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Containing the First Principles of Geometrie

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

Author: Robert Record

Release date: July 5, 2010 [eBook #33093]
Most recently updated: January 6, 2021

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK THE PATH-WAY
TO KNOWLEDG, CONTAINING THE FIRST PRINCIPLES OF
GEOMETRIE ***

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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

T O T H E G E N T L E R E A D E R.
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 secõnde
thing 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 ye 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 ye 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
wtout 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, yt 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.

T H E P R E F A C E,
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

G E O M E T R Y.
E O M E T RY T E A C H E T H T H E
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
A lyne. other prickes betweene euerye two of these, 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 A Straight lyne, is the shortest that maye be drawenne between
yne.
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
an Angle. lyne, but rather twoo lynes: 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
A righte square angle, and a blunte angle. The square angle, whiche is
angle.
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.
A sharpe A sharpe angle is so called, because it Right angles.
corner.
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 A blunt or brode corner, is greater then is a square angle, and
angle.
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
A platte all yet be called by one propre name, Platte formes, and thei
forme.
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 A plaine platte is that, whiche is made al equall in height, so
platte.
that the middle partes nother bulke vp, nother shrink down more
then the bothe endes.
A crooked For whan the one parte is higher then the other, then is it named
platte.
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
A bodie. platte forme, so of manie plattes is made a bodie, whiche
Depenesse. conteigneth Lengthe, bredth, and 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.
Cubike. Now all bodies haue platte formes for their boundes, so in a dye
Asheler. (whiche is called a cubike bodie) by geomatricians, 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.
But because you shall not A dye or ashler. A globe.
muse what I dooe call

A bounde. a bound, I mean therby a
generall name, betokening
the beginning, end and
side, of any forme.
Forme, A forme, figure, or shape,
Fygure.
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 ye 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
A centre can be made by cõpasse without it) then is it called 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
A ground line. flatte (whiche is named the ground line)
and an other commeth downe on it, and is
A perpen­‐ called a perpendiculer or plũme lyne, as in this example you
dicular.
may see. where .A.B. is the grounde line, and C.D. the plumbe
A plume lyne.
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.
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

Parallelys other, for and if they be nerer at one ende
Gemowe then at the other, then are they no paralleles,
ynes.
but maie bee called bought lynes, and loe
here exaumples of them bothe.
I haue added also paralleles tortuouse,
whiche bowe cõtrarie waies with their two
endes: and paralleles circular, whiche be lyke
vnperfecte compasses: for if
Concen­trikes they bee whole circles, then parallelis. bought lines
are they called cõcentrikes,
that is to saie, circles drawẽ
on one centre.
parallelis:
Here might I note the error of
circular.
good Albert Durer, which
affirmeth that no
Concen-
perpendicular lines can be
trikes.
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.
A twine line. And to returne to my matter. an other fashioned line is there,
which is named a twine or twist line, and it goeth as a wreyth
A spirall line. about some other bodie. And an other sorte of lines is there, that
A worme line. is called a spirall 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 spirail lyne

A
twiste
lyne.

A tuch line. 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.
A corde,
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.
Matche
corners 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.
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.
Circum­‐
ference. And the line that encloseth the whole compasse, is called the
circumference.
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, is
called the semidiameter, or halfe diameter.
A 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,

Semi­‐ and as this exaumple sheweth: where A. is the corde. And the
diameter.
compassed line that aunswereth to it, is called
A cord, or a
string­lyne.
an arche lyne, or a bowe lyne, whiche here
An archline marked with B. and the diameter with C.
A bowline.
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, the one the
A cantle greatter cantle as E. and the other the lesser cantle, as D. And if
A semye­circle it be parted iuste by the centre (as you see in F.) 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
A nooke the circumference (as G. is) and then maie it
cantle
be called a nooke cantle, and if it be not parted
A nooke. from the reste of the circle (as you see in H.)
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
An egge Geometricians dooe call an egge forme, because it doeth
fourme.
represent the figure and shape of an egge duely proportioned
(as this figure sheweth) hauyng the one ende greate then the
other.
A tunne or
barrel form

A tunne forme. For if it be lyke the figure of An egge forme
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
An axtre or crosse these figures, and that is called an axelyne, or axtre.
axe lyne.
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, wch 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
An yey tunne forme is made of one lyne, and that figure is named an
fourme
yey fourme.
A triangle
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 wt 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, and Latine
men æequilaterum: and in english it may be
ἰσόπλευρομ. called a threlike triangle, other els two sydes bee equall and the
ισόσκελεσ. thyrd vnequall, which the Greekes call Isosceles, 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 yt 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
σκαλενὄμ. blunte angle, and O, all sharpe angles 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.

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

Quadrãgle say, a square quadrate, whose sides bee all
A square equall, and al the angles square, as you se here
quadrate.
in this figure Q. The second kind is called a
A longe long square, whose foure corners be all square,
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.
A losenge
The thyrd kind is called losenges or diamondes,
A diamõd. 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,
A losenge and therfore ar they named losengelike or
yke.
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
A squyre. this tyme I wyll make an ende, 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.
And thus I make an eand to speake of The globe as is before.
platte formes, and will briefelye saye
somwhat touching the figures of bodeis
which partly haue one platte forme for
their bound, and yt 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
A rounde expresse: or els it is called a rounde spire, or stiple fourme, as
spier.
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.

T H E F Y R S T C O N C LV S I O N .
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.

T H E .II. C O N C L V S I O N
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.

T H E III. C O N C L .
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 yt 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 ye middle of ye arch, and so ye angle is
diuided into ij. equall partes.
Example.
Let the triãgle be A.B.C, thẽ set I one foot of ye cõpasse in B, and with the
other I draw ye 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.

T H E IIII. C O N C L .
To deuide any measurable line into ij. equall partes.
Open your compasse to the iust lẽgth of ye line. And thẽ set one foote
steddely at the one ende of the line, & wt the other fote draw an arch of a
circle against ye 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.

T H E F I F T C O N C LV S I O N .
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 ye 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.

T H E .VI. C O N C L V S I O N .
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 yt 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.

T H E .VII. C O N C L V S I O N .
To make a plumbe lyne on any porcion of a circle, and that on the
vtter or inner bughte.
Mark first the prick where ye 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.

T H E VIII. C O N C L V S Y O N .
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.

T H E IX. C O N C L V S I O N .
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.

T H E X. C O N C L V S I O N .
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 ye
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.

T H E XI. C O N C L V S I O N .
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.

T H E .XII. C O N C L V S I O N .

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.

T H E XIII. C O N C L V S I O N .
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.

T H E XIIII. C O N C L V S I O N .
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.

T H E .XV. C O N C L V S I O N .
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 ye 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 ye .xxxi.
theoreme.

T H E .XVI. C O N C L V S I O N .
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 ye 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.

T H E XVII. C O N C L V S I O N .
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 yt 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 ye other did before, as in
the exãple of F.G. But and if ye 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 ye 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.

T H E XVIII. C O N C L V S I O N .
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, (wch was
assigned vnto me) is so diuided in this point F, yt ye square of ye hole line
A.B, & of the one portiõ (yt 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.

T H E .XIX. C O N C L V S I O N .
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.

T H E .XX. C O N C L V S I O N .
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.

T H E .XXI. C O N C L V S I O N .
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 ye 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.

T H E .XXII. C O N C L V S I O N .
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.

T H E XXIII. C O N C L V S I O N .
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 ye 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 wch shal enclose those iij.
pricks in the edg of his circũferẽce & thus haue you attained ye vse of this
cõclusiõ.

T H E XXIIII. C O N C L V S I O N .
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 ye prick assigned A, opẽ
your cõpas now of such widenes, yt the one foote may be set in E, wch is ye

cẽtre of ye circle, & ye other in A, wch is ye 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 ye inner circle (wch 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.

T H E XXV. C O N C L V S I O N .
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 yt 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.

T H E XXVI. C O N C L V S I O N .
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.

T H E XXVII. C O N C L V S I O N .
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 C. a quadrate
in a triangle. in a circle.

B. a triangle D. a circle in
in a circle. 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.

T H E XXVIII. C O N C L V S I O N .
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.

T H E XXIX. C O N C L V S I O N .
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 wt 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 yt 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.

T H E XXX. C O N C L V S I O N .
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.

T H E XXXI. C O N C L V S I O N .
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 ye ãgle assigned as youre selfe mai easily gesse.
Example.
First for example of a sharpe ãgle let A.
stãd & B.C shal be ye 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 yt 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.

T H E XXXII. C O N C L V S I O N .
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.

T H E XXXIII. C O N C L V S I O N .
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.

T H E XXXIIII. C O N C L V S I O N .
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.

T H E XXXV. C O N C L V S I O N .
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.

T H E XXXVI. C O N C L V S I O N .
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.

T H E XXXVII. C O N C L V S I O N .

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.

T H E XXXVIII. C O N C L V S I O N .
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.

T H E XXXIX. C O N C L V S I O N .
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.

T H E XL. C O N C L V S I O N .
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.

T H E XLI. C O N C L V S I O N
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.

T H E XLII. C O N C L V S I O N .
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.

T H E XLIII. C O N C L V S I O N .
To make a siseangle of equall sides, and equall angles about any
circle assigned.

T H E XLIIII. C O N C L V S I O N .
To make a circle in any siseangle appointed, of equall sides and
equal angles.

T H E XLV. C O N C L V S I O N .
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.

T H E XLVI. C O N C L V S I O N .
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.
F I N I S.

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.

T H E P R E FA C E V N T O
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 yt 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 ye 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 yt 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.

T H E 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.
T H E 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.
T H E arte of measuryng by the astronomers staffe, and by the astronomers
ryng, and the form of makyng them both.
T H E 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.
T H E 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.
B E S I D E 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.

I.

That frõ any pricke to one other, there may be
drawen a right line.
s for example A. being the one pricke, and B.
the other, you maye drawe betwene them from the one to the
other, that is to say, frome A. vnto B, and from B. to A.
II.

That any right line of measurable length may be
drawen forth longer, and straight.
Example of A.B, which as it is a line of measurable
lengthe, so may it be drawen forth farther, as for
example vnto C, and that in true streightenes without
crokinge.

III.

That vpon any centre, there may be made a circle
of anye quãtitee that a man wyll.
Let the centre be set to be A, what shal hinder a man to
drawe a circle aboute it, of what quantitee that he
lusteth, as you se the forme here: other bygger or lesse,
as it shall lyke him to doo:

That all right angles be equall eche to other.
Set for an example A. and B, of which two though A. seme
the greatter angle to some men of small experience, it
happeneth only bicause that the lines aboute A, are longer
thẽ the lines about B, as you may proue by drawing them
longer, for so that B. seme the greater angle yf you make his
lines longer then the lines that make the angle A. And to
proue it by demonstration, I say thus. If any ij. right corners be not equal,
then one right corner is greater then an other, but that corner which is
greatter then a right angle, is a blunt corner (by his definition) so must one
corner be both a right corner and a blunt corner also, which is not
possible: And againe: the lesser right corner must be a sharpe corner, by
his definition, bicause it is lesse then a right angle. which thing is
impossible. Therefore I conclude that all right angles be equall.

Yf one right line do crosse two other right lines, and make ij. inner
corners of one side lesser thẽ ij. righte corners, it is certaine, that
if those two lines be drawen forth right on that side that the sharpe
inner corners be, they wil at lẽgth mete togither, and crosse on an
other.
The ij. lines beinge as A.B. and C.D, and the third line crossing them as
dooth heere E.F, making ij inner cornes (as ar G.H.) lesser then two right
corners, sith ech of them is lesse then a right corner, as your eyes maye

iudge, then say I, if those ij. lines A.B. and C.D.
be drawen in lengthe on that side that G. and H.
are, the will at length meet and crosse one an
other.

Two right lines make no platte forme.
A platte forme, as you harde before, hath bothe length
and bredthe, and is inclosed with lines as with his
boundes, but ij. right lines cannot inclose al the bondes
of any platte forme. Take for an example firste these two
right lines A.B. and A.C. whiche meete togither in A, but
yet cannot be called a platte forme, bicause there is no
bond from B. to C, but if you will drawe a line betwene
them twoo, that is frome B. to C, then will it be a platte
forme, that is to say, a triangle, but then are there iij.
lines, and not only ij. Likewise may you say of D.E. and
F.G, whiche doo make a platte forme, nother yet can they
make any without helpe of two lines more, whereof the
one must be drawen from D. to F, and the other frome E.
to G, and then will it be a longe rquare. So then of two right lines can bee
made no platte forme. But of ij. croked lines be made a platte forme, as you
se in the eye form. And also of one right line, & one croked line, maye a
platte fourme bee made, as the semicircle F. doothe sette forth.

Certayn common sentences manifest to

sence, and acknowledged of all men.

The firste common sentence.

What so euer things be equal to one other thinge, those same bee
equall betwene them selues.
Examples therof you may take both in
greatnes and also in numbre. First (though it
pertaine not proprely to geometry, but to
helpe the vnderstandinge of the rules, whiche
may bee wrought by bothe artes) thus may
you perceaue. If the summe of monnye in my
purse, and the mony in your purse be equall
eche of them to the mony that any other man
hathe, then must needes your mony and mine
be equall togyther. Likewise, if anye ij. quantities, as A. and B, be equal to
an other, as vnto C, then muste nedes A. and B. be equall eche to other, as
A. equall to B, and B. equall to A, whiche thinge the better to perceaue,
tourne these quantities into numbre, so shall A. and B. make sixteene, and
C. as many. As you may perceaue by multipliyng the numbre of their sides
togither.

The seconde common sentence.
And if you adde equall portions to thinges that be equall, what so
amounteth of them shall be equall.
Example, Yf you and I haue like summes of mony, and then receaue eche of
vs like summes more, then our summes wil be like styll. Also if A. and B.
(as in the former example) bee equall, then by adding an equal portion to
them both, as to ech of them, the quarter of A. (that is foure) they will be
equall still.

The thirde common sentence.
And if you abate euen portions from things that are equal, those
partes that remain shall be equall also.
This you may perceaue by the last example. For that that was added there,
is subtracted heere. and so the one doothe approue the other.

The fourth common sentence.
If you abate equalle partes from vnequal thinges, the remainers
shall be vnequall.
As bicause that a hundreth and eight and forty be vnequal if I take tenne
from them both, there will remaine nynetye and eight and thirty, which are
also vnequall. and likewise in quantities it is to be iudged.

The fifte common sentence.
When euen portions are added to vnequalle thinges, those that
amounte shalbe vnequall.
So if you adde twenty to fifty, and lyke ways to nynty, you shall make
seuenty and a hundred and ten whiche are no lesse vnequall, than were fifty
and nynty.

The syxt common sentence.
If two thinges be double to any other, those same two thinges are
equal togither.
Bicause A. and B. are eche of them double to C, therefore must A. and B.
nedes be equall togither. For as v. times viij. maketh xl. which is double to
iiij. times v, that is xx so iiij. times x, likewise is double to xx. (for it maketh
fortie) and therefore muste neades be equall to forty.

The seuenth common sentence.
If any two thinges be the halfes of one other thing, then are thei .ij.
equall togither.
So are D. and C. in the laste example equal togyther, bicause they are eche
of them the halfe of A. other of B, as their numbre declareth.

The eyght common sentence.
If any one quantitee be laide on an other, and thei agree, so that
the one excedeth not the other, then are they equall togither.
As if this figure A.B.C, be layed on that other D.E.F,
so that A. be layed to D, B. to E, and C. to F, you
shall see them agre in sides exactlye and the one not
to excede the other, for the line A.B. is equall to D.E,
and the third lyne C.A, is equall to F.D so that
eueryside in the one is equall to some one side of the
other. Wherfore it is playne, that the two triangles are
equall togither.

The nynth common sentence.
Euery whole thing is greater than any of his partes.

This sentence nedeth none example. For the thyng is more playner then any
declaration, yet considering that other common sentence that foloweth
nexte that.

The tenthe common sentence.
Euery whole thinge is equall to all his partes taken togither.
It shall be mete to expresse both wt one example, for of
thys last sentence many mẽ at the first hearing do make
a doubt. Therfore as in this example of the circle deuided
into sũdry partes it doeth appere that no parte can be so
great as the whole circle, (accordyng to the meanyng of
the eight sentence) so yet it is certain, that all those eight
partes together be equall vnto the whole circle. And this
is the meanyng of that common sentence (whiche many vse, and fewe do
rightly vnderstand) that is, that All the partes of any thing are nothing els,
but the whole. And contrary waies: The whole is nothing els, but all his
partes taken togither. whiche saiynges some haue vnderstand to meane
thus: that all the partes are of the same kind that the whole thyng is: but
that that meanyng is false, it doth plainly appere by this figure
A.B, whose partes A. and B, are triangles, and the whole
figure is a square, and so are they not of one kind. But and if
they applie it to the matter or substance of thinges (as some do)
then it is most false, for euery compound thyng is made of
partes of diuerse matter and substance. Take for example a man, a house,
a boke, and all other compound thinges. Some vnderstand it thus, that the
partes all together can make none other forme, but that that the whole doth
shewe, whiche is also false, for I maie make fiue hundred diuerse figures of
the partes of some one figure, as you shall better perceiue in the third boke.
And in the meane seasõ take for an exãple this square figure following
A.B.C.D, wch is deuided but in two parts, and yet (as you se) I haue made
fiue figures more beside the firste, with onely diuerse ioynyng of those two
partes. But of this shall I speake more largely in an other place. In the
mean season content your self with these principles, whiche are certain of
the chiefe groundes wheron all demonstrations mathematical are fourmed,

of which though the moste parte seeme so
plaine, that no childe doth doubte of them,
thinke not therfore that the art vnto whiche
they serue, is simple, other childishe, but
rather consider, howe certayne the profes of
that arte is, yt hath for his groũdes soche
playne truthes, & as I may say, suche
vndowbtfull and sensible principles, And
this is the cause why all learned menne
dooth approue the certenty of geometry, and
cõsequently of the other artes mathematical,
which haue the grounds (as Arithmeticke,
musike and astronomy) aboue all other artes
and sciences, that be vsed amõgest men.
Thus muche haue I sayd of the first
principles, and now will I go on with the theoremes, whiche I do only by
examples declare, minding to reserue the proofes to a peculiar boke which I
will then set forth, when I perceaue this to be thankfully taken of the
readers of it.

The theoremes of Geometry brieflye

declared by shorte examples.

The firste Theoreme.

When .ij. triangles be so drawen, that the one of thẽ hath ij. sides
equal to ij sides of the other triangle, and that the angles enclosed
with those sides, bee equal also in bothe triangles, then is the
thirde side likewise equall in them. And the whole triangles be of
one greatnes, and euery angle in the one equall to his matche
angle in the other, I meane those angles that be inclosed with like
sides.
Example.
This triangle A.B.C. hath ij. sides (that is to say)
C.A. and C.B, equal to ij. sides of the other
triangle F.G.H, for A.C. is equall to F.G, and B.C.
is equall to G.H. And also the angle C. contayned
beetweene F.G, and G.H, for both of them answere
to the eight parte of a circle. Therfore doth it
remayne that A.B. whiche is the thirde lyne in the
firste triangle, doth agre in lengthe with F.H, wch is
the third line in ye secõd triãgle & ye hole triãgle.
A.B.C. must nedes be equal to ye hole triangle
F.G.H. And euery corner equall to his match, that is to say, A. equall to F,
B. to H, and C. to G, for those bee called match corners, which are
inclosed with like sides, other els do lye against like sides.

The second Theoreme.
In twileke triangles the ij. corners that be about the groũd line, are
equal togither. And if the sides that be equal, be drawẽ out in lẽgth

thẽ wil the corners that are vnder the ground line, be equal also
togither.
Example
A.B.C. is a twileke triangle, for the one side A.C, is equal
to the other side B.C. And therfore I saye that the inner
corners A. and B, which are about the ground lines, (that
is A.B.) be equall togither. And farther if C.A. and C.B.
bee drawen forthe vnto D. and E. as you se that I haue
drawen them, then saye I that the two vtter angles vnder
A. and B, are equal also togither: as the theorem said.
The profe wherof, as of al the rest, shal apeare in Euclide,
whome I intende to set foorth in english with sondry new additions, if I may
perceaue that it wilbe thankfully taken.

The thirde Theoreme.
If in annye triangle there bee twoo angles equall togither, then
shall the sides, that lie against those angles, be equal also.
Example.
This triangle A.B.C. hath two corners equal eche to other,
that is A. and B, as I do by supposition limite, wherfore it
foloweth that the side A.C, is equal to that other side B.C,
for the side A.C, lieth againste the angle B, and the side
B.C, lieth against the angle A.

The fourth Theoreme.
When two lines are drawen frõ the endes of anie one line, and meet
in anie pointe, it is not possible to draw two other lines of like
lengthe ech to his match that shal begĩ at the same pointes, and
end in anie other pointe then the twoo first did.

Example.
The first line is A.B, on which I haue erected two other
lines A.C, and B.C, that meete in the pricke C,
wherefore I say, it is not possible to draw ij. other lines
from A. and B. which shal mete in one point (as you se
A.D. and B.D. mete in D.) but that the match lines
shalbe vnequal, I mean by match lines, the two lines on
one side, that is the ij. on the right hand, or the ij. on
the lefte hand, for as you se in this example A.D. is
longer thẽ A.C, and B.C. is longer then B.D. And it is
not possible, that A.C. and A.D. shall bee of one lengthe, if B.D. and B.C.
bee like longe. For if one couple of matche lines be equall (as the same
example A.E. is equall to A.C. in length) then must B.E. needes be vnequall
to B.C. as you see, it is here shorter.

The fifte Theoreme.
If two triãgles haue there ij. sides equal one to an other, and their
groũd lines equal also, then shall their corners, whiche are
contained betwene like sides, be equall one to the other.
Example.
Because these two triangles A.B.C, and D.E.F. haue
two sides equall one to an other. For A.C. is equall
to D.F, and B.C. is equall to E.F, and again their
groũd lines A.B. and D.E. are lyke in length,
therfore is eche angle of the one triangle equall to
ech angle of the other, comparyng together those
angles that are contained within lyke sides, so is A.
equall to D, B. to E, and C. to F, for they are contayned within like sides, as
before is said.

The sixt Theoreme.

When any right line standeth on an other, the ij. angles that thei
make, other are both right angles, or els equall to .ij. righte angles.
Example.
A.B. is a right line, and on it there doth light
another right line, drawen from C. perpendicularly
on it, therefore saie I, that the .ij. angles that thei
do make, are .ij. right angles as maie be iudged by
the definition of a right angle. But in the second
part of the example, where A.B. beyng still the
right line, on which D. standeth in slope wayes, the
two angles that be made of them are not righte
angles, but yet they are equall to two righte angles, for so muche as the one
is to greate, more then a righte angle, so muche iuste is the other to little,
so that bothe togither are equall to two right angles, as you maye perceiue.

The seuenth Theoreme.
If .ij. lines be drawen to any one pricke in an other lyne, and those
.ij. lines do make with the fyrst lyne, two right angles, other suche
as be equall to two right angles, and that towarde one hande, than
those two lines doo make one streyght lyne.
Example.
A.B. is a streyght lyne, on which there doth lyght
two other lines one frome D, and the other frome C,
but considerynge that they meete in one pricke E,
and that the angles on one hand be equal to two
right corners (as the laste theoreme dothe declare)
therfore maye D.E. and E.C. be counted for one
ryght lyne.

The eight Theoreme.

When two lines do cut one an other crosseways they do make their
matche angles equall.
Example.
What matche angles are, I haue tolde you in the definitions
of the termes. And here A, and B. are matche corners in
this example, as are also C. and D, so that the corner A, is
equall to B, and the angle C, is equall to D.

The nynth Theoreme.
Whan so euer in any triangle the line of one side is drawen forthe
in lengthe, that vtter angle is greater than any of the two inner
corners, that ioyne not with it.
Example.
The triangle A.D.C hathe hys grounde lyne A.C.
drawen forthe in lengthe vnto B, so that the vtter
corner that it maketh at C, is greater then any of
the two inner corners that lye againste it, and
ioyne not wyth it, whyche are A. and D, for they
both are lesser then a ryght angle, and be sharpe
angles, but C. is a blonte angle, and therfore
greater then a ryght angle.

The tenth Theoreme.
In euery triangle any .ij. corners, how so euer you take thẽ, ar lesse
thẽ ij. right corners.
Example.
In the firste triangle E, whiche is a threlyke, and therfore hath all his angles
sharpe, take anie twoo corners that you will, and you shall perceiue that

they be lesser then ij. right corners, for in euery triangle
that hath all sharpe corners (as you see it to be in this
example) euery corner is lesse then a right corner. And
therfore also euery two corners must nedes be lesse then
two right corners. Furthermore in that other triangle
marked with M, whiche hath .ij. sharpe corners and one
right, any .ij. of them also are lesse then two right angles.
For though you take the right corner for one, yet the other
whiche is a sharpe corner, is lesse then a right corner. And
so it is true in all kindes of triangles, as you maie perceiue
more plainly by the .xxij. Theoreme.

The .xi. Theoreme.
In euery triangle, the greattest side lieth against the greattest
angle.
Example.
As in this triangle A.B.C, the greattest angle is C.
And A.B. (whiche is the side that lieth against it) is
the greatest and longest side. And contrary waies, as
A.C. is the shortest side, so B. (whiche is the angle
liyng against it) is the smallest and sharpest angle,
for this doth folow also, that is the longest side lyeth
against the greatest angle, so it that foloweth

The twelft Theoreme.
In euery triangle the greattest angle lieth against the longest side.
For these ij. theoremes are one in truthe.

The thirtenth theoreme.

In euerie triangle anie ij. sides togither how so euer you take them,
are longer thẽ the thirde.
For example you shal take this triangle A.B.C. which
hath a very blunt corner, and therfore one of his sides
greater a good deale then any of the other, and yet the
ij. lesser sides togither ar greater then it. * And if it bee
so in a blunte angeled triangle, it must nedes be true in
all other, for there is no other kinde of triangles that
hathe the one side so greate aboue the other sids, as
thei yt haue blunt corners.

The fourtenth theoreme.
If there be drawen from the endes of anie side of a triangle .ij. lines
metinge within the triangle, those two lines shall be lesse then the
other twoo sides of the triangle, but yet the corner that thei make,
shall bee greater then that corner of the triangle, whiche standeth
ouer it.
Example.
A.B.C. is a triangle. on whose ground line A.B.
there is drawen ij. lines, from the ij. endes of it,
I say from A. and B, and they meete within the
triangle in the pointe D, wherfore I say, that as
those two lynes A.D. and B.D, are lesser then A.C.
and B.C, so the angle D, is greatter then the angle
C, which is the angle against it.

The fiftenth Theoreme.
If a triangle haue two sides equall to the two sides of an other
triangle, but yet the ãgle that is contained betwene those sides,

greater then the like angle in the other triangle, then is his grounde
line greater then the grounde line of the other triangle.
Example.
A.B.C. is a triangle, whose sides A.C. and B.C, are equall to E.D. and D.F,
the two sides of the triangle D.E.F, but bicause the angle in D, is greatter
then the angle C. (whiche are the ij. angles contayned betwene the equal
lynes) therfore muste the ground line E.F. nedes bee greatter thenne the
grounde line A.B, as you se plainely.

The xvi. Theoreme.
If a triangle haue twoo sides equalle to the two sides of an other
triangle, but yet hathe a longer ground line thẽ that other triangle,
then is his angle that lieth betwene the equall sides, greater thẽ the
like corner in the other triangle.
Example.
This Theoreme is nothing els, but the sentence of the last Theoreme turned
backward, and therfore nedeth none other profe nother declaration, then
the other example.

The seuententh Theoreme.

If two triangles be such sort, that two angles of the one be equal to
ij. angles of the other, and that one side of the one be equal to on
side of the other, whether that side do adioyne to one of the equall
corners, or els lye againste one of them, then shall the other twoo
sides of those triangles bee equalle togither, and the thirde corner
also shall be equall in those two triangles.
Example.
Bicause that A.B.C, the one triangle hath two
corners A. and B, equal to D.E, that are twoo
corners of the other triangle. D.E.F. and that they
haue one side in theym bothe equall, that is A.B,
which is equall to D.E, therefore shall both the other
ij. sides be equall one to an other, as A.C. and B.C.
equall to D.F. and E.F, and also the thirde angle in them both shal be
equall, that is, the angle C. shal be equall to the angle F.

The eightenth Theoreme.
When on ij. right lines ther is drawen a third right line crosse
waies, and maketh .ij. matche corners of the one line equall to the
like twoo matche corners of the other line, then ar those two lines
gemmow lines, or paralleles.
Example.
The .ij. fyrst lynes are A.B. and C.D, the thyrd lyne
that crosseth them is E.F. And bycause that E.F.
maketh ij. matche angles with A.B, equall to .ij.
other lyke matche angles on C.D, (that is to say
E.G, equall to K.F, and M.N. equall also to H.L.)
therfore are those ij. lynes A.B. and C.D. gemow
lynes, vnderstand here by lyke matche corners,
those that go one way as doth E.G, and K.F, lyke ways N.M, and H.L, for

as E.G. and H.L, other N.M. and K.F. go not one waie, so be not they lyke
match corners.

The nyntenth Theoreme.
When on two right lines there is drawen a thirde right line
crossewaies, and maketh the ij. ouer corners towarde one hande
equall togither, then ar those .ij. lines paralleles. And in like maner
if two inner corners toward one hande, be equall to .ii. right
angles.
Example.
As the Theoreme dothe speake of .ij. ouer angles, so muste you vnderstande
also of .ij. nether angles, for the iudgement is lyke in bothe. Take for
example the figure of the last theoreme, where A.B, and C.D, be called
paralleles also, bicause E. and K, (whiche are .ij. ouer corners) are equall,
and lykewaies L. and M. And so are in lyke maner the nether corners N.
and H, and G. and F. Nowe to the seconde parte of the theoreme, those .ij.
lynes A.B. and C.D, shall be called paralleles, because the ij. inner
corners. As for example those two that bee toward the right hande (that is
G. and L.) are equall (by the fyrst parte of this nyntenth theoreme) therfore
muste G. and L. be equall to two ryght angles.

The xx. Theoreme.
When a right line is drawen crosse ouer .ij. right gemow lines, it
maketh .ij. matche corners of the one line, equall to two matche
corners of the other line, and also bothe ouer corners of one hande
equall togither, and bothe nether corners like waies, and more ouer
two inner corners, and two vtter corners also towarde one hande,
equall to two right angles.
Example.

Bycause A.B. and C.D, (in the laste figure) are paralleles, therefore the two
matche corners of the one lyne, as E.G. be equall vnto the .ij. matche
corners of the other line, that is K.F, and lykewaies M.N, equall to H.L.
And also E. and K. bothe ouer corners of the lefte hande equall togyther,
and so are M. and L, the two ouer corners on the ryghte hande, in lyke
maner N. and H, the two nether corners on the lefte hande, equall eche to
other, and G. and F. the two nether angles on the right hande equall
togither.
¶ Farthermore yet G. and L. the .ij. inner angles on the right hande bee
equall to two right angles, and so are M. and F. the .ij. vtter angles on the
same hande, in lyke manner shall you say of N. and K. the two inner
corners on the left hand. and of E. and H. the two vtter corners on the same
hande. And thus you see the agreable sentence of these .iii. theoremes to
tende to this purpose, to declare by the angles how to iudge paralleles, and
contrary waies howe you may by paralleles iudge the proportion of the
angles.

The xxi. Theoreme.
What so euer lines be paralleles to any other line, those same be
paralleles togither.
Example.
A.B. is a gemow line, or a parallele vnto C.D. And E.F,
lykewaies is a parallele vnto C.D. Wherfore it foloweth,
that A.B. must nedes bee a parallele vnto E.F.

The .xxij. theoreme.
In euery triangle, when any side is drawen forth in length, the vtter
angle is equall to the ij. inner angles that lie againste it. And all iij.
inner angles of any triangle are equall to ij. right angles.
Example.

The triangle beeyng A.D.E. and the syde A.E.
drawen foorthe vnto B, there is made an vtter
corner, whiche is C, and this vtter corner C, is
equall to bother the inner corners that lye agaynst
it, whyche are A. and D. And all thre inner
corners, that is to say, A.D. and E, are equall to
two ryght corners, whereof it foloweth, that all the
three corners of any one triangle are equall to all
the three corners of euerye other triangle. For what so euer thynges are
equalle to anny one thyrde thynge, those same are equalle togitther, by the
fyrste common sentence, so that bycause all the .iij. angles of euery triangle
are equall to two ryghte angles, and all ryghte angles bee equall togyther
(by the fourth request) therfore must it nedes folow, that all the thre corners
of euery triangle (accomptyng them togyther) are equall to iij. corners of
any triangle, taken all togyther.

The .xxiii. theoreme.
When any ij. right lines doth touche and couple .ij. other righte
lines, whiche are equall in length and paralleles, and if those .ij.
lines bee drawen towarde one hande, then are thei also equall
together, and paralleles.
Example.
A.B. and C.D. are ij. ryght lynes and paralleles and
equall in length, and they ar touched and ioyned
togither by ij. other lynes A.C. and B.D, this beyng
so, and A.C. and B.D. beyng drawen towarde one
syde (that is to saye, bothe towarde the lefte hande)
therefore are A.C. and B.D. bothe equall and also
paralleles.

The .xxiiij. theoreme.

In any likeiamme the two contrary sides ar equall togither, and so
are eche .ij. contrary angles, and the bias line that is drawen in it,
dothe diuide it into two equall portions.
Example.
Here ar two likeiammes ioyned togither, the one is
a longe square A.B.E, and the other is a
losengelike D.C.E.F. which ij. likeiammes ar
proued equall togither, bycause they haue one
ground line, that is, F.E, And are made betwene
one payre of gemow lines, I meane A.D. and E.H.
By this Theoreme may you know the arte of the
righte measuringe of likeiammes, as in my booke of measuring I wil more
plainly declare.

There is no xxv. (25th) theorem.

The xxvi. Theoreme.
All likeiammes that haue equal grounde lines and are drawen
betwene one paire of paralleles, are equal togither.
Example.
Fyrste you muste marke the difference betwene this Theoreme and the laste,
for the laste Theoreme presupposed to the diuers likeiammes one ground
line common to them, but this theoreme doth presuppose a diuers ground
line for euery likeiamme, only meaning them to be equal in length, though
they be diuers in numbre. As for example. In the last figure ther are two
parallels, A.D. and E.H, and betwene them are drawen thre likeiammes, the
firste is, A.B.E.F, the second is E.C.D.F, and the thirde is C.G.H.D. The
firste and the seconde haue one ground line, (that is E.F.) and therfore in so
muche as they are betwene one paire of paralleles, they are equall
accordinge to the fiue and twentye Theoreme, but the thirde likeiamme that

is C.G.H.D. hathe his grounde line G.H, seuerall frome the other, but yet
equall vnto it. wherefore the third likeiam is equall to the other two firste
likeiammes. And for a proofe that G.H. being the groũd line of the third
likeiamme, is equal to E.F, whiche is the ground line to both the other
likeiams, that may be thus declared, G.H. is equall to C.D, seynge they are
the contrary sides of one likeiamme (by the foure and twẽty theoreme) and
so are C.D. and E.F. by the same theoreme. Therfore seynge both those
ground lines E.F. and G.H, are equall to one thirde line (that is C.D.) they
must nedes bee equall togyther by the firste common sentence.

The xxvii. Theoreme.
All triangles hauinge one grounde lyne, and standing betwene one
paire of parallels, ar equall togither.
Example.
A.B. and C.F. are twoo gemowe lines, betweene
which there be made two triangles, A.D.E. and
D.E.B, so that D.E, is the common ground line
to them bothe. wherfore it doth folow, that those
two triangles A.D.E. and D.E.B. are equall
eche to other.

The xxviij. Theoreme.
All triangles that haue like long ground lines, and bee made
betweene one paire of gemow lines, are equall togither.
Example.
Example of this Theoreme you may see in the last figure, where as sixe
triangles made betwene those two gemowe lines A.B. and C.F, the first
triangle is A.C.D, the seconde is A.D.E, the thirde is A.D.B, the fourth is
A.B.E, the fifte is D.E.B, and the sixte is B.E.F, of which sixe triangles,
A.D.E. and D.E.B. are equall, bicause they haue one common grounde line.

And so likewise A.B.E. and A.B.D, whose commen grounde line is A.B, but
A.C.D. is equal to B.E.F, being both betwene one couple of parallels, not
bicause thei haue one ground line, but bicause they haue their ground lines
equall, for C.D. is equall to E.F, as you may declare thus. C.D, is equall to
A.B. (by the foure and twenty Theoreme) for thei are two contrary sides of
one lykeiamme. A.C.D.B, and E.F by the same theoreme, is equall to A.B,
for thei ar the two ye contrary sides of the likeiamme, A.E.F.B, wherfore
C.D. must needes be equall to E.F. like wise the triangle A.C.D, is equal to
A.B.E, bicause they ar made betwene one paire of parallels and haue their
groundlines like, I meane C.D. and A.B. Againe A.D.E, is equal to eche of
them both, for his ground line D.E, is equall to A.B, inso muche as they are
the contrary sides of one likeiamme, that is the long square A.B.D.E. And
thus may you proue the equalnes of all the reste.

The xxix. Theoreme.
Al equal triangles that are made on one grounde line, and rise one
waye, must needes be betwene one paire of parallels.
Example.
Take for example A.D.E, and D.E.B, which (as the xxvij. conclusion dooth
proue) are equall togither, and as you see, they haue one ground line D.E.
And againe they rise towarde one side, that is to say, vpwarde toward the
line A.B, wherfore they must needes be inclosed betweene one paire of
parallels, which are heere in this example A.B. and D.E.

The thirty Theoreme.
Equal triangles that haue their ground lines equal, and be drawẽ
toward one side, ar made betwene one paire of paralleles.
Example.
The example that declared the last theoreme, maye well serue to the
declaracion of this also. For those ij. theoremes do diffre but in this one

pointe, that the laste theoreme meaneth of triangles, that haue one ground
line common to them both, and this theoreme dothe presuppose the grounde
lines to bee diuers, but yet of one length, as A.C.D, and B.E.F, as they are
ij. equall triangles approued, by the eighte and twentye Theorem, so in the
same Theorem it is declared, yt their groũd lines are equall togither, that is
C.D, and E.F, now this beeynge true, and considering that they are made
towarde one side, it foloweth, that they are made betwene one paire of
parallels when I saye, drawen towarde one side, I meane that the triangles
must be drawen other both vpward frome one parallel, other els both
downward, for if the one be drawen vpward and the other downward, then
are they drawen betwene two paire of parallels, presupposinge one to bee
drawen by their ground line, and then do they ryse toward contrary sides.

The xxxi. theoreme.
If a likeiamme haue one ground line with a triangle, and be
drawen betwene one paire of paralleles, then shall the likeiamme
be double to the triangle.
Example.
A.H. and B.G. are .ij. gemow lines, betwene
which there is made a triangle B.C.G, and a
lykeiamme, A.B.G.C, whiche haue a grounde
lyne, that is to saye, B.G. Therfore doth it folow
that the lyke iamme A.B.G.C. is double to the
triangle B.C.G. For euery halfe of that
lykeiamme is equall to the triangle, I meane
A.B.F.E. other F.E.C.G. as you may coniecture by the .xi. conclusion
geometrical.
And as this Theoreme dothe speake of a triangle and likeiamme that haue
one groundelyne, so is it true also, yf theyr groundelynes bee equall, though
they bee dyuers, so that thei be made betwene one payre of paralleles. And
hereof may you perceaue the reason, why in measuryng the platte of a
triangle, you must multiply the perpendicular lyne by halfe the grounde
lyne, or els the hole grounde lyne by halfe the perpendicular, for by any of

these bothe waies is there made a lykeiamme equall to halfe suche a one as
shulde be made on the same hole grounde lyne with the triangle, and
betweene one payre of paralleles. Therfore as that lykeiamme is double to
the triangle, so the halfe of it, must needes be equall to the triangle.
Compare the .xi. conclusion with this theoreme.

The .xxxij. Theoreme.
In all likeiammes where there are more than one made aboute one
bias line, the fill squares of euery of them must nedes be equall.
Example.
Fyrst before I declare the examples, it
shal be mete to shew the true
Bias lyne. vnderstãdyng of this theorem. Therfore
by the Bias line, I meane that lyne,
whiche in any square figure dooth runne
from corner to corner. And euery square
which is diuided by that bias line into
equall halues from corner to corner (that
is to say, into .ij. equall triangles) those
be counted to stande aboute one bias line, and the other squares,
whiche touche that bias line, with one of their corners onely,
Fyll squares. those doo I call Fyll squares, accordyng to the greke name,
ἀναπλη­ρώ­‐ which is anapleromata, and called in latin supplementa, bycause
ματα that they make one generall square, includyng and enclosyng
the other diuers squares, as in this exãple H.C.E.N. is one
square likeiamme, and L.M.G.C. is an other, whiche bothe are
made aboute one bias line, that is N.M, than K.L.H.C. and
C.E.F.G. are .ij. fyll squares, for they doo fyll vp the sydes of the
.ij. fyrste square lykeiammes, in suche sorte, that all them foure
is made one greate generall square K.M.F.N.
Nowe to the sentence of the theoreme, I say, that the .ij. fill squares,
H.K.L.C. and C.E.F.G. are both equall togither, (as it shall bee declared in
the booke of proofes) bicause they are the fill squares of two likeiammes

made aboute one bias line, as the exaumple sheweth. Conferre the twelfthe
conclusion with this theoreme.

The xxxiij. Theoreme.
In all right anguled triangles, the square of that side whiche lieth
against the right angle, is equall to the .ij. squares of both the other
sides.
Example.
A.B.C. is a triangle, hauing a ryght angle in B.
Wherfore it foloweth, that the square of A.C,
(whiche is the side that lyeth agaynst the right
angle) shall be as muche as the two squares of A.B.
and B.C. which are the other .ij. sides.
¶ By the square of any lyne, you muste vnderstande
a figure made iuste square, hauyng all his iiij. sydes
equall to that line, whereof it is the square, so is
A.C.F, the square of A.C. Lykewais A.B.D. is the
square of A.B. And B.C.E. is the square of B.C. Now by the numbre of the
diuisions in eche of these squares, may you perceaue not onely what the
square of any line is called, but also that the theoreme is true, and
expressed playnly bothe by lines and numbre. For as you see, the greatter
square (that is A.C.F.) hath fiue diuisions on eche syde, all equall togyther,
and those in the whole square are twenty and fiue. Nowe in the left square,
whiche is A.B.D. there are but .iij. of those diuisions in one syde, and that
yeldeth nyne in the whole. So lykeways you see in the meane square A.C.E.
in euery syde .iiij. partes, whiche in the whole amount vnto sixtene. Nowe
adde togyther all the partes of the two lesser squares, that is to saye,
sixtene and nyne, and you perceyue that they make twenty and fiue, whyche
is an equall numbre to the summe of the greatter square.
By this theoreme you may vnderstand a redy way to know the syde of any
ryght anguled triangle that is vnknowen, so that you knowe the lengthe of
any two sydes of it. For by tournynge the two sydes certayne into theyr

squares, and so addynge them togyther, other subtractynge the one from the
other (accordyng as in the vse of these theoremes I haue sette foorthe) and
then fyndynge the roote of the square that remayneth, which roote (I meane
the syde of the square) is the iuste length of the vnknowen syde, whyche is
sought for. But this appertaineth to the thyrde booke, and therefore I wyll
speake no more of it at this tyme.

The xxxiiij. Theoreme.
If so be it, that in any triangle, the square of the one syde be equall
to the .ij. squares of the other .ij. sides, than must nedes that corner
be a right corner, which is conteined betwene those two lesser
sydes.
Example.
As in the figure of the laste Theoreme, bicause A.C, made in square, is
asmuch as the square of A.B, and also as the square of B.C. ioyned bothe
togyther, therefore the angle that is inclosed betwene those .ij. lesser lynes,
A.B. and B.C. (that is to say) the angle B. whiche lieth against the line
A.C, must nedes be a ryght angle. This theoreme dothe so depende of the
truthe of the laste, that whan you perceaue the truthe of the one, you can
not iustly doubt of the others truthe, for they conteine one sentence,
contrary waies pronounced.

The .xxxv. theoreme.
If there be set forth .ij. right lines, and one of them parted into
sundry partes, how many or few so euer they be, the square that is
made of those ij. right lines proposed, is equal to all the squares,
that are made of the vndiuided line, and euery parte of the diuided
line.
Example.

The ij. lines proposed ar A.B. and C.D, and the lyne
A.B. is deuided into thre partes by E. and F. Now
saith this theoreme, that the square that is made of
those two whole lines A.B. and C.D, so that the line
A.B. stãdeth for the lẽgth of the square, and the
other line C.D. for the bredth of the same. That
square (I say) wil be equall to all the squares that be
made, of the vndiueded lyne (which is C.D.) and
euery portion of the diuided line. And to declare that particularly, Fyrst I
make an other line G.K, equall to the line .C.D, and the line G.H. to be
equal to the line A.B, and to bee diuided into iij. like partes, so that G.M. is
equall to A.E, and M.N. equal to E.F, and then muste N.H. nedes remaine
equall to F.B. Then of those ij. lines G.K, vndeuided, and G.H. which is
deuided, I make a square, that is G.H.K.L, In which square if I drawe
crosse lines frome one side to the other, according to the diuisions of the
line G.H, then will it appear plaine, that the theoreme doth affirme. For the
first square G.M.O.K, must needes be equal to the square of the line C.D,
and the first portiõ of the diuided line, which is A.E, for bicause their sides
are equall. And so the seconde square that is M.N.P.O, shall be equall to
the square of C.D, and the second part of A.B, that is E.F. Also the third
square which is N.H.L.P, must of necessitee be equal to the square of C.D,
and F.B, bicause those lines be so coupeled that euery couple are equall in
the seuerall figures. And so shal you not only in this example, but in all
other finde it true, that if one line be deuided into sondry partes, and an
other line whole and vndeuided, matched with him in a square, that square
which is made of these two whole lines, is as muche iuste and equally, as
all the seuerall squares, whiche bee made of the whole line vndiuided, and
euery part seuerally of the diuided line.

The xxxvi. Theoreme.
If a right line be parted into ij. partes, as chaunce may happe, the
square that is made of the whole line, is equall to bothe the squares
that are made of the same line, and the twoo partes of it seuerally.
Example.

The line propounded beyng A.B. and deuided, as
chaunce happeneth, in C. into ij. vnequall partes, I say
that the square made of the hole line A.B, is equal to the
two squares made of the same line with the twoo partes
of itselfe, as with A.C, and with C.B, for the square
D.E.F.G. is equal to the two other partial squares of
D.H.K.G and H.E.F.K, but that the greater square is
equall to the square of the whole line A.B, and the
partiall squares equall to the squares of the second
partes of the same line ioyned with the whole line, your eye may iudg
without muche declaracion, so that I shall not neede to make more
exposition therof, but that you may examine it, as you did in the laste
Theoreme.

The xxxvij. Theoreme.
If a right line be deuided by chaunce, as it maye happen, the
square that is made of the whole line, and one of the partes of it
which soeuer it be, shal be equall to that square that is made of the
ij. partes ioyned togither, and to an other square made of that part,
which was before ioyned with the whole line.
Example.
The line A.B. is deuided in C. into twoo partes,
though not equally, of which two partes for an
example I take the first, that is A.C, and of it I
make one side of a square, as for example D.G.
accomptinge those two lines to be equall, the
other side of the square is D.E, whiche is equall
to the whole line A.B.
Now may it appeare, to your eye, that the great
square made of the whole line A.B, and of one of his partes that is A.C,
(which is equall with D.G.) is equal to two partiall squares, whereof the
one is made of the saide greatter portion A.C, in as muche as not only D.G,

beynge one of his sides, but also D.H. beinge the other side, are eche of
them equall to A.C. The second square is H.E.F.K, in which the one side
H.E, is equal to C.B, being the lesser parte of the line, A.B, and E.F. is
equall to A.C. which is the greater parte of the same line. So that those two
squares D.H.K.G and H.E.F.K, bee bothe of them no more then the greate
square D.E.F.G, accordinge to the wordes of the Theoreme afore saide.

The xxxviij. Theoreme.
If a righte line be deuided by chaunce, into partes, the square that
is made of that whole line, is equall to both the squares that ar
made of eche parte of the line, and moreouer to two squares made
of the one portion of the diuided line ioyned with the other in
square.
Example.
The labels A and B were transposed in the illustration as an
alternative to transposing all occurrences of A.C and C.B in the text.

Lette the diuided line bee A.B, and parted in C, into
twoo partes: Nowe saithe the Theoreme, that the
square of the whole lyne A.B, is as mouche iuste as
the square of A.C, and the square of C.B, eche by it
selfe, and more ouer by as muche twise, as A.C. and C.B. ioyned in one
square will make. For as you se, the great square D.E.F.G, conteyneth in
hym foure lesser squares, of whiche the first and the greatest is N.M.F.K,
and is equall to the square of the lyne A.C. The second square is the lest of
them all, that is D.H.L.N, and it is equall to the square of the line C.B.
Then are there two other longe squares both of one bygnes, that is
H.E.N.M. and L.N.G.K, eche of them both hauyng .ij. sides equall to A.C,
the longer parte of the diuided line, and there other two sides equall to C.B,
beeyng he shorter parte of the said line A.B.
So is that greatest square, beeyng made of the hole lyne A.B, equal to the ij.
squares of eche of his partes seuerally, and more by as muche iust as .ij.

longe squares, made of the longer portion of the diuided lyne ioyned in
square with the shorter parte of the same diuided line, as the theoreme
wold. And as here I haue put an example of a lyne diuided into .ij. partes,
so the theoreme is true of all diuided lines, of what number so euer the
partes be, foure, fyue, or syxe. etc.
This theoreme hath great vse, not only in geometrie, but also in arithmetike,
as herafter I will declare in conuenient place.

The .xxxix. theoreme.
If a right line be deuided into two equall partes, and one of these
.ij. partes diuided agayn into two other partes, as happeneth the
longe square that is made of the thyrd or later part of that diuided
line, with the residue of the same line, and the square of the
mydlemoste parte, are bothe togither equall to the square of halfe
the firste line.
Example.
The line A.B. is diuided into ij. equal partes in
C, and that parte C.B. is diuided agayne as
hapneth in D. Wherfore saith the Theorem that
the long square made of D.B. and A.D, with the
square of C.D. (which is the mydle portion)
shall bothe be equall to the square of half the
lyne A.B, that is to saye, to the square of A.C,
or els of C.D, which make all one. The long
square F.G.N.O. whiche is the longe square that the theoreme speaketh of,
is made of .ij. long squares, wherof the fyrst is F.G.M.K, and the seconde is
K.N.O.M. The square of the myddle portion is L.M.O.P. and the square of
the halfe of the fyrste lyne is E.K.Q.L. Nowe by the theoreme, that longe
square F.G.N.O, with the iuste square L.M.O.P, muste bee equall to the
greate square E.K.Q.L, whyche thynge bycause it seemeth somewhat
difficult to vnderstande, althoughe I intende not here to make
demonstrations of the Theoremes, bycause it is appoynted to be done in the

newe edition of Euclide, yet I wyll shew you brefely how the equalitee of the
partes doth stande. And fyrst I say, that where the comparyson of equalitee
is made betweene the greate square (whiche is made of halfe the line A.B.)
and two other, where of the fyrst is the longe square F.G.N.O, and the
second is the full square L.M.O.P, which is one portion of the great square
all redye, and so is that longe square K.N.M.O, beynge a parcell also of the
longe square F.G.N.O, Wherfore as those two partes are common to bothe
partes compared in equalitee, and therfore beynge bothe abated from eche
parte, if the reste of bothe the other partes bee equall, than were those
whole partes equall before: Nowe the reste of the great square, those two
lesser squares beyng taken away, is that longe square E.N.P.Q, whyche is
equall to the long square F.G.K.M, beyng the rest of the other parte. And
that they two be equall, theyr sydes doo declare. For the longest lynes that
is F.K and E.Q are equall, and so are the shorter lynes, F.G, and E.N, and
so appereth the truthe of the Theoreme.

The .xl. theoreme.
If a right line be diuided into .ij. euen partes, and an other right
line annexed to one ende of that line, so that it make one righte line
with the firste. The longe square that is made of this whole line so
augmented, and the portion that is added, with the square of halfe
the right line, shall be equall to the square of that line, whiche is
compounded of halfe the firste line, and the parte newly added.
Example.
The fyrst lyne propounded is A.B, and it is
diuided into ij. equall partes in C, and an other
ryght lyne, I meane B.D annexed to one ende of
the fyrste lyne.
Nowe say I, that the long square A.D.M.K, is
made of the whole lyne so augmẽted, that is A.D,
and the portiõ annexed, yt is D.M, for D.M is
equall to B.D, wherfore yt long square A.D.M.K, with the square of halfe

the first line, that is E.G.H.L, is equall to the great square E.F.D.C. whiche
square is made of the line C.D. that is to saie, of a line compounded of
halfe the first line, beyng C.B, and the portion annexed, that is B.D. And it
is easyly perceaued, if you consyder that the longe square A.C.L.K.
(whiche onely is lefte out of the great square) hath another longe square
equall to hym, and to supply his steede in the great square, and that is
G.F.M.H. For their sydes be of lyke lines in length.

The xli. Theoreme.
If a right line bee diuided by chaunce, the square of the same
whole line, and the square of one of his partes are iuste equall to
the lõg square of the whole line, and the sayde parte twise taken,
and more ouer to the square of the other parte of the sayd line.
Example.
A.B. is the line diuided in C. And D.E.F.G, is the square
of the whole line, D.H.K.M. is the square of the lesser
portion (whyche I take for an example) and therfore
must bee twise reckened. Nowe I saye that those ij.
squares are equall to two longe squares of the whole line
A.B, and his sayd portion A.C, and also to the square of
the other portion of the sayd first line, whiche portion is
C.B, and his square K.N.F.L. In this theoreme there is
no difficultie, if you cõsyder that the litle square D.H.K.M. is .iiij. tymes
reckened, that is to say, fyrst of all as a parte of the greatest square, whiche
is D.E.F.G. Secondly he is rekned by him selfe. Thirdely he is accompted as
parcell of the long square D.E.N.M, And fourthly he is taken as a part of
the other long square D.H.L.G, so that in as muche as he is twise reckened
in one part of the comparisõ of equalitee, and twise also in the second
parte, there can rise none occasion of errour or doubtfulnes therby.

The xlij. Theoreme.

If a right line be deuided as chance happeneth the iiij. long
squares, that may be made of that whole line and one of his partes
with the square of the other part, shall be equall to the square that
is made of the whole line and the saide first portion ioyned to him
in lengthe as one whole line.
Example.
The firste line is A.B, and is deuided by C. into two
vnequall partes as happeneth. The long square of
yt, and his lesser portion A.C, is foure times
drawen, the first is E.G.M.K, the seconde is
K.M.Q.O, the third is H.K.R.S, and the fourthe is
K.L.S.T. And where as it appeareth that one of the
little squares (I meane K.L.P.O) is reckened twise,
ones as parcell of the second long square and
agayne as parte of the thirde long square, to auoide
ambiguite, you may place one insteede of it, an
other square of equalitee, with it. that is to saye,
D.E.K.H, which was at no tyme accompting as parcell of any one of them,
and then haue you iiij. long squares distinctly made of the whole line A.B,
and his lesser portion A.C. And within them is there a greate full square
P.Q.T.V. whiche is the iust square of B.C, beynge the greatter portion of the
line A.B. And that those fiue squares doo make iuste as muche as the whole
square of that longer line D.G, (whiche is as longe as A.B, and A.C. ioyned
togither) it may be iudged easyly by the eye, sith that one greate square
doth comprehẽd in it all the other fiue squares, that is to say, foure long
squares (as is before mencioned) and one full square. which is the intent of
the Theoreme.

The xliij. Theoreme.
If a right line be deuided into ij. equal partes first, and one of those
parts again into other ij. parts, as chaũce hapeneth, the square that
is made of the last part of the line so diuided, and the square of the

residue of that whole line, are double to the square of halfe that
line, and to the square of the middle portion of the same line.
Example.
The line to be deuided is A.B, and is parted in C. into
two equall partes, and then C.B, is deuided againe into
two partes in D, so that the meaninge of the Theoreme,
is that the square of D.B. which is the latter parte of
the line, and the square of A.D, which is the residue of
the whole line. Those two squares, I say, ar double to
the square of one halfe of the line, and to the square of
C.D, which is the middle portion of those thre
diuisions. Which thing that you maye more easilye
perceaue, I haue drawen foure squares, whereof the
greatest being marked with E. is the square of A.D.
The next, which is marked with G, is the square of halfe the line, that is, of
A.C, And the other two little squares marked with F. and H, be both of one
bignes, by reason that I did diuide C.B. into two equall partes, so that you
amy take the square F, for the square of D.B, and the square H, for the
square of C.D. Now I thinke you doubt not, but that the square E. and the
square F, ar double so much as the square G. and the square H, which
thing the easyer is to be vnderstande, bicause that the greate square hath in
his side iij. quarters of the firste line, which multiplied by itselfe maketh
nyne quarters, and the square F. containeth but one quarter, so that bothe
doo make tenne quarters.
Then G. contayneth iiij. quarters, seynge his side containeth
twoo, and H. containeth but one quarter, whiche both make
but fiue quarters, and that is but halfe of tenne.
Whereby you may easylye coniecture,
that the meanynge of the the-
oreme is verified in the
figures of this ex-
ample.

The xliiij. Theoreme.
If a right line be deuided into ij. partes equally, and an other
portion of a righte lyne annexed to that firste line, the square of
this whole line so compounded, and the square of the portion that
is annexed, ar doule as much as the square of the halfe of the firste
line, and the square of the other halfe ioyned in one with the
annexed portion, as one whole line.
Example.
The line is A.B, and is diuided firste into twoo
equal partes in C, and thẽ is there annexed to it an
other portion whiche is B.D. Now saith the
Theoreme, that the square of A.D, and the square
of B.D, ar double to the square of A.C, and to the
square of C.D. The line A.B. cõtaining four
partes, then must needes his halfe containe ij.
partes of such partes I suppose B.D. (which is the
ãnexed line) to containe thre, so shal the hole line
cõprehend vij. parts, and his square xlix. parts,
where vnto if you ad ye square of the annexed lyne,
whiche maketh nyne, than those bothe doo yelde, lviij. whyche must be
double to the square of the halfe lyne with the annexed portion. The halfe
lyne by it selfe conteyneth but .ij. partes, and therfore his square dooth
make foure. The halfe lyne with the annexed portion conteyneth fiue, and
the square of it is .xxv, now put foure to .xxv, and it maketh iust .xxix, the
euen halfe of fifty and eight, wherby appereth the truthe of the theoreme.

The .xlv. theoreme.
In all triangles that haue a blunt angle, the square of the side that
lieth against the blunt angle, is greater than the two squares of the
other twoo sydes, by twise as muche as is comprehended of the one

of those .ij. sides (inclosyng the blunt corner) and the portion of
the same line, beyng drawen foorth in lengthe, which lieth betwene
the said blunt corner and a perpendicular line lightyng on it, and
drawen from one of the sharpe angles of the foresayd triangle.
Example.
For the declaration of this theoreme and the next also, whose vse are
wonderfull in the practise of Geometrie, and in measuryng especially, it
shall be nedefull to declare that euery triangle that hath no ryght angle as
those whyche are called (as in the boke of practise is declared) sharp
cornered triangles, and blunt cornered triangles, yet may they be brought
to haue a ryght angle, eyther by partyng them into two lesser triangles, or
els by addyng an other triangle vnto them, whiche may be a great helpe for
the ayde of measuryng, as more largely shall be sette foorthe in the boke of
measuryng. But for this present place, this forme wyll I vse, (whiche Theon
also vseth) to adde one triangle vnto an other, to bryng the blunt cornered
triangle into a ryght angled triangle, whereby the proportion of the squares
of the sides in suche a blunt cornered triangle may the better bee knowen.
Fyrst therfore I sette foorth the triangle
A.B.C, whose corner by C. is a blunt
corner as you maye well iudge, than to
make an other triangle of yt with a ryght
angle, I must drawe forth the side B.C.
vnto D, and frõ the sharp corner by A.
I brynge a plumbe lyne or perpẽdicular
on D. And so is there nowe a newe
triangle A.B.D. whose angle by D. is a
right angle. Nowe accordyng to the
meanyng of the Theoreme, I saie, that in
the first triangle A.B.C, because it hath a
blunt corner at C, the square of the line
A.B. whiche lieth against the said blunte
corner, is more then the square of the line
A.C, and also of the lyne B.C, (whiche
inclose the blunte corner) by as muche as

will amount twise of the line B.C, and that portion D.C. whiche lieth
betwene the blunt angle by C, and the perpendicular line A.D.
The square of the line A.B, is the great square marked with E. The square
of A.C, is the meane square marked with F. The square of B.C, is the least
square marked with G. And the long square marked with K, is sette in
steede of two squares made of B.C, and C.D. For as the shorter side is the
iuste lengthe of C.D, so the other longer side is iust twise so longe as B.C,
Wherfore I saie now accordyng to the Theoreme, that the greatte square E,
is more then the other two squares F. and G, by the quantitee of the longe
square K, wherof I reserue the profe to a more conuenient place, where I
will also teache the reason howe to fynde the lengthe of all suche
perpendicular lynes, and also of the line that is drawen betweene the blunte
angle and the perpendicular line, with sundrie other very pleasant
conclusions.
Labeling of rectangle K is conjectural. Other configurations of B, C and D will also fit, but the
printed illustration (B and D on the right, nothing on the left) will not. The text requires a
rectangle with BC as one side and CD as the other, doubled as in the next illustration (H.K).

The .xlvi. Theoreme.
In sharpe cornered triangles, the square of anie side that lieth
against a sharpe corner, is lesser then the two squares of the other
two sides, by as muche as is comprised twise in the long square of
that side, on whiche the perpendicular line falleth, and the portion
of that same line, liyng betweene the perpendicular, and the
foresaid sharpe corner.
Example.
Fyrst I sette foorth the triangle A.B.C, and in yt I draw a plũbe line from
the angle C. vnto the line A.B, and it lighteth in D. Nowe by the theoreme
the square of B.C. is not so muche as the square of the other two sydes, that
of B.A. and of A.C. by as muche as is twise conteyned in the lõg square
made of A.B, and A.D, A.B. beyng the line or syde on which the

perpendicular line falleth, and A.D.
beeyng that portion of the same line
whiche doth lye betwene the
perpendicular line, and the sayd sharpe
angle limitted, whiche angle is by A.
For declaration of the figures, the square
marked with E. is the square of B.C,
whiche is the syde that lieth agaynst the
sharpe angle, the square marked with G.
is the square of A.B, and the square
marked with F. is the square of A.C, and
the two longe squares marked with H.K,
are made of the hole line A.B, and one of
his portions A.D. And truthe it is that the
square E. is lesser than the other two
squares C. and F. by the quantitee of
those two long squares H. and K. Wherby
you may consyder agayn, an other proportion of equalitee, that is to saye,
that the square E. with the twoo longsquares H.K, are iuste equall to the
other twoo squares C. and F. And so maye you make, as it were an other
theoreme. That in al sharpe cornered triangles, where a perpendicular line is
drawen frome one angle to the side that lyeth againste it, the square of anye
one side, with the ij. longesquares made at that hole line, whereon the
perpendicular line doth lighte, and of that portion of it, which ioyneth to
that side whose square is all ready taken, those thre figures, I say, are equall
to the ij. squares, of the other ij. sides of the triangle. In whiche you muste
vnderstand, that the side on which the perpendiculare falleth, is thrise vsed,
yet is his square but ones mencioned, for twise he is taken for one side of
the two long squares. And as I haue thus made as it were an other theoreme
out of this fourty and sixe theoreme, so mighte I out of it, and the other that
goeth nexte before, make as manny as woulde suffice for a whole booke, so
that when they shall bee applyed to practise, and consequently to expresse
their benefite, no manne that hathe not well wayde their wonderfull
commoditee, would credite the possibilitie of their wonderfull vse, and
large ayde in knowledge. But all this wyll I remitte to a place conuenient.

The xlvij. Theoreme.
If ij. points be marked in the circumferẽce of a circle, and a right
line drawen frome the one to the other, that line must needes fal
within the circle.
Example.
The circle is A.B.C.D, the ij. poinctes are A.B, the righte
line that is drawenne frome the one to the other, is the
line A.B, which as you see, must needes lyghte within the
circle. So if you putte the pointes to be A.D, or D.C, or
A.C, other B.C, or B.D, in any of these cases you see,
that the line that is drawen from the one pricke to the
other dothe euermore run within the edge of the circle,
els canne it be no right line. How be it, that a croked line, especially being
more croked then the portion of the circumference, maye bee drawen from
pointe to pointe withoute the circle. But the theoreme speaketh only of right
lines, and not of croked lines.

The xlviij. Theoreme.
If a righte line passinge by the centre of a circle, doo crosse an
other right line within the same circle, passinge beside the centre,
if he deuide the saide line into twoo equal partes, then doo they
make all their angles righte. And contrarie waies, if they make all
their angles righte, then doth the longer line cutte the shorter in
twoo partes.
Example.
The circle is A.B.C.D, the line that passeth by the centre, is A.E.C, the line
that goeth beside the centre is D.B. Nowe saye I, that the line A.E.C, dothe
cutte that other line D.B. into twoo iuste partes, and therefore all their four
angles ar righte angles. And contrarye wayes, bicause all their angles are

righte angles, therfore it muste be true, that the
greater cutteth the lesser into two equal partes,
accordinge as the Theoreme would.

The xlix. Theoreme.
If twoo right lines drawen in a circle doo crosse one an other, and
doo not passe by the centre, euery of them dothe not deuide the
other into equall partions.
Example.
The circle is A.B.C.D, and the centre is E, the one line
A.C, and the other is B.D, which two lines crosse one
an other, but yet they go not by the centre, wherefore
accordinge to the woordes of the theoreme, eche of
theim doth cuytte the other into equall portions. For as
you may easily iudge, A.C. hath one portiõ lõger and
an other shorter, and so like wise B.D. Howbeit, it is
not so to be vnderstãd, but one of them may be deuided into ij. euẽ parts,
but bothe to bee cutte equally in the middle, is not possible, onles both
passe through the cẽtre, therfore much rather whẽ bothe go beside the
centre, it can not be that eche of theym shoulde be iustely parted into ij.
euen partes.

The L. Theoreme.
If two circles crosse and cut one an other, then haue not they both
one centre.
Example.

This theoreme seemeth of it selfe so manifest, that it
neadeth nother demonstration nother declaraciõ. Yet
for the plaine vnderstanding of it, I haue sette forthe a
figure here, where ij. circles be drawẽ, so that one of
them doth crosse the other (as you see) in the pointes
B. and G, and their centres appear at the firste sighte
to bee diuers. For the centre of the one is F, and the
centre of the other is E, which diffre as farre asondre
as the edges of the circles, where they bee most distaunte in sonder.

The Li. Theoreme.
If two circles be so drawen, that one of them do touche the other,
then haue they not one centre.
Example.
There are two circles made, as you see, the one is
A.B.C, and hath his centre by G, the other is B.D.E,
and his centre is by F, so that it is easy enough to
perceaue that their centres doe dyffer as muche a
sonder, as the halfe diameter of the greater circle is
lõger then the half diameter of the lesser circle. And
so must it needes be thought and said of all other circles in lyke kinde.

The .lij. theoreme.
If a certaine pointe be assigned in the diameter of a circle, distant
from the centre of the said circle, and from that pointe diuerse
lynes drawen to the edge and circumference of the same circle, the
longest line is that whiche passeth by the centre, and the shortest is
the residew of the same line. And of al the other lines that is euer
the greatest, that is nighest to the line, which passeth by the centre.
And cõtrary waies, that is the shortest, that is farthest from it. And

amongest thẽ all there can be but onely .ij. equall together, and
they must nedes be so placed, that the shortest line shall be in the
iust middle betwixte them.
Example.
The circle is A.B.C.D.E.H, and his centre is F, the
diameter is A.E, in whiche diameter I haue taken a
certain point distaunt from the centre, and that pointe
is G, from which I haue drawen .iiij. lines to the
circumference, beside the two partes of the diameter,
whiche maketh vp vi. lynes in all. Nowe for the
diuersitee in quantitie of these lynes, I saie accordyng
to the Theoreme, that the line whiche goeth by the
centre is the longest line, that is to saie, A.G, and the
residewe of the same diameter beeyng G.E, is the shortest lyne. And of all
the other that lyne is longest, that is neerest vnto that parte of the diameter
whiche gooeth by the centre, and that is shortest, that is farthest distant
from it, wherefore I saie, that G.B, is longer then G.C, and therfore muche
more longer then G.D, sith G.C, also is longer then G.D, and by this maie
you soone perceiue, that it is not possible to drawe .ij. lynes on any one side
of the diameter, whiche might be equall in lengthe together, but on the one
side of the diameter maie you easylie make one lyne equall to an other, on
the other side of the same diameter, as you see in this example G.H, to bee
equall to G.D, betweene whiche the lyne G.E, (as the shortest in all the
circle) doothe stande euen distaunte from eche of them, and it is the precise
knoweledge of their equalitee, if they be equally distaunt from one halfe of
the diameter. Where as contrary waies if the one be neerer to any one halfe
of the diameter then the other is, it is not possible that they two may be
equall in lengthe, namely if they dooe ende bothe in the circumference of
the circle, and be bothe drawen from one poynte in the diameter, so that the
saide poynte be (as the Theoreme doeth suppose) somewhat distaunt from
the centre of the said circle. For if they be drawen from the centre, then
must they of necessitee be all equall, howe many so euer they bee, as the
definition of a circle dooeth importe, withoute any regarde how neere so
euer they be to the diameter, or how distante from it. And here is to be
noted, that in this Theoreme, by neerenesse and distaunce is vnderstand the

nereness and distaunce of the extreeme partes of those lynes where they
touche the circumference. For at the other end they do all meete and
touche.

The .liij. Theoreme.
If a pointe bee marked without a circle, and from it diuerse lines
drawen crosse the circle, to the circumference on the other side, so
that one of them passe by the centre, then that line whiche passeth
by the centre shall be the loongest of them all that crosse the circle.
And of the other lines those are longest, that be nexte vnto it that
passeth by the centre. And those ar shortest, that be farthest distant
from it. But among those partes of those lines, whiche ende in the
outewarde circumference, that is most shortest, whiche is parte of
the line that passeth by the centre, and amongeste the othere eche,
of thẽ, the nerer they are vnto it, the shorter they are, and the
farther from it, the longer they be. And amongest them all there can
not be more then .ij. of any one lẽgth, and they two muste be on the
two contrarie sides of the shortest line.
Example.
Take the circle to be A.B.C, and the point assigned
without it to be D. Now say I, that if there be drawen
sundrie lines from D, and crosse the circle, endyng in
the circumference on the cõtrary side, as here you
see, D.A, D.E, D.F, and D.B, then of all these lines
the longest must needes be D.A, which goeth by the
centre of the circle, and the nexte vnto it, that is D.E,
is the longest amongest the rest. And contrarie waies,
D.B, is the shorteste, because it is farthest distaunt
from D.A. And so maie you iudge of D.F, because it
is nerer vnto D.A, then is D.B, therefore is it longer
then D.B. And likewaies because it is farther of from

D.A, then is D.E, therfore is it shorter then D.E. Now for those partes of the
lines whiche bee withoute the circle (as you see) D.C, is the shortest.
because it is the parte of that line which passeth by the centre, And D.K, is
next to it in distance, and therefore also in shortnes, so D.G, is farthest
from it in distance, and therfore is the longest of them. Now D.H, beyng
nerer then D.G, is also shorter then it, and beynge farther of, then D.K, is
longer then it. So that for this parte of the theoreme (as I think) you do
plainly perceaue the truthe thereof, so the residue hathe no difficulte. For
seing that the nearer any line is to D.C, (which ioyneth with the diameter)
the shorter it is and the farther of from it, the longer it is. And seyng two
lynes can not be of like distaunce beinge bothe on one side, therefore if they
shal be of one lengthe, and consequently of one distaunce, they must needes
bee on contrary sides of the saide line D.C. And so appeareth the meaning
of the whole Theoreme.
And of this Theoreme dothe there folowe an other lyke. whiche you maye
calle other a theoreme by it selfe, or else a Corollary vnto this laste
theoreme, I passe not so muche for the name. But his sentence is this: when
so euer any lynes be drawen frome any pointe, withoute a circle, whether
they crosse the circle, or eande in the utter edge of his circumference, those
two lines that bee equally distaunt from the least line are equal togither, and
contrary waies, if they be equall togither, they ar also equally distant from
that least line.
For the declaracion of this proposition, it shall not need to vse any other
example, then that which is brought for the explication of this laste
theoreme, by whiche you may without any teachinge easyly perceaue both
the meanyng and also the truth of this proposition.

The Liiij. Theoreme.
If a point be set forthe in a circle, and frõ that pointe vnto the
circumference many lines drawen, of which more then two are
equal togither, then is that point the centre of that circle.
Example.

The circle is A.B.C, and within it I haue sette fourth
for an example three prickes, which are D.E. and F,
from euery one of them I haue drawẽ (at the leaste)
iiij. lines vnto the circumference of the circle but
frome D, I haue drawen more, yet maye it appear
readily vnto your eye, that of all the lines whiche be
drawen from E. and F, vnto the circumference, there
are but twoo equall, and more can not bee, for G.E.
nor E.H. hath none other equal to theim, nor canne
not haue any beinge drawen from the same point E. No more can L.F, or
F.K, haue anye line equall to either of theim, beinge drawen from the same
pointe F. And yet from either of those two poinctes are there drawen twoo
lines equall togither, as A.E, is equall to E.B, and B.F, is equall to F.C, but
there can no third line be drawen equall to either of these two couples, and
that is by reason that they be drawen from a pointe distaunte from the
centre of the circle. But from D, althoughe there be seuen lines drawen, to
the circumference, yet all bee equall, bicause it is the centre of the circle.
And therefore if you drawe neuer so mannye more from it vnto the
circumference, all shall be equal, so that this is the priuilege (as it were of
the centre) and therfore no other point can haue aboue two equal lines
drawen from it vnto the circumference. And from all pointes you maye
drawe ij. equall lines to the circumference of the circle, whether that pointe
be within the circle or without it.

The lv. Theoreme.
No circle canne cut an other circle in more pointes then two.
Example.
The first circle is A.B.F.E, the second circle is B.C.D.E, and they crosse
one an other in B. and in E, and in no more pointes. Nother is it possible
that they should, but other figures ther be, which maye cutte a circle in
foure partes, as you se in this exãple. Where I haue set forthe one tunne
forme, and one eye forme, and eche of them cutteth euery of their two
circles into foure partes. But as they be irregulare formes, that is to saye,
suche formes as haue no precise measure nother proportion in their

draughte, so can there scarcely be made any certaine
theorem of them. But circles are regulare formes, that
is to say, such formes as haue in their protracture a
iuste and certaine proportion, so that certain and
determinate truths may be affirmed of them, sith they
ar vniforme and vnchaungable.

The lvi. Theoreme.
If two circles be so drawen, that the one be within the other, and
that they touche one an other: If a line bee drawen by bothe their
centres, and so forthe in lengthe, that line shall runne to that
pointe, where the circles do touche.
Example.
The one circle, which is the greattest and vttermost is
A.B.C, the other circle that is ye lesser, and is drawen
within the firste, is A.D.E. The cẽtre of the greater
circle is F, and the centre of the lesser circle is G, the
pointe where they touche is A. And now you may see
the truthe of the theoreme so plainely, that it needeth
no farther declaracion. For you maye see, that
drawinge a line from F. to G, and so forth in lengthe, vntill it come to the
circumference, it wyll lighte in the very poincte A, where the circles touche
one an other.

The Lvij. Theoreme.
If two circles bee drawen so one withoute an other, that their edges
doo touche and a right line bee drawnenne frome the centre of the

one to the centre of the other, that line shall passe by the place of
their touching.
Example.
The firste circle is A.B.E, and his centre is K,
The secõd circle is D.B.C, and his cẽtre is H, the
point wher they do touch is B. Nowe doo you se
that the line K.H, whiche is drawen from K, that
is centre of the firste circle, vnto H, beyng centre
of the second circle, doth passe (as it must nedes
by the pointe B,) whiche is the verye poynte
wher they do to touche together.

The .lviij. theoreme.
One circle can not touche an other in more pointes then one,
whether they touche within or without.
Example.
For the declaration of this Theoreme, I haue drawen
iiij. circles, the first is A.B.C, and his centre H. the
second is A.D.G, and his centre F. the third is L.M,
and his centre K. the .iiij. is D.G.L.M, and his
centre E. Nowe as you perceiue the second circle
A.D.G, toucheth the first in the inner side, in so much
as it is drawen within the other, and yet it toucheth
him but in one point, that is to say in A, so lykewaies
the third circle L.M, is drawen without the firste circle
and toucheth hym, as you maie see, but in one place. And now as for the
.iiij. circle, it is drawen to declare the diuersitie betwene touchyng and
cuttyng, or crossyng. For one circle maie crosse and cutte a great many
other circles, yet can be not cutte any one in more places then two, as the
fiue and fiftie Theoreme affirmeth.

The .lix. Theoreme.
In euerie circle those lines are to be counted equall, whiche are in
lyke distaunce from the centre, And contrarie waies they are in lyke
distance from the centre, whiche be equall.
Example.
In this figure you see firste the circle drawen, whiche is
A.B.C.D, and his centre is E. In this circle also there
are drawen two lines equally distaunt from the centre,
for the line A.B, and the line D.C, are iuste of one
distaunce from the centre, whiche is E, and therfore
are they of one length. Again thei are of one lengthe
(as shall be proued in the boke of profes) and therefore their distaunce from
the centre is all one.

The lx. Theoreme.
In euerie circle the longest line is the diameter, and of all the other
lines, thei are still longest that be nexte vnto the centre, and they be
the shortest, that be farthest distaunt from it.
Example.
In this circle A.B.C.D, I haue drawen first the
diameter, whiche is A.D, whiche passeth (as it must)
by the centre E, Then haue I drawen ij. other lines as
M.N, whiche is neerer the centre, and F.G, that is
farther from the centre. The fourth line also on the
other side of the diameter, that is B.C, is neerer to
the centre then the line F.G, for it is of lyke distance
as is the lyne M.N. Nowe saie I, that A.D, beyng the
diameter, is the longest of all those lynes, and also of
any other that maie be drawen within that circle, And the other line M.N, is
longer then F.G. Also the line F.G, is shorter then the line B.C, for because

it is farther from the centre then is the lyne B.C. And thus maie you iudge of
al lines drawen in any circle, how to know the proportion of their length, by
the proportion of their distance, and contrary waies, howe to discerne the
proportion of their distance by their lengthes, if you knowe the proportion
of their length. And to speake of it by the waie, it is a maruaylouse thyng to
consider, that a man maie knowe an exacte proportion betwene two
thynges, and yet can not name nor attayne the precise quantitee of those
two thynges, As for exaumple, If two squares be sette foorthe, whereof the
one containeth in it fiue square feete, and the other contayneth fiue and
fortie foote, of like square feete, I am not able to tell, no nor yet anye
manne liuyng, what is the precyse measure of the sides of any of those .ij.
squares, and yet I can proue by vnfallible reason, that their sides be in a
triple proportion, that is to saie, that the side of the greater square (whiche
containeth .xlv. foote) is three tymes so long iuste as the side of the lesser
square, that includeth but fiue foote. But this seemeth to be spoken out of
ceason in this place, therfore I will omitte it now, reseruyng the exacter
declaration therof to a more conuenient place and time, and will procede
with the residew of the Theoremes appointed for this boke.

The .lxi. Theoreme.
If a right line be drawen at any end of a diameter in perpendicular
forme, and do make a right angle with the diameter, that right line
shall light without the circle, and yet so iointly knitte to it, that it is
not possible to draw any other right line betwene that saide line
and the circumferẽce of the circle. And the angle that is made in
the semicircle is greater then any sharpe angle that may be made
of right lines, but the other angle without, is lesser then any that
can be made of right lines.
Example.
In this circle A.B.C, the diameter is A.C, the perpendicular line, which
maketh a right angle with the diameter, is C.A, whiche line falleth without
the circle, and yet ioyneth so exactly vnto it, that it is not possible to draw

an other right line betwene the circumference of the
circle and it, whiche thyng is so plainly seene of the
eye, that it needeth no farther declaracion. For
euery man wil easily consent, that betwene the
croked line A.F, (whiche is a parte of the
circumferẽce of the circle) and A.E (which is the
said perpẽdicular line) there can none other line bee
drawen in that place where they make the angle.
Nowe for the residue of the theoreme. The angle
D.A.B, which is made in the semicircle, is greater
then anye sharpe angle that may bee made of ryghte lines. and yet is it a
sharpe angle also, in as much as it is lesser then a right angle, which is the
angle E.A.D, and the residue of that right angle, which lieth without the
circle, that is to saye, E.A.B, is lesser then any sharpe angle that can be
made of right lines also. For as it was before rehersed, there canne no right
line be drawen to the angle, betwene the circumference and the right line
E.A. Then must it needes folow, that there can be made no lesser angle of
righte lines. And againe, if ther canne be no lesser then the one, then doth it
sone appear, that there canne be no greater then the other, for they twoo
doo make the whole right angle, so that if anye corner coulde be made
greater then the one parte, then shoulde the residue bee lesser then the
other parte, so that other bothe partes muste be false, or els bothe graunted
to be true.

The lxij. Theoreme.
If a right line doo touche a circle, and an other right line drawen
frome the centre of the circle to the pointe where they touche, that
line whiche is drawenne frome the centre, shall be a perpendicular
line to the touch line.
Example.
The circle is A.B.C, and his centre is F. The touche line is D.E, and the
point wher they touch is C. Now by reason that a right line is drawen frome
the centre F. vnto C, which is the point of the touche, therefore saith the

theoreme, that the sayde line F.C, muste needes
bee a perpendicular line vnto the touche line
D.E.

The lxiij. Theoreme.
If a righte line doo touche a circle, and an other right line be
drawen from the pointe of their touchinge, so that it doo make
righte corners with the touche line, then shal the centre of the
circle bee in that same line, so drawen.
Example.
The circle is A.B.C, and the centre of it is G.
The touche line is D.C.E, and the pointe where
it toucheth, is C. Nowe it appeareth manifest,
that if a righte line be drawen from the pointe
where the touch line doth ioine with the circle,
and that the said lyne doo make righte corners
with the touche line, then muste it needes go by
the centre of the circle, and then consequently it must haue the sayde cẽtre
in him. For if the saide line shoulde go beside the centre, as F.C. doth, then
dothe it not make righte angles with the touche line, which in the theoreme
is supposed.

The lxiiij. Theoreme.
If an angle be made on the centre of a circle, and an other angle
made on the circumference of the same circle, and their grounde
line be one common portion of the circumference, then is the angle
on the centre twise so great as the other angle on the circũferẽce.

Example.
The circle is A.B.C.D, and his centre is E: the
angle on the centre is C.E.D, and the angle on the
circumference is C.A.D t their commen ground
line, is C.F.D. Now say I that the angle C.E.D,
whiche is on the centre, is twise so greate as the
angle C.A.D, which is on the circumference.

The lxv. Theoreme.
Those angles whiche be made in one cantle of a circle, must needes
be equal togither.
Example.
Before I declare this theoreme by example, it shall bee
needefull to declare, what is it to be vnderstande by
the wordes in this theoreme. For the sentence canne
not be knowen, onles the uery meaning of the wordes
be firste vnderstand. Therefore when it speaketh of
angles made in one cantle of a circle, it is this to be
vnderstand, that the angle muste touch the
circumference: and the lines that doo inclose that
angle, muste be drawen to the extremities of that line, which maketh the
cantle of the circle. So that if any angle do not touch the circumference, or
if the lines that inclose that angle, doo not ende in the extremities of the
corde line, but ende other in some other part of the said corde, or in the
circumference, or that any one of them do so eande, then is not that angle
accompted to be drawen in the said cantle of the circle. And this promised,
nowe will I cumme to the meaninge of the theoreme. I sette forthe a circle
whiche is A.B.C.D, and his centre E, in this circle I drawe a line D.C,
whereby there ar made two cantels, a more and a lesser. The lesser is
D.E.C, and the geater is D.A.B.C. In this greater cantle I drawe two
angles, the firste is D.A.C, and the second is D.B.C which two angles by
reason they are made bothe in one cantle of a circle (that is the cantle

D.A.B.C) therefore are they both equall. Now doth there appere an other
triangle, whose angle lighteth on the centre of the circle, and that triangle
is D.E.C, whose angle is double to the other angles, as is declared in the
lxiiij. Theoreme, whiche maie stande well enough with this Theoreme, for it
is not made in this cantle of the circle, as the other are, by reason that his
angle doth not light in the circumference of the circle, but on the centre of
it.

The .lxvi. theoreme.
Euerie figure of foure sides, drawen in a circle, hath his two
contrarie angles equall vnto two right angles.
Example.
The circle is A.B.C.D, and the figure of foure sides in it,
is made of the sides B.C, and C.D, and D.A, and A.B.
Now if you take any two angles that be contrary, as the
angle by A, and the angle by C, I saie that those .ij. be
equall to .ij. right angles. Also if you take the angle by
B, and the angle by D, whiche two are also contray,
those two angles are like waies equall to two right
angles. But if any man will take the angle by A, with the angle by B, or D,
they can not be accompted contrary, no more is not the angle by C. estemed
contray to the angle by B, or yet to the angle by D, for they onely be
accompted contrary angles, whiche haue no one line common to them
bothe. Suche is the angle by A, in respect of the angle by C, for there both
lynes be distinct, where as the angle by A, and the angle by D, haue one
common line A.D, and therfore can not be accompted contrary angles, So
the angle by D, and the angle by C, haue D.C, as a common line, and
therefore be not contrary angles. And this maie you iudge of the residewe,
by like reason.

The lxvij. Theoreme.

Vpon one right lyne there can not be made two cantles of circles,
like and vnequall, and drawen towarde one parte.
Example.
Cantles of circles be then called like, when the angles
that are made in them be equall. But now for the
Theoreme, let the right line be A.E.C, on whiche I
draw a cantle of a circle, whiche is A.B.C. Now
saieth the Theoreme, that it is not possible to draw
an other cantle of a circle, whiche shall be vnequall
vnto this first cantle, that is to say, other greatter or
lesser then it, and yet be lyke it also, that is to say, that the angle in the one
shall be equall to the angle in the other. For as in this example you see a
lesser cantle drawen also, that is A.D.C, so if an angle were made in it, that
angle would be greatter then the angle made in the cantle A.B.C, and
therfore can not they be called lyke cantels, but and if any other cantle were
made greater then the first, then would the angle in it be lesser then that in
the firste, and so nother a lesser nother a greater cantle can be made vpon
one line with an other, but it will be vnlike to it also.

The .lxviij. Theoreme.
Lyke cantelles of circles made on equal righte lynes, are equall
together.
Example.
What is ment by like cantles you haue heard before. and it is easie to
vnderstand, that suche figures a called equall, that be of one bygnesse, so
that the one is nother greater nother lesser then the other. And in this kinde
of comparison, they must so agree, that if the one be layed on the other,
they shall exactly agree in all their boundes, so that nother shall excede
other.
Nowe for the example of the Theoreme, I haue set forthe diuers varieties of
cantles of circles, amongest which the first and seconde are made vpõ

equall lines, and ar also both equall and like.
The third couple ar ioyned in one, and be
nother equall, nother like, but expressyng an
absurde deformitee, whiche would folowe if
this Theoreme wer not true. And so in the
fourth couple you maie see, that because they
are not equall cantles, therfore can not they
be like cantles, for necessarily it goeth
together, that all cantles of circles made vpon
equall right lines, if they be like they must be
equall also.

The lxix. Theoreme.
In equall circles, suche angles as be equall are made vpon equall
arch lines of the circumference, whether the angle light on the
circumference, or on the centre.
Example.
Firste I haue sette for an exaumple twoo
equall circles, that is A.B.C.D, whose
centre is K, and the second circle E.F.G.H,
and his centre L, and in eche of thẽ is there
made two angles, one on the circumference,
and the other on the centre of eche circle,
and they be all made on two equall arche
lines, that is B.C.D. the one, and F.G.H. the
other. Now saieth the Theoreme, that if the angle B.A.D, be equall to the
angle F.E.H, then are they made in equall circles, and on equall arch lines
of their circumference. Also if the angle B.K.D, be equal to the angle F.L.H,
then be they made on the centres of equall circles, and on equall arche
lines, so that you muste compare those angles together, whiche are made
both on the centres, or both on the circumference, and maie not conferre
those angles, wherof one is drawen on the circumference, and the other on
the centre. For euermore the angle on the centre in suche sorte shall be

double to the angle on the circumference, as is declared in the three score
and foure Theoreme.

The .lxx. Theoreme.
In equall circles, those angles whiche bee made on equall arche
lynes, are euer equall together, whether they be made on the centre,
or on the circumference.
Example.
This Theoreme doth but conuert the sentence of the last Theoreme before,
and therfore is to be vnderstande by the same examples, for as that saith,
that equall angles occupie equall archelynes, so this saith, that equal arche
lines causeth equal angles, consideringe all other circumstances, as was
taughte in the laste theoreme before, so that this theoreme dooeth affirming
speake of the equalitie of those angles, of which the laste theoreme spake
conditionally. And where the laste theoreme spake affirmatiuely of the
arche lines, this theoreme speaketh conditionally of them, as thus: If the
arche line B.C.D. be equall to the other arche line F.G.H, then is that angle
B.A.D. equall to the other angle F.E.H. Or els thus may you declare it
causally: Bicause the arche line B.C.D, is equal to the other arche line
F.G.H, therefore is the angle B.K.D. equall to the angle F.L.H,
consideringe that they are made on the centres of equall circles. And so of
the other angles, bicause those two arche lines aforesaid ar equal, therfore
the angle D.A.B, is equall to the angle F.E.H, for as muche as they are
made on those equall arche lines, and also on the circumference of equall
circles. And thus these theoremes doo one declare an other, and one verifie
the other.

The lxxi. Theoreme.
In equal circles, equall right lines beinge drawen, doo cutte awaye
equalle arche lines frome their circumferences, so that the greater

arche line of the one is equall to the greater arche line of the other,
and the lesser to the lesser.
Example.
The circle A.B.C.D, is made equall to the
circle E.F.G.H, and the right line B.D. is
equal to the righte line F.H, wherfore it
foloweth, that the ij. arche lines of the circle
A.B.D, whiche are cut from his
circumference by the right line B.D, are
equall to two other arche lines of the circle
E.F.H, being cutte frome his circumference,
by the right line F.H. that is to saye, that the arche line B.A.D, beinge the
greater arch line of the firste circle, is equall to the arche line F.E.H,
beynge the greater arche line of the other circle. And so in like manner the
lesser arche line of the firste circle, beynge B.C.D, is equal to the lesser
arche line of the seconde circle, that is F.G.H.

The lxxij. Theoreme.
In equall circles, vnder equall arche lines the right lines that bee
drawen are equall togither.
Example.
This Theoreme is none other, but the conuersion of the laste Theoreme
beefore, and therefore needeth none other example. For as that did declare
the equalitie of the arche lines, by the equalitie of the righte lines, so dothe
this Theoreme declare the equalnes of the right lines to ensue of the
equalnes of the arche lines, and therefore declareth that right lyne B.D, to
be equal to the other right line F.H, bicause they both are drawen vnder
equall arche lines, that is to saye, the one vnder B.A.D, and thother vnder
F.E.H, and those two arch lines are estimed equall by the theoreme laste
before, and shal be proued in the booke of proofes.

The lxxiij. Theoreme.
In euery circle, the angle that is made in the halfe circle, is a iuste
righte angle, and the angle that is made in a cantle greater then the
halfe circle, is lesser thanne a righte angle, but that angle that is
made in a cantle, lesser then the halfe circle, is greatter then a
right angle. And moreouer the angle of the greater cantle is greater
then a righte angle and the angle of the lesser cantle is lesser then
a right angle.
Example.
In this proposition, it shal be meete to note, that there is a greate diuersite
betwene an angle of a cantle, and an angle made in a cantle, and also
betwene the angle of a semicircle, and ye angle made in a semicircle. Also it
is meet to note yt al angles that be made in ye part of a circle, ar made
other in a semicircle, (which is the iuste half circle) or els in a cantle of the
circle, which cantle is other greater or lesser then the semicircle is, as in
this figure annexed you maye perceaue euerye one of the thinges
seuerallye.
Firste the circle is, as you see, A.B.C.D, and his
centre E, his diameter is A.D, Then is ther a line
drawẽ from A. to B, and so forth vnto F, which is
without the circle: and an other line also frome B.
to D, whiche maketh two cantles of the whole
circle. The greater cantle is D.A.B, and the lesser
cantle is B.C.D, In whiche lesser cantle also there
are two lines that make an angle, the one line is
B.C, and the other line is C.D. Now to showe the
difference of an angle in a cantle, and an angle of a cantle, first for an
example I take the greter cãtle B.A.D, in which is but one angle made, and
that is the angle by A, which is made of a line A.B, and the line A.D, And
this angle is therfore called an angle in a cantle. But now the same cantle
hathe two other angles, which be called the angles of that cantle, so the
twoo angles made of the righte line D.B, and the arche line D.A.B, are the

twoo angles of this cantle, whereof the one is by D, and the other is by B.
Wher you must remẽbre, that the ãgle by D. is made of the right line B.D,
and the arche line D.A. And this angle is diuided by an other right line
A.E.D, which in this case must be omitted as no line. Also the ãgle by B. is
made of the right line D.B, and of the arch line .B.A, & although it be
deuided with ij. other right lines, of wch the one is the right line B.A, &
thother the right line B.E, yet in this case they ar not to be cõsidered. And
by this may you perceaue also which be the angles of the lesser cantle, the
first of thẽ is made of ye right line B.D, & of ye arch line B.C, the secõd is
made of the right line .D.B, & of the arch line D.C. Then ar ther ij. other
lines, wch deuide those ij. corners, yt is the line B.C, & the line C.D, wch ij.
lines do meet in the poynte C, and there make an angle, whiche is called an
angle made in that lesser cantle, but yet is not any angle of that cantle. And
so haue you heard the difference betweene an angle in a cantle, and an
angle of a cantle. And in lyke sorte shall you iudg of the ãgle made in a
semicircle, whiche is distinct frõ the angles of the semicircle. For in this
figure, the angles of the semicircle are those angles which be by A. and D,
and be made of the right line A.D, beeyng the diameter, and of the halfe
circumference of the circle, but by the angle made in the semicircle is that
angle by B, whiche is made of the righte line A.B, and that other right line
B.D, whiche as they mete in the circumference, and make an angle, so they
ende with their other extremities at the endes of the diameter. These thynges
premised, now saie I touchyng the Theoreme, that euerye angle that is
made in a semicircle, is a right angle, and if it be made in any cãtle of a
circle, thẽ must it neds be other a blũt ãgle, or els a sharpe angle, and in no
wise a righte angle. For if the cantle wherein the angle is made, be greater
then the halfe circle, then is that angle a sharpe angle. And generally the
greater the cãtle is, the lesser is the angle comprised in that cantle: and
contrary waies, the lesser any cantle is, the greater is the angle that is
made in it. Wherfore it must nedes folowe, that the angle made in a cantle
lesse then a semicircle, must nedes be greater then a right angle. So the
angle by B, beyng made at the right line A.B, and the righte line B.D, is a
iuste righte angle, because it is made in a semicircle. But the angle made by
A, which is made of the right line A.B, and of the right line A.D, is lesser
then a righte angle, and is named a sharpe angle, for as muche as it is
made in a cantle of a circle, greater then a semicircle. And contrary waies,
the angle by C, beyng made of the righte line B.C, and of the right line

C.D, is greater then a right angle, and is named a blunte angle, because it
is made in a cantle of a circle, lesser then a semicircle. But now touchyng
the other angles of the cantles, I saie accordyng to the Theoreme, that the
.ij. angles of the greater cantle, which are by B. and D, as is before
declared, are greatter eche of them then a right angle. And the angles of the
lesser cantle, whiche are by the same letters B, and D, but be on the other
side of the corde, are lesser eche of them then a right angle, and be therfore
sharpe corners.

The lxxiiij. Theoreme.
If a right line do touche a circle, and from the pointe where they
touche, a righte lyne be drawen crosse the circle, and deuide it, the
angles that the saied lyne dooeth make with the touche line, are
equall to the angles whiche are made in the cantles of the same
circle, on the contrarie sides of the lyne aforesaid.
Example.
The circle is A.B.C.D, and the touche line is E.F.
The pointe of the touchyng is D, from which point I
suppose the line D.B, to be drawen crosse the
circle, and to diuide it into .ij. cantles, wherof the
greater is B.A.D, and the lesser is B.C.D, and in
ech of them an angle is drawen, for in the greater
cantle the angle is by A, and is made of the right
lines B.A, and A.D, in the lesser cantle the angle is
by C, and is made of ye right lines B.C, and C.D.
Now saith the Theoreme that the angle B.D.F, is equall to the angle made
in the cantle on the other side of the said line, that is to saie, in the cantle
B.A.D, so that the angle B.D.F, is equall to the angle B.A.D, because the
angle B.D.F, is on the one side of the line B.D, (whiche is according to the
supposition of the Theoreme drawen crosse the circle) and the angle
B.A.D, is in the cãtle on the other side. Likewaies the angle B.D.E, beyng
on the one side of the line B.D, must be equall to the angle B.C.D, (that is
the ãgle by C,) whiche is made in the cãtle on the other side of the right line

B.D. The profe of all these I do reserue, as I haue often saide, to a
conuenient boke, wherein they shall be all set at large.

The .lxxv. Theoreme.
In any circle when .ij. right lines do crosse one an other, the
likeiamme that is made of the portions of the one line, shall be
equall to the lykeiamme made of the partes of the other lyne.
Because this Theoreme doth serue to many
vses, and wold be wel vnderstande, I haue
set forth .ij. examples of it. In the firste, the
lines by their crossyng do make their
portions somewhat toward an equalitie. In
the second the portiõs of the lynes be very
far frõ an equalitie, and yet in bothe these
and in all other ye Theoreme is true. In the
first exãple the circle is A.B.C.D, in which
thone line A.C, doth crosse thother line
B.D, in ye point E. Now if you do make one
likeiãme or lõgsquare of D.E, & E.B, being
ye .ij. portions of the line D.B, that
longsquare shall be equall to the other
longsquare made of A.E, and E.C, beyng the portions of the other line A.C.
Lykewaies in the second example, the circle is F.G.H.K, in whiche the line
F.H, doth crosse the other line G.K, in the pointe L. Wherfore if you make a
lykeiamme or longsquare of the two partes of the line F.H, that is to saye, of
F.L, and L.H, that longsquare will be equall to an other longsquare made
of the two partes of the line G.K. which partes are G.L, and L.K. Those
longsquares haue I set foorth vnder the circles containyng their sides, that
you maie somewhat whet your own wit in practisyng this Theoreme,
accordyng to the doctrine of the nineteenth conclusion.

The .lxxvi. Theoreme.

If a pointe be marked without a circle, and from that pointe two
right lines drawen to the circle, so that the one of them doe runne
crosse the circle, and the other doe touche the circle onely, the long
square that is made of that whole lyne which crosseth the circle,
and the portion of it, that lyeth betwene the vtter circumference of
the circle and the pointe, shall be equall to the full square of the
other lyne, that onely toucheth the circle.
Example.
The circle is D.B.C, and the pointe without the circle is A, from whiche
pointe there is drawen one line crosse the circle, and that is A.D.C, and an
other lyne is drawn from the said pricke to the marge or edge of the
circumference of the circle, and doeth only touche it, that is the line A.B.
And of that first line A.D.C, you maie perceiue one part of it, whiche is
A.D, to lie without the circle, betweene the vtter circumference of it, and
the pointe assigned, whiche was A. Nowe concernyng the meanyng of the
Theoreme, if you make a longsquare of the whole line A.C, and of that
parte of it that lyeth betwene the circumference and the point, (whiche is
A.D,) that longesquare shall be equall to the full square of the touche line
A.B, accordyng not onely as this figure sheweth, but also the saied
nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the
other.

The lxxvii. Theoreme.
If a pointe be assigned without a circle, and from that pointe .ij.
right lynes be drawen to the circle, so that the one doe crosse the

circle, and the other dooe ende at the circumference, and that the
longsquare of the line which crosseth the circle made with the
portiõ of the same line beyng without the circle betweene the vtter
circumference and the pointe assigned, doe equally agree with the
iuste square of that line that endeth at the circumference, then is
that lyne so endyng on the circumference a touche line vnto that
circle.
Example.
In as muche as this Theoreme is nothyng els but the sentence of the last
Theoreme before conuerted, therfore it shall not be nedefull to vse any
other example then the same, for as in that other Theoreme because the one
line is a touche lyne, therfore it maketh a square iust equal with the
longsquare made of that whole line, whiche crosseth the circle, and his
portion liyng without the same circle. So saith this Theoreme: that if the
iust square of the line that endeth on the circumference, be equall to that
longsquare whiche is made as for his longer sides of the whole line, which
commeth from the pointt assigned, and crosseth the circle, and for his other
shorter sides is made of the portion of the same line, liyng betwene the
circumference of the circle and the pointe assigned, then is that line whiche
endeth on the circumference a right touche line, that is to saie, yf the full
square of the right line A.B, be equall to the longsquare made of the whole
line A.C, as one of his lines, and of his portion A.D, as his other line, then
must it nedes be, that the lyne A.B, is a right touche lyne vnto the circle
D.B.C. And thus for this tyme I make an ende of the Theoremes.
F I N I S,

I M P R I N T E D at London in Poules
churcheyarde, at the signe of the Bra-
senserpent, by Reynold Wolfe.

Cum priuilegio ad imprimen-
dum solum.
A N N O D O M I N I .M.D.L.I.

Title page:
The pathway to
KNOWLEDG,
CONTAI-
NING THE FIRST PRIN-
ciples of Geometrie, as they
may moste aptly be applied vn-
to practice, bothe for vse of

instrumentes Geome-
tricall, and astrono-
micall and
also for proiection of plattes in euerye
kinde, and therfore much ne-
cessary for all sortes of
men.

Dedication:
TO THE MOST NO-
ble and puissaunt prince Edwarde the
sixte by the grace of God, of En-
gland Fraunce and Ireland kynge, de-
fendour of the faithe, and of the
Churche of England and Ire-
lande in earth the su-
preme head.

Title page of Second Book:
THE SECOND BOOKE
OF THE PRINCIPLES
of Geometry, containing certaine
Theoremes, whiche may be cal-
led Approued truthes. And be as
it were the moste certaine
groundes, wheron the
practike cõclusions
of Geometry ar
founded.
[Leaf]
Whervnto are annexed certaine declarations by
examples, for the right vnderstanding of the
same, to the ende that the simple reader

might not iustly cõplain of hardnes
or obscuritee, and for the same
cause ar the demonstra-
tions and iust profes
omitted, vntill a
more conueni-
ent time.
1551.

“Theorems of Geometry”:
The Theoremes of Geometry, before
WHICHE ARE SET FORTHE
certaine grauntable requestes
which serue for demonstrations
Mathematicall.

Transcriber’s Notes
Terminology
Corrections
Illustrations
Notes
Greek

Language and Typography

This information is for readers who know more about geometry than about mid-16th-century
English.
The letters u and v follow the conventional “initial v, non-initial u” pattern except in numbers (xv,
iv). The lower-case j form occurs only as the last digit of a number (ij, xxj); upper-case I and J
share a form, always read as I. Italic double s was printed as an ſ+s ligature, similar to the
German ß; it is shown as simple “ss”. Capital and lower-case w were often used
interchangeably. Words split across line breaks may or may not have a hyphen.
The word “other” is used interchangeably with both “or” and “either”; similarly, “nother” is used
in place of “nor” and “neither”. The expression “an other” is almost always written as two words.
The spellings “then(ne)” and “than(ne)” are used interchangeably; “than(ne)” is rare. The
spelling “liyng”, both by itself and as the end of a longer word, is used consistently.

Terminology

Explanations of these terms are scattered through the book. They are grouped here for
convenience.

right line straight line
gemow (line) parallel
gemew = twin
square quadrilateral
(squire, also means angle square as described under hexagons
squyre) (“siseangles”)
likeiamme parallelogram
iam(me) = jamb = limb, side
longsquare rectangle
touch line tangent
cantle segment of a circle
cantle = slice

Corrections

Unless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were
regularized only where there was a very large disparity between the expected form and the

apparent errors (for example, a thousand “A.B” against a dozen “A,B”), or a flagrant misprint
such as “cnt” for “cut”.
Number forms such as “those. ij. last” or “line. A.B.” were silently regularized to “those .ij. last”
and “line .A.B.” Missing sentence-final periods at the end of a printed line were silently
supplied.
Unusual forms or combinations—especially phrase breaks where a comma is followed by a
capital letter, or a period by a lower-case letter—are unchanged but noted with popups.

Illustrations

A number of illustrations contain errors such as unmarked or mislabeled points (“circle B.C.D”
where only C and D are labeled). Labels added by the transcriber are shown in grey; moved or
transposed labels are shown in red, with mouse-hover explanation.

Notes

1) if they can with their wysedome ouercome all vyces. Of the firste of those three sortes
Text reads “... their rwysedome ... / ... those th ee sortes ...” on consecutive lines:

2) Pagination as shown by signature numbers demands another leaf (two pages) between the
end of the Preface and the beginning of the body text. But no text is missing, and the facsimile
has no blank pages.
3) and yet the ij. lesser sides togither ar greater then it.
Text reads “... yet thr / ... ar greate” at consecutive line-ends:

Greek

All Greek is shown as printed. Errors or anomalies include missing, misplaced or incorrect
diacritics; the letterform σ for ς; and word-final μ (mu) for ν (nu).
ἐίπερ γὰρ ἀδικεῖμ χρὴ, τυραννΐδος περῒ κάλλιστομ ἀδικεῖμ, τ’ ἄλλα δ’ ἐυσεβεῖμ χρεῶμ
eiper gar adikeim chrê, turannidos peri kallistom adikeim, t’ alla d’ eusebeim chreôm.
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.
Philippos Aristotelei chaireim.
esthi moi gegonota huom. pollêm oum tois theois charim echô, ouch houtôs epi tê
gennêsei tou paidos, hôs epi tô kata têm sêm hêlikiam autom gegonenai elpizô gar
autom hupo sou graphenta kai paideuthenta axiom esesthai kai heimôm kai tês tôm
pragmatôm diadochês.
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.
Alezandros Arisotelei eu pratteim.
Ouk orthôs epoiêsas ekdous tous akroamatikous tôm logôm, tini gar dioisomên
hêmeis tôm allôm, ei kath’ hous epaideuthêmen logous, houtoi pantôm esontai
koinoi, egô de bouloi mêm am tais peri ta arista empeiriais, ê tais dunamesi
diapherim. errôso.

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