide line shoulde go beside the centre1 as ,f4,c4 doth1 then dothe it not make righte angles with the touche line1 which in the theoreme is supposed4 ,the lxiiij4 ,theoreme4 ,if an angle be made on the centre of a circle1 and an other angle made on the circumference of the same circle1 and their grounde line be one common portion of the circumference1 then is the angle on the centre twise so great as the other angle on the circumference4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 and his centre is ,e3 the angle on the centre is ,c4,e4,d1 and the angle on the circumference is ,c4,a4,d t their commen ground line1 is ,c4,f4,d4 ,now say ,i that the angle ,c4,e4,d1 whiche is on the centre1 is twise so greate as the angle ,c4,a4,d1 which is on the circumference4 ,the lxv4 ,theoreme4 ,those angles whiche be made in one cantle of a circle1 must needes be equal togither4 ,example4 ,before ,i declare this theoreme by example1 it shall bee needefull to declare1 what is it to be vnderstande by the wordes in this theoreme4 ,for the sentence canne not be knowen1 onles the uery meaning of the wordes be firste vnderstand4 ,therefore when it speaketh of angles made in one cantle of a circle1 it is this to be vnderstand1 that the angle muste touch the circumference3 and the lines that doo inclose that angle1 muste be drawen to the extremities of that line1 which maketh the cantle of the circle4 ,so that if any angle do not touch the circumference1 or if the lines that inclose that angle1 doo not ende in the extremities of the corde line1 but ende other in some other part of the said corde1 or in the circumference1 or that any one of them do so eande1 then is not that angle accompted to be drawen in the said cantle of the circle4 ,and this promised1 nowe will ,i cumme to the meaninge of the theoreme4 ,i sette forthe a circle whiche is ,a4,b4,c4,d1 and his centre ,e1 in this circle ,i drawe a line ,d4,c1 whereby there ar made two cantels1 a more and a lesser4 ,the lesser is ,d4,e4,c1 and the geater is ,d4,a4,b4,c4 ,in this greater cantle ,i drawe two angles1 the firste is ,d4,a4,c1 and the second is ,d4,b4,c which two angles by reason they are made bothe in one cantle of a circle 7that is the cantle ,d4,a4,b4,c7 therefore are they both equall4 ,now doth there appere an other triangle1 whose angle lighteth on the centre of the circle1 and that triangle is ,d4,e4,c1 whose angle is double to the other angles1 as is declared in the lxiiij4 ,theoreme1 whiche maie stande well enough with this ,theoreme1 for it is not made in this cantle of the circle1 as the other are1 by reason that his angle doth not light in the circumference of the circle1 but on the centre of it4 ,illustration ,the 4lxvi4 theoreme4 ,euerie figure of foure sides1 drawen in a circle1 hath his two contrarie angles equall vnto two right angles4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 and the figure of foure sides in it1 is made of the sides ,b4,c1 and ,c4,d1 and ,d4,a1 and ,a4,b4 ,now if you take any two angles that be contrary1 as the angle by ,a1 and the angle by ,c1 ,i saie that those 4ij4 be equall to 4ij4 right angles4 ,also if you take the angle by ,b1 and the angle by ,d1 whiche two are also contray1 those two angles are like waies equall to two right angles4 ,but if any man will take the angle by ,a1 with the angle by ,b1 or ,d1 they can not be accompted contrary1 no more is not the angle by ,c4 estemed contray to the angle by ,b1 or yet to the angle by ,d1 for they onely be accompted contrary angles1 whiche haue no one line common to them bothe4 ,suche is the angle by ,a1 in respect of the angle by ,c1 for there both lynes be distinct1 where as the angle by ,a1 and the angle by ,d1 haue one common line ,a4,d1 and therfore can not be accompted contrary angles1 ,so the angle by ,d1 and the angle by ,c1 haue ,d4,c1 as a common line1 and therefore be not contrary angles4 ,and this maie you iudge of the residewe1 by like reason4 ,the lxvij4 ,theoreme4 ,vpon one right lyne there can not be made two cantles of circles1 like and vnequall1 and drawen towarde one parte4 ,example4 ,illustration ,cantles of circles be then called like1 when the angles that are made in them be equall4 ,but now for the ,theoreme1 let the right line be ,a4,e4,c1 on whiche ,i draw a cantle of a circle1 whiche is ,a4,b4,c4 ,now saieth the ,theoreme1 that it is not possible to draw an other cantle of a circle1 whiche shall be vnequall vnto this first cantle1 that is to say1 other greatter or lesser then it1 and yet be lyke it also1 that is to say1 that the angle in the one shall be equall to the angle in the other4 ,for as in this example you see a lesser cantle drawen also1 that is ,a4,d4,c1 so if an angle were made in it1 that angle would be greatter then the angle made in the cantle ,a4,b4,c1 and therfore can not they be called lyke cantels1 but and if any other cantle were made greater then the first1 then would the angle in it be lesser then that in the firste1 and so nother a lesser nother a greater cantle can be made vpon one line with an other1 but it will be vnlike to it also4 ,the 4lxviij4 ,theoreme4 ,lyke cantelles of circles made on equal righte lynes1 are equall together4 ,example4 ,what is ment by like cantles you haue heard before4 and it is easie to vnderstand1 that suche figures a called equall1 that be of one bygnesse1 so that the one is nother greater nother lesser then the other4 ,and in this kinde of comparison1 they must so agree1 that if the one be layed on the other1 they shall exactly agree in all their boundes1 so that nother shall excede other4 ,illustration ,nowe for the example of the ,theoreme1 ,i haue set forthe diuers varieties of cantles of circles1 amongest which the first and seconde are made vpon equall lines1 and ar also both equall and like4 ,the third couple ar ioyned in one1 and be nother equall1 nother like1 but expressyng an absurde deformitee1 whiche would folowe if this ,theoreme wer not true4 ,and so in the fourth couple you maie see1 that because they are not equall cantles1 therfore can not they be like cantles1 for necessarily it goeth together1 that all cantles of circles made vpon equall right lines1 if they be like they must be equall also4 ,the lxix4 ,theoreme4 ,in equall circles1 suche angles as be equall are made vpon equall arch lines of the circumference1 whether the angle light on the circumference1 or on the centre4 ,example4 ,illustration ,firste ,i haue sette for an exaumple twoo equall circles1 that is ,a4,b4,c4,d1 whose centre is ,k1 and the second circle ,e4,f4,g4,h1 and his centre ,l1 and in eche of them is there made two angles1 one on the circumference1 and the other on the centre of eche circle1 and they be all made on two equall arche lines1 that is ,b4,c4,d4 the one1 and ,f4,g4,h4 the other4 ,now saieth the ,theoreme1 that if the angle ,b4,a4,d1 be equall to the angle ,f4,e4,h1 then are they made in equall circles1 and on equall arch lines of their circumference4 ,also if the angle ,b4,k4,d1 be equal to the angle ,f4,l4,h1 then be they made on the centres of equall circles1 and on equall arche lines1 so that you muste compare those angles together1 whiche are made both on the centres1 or both on the circumference1 and maie not conferre those angles1 wherof one is drawen on the circumference1 and the other on the centre4 ,for euermore the angle on the centre in suche sorte shall be double to the angle on the circumference1 as is declared in the three score and foure ,theoreme4 ,the 4lxx4 ,theoreme4 ,in equall circles1 those angles whiche bee made on equall arche lynes1 are euer equall together1 whether they be made on the centre1 or on the circumference4 ,example4 ,this ,theoreme doth but conuert the sentence of the last ,theoreme before1 and therfore is to be vnderstande by the same examples1 for as that saith1 that equall angles occupie equall archelynes1 so this saith1 that equal arche lines causeth equal angles1 consideringe all other circumstances1 as was taughte in the laste theoreme before1 so that this theoreme dooeth affirming speake of the equalitie of those angles1 of which the laste theoreme spake conditionally4 ,and where the laste theoreme spake affirmatiuely of the arche lines1 this theoreme speaketh conditionally of them1 as thus3 ,if the arche line ,b4,c4,d4 be equall to the other arche line ,f4,g4,h1 then is that angle ,b4,a4,d4 equall to the other angle ,f4,e4,h4 ,or els thus may you declare it causally3 ,bicause the arche line ,b4,c4,d1 is equal to the other arche line ,f4,g4,h1 therefore is the angle ,b4,k4,d4 equall to the angle ,f4,l4,h1 consideringe that they are made on the centres of equall circles4 ,and so of the other angles1 bicause those two arche lines aforesaid ar equal1 therfore the angle ,d4,a4,b1 is equall to the angle ,f4,e4,h1 for as muche as they are made on those equall arche lines1 and also on the circumference of equall circles4 ,and thus these theoremes doo one declare an other1 and one verifie the other4 ,the lxxi4 ,theoreme4 ,in equal circles1 equall right lines beinge drawen1 doo cutte awaye equalle arche lines frome their circumferences1 so that the greater arche line of the one is equall to the greater arche line of the other1 and the lesser to the lesser4 ,example4 ,illustration ,the circle ,a4,b4,c4,d1 is made equall to the circle ,e4,f4,g4,h1 and the right line ,b4,d4 is equal to the righte line ,f4,h1 wherfore it foloweth1 that the ij4 arche lines of the circle ,a4,b4,d1 whiche are cut from his circumference by the right line ,b4,d1 are equall to two other arche lines of the circle ,e4,f4,h1 being cutte frome his circumference1 by the right line ,f4,h4 that is to saye1 that the arche line ,b4,a4,d1 beinge the greater arch line of the firste circle1 is equall to the arche line ,f4,e4,h1 beynge the greater arche line of the other circle4 ,and so in like manner the lesser arche line of the firste circle1 beynge ,b4,c4,d1 is equal to the lesser arche line of the seconde circle1 that is ,f4,g4,h4 ,the lxxij4 ,theoreme4 ,in equall circles1 vnder equall arche lines the right lines that bee drawen are equall togither4 ,example4 ,this ,theoreme is none other1 but the conuersion of the laste ,theoreme beefore1 and therefore needeth none other example4 ,for as that did declare the equalitie of the arche lines1 by the equalitie of the righte lines1 so dothe this ,theoreme declare the equalnes of the right lines to ensue of the equalnes of the arche lines1 and therefore declareth that right lyne ,b4,d1 to be equal to the other right line ,f4,h1 bicause they both are drawen vnder equall arche lines1 that is to saye1 the one vnder ,b4,a4,d1 and thother vnder ,f4,e4,h1 and those two arch lines are estimed equall by the theoreme laste before1 and shal be proued in the booke of proofes4 ,the lxxiij4 ,theoreme4 ,in euery circle1 the angle that is made in the halfe circle1 is a iuste righte angle1 and the angle that is made in a cantle greater then the halfe circle1 is lesser thanne a righte angle1 but that angle that is made in a cantle1 lesser then the halfe circle1 is greatter then a right angle4 ,and moreouer the angle of the greater cantle is greater then a righte angle and the angle of the lesser cantle is lesser then a right angle4 ,example4 ,in this proposition1 it shal be meete to note1 that there is a greate diuersite betwene an angle of a cantle1 and an angle made in a cantle1 and also betwene the angle of a semicircle1 and ye angle made in a semicircle4 ,also it is meet to note yt al angles that be made in ye part of a circle1 ar made other in a semicircle1 7which is the iuste half circle7 or els in a cantle of the circle1 which cantle is other greater or lesser then the semicircle is1 as in this figure annexed you maye perceaue euerye one of the thinges seuerallye4 ,illustration ,firste the circle is1 as you see1 ,a4,b4,c4,d1 and his centre ,e1 his diameter is ,a4,d1 ,then is ther a line drawen from ,a4 to ,b1 and so forth vnto ,f1 which is without the circle3 and an other line also frome ,b4 to ,d1 whiche maketh two cantles of the whole circle4 ,the greater cantle is ,d4,a4,b1 and the lesser cantle is ,b4,c4,d1 ,in whiche lesser cantle also there are two lines that make an angle1 the one line is ,b4,c1 and the other line is ,c4,d4 ,now to showe the difference of an angle in a cantle1 and an angle of a cantle1 first for an example ,i take the greter cantle ,b4,a4,d1 in which is but one angle made1 and that is the angle by ,a1 which is made of a line ,a4,b1 and the line ,a4,d1 ,and this angle is therfore called an angle in a cantle4 ,but now the same cantle hathe two other angles1 which be called the angles of that cantle1 so the twoo angles made of the righte line ,d4,b1 and the arche line ,d4,a4,b1 are the twoo angles of this cantle1 whereof the one is by ,d1 and the other is by ,b4 ,wher you must remembre1 that the angle by ,d4 is made of the right line ,b4,d1 and the arche line ,d4,a4 ,and this angle is diuided by an other right line ,a4,e4,d1 which in this case must be omitted as no line4 ,also the angle by ,b4 is made of the right line ,d4,b1 and of the arch line 4,b4,a1 although it be deuided with ij4 other right lines1 of wch the one is the right line ,b4,a1 thother the right line ,b4,e1 yet in this case they ar not to be considered4 ,and by this may you perceaue also which be the angles of the lesser cantle1 the first of them is made of ye right line ,b4,d1 of ye arch line ,b4,c1 the second is made of the right line 4,d4,b1 of the arch line ,d4,c4 ,then ar ther ij4 other lines1 wch deuide those ij4 corners1 yt is the line ,b4,c1 the line ,c4,d1 wch ij4 lines do meet in the poynte ,c1 and there make an angle1 whiche is called an angle made in that lesser cantle1 but yet is not any angle of that cantle4 ,and so haue you heard the difference betweene an angle in a cantle1 and an angle of a cantle4 ,and in lyke sorte shall you iudg of the angle made in a semicircle1 whiche is distinct from the angles of the semicircle4 ,for in this figure1 the angles of the semicircle are those angles which be by ,a4 and ,d1 and be made of the right line ,a4,d1 beeyng the diameter1 and of the halfe circumference of the circle1 but by the angle made in the semicircle is that angle by ,b1 whiche is made of the righte line ,a4,b1 and that other right line ,b4,d1 whiche as they mete in the circumference1 and make an angle1 so they ende with their other extremities at the endes of the diameter4 ,these thynges premised1 now saie ,i touchyng the ,theoreme1 that euerye angle that is made in a semicircle1 is a right angle1 and if it be made in any cantle of a circle1 then must it neds be other a blunt angle1 or els a sharpe angle1 and in no wise a righte angle4 ,for if the cantle wherein the angle is made1 be greater then the halfe circle1 then is that angle a sharpe angle4 ,and generally the greater the cantle is1 the lesser is the angle comprised in that cantle3 and contrary waies1 the lesser any cantle is1 the greater is the angle that is made in it4 ,wherfore it must nedes folowe1 that the angle made in a cantle lesse then a semicircle1 must nedes be greater then a right angle4 ,so the angle by ,b1 beyng made at the right line ,a4,b1 and the righte line ,b4,d1 is a iuste righte angle1 because it is made in a semicircle4 ,but the angle made by ,a1 which is made of the right line ,a4,b1 and of the right line ,a4,d1 is lesser then a righte angle1 and is named a sharpe angle1 for as muche as it is made in a cantle of a circle1 greater then a semicircle4 ,and contrary waies1 the angle by ,c1 beyng made of the righte line ,b4,c1 and of the right line ,c4,d1 is greater then a right angle1 and is named a blunte angle1 because it is made in a cantle of a circle1 lesser then a semicircle4 ,but now touchyng the other angles of the cantles1 ,i saie accordyng to the ,theoreme1 that the 4ij4 angles of the greater cantle1 which are by ,b4 and ,d1 as is before declared1 are greatter eche of them then a right angle4 ,and the angles of the lesser cantle1 whiche are by the same letters ,b1 and ,d1 but be on the other side of the corde1 are lesser eche of them then a right angle1 and be therfore sharpe corners4 ,the lxxiiij4 ,theoreme4 ,if a right line do touche a circle1 and from the pointe where they touche1 a righte lyne be drawen crosse the circle1 and deuide it1 the angles that the saied lyne dooeth make with the touche line1 are equall to the angles whiche are made in the cantles of the same circle1 on the contrarie sides of the lyne aforesaid4 ,example4 ,illustration ,the circle is ,a4,b4,c4,d1 and the touche line is ,e4,f4 ,the pointe of the touchyng is ,d1 from which point ,i suppose the line ,d4,b1 to be drawen crosse the circle1 and to diuide it into 4ij4 cantles1 wherof the greater is ,b4,a4,d1 and the lesser is ,b4,c4,d1 and in ech of them an angle is drawen1 for in the greater cantle the angle is by ,a1 and is made of the right lines ,b4,a1 and ,a4,d1 in the lesser cantle the angle is by ,c1 and is made of ye right lines ,b4,c1 and ,c4,d4 ,now saith the ,theoreme that the angle ,b4,d4,f1 is equall to the angle made in the cantle on the other side of the said line1 that is to saie1 in the cantle ,b4,a4,d1 so that the angle ,b4,d4,f1 is equall to the angle ,b4,a4,d1 because the angle ,b4,d4,f1 is on the one side of the line ,b4,d1 7whiche is according to the supposition of the ,theoreme drawen crosse the circle7 and the angle ,b4,a4,d1 is in the cantle on the other side4 ,likewaies the angle ,b4,d4,e1 beyng on the one side of the line ,b4,d1 must be equall to the angle ,b4,c4,d1 7that is the angle by ,c17 whiche is made in the cantle on the other side of the right line ,b4,d4 ,the profe of all these ,i do reserue1 as ,i haue often saide1 to a conuenient boke1 wherein they shall be all set at large4 ,the 4lxxv4 ,theoreme4 ,in any circle when 4ij4 right lines do crosse one an other1 the likeiamme that is made of the portions of the one line1 shall be equall to the lykeiamme made of the partes of the other lyne4 ,illustration ,because this ,theoreme doth serue to many vses1 and wold be wel vnderstande1 ,i haue set forth 4ij4 examples of it4 ,in the firste1 the lines by their crossyng do make their portions somewhat toward an equalitie4 ,in the second the portions of the lynes be very far from an equalitie1 and yet in bothe these and in all other ye ,theoreme is true4 ,in the first example the circle is ,a4,b4,c4,d1 in which thone line ,a4,c1 doth crosse thother line ,b4,d1 in ye point ,e4 ,now if you do make one likeiamme or longsquare of ,d4,e1 ,e4,b1 being ye 4ij4 portions of the line ,d4,b1 that longsquare shall be equall to the other longsquare made of ,a4,e1 and ,e4,c1 beyng the portions of the other line ,a4,c4 ,lykewaies in the second example1 the circle is ,f4,g4,h4,k1 in whiche the line ,f4,h1 doth crosse the other line ,g4,k1 in the pointe ,l4 ,wherfore if you make a lykeiamme or longsquare of the two partes of the line ,f4,h1 that is to saye1 of ,f4,l1 and ,l4,h1 that longsquare will be equall to an other longsquare made of the two partes of the line ,g4,k4 which partes are ,g4,l1 and ,l4,k4 ,those longsquares haue ,i set foorth vnder the circles containyng their sides1 that you maie somewhat whet your own wit in practisyng this ,theoreme1 accordyng to the doctrine of the nineteenth conclusion4 ,the 4lxxvi4 ,theoreme4 ,if a pointe be marked without a circle1 and from that pointe two right lines drawen to the circle1 so that the one of them doe runne crosse the circle1 and the other doe touche the circle onely1 the long square that is made of that whole lyne which crosseth the circle1 and the portion of it1 that lyeth betwene the vtter circumference of the circle and the pointe1 shall be equall to the full square of the other lyne1 that onely toucheth the circle4 ,example4 ,the circle is ,d4,b4,c1 and the pointe without the circle is ,a1 from whiche pointe there is drawen one line crosse the circle1 and that is ,a4,d4,c1 and an other lyne is drawn from the said pricke to the marge or edge of the circumference of the circle1 and doeth only touche it1 that is the line ,a4,b4 ,and of that first line ,a4,d4,c1 you maie perceiue one part of it1 whiche is ,a4,d1 to lie without the circle1 betweene the vtter circumference of it1 and the pointe assigned1 whiche was ,a4 ,nowe concernyng the meanyng of the ,theoreme1 if you make a longsquare of the whole line ,a4,c1 and of that parte of it that lyeth betwene the circumference and the point1 7whiche is ,a4,d17 that longesquare shall be equall to the full square of the touche line ,a4,b1 accordyng not onely as this figure sheweth1 but also the saied nyneteenth conclusion dooeth proue1 if you lyste to examyne the one by the other4 ,illustration ,the lxxvii4 ,theoreme4 ,if a pointe be assigned without a circle1 and from that pointe 4ij4 right lynes be drawen to the circle1 so that the one doe crosse the circle1 and the other dooe ende at the circumference1 and that the longsquare of the line which crosseth the circle made with the portion of the same line beyng without the circle betweene the vtter circumference and the pointe assigned1 doe equally agree with the iuste square of that line that endeth at the circumference1 then is that lyne so endyng on the circumference a touche line vnto that circle4 ,example4 ,in as muche as this ,theoreme is nothyng els but the sentence of the last ,theoreme before conuerted1 therfore it shall not be nedefull to vse any other example then the same1 for as in that other ,theoreme because the one line is a touche lyne1 therfore it maketh a square iust equal with the longsquare made of that whole line1 whiche crosseth the circle1 and his portion liyng without the same circle4 ,so saith this ,theoreme3 that if the iust square of the line that endeth on the circumference1 be equall to that longsquare whiche is made as for his longer sides of the whole line1 which commeth from the pointt assigned1 and crosseth the circle1 and for his other shorter sides is made of the portion of the same line1 liyng betwene the circumference of the circle and the pointe assigned1 then is that line whiche endeth on the circumference a right touche line1 that is to saie1 yf the full square of the right line ,a4,b1 be equall to the longsquare made of the whole line ,a4,c1 as one of his lines1 and of his portion ,a4,d1 as his other line1 then must it nedes be1 that the lyne ,a4,b1 is a right touche lyne vnto the circle ,d4,b4,c4 ,and thus for this tyme ,i make an ende of the ,theoremes4 ,,finis1 ,,imprinted at ,london in ,poules churcheyarde1 at the signe of the ,bra- senserpent1 by ,reynold ,wolfe4 ,cum priuilegio ad imprimen- dum solum4 ,,anno ,,domini 4,m4,d4,l4,i4 ,illustration3 ,publisher's ,device 7brazen serpent1 ,,nvm4 ,,xxi47 99 99 99 99 99 99 99 99 99 ,errors and ,inconsistencies3 ,unless otherwise noted1 spelling1 punctuation and capitalization are unchanged4 ,forms were regularized only where there was a very large disparity between the expected form and the apparent errors 7for example1 a thousand ,a4,b against a dozen ,a1,b71 or a flagrant misprint such as cnt for cut4 ,the letters u and v follow the conventional initial v1 non-initial u pattern except in numbers 7xv1 iv74 ,the lower-case j form occurs only as the last digit of a number 7ij1 xxj72 upper-case ,i and ,j share a form1 always read as ,i4 ,capital and lower-case w were often used interchangeably4 ,words split across line breaks may or may not have a hyphen4 ,language3 ,the word other is used interchangeably with both or and either2 similarly1 nother is used in place of nor and neither4 ,the expression an other is almost always written as two words4 ,the spellings then7ne7 and than7ne7 are used interchangeably2 than7ne7 is rare4 ,the spelling liyng1 both by itself and as the end of a longer word1 is used consistently4 ,illustrations3 ,a number of illustrations contain errors such as unmarked or mislabeled points 7circle ,b4,c4,d where only ,c and ,d are labeled74 ,errors of this type are identified in the illustrated ,,html version1 but not in the present text-only file4 ,greek3 ,all ,greek is shown and transliterated as printed4 ,errors or anomalies include missing1 misplaced or incorrect diacritics2 the non-final form of sigma used at the end of words2 and word-final m for n4 ,meaningful ,errors and ,anomalies3 shal be clean extrirped and rooted out spelling unchanged3 intended form not certain new and new causes to pray for your maiestie1 perceiuyng text new and new causes probably intentional ,preface ,and thus for this tyme ,i make an end444 ,body text ,the definitions of the principles of ,,geometry4 ,pagination as shown by signature numbers demands another leaf 7two pages7 between the end of the ,preface and the beginning of the body text4 ,but no text is missing1 and the facsimile has no blank pages4 ,otherlesse then it as you se ,d text unchanged3 intended wording uncertain ,,the 4,,xxii4 ,,conclvsion4 4,,xxi4 ,,the ,,xxxviii4 ,,conclvsion4 ,,xxxvii4 ,the xxvi4 ,theoreme4 ,there is no xxv4 7#be7 theorem4 ,the 4xxxix4 theoreme4 ,the text of the ,example is garbled1 and does not fit the illustration4 ,among other problems1 points ,c and ,d seem to have been switched1 either in the text or in the illustration4 ,the circle is ,a4,b4,c4,d1 and his centre is ,e3 the angle on the centre is ,c4,e4,d1 and the angle on the circumference is ,c4,a4,d t their commen ground line1 is ,c4,f4,d4 cirle printing of ,c4,e4,d unclear3 looks like ,c4,f4,d but center of circle is ,e lone t may be error for ampersand or other punctuation ,c4,f4,d1 ,misprints3 ,aristotle had putte forthe certaine bookes kookes ,and his father king ,dauid ioyneth ,danid those thynges are done by negromancy4 ,and hereof came it that fryer ,bakon was accompted so greate a negromancier spelling unchanged ,for vndoubtedly if they mean vndoudtedly if the that ,godde was alwaies workinge alwaaies all the lines that be drawen to the circumference circumfernece ,an other hath but one compassed hatht whose sydes partlye are all equall as in ,a eqnall as this ,a4 doth partly expresse ,a doth partly eppresse ,to make a threlike triangle on any lyne measurable4 or any and it shall cut the first line in two equall portions4 cnt then open ,i the compass as wide as 4iiij4 partes compaas ,to make a plumbe lyne on any porcion of a circle or any for euery triangle together an equal likeiamme text has to_/gether at line break3 may be meant for two words as you mai se by ,c and ,d1 for ij4 sides of both the trianngles ar parallels4 you maise 444 triangls then will the square of that greater portion portior that line ,i say is a touche line touthe 7that is to saie ,m4 ,g17 error for in ,g8 one perpendicular from ,g4 vnto the side side ,b4,c ,a4,c then must ,i first draw a tuche lyne wust anye of the two lynes contayninge the angle appointed4 nye of draw thence two lines1 one to ,d one to ,a which is made accordinge to the conclusion4 ancordinge you shal perceaue that there will be peceaue a _/ more conueni_/ent time4 text has conui_/ent at line break other in deciding some controuersy decising a great deale the soner somner ,whiche example hath beene vsede hat whan ,i wrote these first connclusions cunclusions suche bokes as ar appoynted as at ,will ,i refuse4 ,willl all right angles be equall eqnall whiche thinge the better to perceaue peceaue whiche ,i do only by examples declare declae about the ground line1 are equal togither groud you shal take this triangle ,a4,b4,c4 which hath a very blunt corner word this illegible veery blunt the one is a longe square ,a4,b4,e printed as shown3 expected form is ,a4,b4,f4,e though they be diuers in numbre4 numbhe proofe that ,g4,h4 being the ground line groud and standing betwene one paire of parallels an for thei ar the two ye contrary sides word ye superfluous8 they haue one ground line ,d4,e4 on ,by the square of any lyne sqnare squares that are made of the same line sane ,the fyrst lyne propounded is ,a4,b propouned hath another longe square equall to hym text has anomalous a _/ nother at line break one of those parts again into other ij4 parts iuto which thing the easyer is to be vnderstande eayser and blunt cornered triangles couered the square marked with ,g4 is the square of ,a4,b with ,c4 ,and from all pointes you maye drawe ij4 equall lines poittes ,if two circles bee drawen so one withoute an other and other drawen frome the centre of the circle to the pointe tge which in the theoreme is supposed4 the 4heoreme and therfore can not they be called lyke cantels ban then would the angle in it be lesser it being cutte frome his circumference1 by the right line ,f4,h4 circumforence ,and in lyke sorte shall you iudg ,and n lyke ,punctuation1 ,spacing1 ,capitalization3 ,phrase breaks where a comma is followed by a capital letter1 or a period by a lower-case letter1 are not individually noted4 ,number forms such as those4 ij4 last or line4 ,a4,b4 were silently regularized to those 4ij4 last and line 4,a4,b4 ,missing sentence-final periods at the end of a printed line were silently supplied4 ,initial u or medial v unchanged3 ioyneth uertuous conuersacion ,,xxv4 ,,conclvsion4444 the whole circle agreynge therevnto4 or eande in the utter edge of his circumference onles the uery meaning of the wordes be firste vnderstand ,geometry teacheth the drawyng1 ,measuring and proporcion capital ,m in original ,and herof commeth that seconnde thing wherin al agree that seconnde 4 thing if they can with their wysedome ouercome all vyces4 ,of the firste of those three sortes text reads 444 their rwysedome 444 _/ 444 th ee sortes 444 on consecutive lines4 ,the extraneous r is directly above the missing or invisible r akroamatikoi4 final 4 missing was he estemed for his wisdom4 final 4 missing wisedom is better then pretious stones 4 yea punctuation unchanged your ,maiesties excellencye1 excellencye11 if his subiectes be riche in substaunce1 sub staunce new and new causes to pray for your maiestie1 perceiuyng maiestie1perceiuyng can be ignorant thereof1 in so much that thereof4 in those thynges are done by negromancy4 spelling unchanged thogh it be but smal1 and thefore not notable4 final 4 missing but boweth any waye1 such are called waye4 such not call it one croked lyne1 but rather cr oked ,now to geue you example of triangles1 triangles11 a portion of a globe as the figur marked wt ,a4 final 4 missing as is the figure ,g4 other ij4 sharp and one blunt these examples 4,m4 ,n1 and ,o4 where ,m4 hath a right angle1 ,n1 a blunte angle 4,m4,n1 and ,o where 444 ,n1 ,a1 blunte as in this example because ,a4,b1 is drawen in length ,a1,b1 the angle ,c1 is called an vtter angle4 final 4 missing diamondlike1 whose figur is noted with ,t4 final 4 missing and partlye vnequall1 as in ,b1 and they in1 ,b and as in this figure is some what expressed4 final 4 missing then draw a line from ,b1 to ,f1 so ,i haue mine intent4 final 4 missing then open ,i the compasse as wyde as ,a4,c1 ,a1,c1 ,f1 this space betwene ,d4 and ,f4 betwene ,d and ,f4 crosse that other first line in 4ij4 places4 final 4 missing ,the arch to be diuided ys ,a4,d4,c1 the corde is ,a4,b4,c1 ,a1,b4,c1 and ther set a mark4 ,then take a long line mark ,then a parallel must be drawen howe you shall doo it4 doo it1 equal to ,a4,b1 as for example ,d4,f4 ,d1,f4 accordyng to the fifth conclusion accordyngt o 7accordyng as the 4viij4 conclusion teacheth7 comma for close parenthesis it hath one angle 7that is ,b4,a4,e47 like to ,d close parenthesis missing therefore ,i wyl by example sette forth ther efore thenne drawe ,i a line also from ,n4 vnto ,c4 vnto ,c is equall vnto ,a4,b1 and so is ,k4,l4 ,k4,l1 you mai se by ,c4 and ,d1 for ij4 sides by ,c and ,d1 accordyng to the lengthe lo the peece that remaineth error for of the8 th e peece this line ,a4,b1 7wch was assigned vnto me7 ,a1,b1 thus haue you attained ye vse of this conclusion4 final 4 missing ,make in a table the like draught ,a table vsing it as it wer a croked ruler4 final 4 missing the lines of diuision ,a4,d1 ,b4,e ,a4 ,d4 ,b4 ,e whose two sides ,a4,c4 and ,b4,c4 are diuided ,b1,c4 enclose those iij prickes4 that centre as you se is ,d prickes that centre 4 as as you se ,b4,a4,d1 and ,b4,c4,e4 ,b4,c1,e4 at this tyme wyll shewe you she1we as youre selfe mai easily gesse4 final 4 missing ,a4,c4 and ,b4,d4 are the two diameters ,a4,c4 and ,b ,d4 ,a4,b4,c4,d4 is the quadrate appointed ,a1,b4,c4,d4 quadrate from angle to angle1 as you se ,a4,c4 ,b ,d4 ,a4,c4 ,b ,d4 ,a4,b4,c4 is the circle1 whiche ,i haue deuided ,a1,b4,c4 ,and from eche pricke ij4 lines drawen line sdrawen ,i would make a circle4 ,therfore ,i drawe circle1 ,therfore of equall sides and equall angles4 angles1 perceaue by the xxxvij4 xxxviij4 xxxix4 and xl4 conclusions xxxviij xxxix4 that be nighest that be 4 nighest learne the demonstrations by harte1 7as somme missing open parenthesis sundrye woorkes partely ended1 and partely to bee ended ,and ,as for example ,a--------,b4 ,a4 being the one pricke ,a being that corner which is greatter then a right angle s in is invisible firste these two right lines ,a4,b4 and ,a4,c4 ,a ,b4 and ,a4,c4 ij4 quantities1 as ,a4 and ,b1 be equal to an other as ,a and ,b1 in an other place4 ,in the mean season in drawen forthe vnto ,d4 and ,e4 vnto ,d and ,e4 ,the thirde ,theoreme4 ,example4 final 4 missing and yet the ij4 lesser sides togither ar greater then it4 text has 444 yet thr 444 ar greate at consecutive line-ends the angle ,c4 7whiche are the ij4 angles contayned missing open parenthesis ,m4,n4 equall also to ,h4,l4 ,h1,l4 therefore are ,a4,c4 and ,b4,d4 bothe equall ,a1,c4 ,a4,d4,e1 and ,d4,e4,b1 which 7as the xxvij4 missing open parenthesis ,the xxxi4 theoreme4 xxxi1 theoreme4 there is made a triangle ,b4,c4,g1 and a lykeiamme ,b4,c ,g1 fyll vp the sydes of the 4ij4 fyrste square lykeiammes 4ij fyrste equall to the 4ij4 squares of both the other sides4 final 4 missing ,the ij4 lines proposed ar ,a4,b4 and ,c4,d ,a ,b4 and ,c4,d for the square ,d4,e4,f4,g4 is equal to the two other partial squares of ,d4,h4,k4,g and ,h4,e4,f4,k1 ,d1,e4,f4,g4 444 ,d4,h4,k ,g ,the xxxvij4 ,theoreme4 xxxvij ,theoreme4 the square that is made of the whole line first t in that invisible 7which is equall with ,d4,g47 ,d1,g4 ,lette the diuided line bee ,a4,b1 and parted in ,c ,a1,b1 the square of the whole lyne ,a4,b1 ,a1,b1 as herafter ,i will declare in conuenient place4 final 4 missing two lesser squares beyng taken away1 close parenthesis for comma the great square1 and that is ,g4,f4,m4,h4 ,g1,f4,m4,h4 two vnequall partes as happeneth4 ,the long square the ,the 4xlvi4 ,theoreme4 ,the4xlvi4 ,the xlvij4 ,theoreme4 xlvij ,theoreme4 and doo not passe by the centre passeby ,for as you may easily iudge1 ,a4,c4 hath one portion ,a ,c4 if they be equally distaunt from one halfe of the diameter second l in equally invisible then it1 and beynge farther of it4 and the second circle is ,b4,c4,d4,e1 and they crosse ,b4,c4,d1,e1 ,the second circle is ,d4,b4,c1 and his centre is ,h ,d1,b4,c1 that is farther from the centre4 ,the fourth centre1 ,the twise so great as the other angle on the circumference4 final 4 missing ,the lesser is ,d4,e4,c1 and the geater is ,d4,a4,b4,c4 ,d4,f4,c1 therefore are they both equall4 final 4 missing ,what is ment by like cantles you haue heard before mentby by the equalitie of the righte lines1 so dothe this ,theoreme so do the this ,also it is meet to note yt al angles that be made ytal whiche maketh two cantles of the whole circle4 circle1 which is made of a line ,a4,b1 and the line ,a4,d1 ,a1,b1 ,i saie accordyng to the ,theoreme1 that the 4ij4 angles 4ij angles so that the angle ,b4,d4,f1 is equall to the angle ,b4,a4,d ,b ,d4,f1 make their portions somewhat toward an equalitie4 final 4 missing doth crosse thother line ,b4,d1 in ye point ,e4 ,b ,d1 whiche was ,a4 ,nowe concernyng the meanyng ,no we