The Path-Way to Knowledg, Containing the First Principles of Geometrie

by Robert Record

[This text] includes characters that require UTF-8 (Unicode) file encoding:

ã ẽ ĩ õ ũ (vowels with overline, shown here as a tilde)

ἐίπερ γὰρ ἀδικεῖμ χρὴ (Greek, mainly in the introduction)

If any of these characters do not display properly—in particular, if the diacritic does not appear directly above the letter—or if the apostrophes and quotation marks in this paragraph appear as garbage, you may have an incompatible browser or unavailable fonts. First, make sure that the browser’s “character set” or “file encoding” is set to Unicode (UTF-8). You may also need to change your browser’s default font.

Unless otherwise noted, spelling, punctuation and capitalization are unchanged. Text in sans-serif type was originally printed in blackletter (“Gothic”, Old English). Typographical errors are marked in the text with mouse-hover popups. Trans­literation of single Greek words and short phrases is shown in the same way; the longer Greek passages from the introduction are given at the end of the e-text.

The book does not have page numbers. Instead, it labeled the recto (odd) pages of the first few leaves of each 8-page signature. These will appear in the right margin as A.i., A.ij., A.iij.... Page numbers in brackets, including all verso (v) pages, were added by the transcriber.

Any problems or explanations too long or complicated to fit into a popup are noted either in a separate paragraph or at the [end of the e-text].

[Full Contents]
[Principles of Geometry]
[Conclusions]
[Axioms]
(“Grauntable Requestes”
and “Common Sentences”)
[Theorems]

Geometries verdicte

All fresshe fine wittes by me are filed,

All grosse dull wittes wishe me exiled:

Thoughe no mannes witte reiect will I,

Yet as they be, I wyll them trye.

---

[ The argumentes of the foure bookes]

The first booke declareth the definitions of the termes and names vsed in Geometry, with certaine of the chiefe grounds whereon the arte is founded. And then teacheth those conclusions, which may serue diuersely in al workes Geometricall.

The second booke doth sette forth the Theoremes, (whiche maye be called approued truthes) seruinge for the due knowledge and sure proofe of all conclusions and workes in Geometrye.

The third booke intreateth of diuers formes, and sondry protractions thereto belonging, with the vse of certain conclusions.

The fourth booke teacheth the right order of measuringe all platte formes, and bodies also, by reson Geometricall.

[Contents]
(added by transcriber)

[Title Page] (above)

[Arguments of the Four Books] (above)

First Book:
[To the Gentle Reader]
[Dedication to King Edward VI]
[Preface to the First Book]
[The Principles of Geometry]
[Conclusions 1–46]

Second Book:
[Title Page]
[Preface to the Second Book]
[Grantable Requests]
[Common Sentences]
[Theorems 1–77]

[Text of Decorative Headers]

[Transcriber’s Notes]

[ TO THE GENTLE READER.]

xcvse me, gentle reder if oughte be amisse, straung paths ar not trodẽ al truly at the first: the way muste needes be comberous, wher none hathe gone before. Where no man hathe geuen light, lighte is it to offend, but when the light is shewed ones, light is it to amende. If my light may so light some other, to espie and marke my faultes, I wish it may so lighten thẽ, that they may voide offence. Of staggeringe and stomblinge, and vnconstaunt turmoilinge: often offending, and seldome amending, such vices to eschewe, and their fine wittes to shew that they may winne the praise, and I to hold the candle, whilest they their glorious works with eloquence sette forth, so cunningly inuented, so finely indited, that my bokes maie seme worthie to occupie no roome. For neither is mi wit so finelie filed, nother mi learning so largly lettred, nother yet mi laiser so quiet and vncõbered, that I maie perform iustlie so learned a laboure or accordinglie to accomplishe so

haulte an enforcement, yet maie I thinke thus: This candle did I light: this light haue I kindeled: that learned men maie se, to practise their pennes, their eloquence to aduaunce, to register their names in the booke of memorie I drew the platte rudelie, whereon thei maie builde, whom god hath indued with learning and liuelihod. For liuing by laboure doth learning so hinder, that learning serueth liuinge, whiche is a peruers trade. Yet as carefull familie shall cease hir cruell callinge, and suffre anie laiser to learninge to repaire, I will not cease from trauaile the pathe so to trade, that finer wittes maie fashion them selues with such glimsinge dull light, a more complete woorke at laiser to finisshe, with inuencion agreable, and aptnes of eloquence.

And this gentle reader I hartelie protest where erroure hathe happened I wisshe it redrest.

---

t is not vnknowen to youre maiestie, moste soueraigne lorde, what great disceptacion hath been amongest the wyttie men of all nacions, for the exacte knoweledge of true felicitie, bothe what it is, and wherein it consisteth: touchynge whiche thyng, their opinions almoste were as many in numbre, as were the persons of them, that either disputed or wrote thereof. But and if the diuersitie of opinions in the vulgar sort for placyng of their felicitie shall be considered also, the varietie shall be found so great, and the opinions so dissonant, yea plainly monsterouse, that no honest witte would vouchesafe to lose time in hearyng thẽ, or rather (as I may saie) no witte is of so exact remembrance, that can consider together the monsterouse multitude of them all. And yet not withstãdyng this repugnant diuersitie, in two thynges do they all agree. First all do agre, that felicitie is and ought to be the stop and end of all their doynges, so that he that hath a full desire to any thyng how so euer it be estemed of other mẽ, yet he estemeth him self happie, if he maie obtain it: and contrary waies vnhappie if he can not attaine it. And therfore do all men put their whole studie to gette that thyng, wherin they haue perswaded them self that felicitie

doth consist. Wherfore some whiche put their felicitie in fedyng their bellies, thinke no pain to be hard, nor no dede to be vnhonest, that may be a meanes to fill that foule panche. Other which put their felicitie in play and ydle pastimes, iudge no time euill spent, that is employed thereabout: nor no fraude vnlawfull that may further their winning. If I should particularly ouerrũne but the common sortes of men, which put their felicitie in their desires, it wold make a great boke of it self. Therfore wyl I let them al go, and conclude as I began, That all men employ their whole endeuour to that thing, wherin thei thinke felicitie to stand. whiche thyng who so listeth to mark exactly, shall be able to espie and iudge the natures of al men, whose conuersaciõ he doth know, though thei vse great dissimulacion to colour their desires, especially whẽ they perceiue other men to mislyke that which thei so much desire: For no mã wold gladly haue his appetite improued. And herof cõmeth that secõndething wherin al agree, that euery man would most gladly win all other men to his sect, and to make thẽ of his opinion, and as far as he dare, will dispraise all other mens iudgemẽtes, and praise his own waies only, onles it be when he dissimuleth, and that for the furtherãce of his own purpose. And this propertie also doth geue great light to the full knowledge of mens natures, which as all men ought to obserue, so princes aboue other haue most cause to mark for sundrie occasions which may lye them on, wherof I shall not nede to speke any farther, consideryng not only the greatnes of wit, and exactnes of iudgement whiche god hath lent vnto your highnes person, but also y^e most graue wisdom and profoũd knowledge of your maiesties most honorable coũcel, by whõ your highnes may so sufficiently vnderstãd all thinges conuenient, that lesse shal it nede to vnderstand by priuate readying, but yet not vtterly to refuse to read as often as occasion may serue, for bokes dare speake, when men feare to displease. But to

returne agayne to my firste matter, if none other good thing maie be lerned at their maners, which so wrõgfully place their felicity, in so miserable a cõditiõ (that while they thinke them selfes happy, their felicitie must nedes seme vnluckie, to be by them so euill placed) yet this may men learn at them, by those .ij. spectacles to espye the secrete natures and dispositions of others, whiche thyng vnto a wise man is muche auailable. And thus will I omit this great tablement of vnhappie hap, and wil come to .iij. other sortes of a better degre, wherof the one putteth felicitie to consist in power and royaltie. The second sorte vnto power annexeth worldly wisdome, thinkyng him full happie, that could attayn those two, wherby he might not onely haue knowledge in all thynges, but also power to bryng his desires to ende. The thyrd sort estemeth true felicitie to consist in wysdom annexed with vertuouse maners, thinkyng that they can take harme of nothyng, if they can with their wysedome[*] ouercome all vyces. Of the firste of those three sortes there hath been a great numbre in all ages, yea many mightie kinges and great gouernoures which cared not greately howe they myght atchieue their pourpose, so that they dyd preuayle: nor did not take any greatter care for gouernance, then to kepe the people in onely feare of them, Whose common sentence was alwaies this: Oderint dum metuant. And what good successe suche menne had, all histories doe report. Yet haue they not wanted excuses: yea Iulius Cæsar (whiche in dede was of the second sorte) maketh a kynde of excuse by his common sentence, for them of that fyrste sorte, for he was euer woonte to saie: ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμ. Whiche sentence I wysshe had neuer been learned out of Grecia. But now to speake of the second sort, of whiche there hathe been verye many also, yet for this present time amongest them all, I wyll take the exaumples of

kynge Phylippe of Macedonie, and of Alexander his sonne, that valiaunt conquerour. First of kinge Phylip it appeareth by his letter sente vnto Aristotle that famous philosopher, that he more delited in the birthe of his sonne, for the hope of learning and good education, that might happen to him by the said Aristotle, then he didde reioyse in the continuaunce of his succession, for these were his wordes and his whole epistle, worthye to bee remembred and registred euery where.

Φΐλιππος Αριστοτέλει χαίρειμ.

ἔσθι μοι γεγονότα ὑομ. πολλὴμ οὖμ τοῖσ θεοῖσ χάριμ ἔχω, ὀυχ ὅυτωσ ἐπῒ τῆ γεννήσει του παιδόσ, ὡσ ἐπῒ τῷ κατὰ τὴμ σὴμ ἡλικῒαμ αὐτόμ γεγονέναι ἐλπΐζω γὰρ αὐτὸμ ὑπὸ σοῦ γραφέντα καὶ παιδευθέντα ἄξιομ ἔσεσθαι καὶ ἑιμῶμ καὶ τῆς τῶμ τραγμάτωμ διαδοχῆσ.

That is thus in sense,

Philip vnto Aristotle sendeth gretyng.

You shall vnderstande, that I haue a sonne borne, for whiche cause I yelde vnto God moste hartie thankes, not so muche for the byrthe of the childe, as that it was his chaunce to be borne in your tyme. For my trust is, that he shall be so brought vp and instructed by you, that he shall become worthie not only to be named our sonne, but also to be the successour of our affayres.

And his good desire was not all vayne, for it appered that Alexander was neuer so busied with warres (yet was he neuer out of moste terrible battaile) but that in the middes thereof he had in remembraunce his studies, and caused in all countreies as he went, all strange beastes,

fowles and fisshes, to be taken and kept for the ayd of that knowledg, which he learned of Aristotle: And also to be had with him alwayes a greate numbre of learned men. And in the moste busye tyme of all his warres against Darius kinge of Persia, when he harde that Aristotle had putte forthe certaine bookes of suche knowledge wherein he hadde before studied, hee was offended with Aristotle, and wrote to hym this letter.

Ἄλέζανδρος Αρισοτέλει εὖ πράττειμ.

Ὂυκ ὀρθῶσ ἐπόιησασ ἐκδοὺσ τοὺσ ἀκροαματικόυσ τῶμ λόγωμ,τΐνι γὰρ διοισομην ἡμεῖσ τῶμ ἄλλωμ, ἐι καθ’ οὕσ ἐπαιδεύθημεν λόγουσ, ὅυτοι πάντωμ ἔσονταιν κοινόι, ἐγὼ δὲ βουλοί μημ ἅμ ταῖσ περι τὰ ἄριστα ἐμπειρΐαισ, ἢ τὰισ δυνάμεσι διαφέριμ. ἔρρωσο. that is

Alexander vnto Aristotle sendeth greeting.

You haue not doone well, to put forthe those bookes of secrete phylosophy intituled, ακροαματικοι. For wherin shall we excell other, yf that knowledge that wee haue studied, shall be made commen to all other men, namely sithe our desire is to excelle other men in experience and knowledge, rather then in power and strength. Farewell.

By whyche lettre it appeareth that hee estemed learninge and knowledge aboue power of men. And the like iudgement did he vtter, when he beheld the state of Diogenes Linicus, adiudginge it the beste state next to his owne, so that he said: If I were not Alexander, I wolde wishe to be Diogenes. Whereby apeareth, how he esteemed learning, and what felicity he putte therin, reputing al the worlde saue him selfe to be inferiour to Diogenes. And bi al coniecturs, Alexander did esteme Diogenes one of them whiche contemned the vaine estimation of the disceitfull world, and put his whole felicity in knowledg of vertue, and practise of the same, though some reporte

that he knew more vertue then he folowed: But whatso euer he was, it appeareth that Socrates and Plato and many other did forsake their liuings and sel away their patrimony, to the intent to seeke and trauaile for learning, which examples I shall not need to repete to your Maiesty, partly for that your highnes doth often reade them and other lyke, and partly sith your maiesty hath at hand such learned schoolemaysters, which can much better thẽ I, declare them vnto your highnes, and that more largely also then the shortenes of thys epistle will permit. But thys may I yet adde, that King Salomon whose renoume spred so farre abroad, was very greatlye estemed for his wonderfull power and exceading treasure, but yet much more was he estemed for his wisdom. And him selfe doth bear witnes, that wisedom is better then pretious stones . yea all thinges that can be desired ar not to be compared to it. But what needeth to alledge one sentence of him, whose bookes altogither do none other thing, then set forth the praise of wisedom & knowledg? And his father king Dauid ioyneth uertuous conuersacion and knowledg togither, as the summe of perfection and chief felicity. Wherfore I maye iustelye conclude, that true felicity doth consist in wisdome and vertu. Then if wisdome be as Cicero defineth it, Diuinarum atq; humanarum rerum scientia, then ought all men to trauail for knowledg in matters both of religion and humaine docrine, if he shall be counted wyse, and able to attaine true felicitie: But as the study of religious matters is most principall, so I leue it for this time to them that better can write of it then I can. And for humaine knowledge thys wil I boldly say, that who soeuer wyll attain true iudgment therein, must not only trauail in y^e knowledg of the tungs, but must also before al other arts, taste of the mathematical sciences, specially Arithmetike and Geometry, without which it is not possible to attayn full knowledg in any art. Which may sufficiẽtly by gathered by Aristotle not õly in his bookes

of demonstration (whiche can not be vnderstand without Geometry) but also in all his other workes. And before him Plato his maister wrote this sentence on his schole house dore. Αγεομέτρητοσ ὀυδὲισ ἐισΐτω.. Let no man entre here (saith he) without knowledg in Geometry. Wherfore moste mighty prince, as your most excellent Maiesty appeareth to be borne vnto most perfect felicity, not only by reasõ that God moued with the long prayers of this realme, did send your highnes as moste comfortable inheritour to the same, but also in that your Maiesty was borne in the time of such skilful schoolmaisters & learned techers, as your highnes doth not a little reioyse in, and profite by them in all kind of vertu & knowledg. Amõgst which is that heauẽly knowledg most worthely to be praised, wherbi the blindnes of errour & superstition is exiled, & good hope cõceiued that al the sedes & fruts therof, with all kindes of vice & iniquite, wherby vertu is hindered, & iustice defaced, shal be clean extrirped and rooted out of this realm, which hope shal increase more and more, if it may appear that learning be estemed & florish within this realm. And al be it the chief learnĩg be the diuine scriptures, which instruct the mind principally, & nexte therto the lawes politike, which most specially defẽd the right of goodes, yet is it not possible that those two can long be wel vsed, if that ayde want that gouerneth health and expelleth sicknes, which thing is done by Physik, & these require the help of the vij. liberall sciences, but of none more then of Arithmetik and Geometry, by which not only great thinges ar wrought touchĩg accõptes in al kinds, & in suruaiyng & measuring of lãdes, but also al arts depend partly of thẽ, & building which is most necessary can not be w^tout them, which thing cõsidering, moued me to help to serue your maiesty in this point as wel as other wais, & to do what mai be in me, y^t not õly thei which studi prĩcipalli for lernĩg, mai haue furderãce bi mi poore help, but

also those whiche haue no tyme to trauaile for exacter knowlege, may haue some helpe to vnderstand in those Mathematicall artes, in whiche as I haue all readye set forth sumwhat of Arithmetike, so god willing I intend shortly to setforth a more exacter worke therof. And in the meane ceason for a taste of Geometry, I haue sette forthe this small introduction, desiring your grace not so muche to beholde the simplenes of the woorke, in comparison to your Maiesties excellencye, as to fauour the edition thereof, for the ayde of your humble subiectes, which shal thinke them selues more and more dayly bounden to your highnes, if when they shall perceaue your graces desyre to haue theym profited in all knowledge and vertue. And I for my poore ability considering your Maiesties studye for the increase of learning generally through all your highenes dominions, and namely in the vniuersities of Oxforde and Camebridge, as I haue an earnest good will as far as my simple seruice and small knowledg will suffice, to helpe toward the satisfiyng of your graces desire, so if I shall perceaue that my seruice may be to your maiesties contẽtacion, I wil not only put forth the other two books, whiche shoulde haue beene sette forth with these two, yf misfortune had not hindered it, but also I wil set forth other bookes of more exacter arte, bothe in the Latine tongue and also in the Englyshe, whereof parte bee all readye written, and newe instrumentes to theym deuised, and the residue shall bee eanded with all possible speede. I was boldened to dedicate this booke of Geometrye vnto your Maiestye, not so muche bycause it is the firste that euer was sette forthe in Englishe, and therefore for the noueltye a straunge presente, but for that I was perswaded, that suche a wyse prince doothe desire to haue a wise sorte of subiectes. For it is a kynges chiefe reioysinge and glory, if his subiectes be riche in substaunce, and wytty in knowledge: and contrarye

waies nothyng can bee more greuouse to a noble kyng, then that his realme should be other beggerly or full of ignoraunce: But as god hath geuen your grace a realme bothe riche in commodities and also full of wyttie men, so I truste by the readyng of wyttie artes (whiche be as the whette stones of witte) they muste needes increase more and more in wysedome, and peraduenture fynde some thynge towarde the ayde of their substaunce, whereby your grace shall haue newe occasion to reioyce, seyng your subiectes to increase in substance or wisdom, or in both. And thei again shal haue new and new causes to pray for your maiestie, perceiuyng so graciouse a mind towarde their benefite. And I truste (as I desire) that a great numbre of gentlemen, especially about the courte, whiche vnderstand not the latin tong, or els for the hardnesse of the mater could not away with other mens writyng, will fall in trade with this easie forme of teachyng in their vulgar tong, and so employe some of their tyme in honest studie, whiche were wont to bestowe most part of their time in triflyng pastime: For vndoubtedly if they mean other your maiesties seruice, other their own wisdome, they will be content to employ some tyme aboute this honest and wittie exercise. For whose encouragemẽt to the intent they maie perceiue what shall be the vse of this science, I haue not onely written somewhat of the vse of Geometrie, but also I haue annexed to this boke the names and brefe argumentes of those other bokes whiche I will set forth hereafter, and that as shortly as it shall appeare vnto your maiestie by coniecture of their diligent vsyng of this first boke, that they wyll vse well the other bokes also. In the meane ceason, and at all times I wil be a continuall petitioner, that god may work in all english hartes an ernest mynde to all honest exercises, wherby thei may serue the better your maiestie and

the realm. And for your highnes I besech the most mercifull god, as he hath most fauourably sent you vnto vs, as our chefe comforter in earthe, so that he will increase your maiestie daiely in all vertue and honor with moste prosperouse successe, and augment in vs your most humble subiectes, true loue to godward, and iust obedience toward your highnes with all reuerence and subiection.

At London the .xxviij. daie of Ianuarie. M. D. L I.

Your maiesties moste humble seruant
and obedient subiect,
Robert Recorde.

---

[ THE PREFACE,]
declaring briefely the commodi-

tes of Geometrye, and the
necessitye thereof.

Eometrye may thinke it selfe to sustaine great iniury, if it shall be inforced other to show her manifold commodities, or els not to prease into the sight of men, and therefore might this wayes answere briefely: Other I am able to do you much good, or els but litle. If I bee able to doo you much good, then be you not your owne friendes, but greatlye your owne enemies to make so little of me, which maye profite you so muche. For if I were as vncurteous as you vnkind, I shuld vtterly refuse to do them any good, which will so curiously put me to the trial and profe of my commodities, or els to suffre exile, and namely sithe I shal only yeld benefites to other, and receaue none againe. But and if you could saye truely, that my benefites be nother many nor yet greate, yet if they bee anye, I doo yelde more to you, then I doo receaue againe of you, and therefore I oughte not to bee repelled of them that loue them selfe, althoughe they loue me not all for my selfe. But as I am in nature a liberall science, so canne I not againste nature contende with your inhumanitye, but muste shewe my selfe liberall euen to myne enemies. Yet this is my comforte againe, that I haue none enemies but them that knowe me not, and therefore may hurte themselues, but can not noye me. Yf they dispraise the thinge that they know not, all wise men will blame them and not credite them, and yf they thinke they knowe me, lette theym shewe one vntruthe and erroure in me, and I wyll geue the victorye.

Yet

can no humayne science saie thus, but I onely, that there is no sparke of vntruthe in me: but all my doctrine and workes are without any blemishe of errour that mans reason can discerne. And nexte vnto me in certaintie are my three systers, Arithmetike, Musike, and Astronomie, whiche are also so nere knitte in amitee, that he that loueth the one, can not despise the other, and in especiall Geometrie, of whiche not only these thre, but all other artes do borow great ayde, as partly hereafter shall be shewed. But first will I beginne with the vnlearned sorte, that you maie perceiue how that no arte can stand without me. For if I should declare how many wayes my helpe is vsed, in measuryng of ground, for medow, corne, and wodde: in hedgyng, in dichyng, and in stackes makyng, I thinke the poore Husband man would be more thankefull vnto me, then he is nowe, whyles he thinketh that he hath small benefite by me. Yet this maie he coniecture certainly, that if he kepe not the rules of Geometrie, he can not measure any ground truely. And in dichyng, if he kepe not a proportion of bredth in the mouthe, to the bredthe of the bottome, and iuste slopenesse in the sides agreable to them bothe, the diche shall be faultie many waies. When he doth make stackes for corne, or for heye, he practiseth good Geometrie, els would thei not long stand: So that in some stakes, whiche stand on foure pillers, and yet made round, doe increase greatter and greatter a good height, and then againe turne smaller and smaller vnto the toppe: you maie see so good Geometrie, that it were very difficult to counterfaite the lyke in any kynde of buildyng. As for other infinite waies that he vseth my benefite, I ouerpasse for shortnesse.

Carpenters, Karuers, Ioyners, and Masons, doe willingly acknowledge that they can worke nothyng without reason of Geometrie, in so muche that they chalenge me as a peculiare science for them. But in that they should do wrong to all other men, seyng euerie kynde of men haue som benefit by me, not only in buildyng, whiche is but other mennes costes, and the arte of Carpenters, Masons, and the other aforesayd, but in their

owne priuate profession, whereof to auoide tediousnes I make this rehersall.

Sith Merchauntes by shippes great riches do winne,

I may with good righte at their seate beginne.

The Shippes on the sea with Saile and with Ore,

were firste founde, and styll made, by Geometries lore.

Their Compas, their Carde, their Pulleis, their Ankers,

were founde by the skill of witty Geometers.

To sette forth the Capstocke, and eche other parte,

wold make a greate showe of Geometries arte.

Carpenters, Caruers, Ioiners and Masons,

Painters and Limners with suche occupations,

Broderers, Goldesmithes, if they be cunning,

Must yelde to Geometrye thankes for their learning.

The Carte and the Plowe, who doth them well marke,

Are made by good Geometrye. And so in the warke

Of Tailers and Shoomakers, in all shapes and fashion,

The woorke is not praised, if it wante proportion.

So weauers by Geometrye hade their foundacion,

Their Loome is a frame of straunge imaginacion.

The wheele that doth spinne, the stone that doth grind,

The Myll that is driuen by water or winde,

Are workes of Geometrye straunge in their trade,

Fewe could them deuise, if they were vnmade.

And all that is wrought by waight or by measure,

without proofe of Geometry can neuer be sure.

Clockes that be made the times to deuide,

The wittiest inuencion that euer was spied,

Nowe that they are common they are not regarded,

The artes man contemned, the woorke vnrewarded.

But if they were scarse, and one for a shewe,

Made by Geometrye, then shoulde men know,

That neuer was arte so wonderfull witty,

So needefull to man, as is good Geometry.

The firste findinge out of euery good arte,

Seemed then vnto men so godly a parte,

That no recompence might satisfye the finder,

But to make him a god, and honoure him for euer.

So Ceres and Pallas, and Mercury also,

Eolus and Neptune, and many other mo,

Were honoured as goddes, bicause they did teache,

Firste tillage and weuinge and eloquent speache,

Or windes to obserue, the seas to saile ouer,

They were called goddes for their good indeuour.

Then were men more thankefull in that golden age:

This yron wolde nowe vngratefull in rage,

Wyll yelde the thy reward for trauaile and paine,

With sclaunderous reproch, and spitefull disdaine.

Yet thoughe other men vnthankfull will be,

Suruayers haue cause to make muche of me.

And so haue all Lordes, that landes do possesse:

But Tennaunted I feare will like me the lesse.

Yet do I not wrong but measure all truely,

All yelde the full right of euerye man iustely.

Proportion Geometricall hath no man opprest,

Yf anye bee wronged, I wishe it redrest.

But now to procede with learned professions, in Logike and Rhetorike and all partes of phylosophy, there neadeth none other proofe then Aristotle his testimony, whiche without Geometry proueth almost nothinge. In Logike all his good syllogismes and demonstrations, hee declareth by the principles of Geometrye. In philosophye, nether motion, nor time, nor ayrye impressions could hee aptely declare, but by the helpe of Geometrye as his woorkes do witnes. Yea the faculties of the minde dothe hee expresse by similitude to figures of Geometrye. And in morall phylosophy he thought that iustice coulde not wel be taught, nor yet well executed without proportion geometricall. And this estimacion of Geometry he maye seeme to haue learned of his maister Plato, which without Geometrye wolde teache nothinge, nother wold admitte any to heare him, except he were experte in Geometry. And what merualle if he so muche estemed geometrye, seinge his opinion was, that Godde was alwaies workinge

by Geometrie? Whiche sentence Plutarche declareth at large. And although Platto do vse the helpe of Geometrye in all the most waighte matter of a common wealth, yet it is so generall in vse, that no small thinges almost can be wel done without it. And therfore saith he: that Geometrye is to be learned, if it were for none other cause, but that all other artes are bothe soner and more surely vnderstand by helpe of it.

What greate help it dothe in physike, Galene doth so often and so copiousely declare, that no man whiche hath redde any booke almoste of his, can be ignorant thereof, in so much that he coulde neuer cure well a rounde vlcere, tyll reason geometricall dydde teache it hym. Hippocrates is earnest in admonyshynge that study of geometrie must prepare the way to physike, as well as to all other artes.

I shoulde seeme somewhat to tedious, if I shoulde recken vp, howe the diuines also in all their mysteries of scripture doo vse healpe of geometrie: and also that lawyers can neuer vnderstande the hole lawe, no nor yet the firste title therof exactly without Geometrie. For if lawes can not well be established, nor iustice duelie executed without geometricall proportion, as bothe Plato in his Politike bokes, and Aristotle in his Moralles doo largely declare. Yea sithe Lycurgus that cheefe lawmaker amongest the Lacedemonians, is moste praised for that he didde chaunge the state of their common wealthe frome the proportion Arithmeticall to a proportion geometricall, whiche without knowledg of bothe he coulde not dooe, than is it easye to perceaue howe necessarie Geometrie is for the lawe and studentes thereof. And if I shall saie preciselie and freelie as I thinke, he is vtterlie destitute of all abilitee to iudge in anie arte, that is not sommewhat experte in the Theoremes of Geometrie.

And

that caused Galene to say of hym selfe, that he coulde neuer perceaue what a demonstration was, no not so muche, as whether there were any or none, tyll he had by geometrie gotten abilitee to vnderstande it, although he heard the beste teachers that were in his tyme. It shuld be to longe and nedelesse

also to declare what helpe all other artes Mathematicall haue by geometrie, sith it is the grounde of all theyr certeintie, and no man studious in them is so doubtful therof, that he shall nede any persuasion to procure credite thereto. For he can not reade .ij. lines almoste in any mathematicall science, but he shall espie the nedefulnes of geometrie. But to auoyde tediousnesse I will make an ende hereof with that famous sentence of auncient Pythagoras, That who so will trauayle by learnyng to attayne wysedome, shall neuer approche to any excellencie without the artes mathematicall, and especially Arithmetike and Geometrie.

And yf I shall somewhat speake of noble men, and gouernours of realmes, howe needefull Geometrye maye bee vnto them, then must I repete all that I haue sayde before, sithe in them ought all knowledge to abounde, namely that maye appertaine either to good gouernaunce in time of peace, eyther wittye pollicies in time of warre. For ministration of good lawes in time of peace Lycurgus example with the testimonies of Plato and Aristotle may suffise. And as for warres, I might thinke it sufficient that Vegetius hath written, and after him Valturius in commendation of Geometry, for vse of warres, but all their woordes seeme to saye nothinge, in comparison to the example of Archimedes worthy woorkes make by geometrie, for the defence of his countrey, to reade the wonderfull praise of his wittie deuises, set foorthe by the most famous hystories of Liuius, Plutarche, and Plinie, and all other hystoriographiers, whyche wryte of the stronge siege of Syracusæ made by that valiant capitayne, and noble warriour Marcellus, whose power was so great, that all men meruayled how that one citee coulde withstande his wonderfull force so longe. But much more woulde they meruaile, if they vnderstode that one man onely dyd withstand all Marcellus strength, and with counter engines destroied his engines to the vtter astonyshment of Marcellus, and all that were with hym. He had inuented suche balastelas that dyd shoote out a hundred dartes at one

shotte, to the great destruction of Marcellus souldiours, wherby a fonde tale was spredde abrode, how that in Syracusæ there was a wonderfull gyant, whiche had a hundred handes, and coulde shoote a hundred dartes at ones. And as this fable was spredde of Archimedes, so many other haue been fayned to bee gyantes and monsters, bycause they dyd suche thynges, whiche farre passed the witte of the common people. So dyd they feyne Argus to haue a hundred eies, bicause they herde of his wonderfull circumspection, and thoughte that as it was aboue their capacitee, so it could not be, onlesse he had a hundred eies. So imagined they Ianus to haue two faces, one lokyng forwarde, and an other backwarde, bycause he coulde so wittily compare thynges paste with thynges that were to come, and so duely pondre them, as yf they were all present. Of like reasõ did they feyn Lynceus to haue such sharp syght, that he could see through walles and hylles, bycause peraduenture he dyd by naturall iudgement declare what cõmoditees myght be digged out of the grounde. And an infinite noumbre lyke fables are there, whiche sprange all of lyke reason.

For what other thyng meaneth the fable of the great gyant Atlas, whiche was ymagined to beare vp heauen on his shulders? but that he was a man of so high a witte, that it reached vnto the skye, and was so skylfull in Astronomie, and coulde tell before hande of Eclipses, and other like thynges as truely as though he dyd rule the sterres, and gouerne the planettes.

So was Eolus accompted god of the wyndes, and to haue theim all in a caue at his pleasure, by reason that he was a wittie man in naturall knowlege, and obserued well the change of wethers, aud was the fyrst that taught the obseruation of the wyndes. And lyke reson is to be geuen of al the old fables.

But to retourne agayne to Archimedes, he dyd also by arte perspectiue (whiche is a parte of geometrie) deuise such glasses within the towne of Syracusæ, that dyd bourne their ennemies shyppes a great way from the towne, whyche was a meruaylous politike thynge. And if I shulde repete the

varietees of suche straunge inuentions, as Archimedes and others haue wrought by geometrie, I should not onely excede the order of a preface, but I should also speake of suche thynges as can not well be vnderstande in talke, without somme knowledge in the principles of geometrie.

But this will I promyse, that if I may perceaue my paynes to be thankfully taken, I wyll not onely write of suche pleasant inuentions, declaryng what they were, but also wil teache howe a great numbre of them were wroughte, that they may be practised in this tyme also. Wherby shallbe plainly perceaued, that many thynges seme impossible to be done, whiche by arte may very well be wrought. And whan they be wrought, and the reason therof not vnderstande, than say the vulgare people, that those thynges are done by negromancy. And hereof came it that fryer Bakon was accompted so greate a negromancier, whiche neuer vsed that arte (by any coniecture that I can fynde) but was in geometrie and other mathematicall sciences so experte, that he coulde dooe by theim suche thynges as were wonderfull in the syght of most people.

Great talke there is of a glasse that he made in Oxforde, in whiche men myght see thynges that were doon in other places, and that was iudged to be done by power of euyll spirites. But I knowe the reason of it to bee good and naturall, and to be wrought by geometrie (sythe perspectiue is a parte of it) and to stande as well with reason as to see your face in cõmon glasse. But this conclusion and other dyuers of lyke sorte, are more mete for princes, for sundry causes, than for other men, and ought not to bee taught commonly. Yet to repete it, I thought good for this cause, that the worthynes of geometry myght the better be knowen, & partly vnderstanding geuen, what wonderfull thynges may be wrought by it, and so consequently how pleasant it is, and how necessary also.

And thus for this tyme I make an end. The reason of som thynges done in this boke, or omitted in the same, you shall fynde in the preface before the Theoremes.[*]

---

[ The definitions of the principles of]

GEOMETRY.

eometry teacheth the drawyng, Measuring and proporcion of figures. but in as muche as no figure can bee drawen, but it muste haue certayne boũdes and inclosures of lines: and euery lyne also is begon and ended at some certaine prycke, fyrst it shal be meete to know these smaller partes of euery figure, that therby the whole figures may the better bee iudged, and distincte in sonder.

A poincte. A Poynt or a Prycke, is named of Geometricians that small and vnsensible shape, whiche hath in it no partes, that is to say: nother length, breadth nor depth. But as their exactnes of definition is more meeter for onlye Theorike speculacion, then for practise and outwarde worke (consideringe that myne intent is to applye all these whole principles to woorke) I thynke meeter for this purpose, to call a poynt or prycke, that small printe of penne, pencyle, or other instrumente, whiche is not moued, nor drawen from his fyrst touche, and therfore hath no notable length nor bredthe: as this example doeth declare. ∴

Where I haue set .iij. prickes, eche of them hauyng both lẽgth and bredth, thogh it be but smal, and thefore not notable.

Nowe of a great numbre of these prickes, is made a Lyne, as you may perceiue by this forme ensuyng. ············ where as I haue set a numbre of prickes, so if you with your pen will set in more other prickes betweene euerye two of these, A lyne. then wil it be a lyne, as here you may see and this lyne, is called of Geometricians, Lengthe withoute breadth.

But

as they in theyr theorikes (which ar only mind workes)

do precisely vnderstand these definitions, so it shal be sufficient for those men, whiche seke the vse of the same thinges, as sense may duely iudge them, and applye to handy workes if they vnderstand them so to be true, that outwarde sense canne fynde none erroure therein.

Of lynes there bee two principall kyndes. The one is called a right or straight lyne, and the other a croked lyne.

A streghte lyne. A Straight lyne, is the shortest that maye be drawenne between two prickes.

A crokyd lyne. And all other lines, that go not right forth from prick to prick, but boweth any waye, such are called Croked lynes as in these examples folowyng ye may se, where I haue set but one forme of a straight lyne, for more formes there be not, but of crooked lynes there bee innumerable diuersities, whereof for examples sum I haue sette here.

A right lyne.

Croked lynes.
Croked lines.

So now you must vnderstand, that euery lyne is drawen betwene twoo prickes, wherof the one is at the beginning, and the other at the ende.

Therefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall you

not call it one croked lyne, but rather twoo lynes: an Angle. in as muche as there is a notable and sensible angle by .A. whiche euermore is made by the meetyng of two seuerall lynes. And likewayes shall you iudge of this figure, whiche is made of two lines, and not of one onely.

So that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called an Angle or corner.

Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle. A righte angle. The square angle, whiche is commonly named a right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.

Right angles.

A sharpe corner. A sharpe angle is so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departynge as in a square corner, and if thei be drawen crosse, all fower corners will not be equall.

A blunte angle. A blunt or brode corner, is greater then is a square angle, and his lines do parte more in sonder then in a right angle, of whiche all take these examples.

And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it would

muche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.

Sharpe angles.

Blunte or brode angles.

But now as of many prickes there is made one line, so of diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name, A platte forme. Platte formes, and thei haue bothe length and bredth, but yet no depenesse.

And the boundes of euerie platte forme are lines: as by the examples you maie perceiue.

Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.

A plaine platte. A plaine platte is that, whiche is made al equall in height, so that the middle partes nother bulke vp, nother shrink down more then the bothe endes.

A crooked platte. For whan the one parte is higher then the other, then is it named a Croked platte.

And if it be partlie plaine, and partlie crooked, then is it called a Myxte platte, of all whiche, these are exaumples.

A plaine platte.	A croked platte.	A myxte platte.

And as of many prickes is made a line, and of diuerse lines one platte forme, A bodie. so of manie plattes is made a bodie, whiche conteigneth Lengthe, bredth, and depenesse. Depenesse. By Depenesse I vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesse

of any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.

Now all bodies haue platte formes for their boundes, Cubike. so in a dye (whiche is called a cubike bodie) by geomatricians, Asheler. and an ashler of masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.

A globe. But in a Globe, (whiche is a bodie rounde as a bowle) there is but one platte forme, and one bounde, and these are the exaumples of them bothe.

A dye or ashler.	A globe.

But because you shall not muse what I dooe call a bound, A bounde. I mean therby a generall name, betokening the beginning, end and side, of any forme.

Forme, Fygure. A forme, figure, or shape, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.

Of figures there be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie, a figure is euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.

Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche y^e eye maie discerne distinctly.

And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.

	A lynearic numbre.
	Trianguler numbres
	Longsquare nũbre.
	Iust square numbres
	a threcornered spire.
	A square spire.

And so maie there be infinite formes more, whiche I omitte for this time, cõsidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.

But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by cõpasse without it) then is it called a centre. A centre And thereof doe masons, and other worke menne call that patron, a centre, whereby thei drawe the lines, for iust hewyng of stones for arches, vaultes, and chimneies, because the chefe vse of that patron is wrought by findyng that pricke or centre, from whiche all the lynes are drawen, as in the thirde booke it doeth appere.

Lynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be called

figures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is named A ground line. the ground line) and an other commeth downe on it, and is called A perpen­dicular. A plume lyne. a perpendiculer or plũme lyne, as in this example you may see. where .A.B. is the grounde line, and C.D. the plumbe line.

And

like waies in this figure there are three lines, the grounde lyne whiche is A.B. the plumme line that is A.C. and the bias line, whiche goeth from the one of thẽ to the other, and lieth against the right corner in such a figure whiche is here .C.B.

But

consideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.

Parallelys Gemowe lynes. Paralleles, or gemowe lynes be suche lines as be drawen foorth still in one distaunce, and are no nerer in one place then in an other, for and if they be nerer at one ende then at the other, then are they no paralleles, but maie bee called bought lynes, and loe here exaumples of them bothe.

parallelis.	bought lines
parallelis: circular. Concen- trikes.

I haue added also paralleles tortuouse, whiche bowe cõtrarie waies with their two endes: and paralleles circular, whiche be lyke vnperfecte compasses: for if they bee whole circles, Concen­trikes then are they called cõcentrikes, that is to saie, circles drawẽ on one centre.

Here

might I note the error of good Albert Durer, which affirmeth that no perpendicular lines can be paralleles. which errour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.

And to returne to my matter. A twine line. an other fashioned line is there, which is named a twine or twist line, and it goeth as a wreyth about some other bodie. A spirall line. And an other sorte of lines is there, that is called a spirall line, A worme line. or a worm line, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.

A twiste lyne.		A spirail lyne

A touche lyne, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.

And when that a line doth crosse the edg of the circle, thẽ is it called a cord, as you shall see anon in the speakynge of circles.

In the meane season must I not omit to declare what angles bee called matche corners, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.

Where A. and B. are matche corners, so are C. and D. but not A. and C. nother D. and A.

Nowe will I beginne to speak of figures, that be properly so called, of whiche all be made of diuerse lines, except onely a circle, an egge forme, and a tunne forme, which .iij. haue no angle and haue but one line for their bounde, and an eye fourme whiche is made of one lyne, and hath an angle onely.

A circle

is a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to the circumference are equall all in length, as here you see.

And the line that encloseth the whole compasse, is called the circumference.

A diameter. And all the lines that bee drawen crosse the circle, and goe by the centre, are named diameters, whose halfe, I meane from the center to the circumference

any waie, Semi­diameter. is called the semidiameter, or halfe diameter.

But and if the line goe crosse the circle, and passe beside the centre, then is it called a corde, or a stryng line, as I said before, and as this exaumple sheweth: where A. is the corde. And the compassed line that aunswereth to it, An archline is called an arche lyne, A bowline. or a bowe lyne, whiche here marked with B. and the diameter with C.

But and if that part be separate from the rest of the circle (as in this exãple you see) then ar both partes called cãtelles, A cantle the one the greatter cantle as E. and the other the lesser cantle, as D. And if it be parted iuste by the centre (as you see in F.) A semye­circle then is it called a semicircle, or halfe compasse.

Sometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (as G. is) A nooke cantle and then maie it be called a nooke cantle, and if it be not parted from the reste of the circle (as you see in H.) A nooke. then is it called a nooke plainely without any addicion. And the compassed lyne in it is called an arche lyne, as the exaumple here doeth shewe.

An arche.

Nowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call an An egge fourme. egge forme, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.

A tunne forme.

An egge forme

For if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called a tunne forme, or barrell forme, the right makyng of whiche figures, I wyll declare hereafter in the thirde booke.

An other forme there is, whiche you maie call a nutte forme, and is made of one lyne muche lyke an egge forme, saue that it hath a sharpe angle.

And it chaunceth sometyme that there is a right line drawen crosse these figures, An axtre or axe lyne. and that is called an axelyne, or axtre. Howe be it properly that line that is called an axtre, whiche gooeth throughe the myddell of a Globe, for as a diameter is in a circle, so is an axe lyne or axtre in a Globe,

that lyne that goeth from side to syde, and passeth by the middell of it. And the two poyntes that suche a lyne maketh in the vtter bounde or platte of the globe, are named polis, w^ch you may call aptly in englysh, tourne pointes: of whiche I do more largely intreate, in the booke that I haue written of the vse of the globe.

But to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are the cantle of a circle, and the halfe circle, of which I haue spoken allready. Likewyse the halfe of an egge forme, the cantle of an egge forme, the halfe of a tunne fourme, and the cantle of a tunne fourme, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne, An yey fourme and that figure is named an yey fourme.

The nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generally triangles. for a triangle is nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.

Now to geue you example of triangles, there is one whiche is all of croked lynes, and may be taken fur a portiõ of a globe as the figur marked w^t A.

An other hath two compassed lines and one right lyne, and is as the portiõ of halfe a globe, example of B.

An other hath but one compassed

lyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se in C. Otherlesse then it as you se D, whose right lines make a sharpe corner, or greater then a quadrate, as is F, and then the right lynes of it do make a blunt corner.

Also some triangles haue all righte lynes and they be distincted in sonder by their angles, or corners. for other their corners bee all sharpe, as you see in the figure, E. other ij. sharpe and one blunt, as is the figure G. other ij. sharp and one blunt as in the figure H.

There is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figure E, and that the Greekes doo call Isopleuron, ἰσόπλευρομ. and Latine men æequilaterum: and in english it may be called a threlike triangle, other els two sydes bee equall and the thyrd vnequall, which the Greekes call Isosceles, ισόσκελεσ. the Latine men æquicurio, and in english tweyleke may they be called, as in G, H, and K. For, they may be of iij. kinds that is to say, with one square angle, as is G, or with a blunte corner as H, or with all in sharpe korners, as you see in K.

Further more it may be y^t they haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples .M. N, and O. where M. hath a right angle, N, a blunte angle, and O, all sharpe angles σκαλενὄμ. these the Greekes and latine men do

cal scalena and in englishe theye may be called nouelekes, for thei haue no side equall, or like lõg, to ani other in the same figur. Here it is to be noted, that in a triãgle al the angles bee called innerãgles except ani side bee drawenne forth in lengthe, for then is that fourthe corner caled an vtter corner, as in this exãple because A.B, is drawen in length, therfore the ãgle C, is called an vtter ãgle.

Quadrãgle And thus haue I done with triãguled figures, and nowe foloweth quadrangles, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say, A square quadrate. a square quadrate, whose sides bee all equall, and al the angles square, as you se here in this figure Q. A longe square. The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure .R.

The thyrd kind is called losenges A diamõd. or diamondes, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in .S.

The iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they named A losenge lyke. losengelike or diamõdlike, whose figur is noted with T. Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge, likesides, as Q. and S. and those that haue only the contrary sydes equal, as R. and T. haue, those wyll I call likeiammys, for a difference.

The fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekes trapezia, of Latin mẽ mensulæ and of Arabitians, helmuariphe, they may be called in englishe borde formes, Borde formes. they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they counted vnruled formes, and the other foure kindes onely are counted ruled formes, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.

And nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may call cink-angles, whose sydes partlye are all equall as in A, and those are counted ruled cinkeangles, and partlye vnequall, as in B, and they are called vnruled.

Likewyse shall you iudge of siseangles, which haue sixe corners, septangles, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende, A squyre. and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called a squire, is made of two long squares ioyned togither, as this example sheweth.

The globe as is before.

And thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures of bodeis which partly haue one platte forme for their bound, and y^t iust roũd as a globe hath, or ended long as in an egge, and a tunne fourme, whose pictures are these.

Howe be it you must marke that I meane not the very figure of a tunne, when I saye tunne form, but a figure like a tunne, for a tune fourme,

hath but one plat forme, and therfore must needs be round at the endes, where as a tunne hath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.

Bodies of two plattes, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as this A. doth partly expresse: A rounde spier. or els it is called a rounde spire, or stiple fourme, as in this figure is some what expressed.

Nowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.

But as these formes be harde to be iudged by their pycturs, so I doe entende to passe them ouer with a great number of other formes of bodyes, which afterwarde shall be set forth in the boke of Perspectiue, bicause that without perspectiue knowledge, it is not easy to iudge truly the formes of them in flatte protacture.

And thus I made an ende for this tyme, of the defi-
nitions Geometricall, appertayning to this
parte of practise, and the rest wil
I prosecute as cause shall
serue.

---

[ THE PRACTIKE WORKINGE OF]
sondry conclusions geometrical.

[ THE FYRST CONCLVSION.]
To make a threlike triangle on any lyne measurable.

ake the iuste lẽgth of the lyne with your cõpasse, and stay the one foot of the compas in one of the endes of that line, turning the other vp or doun at your will, drawyng the arche of a circle against the midle of the line, and doo like wise with the same cõpasse vnaltered, at the other end of the line, and wher these ij. croked lynes doth crosse, frome thence drawe a lyne to ech end of your first line, and there shall appear a threlike triangle drawen on that line.

Example.

A.B. is the first line, on which I wold make the threlike triangle, therfore I open the compasse as wyde as that line is long, and draw two arch lines that mete in C, then from C, I draw ij other lines one to A, another to B, and than I haue my purpose.

[ THE .II. CONCLVSION]
If you wil make a twileke or a nouelike triangle on ani certaine line.

Consider fyrst the length that yow will haue the other sides to containe, and to that length open your compasse, and

then worke as you did in the threleke triangle, remembryng this, that in a nouelike triangle you must take ij. lengthes besyde the fyrste lyne, and draw an arche lyne with one of thẽ at the one ende, and with the other at the other end, the exãple is as in the other before.

[ THE III. CONCL.]
To diuide an angle of right lines into ij. equal partes.

First open your compasse as largely as you can, so that it do not excede the length of the shortest line y^t incloseth the angle. Then set one foote of the compasse in the verye point of the angle, and with the other fote draw a compassed arch frõ the one lyne of the angle to the other, that arch shall you deuide in halfe, and thẽ draw a line frõ the ãgle to y^e middle of y^e arch, and so y^e angle is diuided into ij. equall partes.

Example.

Let the triãgle be A.B.C, thẽ set I one foot of y^e cõpasse in B, and with the other I draw y^e arch D.E, which I part into ij. equall parts in F, and thẽ draw a line frõ B, to F, & so I haue mine intẽt.

[ THE IIII. CONCL.]
To deuide any measurable line into ij. equall partes.

Open your compasse to the iust lẽgth of y^e line. And thẽ set one foote steddely at the one ende of the line, & w^t the other fote draw an arch of a circle against y^e midle of the line, both ouer it, and also vnder it, then doo lykewaise

at the other ende of the line. And marke where those arche lines do meet crosse waies, and betwene those ij. pricks draw a line, and it shall cut the first line in two equall portions.

Example.

The lyne is A.B. accordyng to which I open the compasse and make .iiij. arche lines, whiche meete in C. and D, then drawe I a lyne from C, so haue I my purpose.

This conlusion serueth for makyng of quadrates and squires, beside many other commodities, howebeit it maye bee don more readylye by this conclusion that foloweth nexte.

[ THE FIFT CONCLVSION.]
To make a plumme line or any pricke that you will in any right lyne appointed.

Open youre compas so that it be not wyder then from the pricke appoynted in the line to the shortest ende of the line, but rather shorter. Then sette the one foote of the compasse in the first pricke appointed, and with the other fote marke ij. other prickes, one of eche syde of that fyrste, afterwarde open your compasse to the wydenes of those ij. new prickes, and draw from them ij. arch lynes, as you did in the fyrst conclusion, for making of a threlyke triãgle. then if you do mark their crossing, and from it drawe a line to your fyrste pricke, it shall bee a iust plum lyne on that place.

Example.

The lyne is A.B. the prick on whiche I shoulde make the plumme lyne, is C. then open I the compasse as wyde as A.C, and sette one foot in C. and with the other doo I marke out C.A. and C.B, then open I the compasse as wide as A.B, and make ij. arch lines which do crosse in D, and so haue I doone.

Howe bee it, it happeneth so sommetymes, that the

pricke on whiche you would make the perpendicular or plum line, is so nere the eand of your line, that you can not extende any notable length from it to thone end of the line, and if so be it then that you maie not drawe your line lenger frõ that end, then doth this conclusion require a newe ayde, for the last deuise will not serue. In suche case therfore shall you dooe thus: If your line be of any notable length, deuide it into fiue partes. And if it be not so long that it maie yelde fiue notable partes, then make an other line at will, and parte it into fiue equall portiõs: so that thre of those partes maie be found in your line. Then open your compas as wide as thre of these fiue measures be, and sette the one foote of the compas in the pricke, where you would haue the plumme line to lighte (whiche I call the first pricke,) and with the other foote drawe an arche line righte ouer the pricke, as you can ayme it: then open youre compas as wide as all fiue measures be, and set the one foote in the fourth pricke, and with the other foote draw an other arch line crosse the first, and where thei two do crosse, thense draw a line to the poinct where you woulde haue the perpendicular line to light, and you haue doone.

Example.

The line is A.B. and A. is the prick, on whiche the perpendicular line must light. Therfore I deuide A.B. into fiue partes equall, then do I open the compas to the widenesse of three partes (that is A.D.) and let one foote staie in A. and with the other I make an arche line in C. Afterwarde I open the compas as wide as A.B.

(that is as wide as all fiue partes) and set one foote in the .iiij. pricke, which is E, drawyng an arch line with the other foote in C. also. Then do I draw thence a line vnto A, and so haue I doone. But and if the line be to shorte to be parted into fiue partes, I shall deuide it into iij. partes only, as you see the liue F.G, and then make D. an other line (as is K.L.) whiche I deuide into .v. suche diuisions, as F.G. containeth .iij, then open I the compass as wide as .iiij. partes (whiche is K.M.) and so set I one foote of the compas in F, and with the other I drawe an arch lyne toward H, then open I the cõpas as wide as K.L. (that is all .v. partes) and set one foote in G, (that is the iij. pricke) and with the other I draw an arch line toward H. also: and where those .ij. arch lines do crosse (whiche is by H.) thence draw I a line vnto F, and that maketh a very plumbe line to F.G, as my desire was. The maner of workyng of this conclusion, is like to the second conlusion, but the reason of it doth depẽd of the .xlvi. proposiciõ of y^e first boke of Euclide. An other waie yet. set one foote of the compas in the prick, on whiche you would haue the plumbe line to light, and stretche forth thother foote toward the longest end of the line, as wide as you can for the length of the line, and so draw a quarter of a compas or more, then without stirryng of the compas, set one foote of it in the same line, where as the circular line did begin, and extend thother in the circular line, settyng a marke where it doth light, then take half that quantitie more there vnto, and by that prick that endeth the last part, draw a line to the pricke assigned, and it shall be a perpendicular.

Example.

A.B. is the line appointed, to whiche I must make a perpendicular line to light in the pricke assigned, which is A. Therfore doo I set one foote of the compas in A, and extend the other vnto D. makyng a part of a circle,

more then a quarter, that is D.E. Then do I set one foote of the compas vnaltered in D, and stretch the other in the circular line, and it doth light in F, this space betwene D. and F. I deuide into halfe in the pricke G, whiche halfe I take with the compas, and set it beyond F. vnto H, and thefore is H. the point, by whiche the perpendicular line must be drawn, so say I that the line H.A, is a plumbe line to A.B, as the conclusion would.

[ THE .VI. CONCLVSION.]
To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.

Open your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places. Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion.

Example.

C. is the appointed pricke, from whiche vnto the line A.B. I must draw a perpẽdicular. Thefore I open the cõpas so wide, that it may haue one foote in C, and thother to reach ouer the line, and with y^t foote I draw an arch line as you see, betwene A. and B, which arch line I deuide in the middell in the point D. Then drawe I a line from C. to D, and it is perpendicular to the line A.B, accordyng as my desire was.

[ THE .VII. CONCLVSION.]
To make a plumbe lyne on any porcion of a circle, and that on the vtter or inner bughte.

Mark first the prick where y^e plũbe line shal lyght: and prick out of ech side of it .ij. other poinctes equally distant from that first pricke. Then set the one foote of the cõpas in one of those side prickes, and the other foote in the other side pricke, and first moue one of the feete and drawe an arche line ouer the middell pricke, then set the compas steddie with the one foote in the other side pricke, and with the other foote drawe an other arche line, that shall cut that first arche, and from the very poincte of their meetyng, drawe a right line vnto the firste pricke, where you do minde that the plumbe line shall lyghte. And so haue you performed thintent of this conclusion.

Example.

The arche of the circle on whiche I would erect a plumbe line, is A.B.C. and B. is the pricke where I would haue the plumbe line to light. Therfore I meate out two equall distaunces on eche side of that pricke B. and they are A.C. Then open I the compas as wide as A.C. and settyng one of the feete in A. with the other I drawe an arche line which goeth by G. Like waies I set one foote of the compas steddily in C. and with the other I drawe an arche line, goyng by G. also. Now consideryng that G. is the pricke of their meetyng, it shall be also the poinct fro whiche I must drawe the plũbe line. Then draw I a right line from G. to B. and so haue mine intent. Now as A.B.C. hath a plumbe line erected on his

vtter bought, so may I erect a plumbe line on the inner bught of D.E.F, doynge with it as I did with the other, that is to saye, fyrste settyng forthe the pricke where the plumbe line shall light, which is E, and then markyng one other on eche syde, as are D. and F. And then proceding as I dyd in the example before.

[ THE VIII. CONCLVSYON.]
How to deuide the arche of a circle into two equall partes, without measuring the arche.

Deuide the corde of that line info ij. equall portions, and then from the middle prycke erecte a plumbe line, and it shal parte that arche in the middle.

Example.

The arch to be diuided ys A.D.C, the corde is A.B.C, this corde is diuided in the middle with B, from which prick if I erect a plum line as B.D, thẽ will it diuide the arch in the middle, that is to say, in D.

[ THE IX. CONCLVSION.]

To do the same thynge other wise. And for shortenes of worke, if you wyl make a plumbe line without much labour, you may do it with your squyre, so that it be iustly made, for yf you applye the edge of the squyre to the line in which the prick is, and foresee the very corner of the squyre doo touche the pricke. And than frome that corner if you drawe a lyne by the other edge of the squyre, yt will be perpendicular to the former line.

Example.

A.B. is the line, on which I wold make the plumme line, or perpendicular. And therefore I marke the prick, from which the plumbe lyne muste rise, which here is C. Then do I sette one edg of my squyre (that is B.C.) to the line A.B, so at the corner of the squyre do touche C. iustly. And from C. I drawe a line by the other edge of the squire, (which is C.D.) And so haue I made the plumme line D.C, which I sought for.

[ THE X. CONCLVSION.]
How to do the same thinge an other way yet

If so be it that you haue an arche of suche greatnes, that your squyre wyll not suffice therto, as the arche of a brydge or of a house or window, then may you do this. Mete vnderneth the arch where y^e midle of his cord wyl be, and ther set a mark. Then take a long line with a plummet, and holde the line in suche a place of the arch, that the plummet do hang iustely ouer the middle of the corde, that you didde diuide before, and then the line doth shewe you the middle of the arche.

Example.

The arch is A.D.B, of which I trye the midle thus. I draw a corde from one syde to the other (as here is A.B,) which I diuide in the middle in C. Thẽ take I a line with a plummet (that is D.E,) and so hold I the line that the plummet E, dooth hange ouer C, And

then I say that D. is the middle of the arche. And to thentent that my plummet shall point the more iustely, I doo make it sharpe at the nether ende, and so may I trust this woorke for certaine.

[ THE XI. CONCLVSION.]
When any line is appointed and without it a pricke, whereby a parallel must be drawen howe you shall doo it.

Take the iuste measure beetwene the line and the pricke, accordinge to which you shal open your compasse. Thẽ pitch one foote of your compasse at the one ende of the line, and with the other foote draw a bowe line right ouer the pytche of the compasse, lyke-wise doo at the other ende of the lyne, then draw a line that shall touche the vttermoste edge of bothe those bowe lines, and it will bee a true parallele to the fyrste lyne appointed.

Example.

A.B, is the line vnto which I must draw an other gemow line, which muste passe by the prick C, first I meate with my compasse the smallest distance that is from C. to the line, and that is C.F, wherfore staying the compasse at that distaunce, I seete the one foote in A, and with the other foot I make a bowe lyne, which is D, thẽ like wise set I the one foote of the compasse in B, and with the other I make the second bow line, which is E. And then draw I a line, so that it toucheth the vttermost edge of bothe these bowe lines, and that lyne passeth by the pricke C, end is a gemowe line to A.B, as my sekyng was.

[ THE .XII. CONCLVSION.]

To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be longer then the thirde. For this rule is generall, that any two sides of euerie triangle taken together, are longer then the other side that remaineth.

If you do remember the first and seconde conclusions, then is there no difficultie in this, for it is in maner the same woorke. First cõsider the .iij. lines that you must take, and set one of thẽ for the ground line, then worke with the other .ij. lines as you did in the first and second conclusions.

Example.

I haue .iij. A.B. and C.D. and E.F. of whiche I put .C.D. for my ground line, then with my compas I take the length of .A.B. and set the one foote of my compas in C, and draw an arch line with the other foote. Likewaies I take the lẽgth of E.F, and set one foote in D, and with the other foote I make an arch line crosse the other arche, and the pricke of their metyng (whiche is G.) shall be the thirde corner of the triangle, for in all suche kyndes of woorkynge to make a tryangle, if you haue one line drawen, there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee, for two of them must needes be at the two eandes of the lyne that is drawen.

[ THE XIII. CONCLVSION.]

If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined angle, all ready assigned.

Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. If it bee not longe inoughe, thenn lengthen it fyrste, and afterwarde doo as I haue sayde beefore.

Example.

Lette the angle appoynted bee A.B.C, and the corner assigned, B. Farthermore let the lymited line bee D.G, and the pricke assigned D.

Fyrste therefore by drawinge the line A.C, I make the triangle A.B.C.

Then consideringe that D.G, is longer thanne A.B, you shall cut out a line frõ D. toward G, equal to A.B, as for exãple D.F. Thẽ measure oute the other ij. lines and worke with thẽ according as the conclusion with the fyrste also and the second teacheth yow, and then haue you done.

[ THE XIIII. CONCLVSION.]
To make a square quadrate of any righte lyne appoincted.

First make a plumbe line vnto your line appointed, whiche shall light at one of the endes of it, accordyng to the fifth conclusion, and let it be of like length as your first line is, then opẽ your compasse to the iuste length of one of them, and sette one foote of the compasse in the ende of the one line, and with the other foote draw an arche line, there as you thinke that the fowerth corner shall be, after that set the one foote of the same compasse vnsturred, in the eande of the other line, and drawe an other arche line crosse the first archeline, and the poincte that they do crosse in, is the pricke of the fourth corner of the square quadrate which you seke for, therfore draw a line from that pricke to the eande of eche line, and you shall therby haue made a square quadrate.

Example.

A.B. is the line proposed, of whiche I shall make a square quadrate, therefore firste I make a plũbe line vnto it, whiche shall lighte in A, and that plũb line is A.C, then open I my compasse as wide as the length of A.B, or A.C, (for they must be bothe equall) and I set the one foote of thend in C, and with the other I make an arche line nigh vnto D, afterward I set the compas again with one foote in B, and with the other foote I make an arche line crosse the first arche line in D, and from the prick of their crossyng I draw .ij. lines, one to B, and an other to C, and so haue I made the square quadrate that I entended.

[ THE .XV. CONCLVSION.]
To make a likeiãme equall to a triangle appointed, and that in a right lined ãgle limited.

First from one of the angles of the triangle, you shall drawe a gemowe line, whiche shall be a parallele to that syde of the triangle, on whiche you will make that likeiamme. Then on one end of the side of the triangle, whiche lieth against the gemowe lyne, you shall draw forth a line vnto the gemow line, so that one angle that commeth of those .ij. lines be like to the angle which is limited vnto you. Then shall you deuide into ij. equall partes that side of the triangle whiche beareth that line, and from the pricke of that deuision, you shall raise an other line parallele to that former line, and continewe it vnto the first gemowe line, and thẽ of those .ij. last gemowe lynes, and the first gemowe line, with the halfe side of the triangle, is made a lykeiamme equall to the triangle appointed, and hath an angle lyke to an angle limited, accordyng to the conclusion.

Example.

B.C.G, is the triangle appoincted vnto, whiche I muste make an equall likeiamme. And D, is the angle that the likeiamme must haue. Therfore first entendyng to erecte the likeiãme on the one side, that the ground line of the triangle (whiche is B.G.) I do draw a gemow line by C, and make it parallele to the ground line B.G, and that new gemow line is A.H. Then do I raise a line from B. vnto the gemowe line, (whiche line is A.B) and make an angle equall to D, that is the appointed angle (accordyng as the .viij. cõclusion teacheth) and that angle is B.A.E. Then to procede, I doo parte in y^e middle the said groũd line B.G, in the prick F, frõ which prick I draw

to the first gemowe line (A.H.) an other line that is parallele to A.B, and that line is E.F. Now saie I that the likeiãme B.A.E.F, is equall to the triangle B.C.G. And also that it hath one angle (that is B.A.E.) like to D. the angle that was limitted. And so haue I mine intent. The profe of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke, and is the .xxxi. proposition of this second boke of Theoremis, whiche saieth, that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines, and haue their ground line of one length, then is the likeiamme double to the triangle, wherof it foloweth, that if .ij. suche figures so drawen differ in their ground line onely, so that the ground line of the likeiamme be but halfe the ground line of the triangle, then be those .ij. figures equall, as you shall more at large perceiue by the boke of Theoremis, in y^e .xxxi. theoreme.

[ THE .XVI. CONCLVSION.]
To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.

In the last conclusion the sides of your likeiamme wer left to your libertie, though you had an angle appoincted. Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted, which must be the side of the likeiãme. Therfore thus shall you procede. Firste accordyng to the laste conclusion, make a likeiamme in the angle appoincted, equall to the triangle that is assigned. Then with your compasse take the length of your line appointed, and set out two lines of the same length in the second gemowe lines, beginnyng at the one side of the likeiamme, and by those two prickes shall you draw an other gemowe line, whiche shall be parallele to two sides of the likeiamme. Afterward shall you draw .ij. lines more for the accomplishement of your worke, which better shall be

perceaued by a shorte exaumple, then by a greate numbre of wordes, only without example, therefore I wyl by example sette forth the whole worke.

Example.

Fyrst, according to the last conclusion, I make the likeiamme E.F.C.G, equal to the triangle D, in the appoynted angle whiche is E. Then take I the lengthe of the assigned line (which is A.B,) and with my compas I sette forthe the same lẽgth in the ij. gemow lines N.F. and H.G, setting one foot in E, and the other in N, and againe settyng one foote in C, and the other in H. Afterward I draw a line from N. to H, whiche is a gemow lyne, to ij. sydes of the likeiamme. thenne drawe I a line also from N. vnto C. and extend it vntyll it crosse the lines, E.L. and F.G, which both must be drawen forth longer then the sides of the likeiamme. and where that lyne doeth crosse F.G, there I sette M. Nowe to make an ende, I make an other gemowe line, whiche is parallel to N.F. and H.G, and that gemowe line doth passe by the pricke M, and then haue I done. Now say I that H.C.K.L, is a likeiamme equall to the triangle appointed, whiche was D, and is made of a line assigned that is A.B, for H.C, is equall vnto A.B, and so is K.L. The profe of y^e equalnes of this likeiam vnto the triãgle, depẽdeth of the thirty and two Theoreme: as in the boke of Theoremes doth appear, where it is declared, that in al likeiammes, whẽ there are more then one made about one bias line, the filsquares of euery of them muste needes be equall.

[ THE XVII. CONCLVSION.]
To make a likeiamme equal to any right lined figure, and that on an angle appointed.

The readiest waye to worke this conclusion, is to tourn that rightlined figure into triangles, and then for euery triangle together an equal likeiamme, according vnto the eleuen cõclusion, and then to ioine al those likeiammes into one, if their sides happen to be equal, which thing is euer certain, when al the triangles happẽ iustly betwene one pair of gemow lines. but and if they will not frame so, then after that you haue for the firste triangle made his likeiamme, you shall take the lẽgth of one of his sides, and set that as a line assigned, on whiche you shal make the other likeiams, according to the twelft cõclusion, and so shall you haue al your likeiammes with ij. sides equal, and ij. like angles, so y^t you mai easily ioyne thẽ into one figure.

Example.

If the right lined figure be like vnto A, thẽ may it be turned into triangles that wil stãd betwene ij. parallels anye ways, as you mai se by C. and D, for ij. sides of both the triãngles ar parallels. Also if the right lined figure be like vnto E, thẽ wil it be turned into triãgles, liyng betwene two parallels also, as y^e other did before, as in the exãple of F.G. But and if y^e

right lined figure be like vnto H, and so turned into triãgles as you se in K.L.M, wher it is parted into iij triãgles, thẽ wil not all those triangles lye betwen one pair of parallels or gemow lines, but must haue many, for euery triangle must haue one paire of parallels seuerall, yet it maye happen that when there bee three or fower triangles, ij. of theym maye happen to agre to one pair of parallels, whiche thinge I remit to euery honest witte to serche, for the manner of their draught wil declare, how many paires of parallels they shall neede, of which varietee bicause the examples ar infinite, I haue set forth these few, that by them you may coniecture duly of all other like.

Further explicacion you shal not greatly neede, if you remembre what hath ben taught before, and then diligẽtly behold how these sundry figures be turned into triãgles. In the fyrst you se I haue made v. triangles, and four paralleles. in the seconde vij. triangles and foure paralleles. in the thirde thre triãgles, and fiue parallels, in the iiij. you se fiue triãgles & four parallels. in the fift, iiij. triãgles and .iiij. parallels, & in y^e sixt ther ar fiue triãgles & iiij. paralels. Howbeit a mã maye at liberty alter them into diuers formes of triãgles & therefore I

leue it to the discretion of the woorkmaister, to do in al suche cases as he shal thinke best, for by these examples (if they bee well marked) may all other like conclusions be wrought.

[ THE XVIII. CONCLVSION.]

To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.

First deuide your lyne into ij. equal parts, and of the length of one part make a perpendicular to light at one end of your line assigned. then adde a bias line, and make thereof a triangle, this done if you take from this bias line the halfe lengthe of your line appointed, which is the iuste length of your perpendicular, that part of the bias line whiche dothe remayne, is the greater portion of the deuision that you seke for, therefore if you cut your line according to the lengthe of it, then will the square of that greater portion be equall to the square that is made of the whole line and his lesser portion. And contrary wise, the square of the whole line and his lesser parte, wyll be equall to the square of the greater parte.

Example.

A.B, is the lyne assigned. E. is the middle pricke of A.B, B.C. is the plumb line or perpendicular, made of the halfe of A.B, equall to A.E, other B.E, the byas line is C.A, from whiche I cut a peece, that is C.D, equall to C.B, and accordyng to the lengthe lo the peece that remaineth (whiche is D.A,) I doo deuide the line A.B, at whiche diuision I set F. Now say I, that this line A.B, (w^ch was assigned vnto me) is so diuided in this point F, y^t y^e square of y^e hole line A.B, & of the one portiõ (y^t is F.B, the

lesser part) is equall to the square of the other parte, whiche is F.A, and is the greater part of the first line. The profe of this equalitie shall you learne by the .xl. Theoreme.

There are two ways to make this Example work:
—transpose E and F in the illustration, and change one occurrence of E to F in the text, or:
—keep the illustration as printed, and transpose all other occurrences of E and F in the text.

[ THE .XIX. CONCLVSION.]
To make a square quadrate equall to any right lined figure appoincted.

First make a likeiamme equall to that right lined figure, with a right angle, accordyng to the .xi. conclusion, then consider the likeiamme, whether it haue all his sides equall, or not: for yf they be all equall, then haue you doone your conclusion. but and if the sides be not all equall, then shall you make one right line iuste as long as two of those vnequall sides, that line shall you deuide in the middle, and on that pricke drawe half a circle, then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme, and from that pointe of diuision shall you erecte a perpendicular, which shall touche the edge of the circle. And that perpendicular shall be the iuste side of the square quadrate, equall both to the lykeiamme, and also to the right lined figure appointed, as the conclusion willed.

Example.

K, is the right lined figure appointed, and B.C.D.E, is the likeiãme, with right angles equall vnto K, but because that this likeiamme is not a square quadrate, I must turne it into such one after this sort, I shall make one right line, as long as .ij. vnequall sides of the likeiãme, that line here is F.G, whiche is equall to B.C, and C.E. Then part I that line in the middle in the

pricke M, and on that pricke I make halfe a circle, accordyng to the length of the diameter F.G. Afterward I cut awaie a peece from F.G, equall to C.E, markyng that point with H. And on that pricke I erecte a perpendicular H.K, whiche is the iust side to the square quadrate that I seke for, therfore accordyng to the doctrine of the .x. conclusion, of the lyne I doe make a square quadrate, and so haue I attained the practise of this conclusion.

[ THE .XX. CONCLVSION.]
When any .ij. square quadrates are set forth, how you maie make one equall to them bothe.

First drawe a right line equall to the side of one of the quadrates: and on the ende of it make a perpendicular, equall in length to the side of the other quadrate, then drawe a byas line betwene those .ij. other lines, makyng thereof a right angeled triangle. And that byas lyne wyll make a square quadrate, equall to the other .ij. quadrates appointed.

Example.

A.B. and C.D, are the two square quadrates appointed, vnto which I must make one equall square quadrate. First therfore I dooe make a righte line E.F, equall to one of the sides of the square quadrate A.B. And on the one end of it I make a plumbe line E.G, equall to the side of the other quadrate D.C. Then drawe I a byas line G.F, which beyng made the side of a quadrate

(accordyng to the tenth conclusion) will accomplishe the worke of this practise: for the quadrate H. is muche iust as the other two. I meane A.B. and D.C.

[ THE .XXI. CONCLVSION.]
When any two quadrates be set forth, howe to make a squire about the one quadrate, whiche shall be equall to the other quadrate.

Determine with your selfe about whiche quadrate you wil make the squire, and drawe one side of that quadrate forth in lengte, accordyng to the measure of the side of the other quadrate, whiche line you maie call the grounde line, and then haue you a right angle made on this line by an other side of the same quadrate: Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion, by makyng of a byas line, and that byas lyne will performe the worke of your desire. For if you take the length of that byas line with your compasse, and then set one foote of the compas in the farthest angle of the first quadrate (whiche is the one ende of the groundline) and extend the other foote on the same line, accordyng to the measure of the byas line, and of that line make a quadrate, enclosyng y^e first quadrate, then will there appere the forme of a squire about the first quadrate, which squire is equall to the second quadrate.

Example.

The first square quadrate is A.B.C.D, and the seconde is E. Now would I make a squire about the quadrate A.B.C.D, whiche shall bee equall vnto the quadrate E.

Therfore

first I draw the line A.D, more in length, accordyng to the measure of the side of E, as you see, from D. vnto F, and so the hole line of bothe these seuerall sides is A.F, thẽ make I a byas line from C, to F, whiche byas line is the measure of this woorke. wherefore I open my compas accordyng to the length of that byas line C.F, and set the one compas foote in A, and extend thother foote of the compas toward F, makyng this pricke G, from whiche I erect a plumbeline G.H, and so make out the square quadrate A.G.H.K, whose sides are equall eche of them to A.G. And this square doth contain the first quadrate A.B.C.D, and also a squire G.H.K, whiche is equall to the second quadrate E, for as the last conclusion declareth, the quadrate A.G.H.K, is equall to bothe the other quadrates proposed, that is A.B.C.D, and E. Then muste the squire G.H.K, needes be equall to E, consideryng that all the rest of that great quadrate is nothyng els but the quadrate self, A.B.C.D, and so haue I thintent of this conclusion.

[ THE .XXII. CONCLVSION.]
To find out the cẽtre of any circle assigned.

Draw a corde or stryngline crosse the circle, then deuide into .ij. equall partes, both that corde, and also the bowe line, or arche line, that serueth to that corde, and from the prickes of those diuisions, if you drawe an other line crosse the circle, it must nedes passe by the centre. Therfore deuide that line in the middle, and that middle pricke is the centre of the circle proposed.

Example.

Let the circle be A.B.C.D, whose centre I shall seke. First therfore I draw a corde crosse the circle, that is A.C. Then do I deuide that corde in the middle, in E, and likewaies also do I deuide his arche line A.B.C, in the middle, in the pointe B. Afterward I drawe a line from B. to E, and so crosse the

circle, whiche line is B.D, in which line is the centre that I seeke for. Therefore if I parte that line B.D, in the middle in to two equall portions, that middle pricke (which here is F) is the verye centre of the sayde circle that I seke. This conclusion may other waies be wrought, as the moste part of conclusions haue sondry formes of practise, and that is, by makinge thre prickes in the circũference of the circle, at liberty where you wyll, and then findinge the centre to those thre pricks, Which worke bicause it serueth for sondry vses, I think meet to make it a seuerall conclusion by it selfe.

[ THE XXIII. CONCLVSION.]
To find the commen centre belongyng to anye three prickes appointed, if they be not in an exacte right line.

It is to be noted, that though euery small arche of a greate circle do seeme to be a right lyne, yet in very dede it is not so, for euery part of the circumference of al circles is compassed, though in litle arches of great circles the eye cannot discerne the crokednes, yet reason doeth alwais declare it, therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle. But and if they be not in a right line how so euer they stande, thus shall you find their cõmon centre. Opẽ your compas so wide, that it be somewhat more then the

halfe distance of two of those prickes. Then sette the one foote of the compas in the one pricke, and with the other foot draw an arche lyne toward the other pricke, Then againe putte the foot of your compas in the second pricke, and with the other foot make an arche line, that may crosse the firste arch line in ij. places. Now as you haue done with those two pricks, so do with the middle pricke, and the thirde that remayneth. Then draw ij. lines by the poyntes where those arche lines do crosse, and where those two lines do meete, there is the centre that you seeke for.

Example

The iij. prickes I haue set to be A.B, and C, whiche I wold bring into the edg of one common circle, by finding a centre cõmen to them all, fyrst therefore I open my cõpas, so that thei occupye more then y^e halfe distance betwene ij. pricks (as are A.B.) and so settinge one foote in A. and extendinge the other toward B, I make the arche line D.E. Likewise settĩg one foot in B, and turninge the other toward A, I draw an other arche line that crosseth the first in D. and E. Then from D. to E, I draw a right lyne D.H. After this I open my cõpasse to a new distance, and make ij. arche lines betwene B. and C, whiche crosse one the other in F. and G, by whiche two pointes I draw an other line, that is F.H. And bycause that the lyne D.H. and the lyne F.H. doo meete in H, I saye that H. is the centre that serueth to those iij. prickes. Now therfore if you set one foot of your compas in H, and extend the other to any of the iij. pricks, you may draw a circle w^ch shal enclose those iij. pricks in the edg of his circũferẽce & thus haue you attained y^e vse of this cõclusiõ.

[ THE XXIIII. CONCLVSION.]
To drawe a touche line onto a circle, from any poincte assigned.

Here must you vnderstand that the pricke must be without the circle, els the conclusion is not possible. But the pricke or poinct beyng without the circle, thus shall you procede: Open your compas, so that the one foote of it maie be set in the centre of the circle, and the other foote on the pricke appoincted, and so draw an other circle of that largenesse about the same centre: and it shall gouerne you certainly in makyng the said touche line. For if you draw a line frõ the pricke appointed vnto the centre of the circle, and marke the place where it doeth crosse the lesser circle, and from that poincte erect a plumbe line that shall touche the edge of the vtter circle, and marke also the place where that plumbe line crosseth that vtter circle, and from that place drawe an other line to the centre, takyng heede where it crosseth the lesser circle, if you drawe a plumbe line from that pricke vnto the edge of the greatter circle, that line I say is a touche line, drawen from the point assigned, according to the meaning of this conclusion.

Example.

Let the circle be called B.C.D, and his cẽtre E, and y^e prick assigned A, opẽ your cõpas now of such widenes, y^t the one foote may be set in E, w^ch is y^e cẽtre of y^e circle, & y^e other in A, w^ch is y^e pointe assigned, & so make an other greter circle (as here is A.F.G) thẽ draw a line from A. vnto E, and wher that line doth cross y^e inner circle (w^ch heere is in the prick B.) there erect a plũb line vnto the line. A.E. and let that plumb line touch the vtter circle, as it doth here in the point F, so shall B.F. bee that plumbe lyne. Then from F. vnto E.

drawe an other line whiche shal be F.E, and it will cutte the inner circle, as it doth here in the point C, from which pointe C. if you erect a plumb line vnto A, then is that line A.C, the touche line, whiche you shoulde finde. Not withstandinge that this is a certaine waye to fynde any touche line, and a demonstrable forme, yet more easyly by many folde may you fynde and make any suche line with a true ruler, layinge the edge of the ruler to the edge of the circle and to the pricke, and so drawing a right line, as this example sheweth, where the circle is E, the pricke assigned is A. and the ruler C.D. by which the touch line is drawen, and that is A.B, and as this way is light to doo, so is it certaine inoughe for any kinde of workinge.

[ THE XXV. CONCLVSION.]
When you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreynge therevnto.

First seeke out of the centre of that arche, according to the doctrine of the seuententh conclusion, and then setting one foote of your compas in the centre, and extending the other foot vnto the edge of the arche or peece of the circumference, it is easy to drawe the whole circle.

Example.

A peece of an olde pillar was found, like in forme to thys figure A.D.B. Now to knowe howe muche the cõpasse of the hole piller was, seing by this parte it appereth that it was round, thus shal you do. Make in a table the like draught of y^t circũference by the self patrõ, vsing it as it wer a croked ruler.

Then make .iij. prickes in that arche line, as I haue made, C. D. and E. And then finde out the common centre to them all, as the .xvij. conclusion teacheth. And that cẽtre is here F, nowe settyng one foote of your compas in F, and the other in C. D, other in E, and so makyng a compasse, you haue youre whole intent.

[ THE XXVI. CONCLVSION.]
To finde the centre to any arche of a circle.

If so be it that you desire to find the centre by any other way then by those .iij. prickes, consideryng that sometimes you can not haue so much space in the thyng where the arche is drawen, as should serue to make those .iiij. bowe lines, then shall you do thus: Parte that arche line into two partes, equall other vnequall, it maketh no force, and vnto ech portion draw a corde, other a stringline. And then accordyng as you dyd in one arche in the .xvi. conclusion, so doe in bothe those arches here, that is to saie, deuide the arche in the middle, and also the corde, and drawe then a line by those two deuisions, so then are you sure that that line goeth by the centre. Afterward do lykewaies with the other arche and his corde, and where those .ij. lines do crosse, there is the centre, that you seke for.

Example.

The arche of the circle is A.B.C, vnto whiche I must seke

a centre, therfore firste I do deuide it into .ij. partes, the one of them is A.B, and the other is B.C. Then doe I cut euery arche in the middle, so is E. the middle of A.B, and G. is the middle of B.C. Likewaies, I take the middle of their cordes, whiche I mark with F. and H, settyng F. by E, and H. by G. Then drawe I a line from E. to F, and from G. to H, and they do crosse in D, wherefore saie I, that D. is the centre, that I seke for.

[ THE XXVII. CONCLVSION.]
To drawe a circle within a triangle appoincted.

For this conclusion and all other lyke, you muste vnderstande, that when one figure is named to be within an other, that it is not other waies to be vnderstande, but that eyther euery syde of the inner figure dooeth touche euerie corner of the other, other els euery corner of the one dooeth touche euerie side of the other. So I call that triangle drawen in a circle, whose corners do touche the circumference of the circle. And that circle is contained in a triangle, whose circumference doeth touche iustely euery side of the triangle, and yet dooeth not crosse ouer any side of it. And so that quadrate is called properly to be drawen in a circle, when all his fower angles doeth touche the edge of the circle, And that circle is drawen in a quadrate, whose circumference doeth touche euery side of the quadrate, and lykewaies of other figures.

Examples are these. A.B.C.D.E.F.
A. is a circle in a triangle.	C. a quadrate in a circle.
B. a triangle in a circle.	D. a circle in a quadrate.

In these .ij. last figures E. and F, the circle is not named to be drawen in a triangle, because it doth not touche the sides of the triangle, neither is the triangle coũted to be drawen in the circle, because one of his corners doth not touche the circumference of the circle, yet (as you see) the circle is within the triangle, and the triangle within the circle, but nother of them is properly named to be in the other. Now to come to the conclusion. If the triangle haue all .iij. sides lyke, then shall you take the middle of euery side, and from the contrary corner drawe a right line vnto that poynte, and where those lines do crosse one an other, there is the centre. Then set one foote of the compas in the centre and stretche out the other to the middle pricke of any of the sides, and so drawe a compas, whiche shall touche euery side of the triangle, but shall not passe with out any of them.

Example.

The triangle is A.B.C, whose sides I do part into .ij. equall partes, eche by it selfe in these pointes D.E.F, puttyng F. betwene A.B, and D. betwene B.C, and E. betwene A.C. Then draw I a line from C. to F, and an other from A. to D, and the third from B. to E.

And

where all those lines do mete (that is to saie M. G,) I set the one foote of my compasse, because it is the common centre, and so drawe a circle accordyng to the distaunce of any of the sides of the triangle. And then find I that circle to agree iustely to all the sides of the triangle, so that the circle is iustely made in the triangle, as the conclusion did purporte. And this is euer true, when the triangle hath all thre sides equall, other at the least .ij. sides lyke long. But in the other kindes of triangles you must deuide euery angle in the middle, as the third conclusion teaches you.

And so drawe lines frõ eche angle to their middle pricke. And where those lines do crosse, there is the common centre, from which you shall draw a perpendicular to one of the sides. Then sette one foote of the compas in that centre, and stretche the other foote accordyng to the lẽgth of the perpendicular, and so drawe your circle.

Example.

The triangle is A.B.C, whose corners I haue diuided in the middle with D.E.F, and haue drawen the lines of diuision A.D, B.E, and C.F, which crosse in G, therfore shall G. be the common centre. Then make I one perpẽdicular from G. vnto the side B.C, and that

is G.H. Now sette I one fote of the compas in G, and extend the other foote vnto H. and so drawe a compas, whiche wyll iustly answere to that triãgle according to the meaning of the conclusion.

[ THE XXVIII. CONCLVSION.]
To drawe a circle about any triãgle assigned.

Fyrste deuide two sides of the triangle equally in half and from those ij. prickes erect two perpendiculars, which muste needes meet in crosse, and that point of their meting is the centre of the circle that must be drawen, therefore sette one foote of the compasse in that pointe, and extend the other foote to one corner of the triangle, and so make a circle, and it shall touche all iij. corners of the triangle.

Example.

A.B.C. is the triangle, whose two sides A.C. and B.C. are diuided into two equall partes in D. and E, settyng D. betwene B. and C, and E. betwene A. and C. And from eche of those two pointes is ther erected a perpendicular (as you se D.F, and E.F.) which mete, and crosse in F, and stretche forth the other foot of any corner of the triangle, and so make a circle, that circle shal touch euery corner of the triangle, and shal enclose the whole triangle, accordinge, as the conclusion willeth.

An other way to do the same.

And yet an other waye may you doo it, accordinge as you learned in the seuententh conclusion, for if you call the three

corners of the triangle iij. prickes, and then (as you learned there) yf you seeke out the centre to those three prickes, and so make it a circle to include those thre prickes in his circumference, you shall perceaue that the same circle shall iustelye include the triangle proposed.

Example.

A.B.C. is the triangle, whose iij. corners I count to be iij. pointes. Then (as the seuentene conclusion doth teache) I seeke a common centre, on which I may make a circle, that shall enclose those iij prickes. that centre as you se is D, for in D. doth the right lines, that passe by the angles of the arche lines, meete and crosse. And on that centre as you se, haue I made a circle, which doth inclose the iij. angles of the triãgle, and consequentlye the triangle itselfe, as the conclusion dydde intende.

[ THE XXIX. CONCLVSION.]
To make a triangle in a circle appoynted whose corners shal be equall to the corners of any triangle assigned.

When I will draw a triangle in a circle appointed, so that the corners of that triangle shall be equall to the corners of any triangle assigned, then must I first draw a tuche lyne vnto that circle, as the twenty conclusion doth teach, and in the very poynte of the touche muste I make an angle, equall to one angle of the triangle, and that inwarde toward the circle: likewise in the same pricke must I make an other angle w^t the other halfe of the touche line, equall to an other corner of the triangle appointed, and then betwen those two corners

will there resulte a third angle, equall to the third corner of that triangle. Nowe where those two lines that entre into the circle, doo touche the circumference (beside the touche line) there set I two prickes, and betwene them I drawe a thyrde line. And so haue I made a triangle in a circle appointed, whose corners bee equall to the corners of the triangle assigned.

Example.

A.B.C, is the triangle appointed, and F.G.H. is the circle, in which I muste make an other triangle, with lyke angles to the angles of A.B.C. the triangle appointed. Therefore fyrst I make the touch lyne D.F.E. And then make I an angle in F, equall to A, whiche is one of the angles of the triangle. And the lyne that maketh that angle with the touche line, is F.H, whiche I drawe in lengthe vntill it touche the edge of the circle. Then againe in the same point F, I make an other corner equall to the angle C. and the line that maketh that corner with the touche line, is F.G. whiche also I drawe foorthe vntill it touche the edge of the circle. And then haue I made three angles vpon that one touch line, and in y^t one point F, and those iij. angles be equall to the iij. angles of the triangle assigned, whiche thinge doth plainely appeare, in so muche as they bee equall

to ij. right angles, as you may gesse by the fixt theoreme. And the thre angles of euerye triangle are equill also to ij. righte angles, as the two and twenty theoreme dothe show, so that bicause they be equall to one thirde thinge, they must needes be equal togither, as the cõmon sentence saith. Thẽ do I draw a line frome G. to H, and that line maketh a triangle F.G.H, whole angles be equall to the angles of the triangle appointed. And this triangle is drawn in a circle, as the conclusion didde wyll. The proofe of this conclusion doth appeare in the seuenty and iiij. Theoreme.

[ THE XXX. CONCLVSION.]
To make a triangle about a circle assigned which shall haue corners, equall to the corners of any triangle appointed.

First draw forth in length the one side of the triangle assigned so that therby you may haue ij. vtter angles, vnto which two vtter angles you shall make ij. other equall on the centre of the circle proposed, drawing thre halfe diameters frome the circumference, whiche shal enclose those ij. angles, thẽ draw iij. touche lines which shall make ij. right angles, eche of them with one of those semidiameters. Those iij. lines will make a triangle equally cornered to the triangle assigned, and that triangle is drawẽ about a circle apointed, as the cõclusiõ did wil.

Example.

A.B.C, is the triangle assigned, and G.H.K, is the circle appointed, about which I muste make a triangle hauing equall angles to the angles of that triangle A.B.C. Fyrst therefore I draw A.C. (which is one of the sides of the triangle) in length that there may appeare two vtter angles in that triangle, as you se B.A.D, and B.C.E.

Then

drawe I in the circle appointed a semidiameter, which is here H.F, for F. is the cẽtre of the circle G.H.K. Then make I on that centre an angle equall to the vtter angle B.A.D, and that angle is H.F.K. Like waies on the same cẽtre by drawyng an other semidiameter, I make an other angle H.F.G, equall to the second vtter angle of the triangle, whiche is B.C.E. And thus haue I made .iij. semidiameters in the circle appointed. Then at the ende of eche semidiameter, I draw a touche line, whiche shall make righte angles with the semidiameter. And those .iij. touch lines mete, as you see, and make the trianagle L.M.N, whiche is the triangle that I should make, for it is drawen about a circle assigned, and hath corners equall to the corners of the triangle appointed, for the corner M. is equall to C. Likewaies L. to A, and N. to B, whiche thyng you shall better perceiue by the vi. Theoreme, as I will declare in the booke of proofes.

[ THE XXXI. CONCLVSION.]

To make a portion of a circle on any right line assigned, whiche shall conteine an angle equall to a right lined angle appointed.

The angle appointed, maie be a sharpe angle, a right angle, other a blunte angle, so that the worke must be diuersely handeled

according to the diuersities of the angles, but consideringe the hardenes of those seuerall woorkes, I wyll omitte them for a more meter time, and at this tyme wyll shewe you one light waye which serueth for all kindes of angles, and that is this. When the line is proposed, and the angle assigned, you shall ioyne that line proposed so to the other twoo lines contayninge the angle assigned, that you shall make a triangle of theym, for the easy dooinge whereof, you may enlarge or shorten as you see cause, anye of the two lynes contayninge the angle appointed. And when you haue made a triangle of those iij. lines, then accordinge to the doctrine of the seuẽ and twẽty coclusiõ, make a circle about that triangle. And so haue you wroughte the request of this conclusion. Whyche yet you maye woorke by the twenty and eight conclusion also, so that of your line appointed, you make one side of the triãgle be equal to y^e ãgle assigned as youre selfe mai easily gesse.

Example.

First for example of a sharpe ãgle let A. stãd & B.C shal be y^e lyne assigned. Thẽ do I make a triangle, by adding B.C, as a thirde side to those other ij. which doo include the ãgle assigned, and that triãgle is D.E.F, so y^t E.F.

is the line appointed, and D. is the angle assigned. Then doo I drawe a portion of a circle about that triangle, from the one ende of that line assigned vnto the other, that is to saie, from E. a long by D. vnto F, whiche portion is euermore greatter then the halfe of the circle, by reason that the angle is a sharpe angle. But if the angle be right (as in the second exaumple you see it) then shall the portion of the circle that containeth that angle, euer more be the iuste halfe of a circle. And when the angle is a blunte angle, as the thirde exaumple dooeth propounde, then shall the portion of the circle euermore be lesse then the halfe circle. So in the seconde example, G. is the right angle assigned, and H.K. is the lyne appointed, and L.M.N. the portion of the circle aunsweryng thereto. In the third exaumple, O. is the blunte corner assigned, P.Q. is the line, and R.S.T. is the portion of the circle, that containeth that blũt corner, and is drawen on R.T. the line appointed.

[ THE XXXII. CONCLVSION.]
To cutte of from a circle appointed, a portion containyng an angle equall to a right lyned angle assigned.

When the angle and the circle are assigned, first draw a touch line vnto that circle, and then drawe an other line from the pricke of the touchyng to one side of the circle, so that thereby those two lynes do make an angle equall to the angle assigned. Then saie I that the portion of the circle of the contrarie side to the angle drawen, is the parte that you seke for.

Example.

A. is the angle appointed, and D.E.F. is the circle assigned, frõ which I must cut away a portiõ that doth contain an angle

equall to this angle A. Therfore first I do draw a touche line to the circle assigned, and that touch line is B.C, the very pricke of the touche is D, from whiche D. I drawe a lyne D.E, so that the angle made of those two lines be equall to the angle appointed. Then say I, that the arch of the circle D.F.E, is the arche that I seke after. For if I doo deuide that arche in the middle (as here is done in F.) and so draw thence two lines, one to D, and the other to E, then will the angle F, be equall to the angle assigned.

[ THE XXXIII. CONCLVSION.]
To make a square quadrate in a circle assigned.

Draw .ij. diameters in the circle, so that they runne a crosse, and that they make .iiij. right angles. Then drawe .iiij. lines, that may ioyne the .iiij. ends of those diameters, one to an other, and then haue you made a square quadrate in the circle appointed.

Example.

A.B.C.D. is the circle assigned, and A.C. and B.D. are the two diameters which crosse in the centre E, and make .iiij. right corners. Then do I make fowre other lines, that is A.B, B.C, C.D, and D.A, which do ioyne together the fowre endes of the ij. diameters. And so is the square

quadrate made in the circle assigned, as the conclusion willeth.

[ THE XXXIIII. CONCLVSION.]
To make a square quadrate aboute annye circle assigned.

Drawe two diameters in crosse waies, so that they make foure righte angles in the centre. Then with your compasse take the length of the halfe diameter, and set one foote of the compas in eche end of the compas, so shall you haue viij. archelines. Then yf you marke the prickes wherin those arch lines do crosse, and draw betwene those iiij. prickes iiij right lines, then haue you made the square quadrate accordinge to the request of the conclusion.

Example.

A.B.C. is the circle assigned in which first I draw two diameters, in crosse waies, making iiij. righte angles, and those ij. diameters are A.C. and B.D. Then sette I my compasse (whiche is opened according to the semidiameter of the said circle) fixing one foote in the end of euery semidiameter, and drawe with the other foote twoo arche lines, one on euery side. As firste, when I sette the one foote in A,

then with the other foote I doo make twoo arche lines, one in E, and an other in F. Then sette I the one foote of the compasse in B, and drawe twoo arche lines F. and G. Like wise setting the compasse foote in C, I drawe twoo other arche lines, G. and H, and on D. I make twoo other, H. and E. Then frome the crossinges of those eighte arche lines I drawe iiij. straighte lynes, that is to saye, E.F, and F.G, also G.H, and H.E, whiche iiij. straighte lynes do make the square quadrate that I should draw about the circle assigned.

[ THE XXXV. CONCLVSION.]
To draw a circle in any square quadrate appointed.

Fyrste deuide euery side of the quadrate into twoo equall partes, and so drawe two lynes betwene eche two contrary poinctes, and where those twoo lines doo crosse, there is the centre of the circle. Then sette the foote of the compasse in that point, and stretch forth the other foot, according to the length of halfe one of those lines, and so make a compas in the square quadrate assigned.

Example.

A.B.C.D. is the quadrate appointed, in whiche I muste make a circle. Therefore first I do deuide euery side in ij. equal partes, and draw ij. lines acrosse, betwene eche ij. cõtrary prickes, as you se E.G, and F.H, whiche mete in K, and therfore shal K, be the centre of the circle. Then do I set one foote of the compas in K. and opẽ the other as wide as K.E, and so draw a circle, which is made accordinge to the conclusion.

[ THE XXXVI. CONCLVSION.]
To draw a circle about a square quadrate.

Draw ij. lines betwene the iiij. corners of the quadrate, and where they mete in crosse, ther is the centre of the circle that you seeke for. Thẽ set one foot of the compas in that centre, and extend the other foote vnto one corner of the quadrate, and so may you draw a circle which shall iustely inclose the quadrate proposed.

Example.

A.B.C.D. is the square quadrate proposed, about which I must make a circle. Therfore do I draw ij. lines crosse the square quadrate from angle to angle, as you se A.C. & B.D. And where they ij. do crosse (that is to say in E.) there set I the one foote of the compas as in the centre, and the other foote I do extend vnto one angle of the quadrate, as for exãple to A, and so make a compas, whiche doth iustly inclose the quadrate, according to the minde of the conclusion.

[ THE XXXVII. CONCLVSION.]

To make a twileke triangle, whiche shall haue euery of the ij. angles that lye about the ground line, double to the other corner.

Fyrste make a circle, and deuide the circumference of it into fyue equall partes. And thenne drawe frome one pricke (which you will) two lines to ij. other prickes, that is to say to the iij. and iiij. pricke, counting that for the first, wherhence you drewe both those lines, Then drawe the thyrde lyne to make a triangle with those other twoo, and you haue doone according to the conclusion, and haue made a twelike triãgle,

whose ij. corners about the grounde line, are eche of theym double to the other corner.

At no point in this or the accompanying book does the author show how to divide a circle into five.

Example.

A.B.C. is the circle, whiche I haue deuided into fiue equal portions. And from one of the prickes (which is A,) I haue drawẽ ij. lines, A.B. and B.C, whiche are drawen to the third and iiij. prickes. Then draw I the third line C.B, which is the grounde line, and maketh the triangle, that I would haue, for the ãgle C. is double to the angle A, and so is the angle B. also.

[ THE XXXVIII. CONCLVSION.]
To make a cinkangle of equall sides, and equall corners in any circle appointed.

Deuide the circle appointed into fiue equall partes, as you didde in the laste conclusion, and drawe ij. lines from euery pricke to the other ij. that are nexte vnto it. And so shall you make a cinkangle after the meanynge of the conclusion.

Example.

Yow se here this circle A.B.C.D.E. deuided into fiue equall portions. And from eche pricke ij. lines drawen to the other ij. nexte prickes, so from A. are drawen ij. lines, one to B, and the other to E, and so from C. one to B. and an other

to D, and likewise of the reste. So that you haue not only learned hereby how to make a sinkangle in anye circle, but also how you shal make a like figure spedely, whanne and where you will, onlye drawinge the circle for the intente, readylye to make the other figure (I meane the cinkangle) thereby.

[ THE XXXIX. CONCLVSION.]
How to make a cinkangle of equall sides and equall angles about any circle appointed.

Deuide firste the circle as you did in the last conclusion into fiue equall portions, and draw fiue semidiameters in the circle. Then make fiue touche lines, in suche sorte that euery touche line make two right angles with one of the semidiameters. And those fiue touche lines will make a cinkangle of equall sides and equall angles.

Example.

A.B.C.D.E. is the circle appointed, which is deuided into fiue equal partes. And vnto euery prycke is drawẽ a semidiameter, as you see. Then doo I make a touche line in the pricke B, whiche is F.G, making ij. right angles

with the semidiameter B, and lyke waies on C. is made G.H, on D. standeth H.K, and on E, is set K.L, so that of those .v. touche lynes are made the .v. sides of a cinkeangle, accordyng to the conclusion.

An other waie.

Another waie also maie you drawe a cinkeangle aboute a circle, drawyng first a cinkeangle in the circle (whiche is an easie thyng to doe, by the doctrine of the .xxxvij. conclusion) and then drawing .v. touche lines whiche shall be iuste paralleles to the .v. sides of the cinkeangle in the circle, forseeyng that one of them do not crosse ouerthwarte an other and then haue you done. The exaumple of this (because it is easie) I leaue to your owne exercise.

[ THE XL. CONCLVSION.]
To make a circle in any appointed cinkeangle of equall sides and equall corners.

Drawe a plumbe line from any one corner of the cinkeangle, vnto the middle of the side that lieth iuste against that angle. And do likewaies in drawyng an other line from some other corner, to the middle of the side that lieth against that corner also. And those two lines wyll meete in crosse in the pricke of their crossyng, shall you iudge the centre of the circle to be. Therfore set one foote of the compas in that pricke, and extend the other to the end of the line that toucheth the middle of one side, whiche you liste, and so drawe a circle. And it shall be iustly made in the cinkeangle, according to the conclusion.

Example.

The cinkeangle assigned is A.B.C.D.E, in whiche I muste

make a circle, wherefore I draw a right line from the one angle (as from B,) to the middle of the contrary side (whiche is E. D,) and that middle pricke is F. Then lykewaies from an other corner (as from E) I drawe a right line to the middle of the side that lieth against it (whiche is B.C.) and that pricke is G. Nowe because that these two lines do crosse in H, I saie that H. is the centre of the circle, whiche I would make. Therfore I set one foote of the compasse in H, and extend the other foote vnto G, or F. (whiche are the endes of the lynes that lighte in the middle of the side of that cinkeangle) and so make I the circle in the cinkangle, right as the cõclusion meaneth.

[ THE XLI. CONCLVSION]
To make a circle about any assigned cinkeangle of equall sides, and equall corners.

Drawe .ij. lines within the cinkeangle, from .ij. corners to the middle on tbe .ij. contrary sides (as the last conclusion teacheth) and the pointe of their crossyng shall be the centre of the circle that I seke for. Then sette I one foote of the compas in that centre, and the other foote I extend to one of the angles of the cinkangle, and so draw I a circle about the cinkangle assigned.

Example.

A.B.C.D.E, is the cinkangle assigned, about which I would make a circle. Therfore I drawe firste of all two lynes (as you see) one frõ E. to G, and the other frõ C. to F, and because thei do

meete in H, I saye that H. is the centre of the circle that I woulde haue, wherfore I sette one foote of the compasse in H. and extende the other to one corner (whiche happeneth fyrste, for all are like distaunte from H.) and so make I a circle aboute the cinkeangle assigned.

An other waye also.

Another waye maye I do it, thus presupposing any three corners of the cinkangle to be three prickes appointed, vnto whiche I shoulde finde the centre, and then drawinge a circle touchinge them all thre, accordinge to the doctrine of the seuentene, one and twenty, and two and twenty conclusions. And when I haue founde the centre, then doo I drawe the circle as the same conclusions do teache, and this forty conclusion also.

[ THE XLII. CONCLVSION.]
To make a siseangle of equall sides, and equall angles, in any circle assigned.

Yf the centre of the circle be not knowen, then seeke oute the centre according to the doctrine of the sixtenth conclusion. And with your compas take the quantitee of the semidiameter iustly. And then sette one foote in one pricke of the

circũference of the circle, and with the other make a marke in the circumference also towarde both sides. Then sette one foote of the compas stedily in eche of those new prickes, and point out two other prickes. And if you haue done well, you shal perceaue that there will be but euen sixe such diuisions in the circumference. Whereby it dothe well appeare, that the side of anye sisangle made in a circle, is equalle to the semidiameter of the same circle.

Example.

The circle is B.C.D.E.F.G, whose centre I finde to bee A. Therefore I sette one foote of the compas in A, and do extẽd the other foote to B, thereby takinge the semidiameter. Then sette I one foote of the compas vnremoued in B, and marke with the other foote on eche side C. and G. Then from C. I marke D, and frõ D, E: from E. marke I F. And then haue I but one space iuste vnto G. and so haue I made a iuste siseangle of equall sides and equall angles, in a circle appointed.

[ THE XLIII. CONCLVSION.]
To make a siseangle of equall sides, and equall angles about any circle assigned.

[ THE XLIIII. CONCLVSION.]
To make a circle in any siseangle appointed, of equall sides and equal angles.

[ THE XLV. CONCLVSION.]
To make a circle about any sise angle limited of equall sides and equall angles.

Bicause you maye easily coniecture the makinge of these figures by that that is saide before of cinkangles, only consideringe that there is a difference in the numbre of sides, I thought beste to leue these vnto your owne deuice, 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 conclusions. For thoughe it seeme one thing to make a siseangle in a circle, and to make a circle about a siseangle, yet shall you perceaue, that is not one thinge, nother are those twoo conclusions wrought one way. Likewaise shall you thinke of those other two conclusions. To make a siseangle about a circle, and to make a circle in a siseangle, thoughe the figures be one in fashion, when they are made, yet are they not one in working, as you may well perceaue by the xxxvij. xxxviij. xxxix. and xl. conclusions, in whiche the same workes are taught, touching a circle and a cinkangle, yet this muche wyll I saye, for your helpe in working, that when you shall seeke the centre in a siseangle (whether it be to make a circle in it other about it) you shall drawe the two crosselines, from one angle to the other angle that lieth againste it, and not to the middle of any side, as you did in the cinkangle.

[ THE XLVI. CONCLVSION.]
To make a figure of fifteene equall sides and angles in any circle appointed.

This rule is generall, that how many sides the figure shall

haue, that shall be drawen in any circle, into so many partes iustely muste the circles bee deuided. And therefore it is the more easier woorke commonly, to drawe a figure in a circle, then to make a circle in an other figure. Now therefore to end this conclusion, deuide the circle firste into fiue partes, and
then eche of them into three partes againe: Or els
first deuide it into three partes, and then ech
of thẽ into fiue other partes, as you
list, and canne most readilye.
Then draw lines betwene
euery two prickes
that be nighest
togither, and
ther wil appear rightly drawẽ the figure, of fiftene sides, and
angles equall. And so do with any other figure
of what numbre of sides so euer it bee.

FINIS.

---

---

If truthe maie trie it selfe,

By Reasons prudent skyll,

If reason maie preuayle by right,

And rule the rage of will,

I dare the triall byde,

For truthe that I pretende.

And though some lyst at me repine,

Iuste truthe shall me defende.

---

[ THE PREFACE VNTO]
the Theoremes.

Doubt not gentle reader, but as my argument is straunge and vnacquainted with the vulgare toungue, so shall I of many men be straungly talked of, and as straungly iudged. Some men will saye peraduenture, I mighte haue better imployed my tyme in some pleasaunte historye, comprisinge matter of chiualrye. Some other wolde more haue preised my trauaile, if I hadde spente the like time in some morall matter, other in deciding some controuersy of religion. And yet some men (as I iudg) will not mislike this kind of mater, but then will they wishe that I had vsed a more certaine order in placinge bothe the Propositions and Theoremes, and also a more exacter proofe of eche of theim bothe, by demonstrations mathematicall. Some also will mislike my shortenes and simple plainesse, as other of other affections diuersely shall espye somwhat that they shall thinke blame worthy, and shal misse somewhat, that thei wold with to haue bene here vsed, so that euerie manne shall giue his verdicte of me according to his phantasie, vnto whome ioinctly, I make this my firste answere: that as they ar many and in opinions verie diuers, so were it scarse possible to please them all with anie one argumente, of what kinde so euer it were. And for my seconde aunswere, I saye thus. That if annye one argumente mighte please them all, then should thei be thankfull vnto me for this kind of matter. For nother is there anie matter more straunge in the englishe tungue, then this whereof neuer booke was written before now, in that tungue, and therefore oughte to delite all them, that desire to vnderstand strange matters, as most men commonlie doo. And againe the practise is so pleasaunt in vsinge, and so profitable in appliynge, that who so euer dothe

delite in anie of bothe, ought not of right to mislike this arte. And if any manne shall like the arte welle for it selfe, but shall mislyke the fourme that I haue vsed in teachyng of it, to hym I shall saie, Firste, that I dooe wishe with hym that some other man, whiche coulde better haue doone it, hadde shewed his good will, and vsed his diligence in suche sorte, that I myght haue bene therby occasioned iustely to haue left of my laboure, or after my trauaile to haue suppressed my bookes. But sithe no manne hath yet attempted the like, as far as I canne learne, I truste all suche as bee not exercised in the studie of Geometrye, shall finde greate ease and furtheraunce by this simple, plaine, and easie forme of writinge. And shall perceaue the exacte woorkes of Theon, and others that write on Euclide, a great deale the soner, by this blunte delineacion afore hande to them taughte. For I dare presuppose of them, that thing which I haue sette in my selfe, and haue marked in others, that is to saye, that it is not easie for a man that shall trauaile in a straunge arte, to vnderstand at the beginninge bothe the thing that is taught and also the iuste reason whie it is so. And by experience of teachinge I haue tried it to bee true, for whenne I haue taughte the proposition, as it is imported in meaninge, and annexed the demonstration with all, I didde perceaue that it was a greate trouble and a painefull vexacion of mynde to the learner, to comprehend bothe those thinges at ones. And therfore did I proue firste to make them to vnderstande the sence of the propositions, and then afterward did they conceaue the demonstrations muche soner, when they hadde the sentence of the propositions first ingrafted in their mindes. This thinge caused me in bothe these bookes to omitte the demonstrations, and to vse onlye a plaine forme of declaration, which might best serue for the firste introduction. Whiche example hath beene vsed by other learned menne before nowe, for not only Georgius Ioachimus Rheticus, but also Boetius that wittye clarke did set forth some whole books of Euclide, without any demonstration or any

other declaratiõ at al. But & if I shal hereafter perceaue that it maie be a thankefull trauaile to sette foorth the propositions of geometrie with demonstrations, I will not refuse to dooe it, and that with sundry varietees of demonstrations, bothe pleasaunt and profitable also. And then will I in like maner prepare to sette foorth the other bookes, whiche now are lefte vnprinted, by occasion not so muche of the charges in cuttyng of the figures, as for other iuste hynderances, whiche I truste hereafter shall bee remedied. In the meane season if any man muse why I haue sette the Conclusions beefore the Teoremes, seynge many of the Theoremes seeme to include the cause of some of the conclusions, and therfore oughte to haue gone before them, as the cause goeth before the effecte. Here vnto I saie, that although the cause doo go beefore the effect in order of nature, yet in order of teachyng the effect must be fyrst declared, and than the cause therof shewed, for so that men best vnderstãd things First to lerne that such thinges ar to be wrought, and secondarily what thei ar, and what thei do import, and thã thirdly what is the cause therof. An other cause why y^t the theoremes be put after the cõclusions is this, whã I wrote these first cõnclusions (which was .iiiij. yeres passed) I thought not then to haue added any theoremes, but next vnto y^e cõclusiõs to haue taught the order how to haue applied thẽ to work, for drawing of plottes & such like vses. But afterward cõsidering the great cõmoditie y^t thei serue for, and the light that thei do geue to all sortes of practise geometricall, besyde other more notable benefites, whiche shall be declared more specially in a place conuenient, I thoughte beste to geue you some taste of theym, and the pleasaunt contemplation of suche geometrical propositions, which might serue diuerselye in other bookes for the demonstrations and proofes of all Geometricall woorkes. And in theim, as well as in the propositions, I haue drawen in the Linearie examples many tymes more lynes, than be spoken of in the explication of them, whiche is doone to this intent, that yf any manne lyst to learne the demonstrations by harte, (as somme

learned men haue iudged beste to doo) those same men should find the Linearye exaumples to serue for this purpose, and to wante no thyng needefull to the iuste proofe, whereby this booke may bee wel approued to be more complete then many men wolde suppose it.

And thus for this tyme I wyll make an ende without any larger declaration of the commoditiees of this arte, or any farther answeryng to that may bee obiected agaynst my handelyng of it, wyllyng them that myslike it, not to medle with it: and vnto those that will not disdaine the studie of it, I promise all suche aide as I shall be able to shewe for their farther procedyng both in the same, and in all other commoditees that thereof maie ensue. And for their incouragement I haue here annexed the names and brefe argumentes of suche bookes, as I intende (God willyng) shortly to sette forth, if I shall perceaue that my paynes maie profyte other, as my desyre is.

The brefe argumentes of suche bokes as ar appoynted shortly to be set forth by the author herof.

The seconde part of Arithmetike, teachyng the workyng by fractions, with extraction of rootes both square and cubike: And declaryng the rule of allegation, with sundrye plesaunt exaumples in metalles and other thynges. Also the rule of false position, with dyuers examples not onely vulgar, but some appertaynyng to the rule of Algeber, applied vnto quantitees partly rationall, and partly surde.

The arte of Measuryng by the quadrate geometricall, and the disorders committed by vsyng the same, not only reueled but reformed also (as muche as to the instrument pertayneth) by the deuise of a new quadrate newely inuented by the author hereof.

The arte of measuryng by the astronomers staffe, and by the astronomers ryng, and the form of makyng them both.

The arte of makyng of Dials, bothe for the daie and the nyght, with certayn new formes of fixed dialles for the moon

and other for the sterres, whiche may bee sette in glasse windowes to serue by daie and by night. And howe you may by those dialles knowe in what degree of the Zodiake not only the sonne, but also the moone is. And how many howrs old she is. And also by the same dial to know whether any eclipse shall be that moneth, of the sonne or of the moone.

The makyng and vse of an instrument, wherby you maye not onely measure the distance at ones of all places that you can see togyther, howe muche eche one is from you, and euery one from other, but also therby to drawe the plotte of any countreie that you shall come in, as iustely as maie be, by mannes diligence and labour.

The vse bothe of the Globe and the Sphere, and therin also of the arte of Nauigation, and what instrumentes serue beste thervnto, and of the trew latitude and longitude of regions and townes.

Euclides woorkes in foore partes, with diuers demonstrations Arithmeticall and Geometricall or Linearie. The fyrst parte of platte formes. The second of numbres and quantitees surde or irrationall. The third of bodies and solide formes. The fourthe of perspectiue, and other thynges thereto annexed.

BESIDE these I haue other sundrye woorkes partely ended, and partely to bee ended, Of the peregrination of man, and the originall of al nations, The state of tymes, and mutations of realmes, The image of a perfect common welth, with diuers other woorkes in naturall sciences, Of the wonderfull workes and effectes in beastes, plantes, and minerals, of whiche at this tyme, I will omitte the argumentes, beecause thei doo appertaine littel to this arte, and handle other matters in an other sorte.

To haue, or leaue,

Nowe maie you chuse,

No paine to please,

Will I refuse.

---