1

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2    A touche lyne , is a line that runneth a long by the edge
3of a circle, onely touching it, but doth
4not crosse the circumference of it, as in
5this exaumple you maie see.
6    And when that a line doth crosse the
7edg of the circle, the~ is it called a cord ,
8as you shall see anon in the speakynge
9of circles.
10    In the meane season must I not omit
11to declare what angles bee called matche corners , that is to
12saie, suche as stande directly one against the other, when twoo
13lines be drawen acrosse, as here
14appereth.
15    Where A. and B. are matche corners,
16so are C. and D. but not A.
17and C. nother D. and A.
18    Nowe will I beginne to speak
19of figures, that be properly so called,
20of whiche all be made of diuerse
21lines, except onely a circle,
22an egge forme, and a tunne forme,
23which .iij. haue no angle and haue
24but one line for their bounde, and an eye fourme whiche is
25made of one lyne, and hath an angle onely.
26A circle is a figure made and enclosed with one line, and hath
27in the middell of it a pricke or centre, from whiche all the
28lines that be drawen to the circumfernece are equall all in
29length, as here you see.
30    And the line that encloseth the
31whole compasse, is called the
32circumference .
33    And all the lines that bee drawen
34crosse the circle, and goe by the centre,
35are named diameters , whose halfe, I
36meane from the center to the circumference

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37any waie, is called the semidiameter , or halfe
38diameter .
39    But and if the line goe crosse the circle,
40and passe beside the centre, then is
41it called a corde , or a stryng line ,
42as I said before, and as this exaumple
43sheweth: where A. if the corde.
44And the compassed line that aunswereth
45to it, is called an arche lyne , or
46a bowe lyne , whiche here is marked
47with B. and the diameter with C.
48    But and if that
49part be separate
50from the rest of
51the circle (as in
52this exa~ple you
53see) then ar both
54partes called ca~telles,
55the one
56the greatter cantle , as E. and the other the lesser cantle ,
57as D. And if it be parted iuste by the centre (as you see in F.)
58then is it called a semicircle , or halfe compasse .
59    Sometimes it happeneth that a cantle is cutte out with two
60lynes drawen from the centre to the circumference (as G. is)
61and then maie it be called a nooke cantle ,
62and if it be not parted from the reste
63of the circle (as you see in H.) then is it
64called a nooke plainely without any
65addicion. And the compassed lyne in it
66is called an arche lyne , as the exaumple
67here doeth shewe.

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68 Nowe haue you heard as touchyng
69circles, meetely sufficient instruction,
70so that it should seme nedeles to speake
71any more of figures in that kynde, saue
72that there doeth yet remaine ij. formes
73of an imperfecte circle, for it is lyke a
74circle that were brused, and thereby
75did runne out endelong one waie, whiche
76forme Geometricians dooe call an
77egge forme , because it doeth
78represent the figure and shape of
79an egge duely proportioned (as
80this figure sheweth) hauyng the
81one ende greater then the other.
82    For if it be lyke the figure of a circle pressed in length, and
83bothe endes lyke bygge, then is it called a tunne forme , or
84barrell forme , the right makyng of whiche figures, I wyll
85declare hereafter in the thirde booke.
86    An other forme there is, whiche you maie call a nutte forme,
87and is made of one lyne muche lyke an egge forme, saue that it
88hath a sharpe angle.
89    And it chaunceth sometyme that there is a right line drawen
90crosse these figures, and that is called an axelyne , or axtre .
91Howebeit properly that line that is called an axtre ,
92whiche gooeth thoroughe the myddell of a Globe, for as a
93diameter is in a circle, so is an axe lyne or axtre in a Globe,

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94that lyne that goeth from side to syde, and passeth by the middell
95of it. And the two poyntes that suche a lyne maketh in
96the vtter bounde or platte of the globe, are named polis , w^ch
97you may call aptly in englysh, tourne pointes : of whiche I
98do more largely intreate, in the booke that I haue written of
99the vse of the globe.
100    But to returne to the diuersityes of figures that remayne
101vndeclared, the most simple of them ar such ones as be made
102but of two lynes, as are the cantle of a circle , and the halfe
103circle , of which I haue spoken allready. Likewyse the halfe
104of an egge forme , the cantle of an egge forme , the halfe
105of a tunne fourme , and the cantle of a tunne fourme ,
106and besyde these a figure moche like to a tunne fourme, saue
107that it is sharp couered at both the endes,
108and therfore doth consist of twoo
109lynes, where a tunne forme is made
110of one lyne, and that figure is named
111an yey fourme .
112    The nexte kynd of figures are those
113that be made of .iij. lynes other be all right lynes, all crooked
114lynes, other some right and some crooked. But what fourme
115so euer they be of, they are named generally triangles. for a triangle
116is nothinge els to say, but a figure of three corners.
117And thys is a generall rule, looke how
118many lynes any figure hath, so mannye
119corners it hath also, yf it bee a platte
120forme, and not a bodye. For a bodye
121hath dyuers lynes metyng sometime
122in one corner.
123    Now to geue you example of triangles,
124there is one whiche is all of croked
125lynes, and may be taken fur a portio~
126of a globe as the figur marked w^t A
127    An other hath two compassed lines
128and one right lyne, and is as the portion
129of halfe a globe, example of B.
130    An other hatht but one compassed

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131lyne, and is the quarter of a circle, named a
132quadrate, and the ryght lynes make a right corner,
133as you se in C. Other lesse then it as you
134se D, whose right lines make a sharpe corner,
135or greater then a quadrate, as is F, and then
136the right lynes of it do make a blunt corner.
137    Also some triangles haue all righte lynes
138and they be distincted in sonder by their angles,
139or corners. for other their corners bee
140all sharpe, as you see in the figure, E. other ij.
141sharpe and one right square, as is the figure G
142other ij. sharp and one blunt as in the figure H
143    There is also an other distinction of the
144names of triangles, according to their sides,
145whiche other be all equal as in the figure E,
146and that the Greekes doo call Isopleuron ,
147and Latine men aequilaterium : and
148in english it may be called a threlike
149triangle , other els two sydes bee equall
150and the thyrd vnequall, which
151the Greekes call Isosceles , the Latine
152men aequicurio , and in english
153tweyleke may they be called, as in G,
154H, and K. For, they may be of iij. kinds
155that is to say, with one square angle, as
156is G, or with a blunte corner as H, or
157with all in sharpe korners, as you see
158in K.
159    Further more it may be y^t they haue
160neuer a one syde equall to an other,
161and they be in iij kyndes also distinct
162lyke the twilekes, as you maye perceaue
163by these examples. M. N, and O
164where M. hath a right angle, N, A,
165blunte angle, and O, all sharpe angles
166these the Greekes and latine men do

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167cal scalena
168and in englishe
169theye
170may be called
171nouelekes ,
172for thei
173haue no side
174equall, or
175like lo~g, to ani other in the same figur.
176Here it is to be noted, that in a tria~gle
177al the angles bee called innera~gles
178except ani side
179bee drawenne
180forth in lengthe,
181for then
182is that fourthe
183corner caled an
184vtter corner ,
185as in this exa~ple
186because A, B, is drawen in length, therfore
187the a~gle C, is called an vtter a~gle
188    And thus haue I done with tria~guled
189figures, and nowe foloweth quadrangles ,
190which are figures of iiij. corners
191and of iiij. lines also, of whiche there
192be diuers kindes, but chiefely
193v. that is to say, a square
194quadrate , whose sides bee
195all equall, and al the angles
196square, as you se here in this
197figure Q. The second kind
198is called a long square, whose foure corners
199be all square, but the sides are not
200equall eche to other, yet is euery side
201equall to that other that is against it, as
202you maye perceaue in this figure R.

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203    The thyrd kind is called losenges
204or diamondes , whose sides bee all equall,
205but it hath neuer a square corner,
206for two of them be sharpe, and the
207other two be blunt, as appeareth in, S.
208    The iiij. sorte are like vnto losenges,
209saue that they are longer one waye, and
210their sides be not equal, yet ther corners
211are like the corners of a losing, and therfore
212ar they named losengelike or diamo~dlike ,
213whose figur is noted with T
214Here shal you marke that al those squares
215which haue their sides al equal, may
216be called also for easy vnderstandinge,
217likesides , as Q. and S. and those that
218haue only the contrary sydes equal, as
219R. and T. haue, those wyll I call likeiammys ,
220for a difference.
221    The fift sorte
222doth containe all
223other fashions of
224foure cornered figurs,
225and ar called
226of the Grekes trapezia ,
227of Latin me~
228mensulae and of Arabitians, helmuariphe ,
229they may be called in englishe
230borde formes , they haue no syde equall
231to an other as these examples shew, neither keepe they
232any rate in their corners, and therfore are they counted vnruled
233formes , and the other foure kindes onely are counted
234ruled formes , in the kynde of quadrangles. Of these vnruled
235formes ther is no numbre, they are so mannye and so dyuers,
236yet by arte they may be changed into other kindes of figures,
237and therby be brought to measure and proportion, as
238in the thirtene conclusion is partly taught, but more plainly
239in my booke of measuring you may see it.

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240 And nowe to make an eande of the
241dyuers kyndes of figures, there dothe
242folowe now figures of .v. sydes, other
243v. corners, which we may call cinkangles ,
244whose sydes partlye are all equall
245as in A, and those are counted
246ruled cinkeangles . and partlye vnequall
247as in, B and they are called vnruled .
248    Likewyse shall you iudge of fifeanglessiseangles ,
249which haue fixesixe corners, septangles ,
250which haue seuen angles, and so forth, for as mannye
251numbres as there maye be of sydes and angles, so manye diuers
252
253    kindes be there of figures, vnto which yow shall geue
254names according to the numbre of their sides and angles, of
255whiche for this tyme I wyll make an
256ende, and wyll sette forthe on example
257of a syseangle, which I had almost forgotten,
258and that is it, whose vse commeth
259often in Geometry, and is called a
260squire , is made of two long squares ioyned
261togither, as this example sheweth.
262    And thus I make an eand to speake of
263platte formes, and will briefelye saye
264somwhat touching the figures of bodeis
265which partly haue one platte forme
266for their bound, and y^t iust rou~d
267as a globe hath, or ended long
268as in an egge , and a tunne
269fourme , whose pictures are
270these.
271    Howebeit you must marke
272that I meane not the very figure
273of a tunne, when I saye
274tunne form, but a figure like
275a tunne, for a tune fourme ,

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276hath but one plat forme, and therfore must needs be round at
277the endes, where as a tunne hath thre platte formes, and is
278flatte at eche end, as partly these pictures do shewe.
279    Bodies of two plattes are other cantles or halues of
280those other bodies, that haue but one platte forme, or els
281they are lyke in fvorme to two such cantles ioyned togither
282as this A doth partly eppresse: or els
283it is called a rounde spire , or triple
284fourme , as in this figure is some
285what expressed
286    Nowe of three plattes there are
287made certain figures of bodyes, as the
288cantels and halues of all bodyes that
289haue but ij. plattys, and also the halues
290of halfe globys and canteles of
291a globe. Lykewyse a rounde piller,
292and a spyre made of a rounde spyre,
293slytte in ij. partes long ways.
294    But as these formes be harde to be iudged by their pycturs,
295so I doe entende to passe them ouer with a great number of
296other formes of bodyes, which afterwarde shall be set forth
297in the boke of Perspectiue, bicause that without perspectiue
298knowledge, it is not easy to iudge truly the formes of them in
299flatte protacture.
300    And thus I make an ende for this tyme, of the definitions
301Geometricall, appertayning to this
302parte of practise, and the rest wil
303I prosecute as cause shall
304serue.
305

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306

SONDRY CONCLUSIONS GEOMETRICAL.

307

THE FYRST CONCLVSION.

308

TO MAKE A THRELIKE TRIANGLE OR ANY LYNE
309MEASURABLE.

310
311    Take the iuste
312le~gth of the lyne with your co~passe,
313and stay the one foot of the compas
314in one of the endes of that line, turning
315the other vp or doun at your
316will, drawyng the arche of a circle
317against the
318midle of the
319line, and doo like wise with the same
320co~passe vnaltered, at the other end of
321the line, and wher these ij. croked lynes
322doth crosse, frome thence drawe a
323lyne to echend of your first line, and
324there shall appear a threlike triangle
325drawen on that line.
326Example.
327A.B. is the first line, on which I wold
328make the threlike triangle, therfore I
329open the compasse as wyde as that line
330is long, and draw two arch lines that
331mete in C, then from C. I draw ij other
332lines one to A, another to B, and than
333I haue my purpose.
334
335

THE. II CONCLVSION.

336

IF YOU WIL MAKE A TWILIKE OR
337A NOUELIKE TRIANGLE ON ANI CERTAINE
338LINE.

339    Consider fyrst the length that yow will haue the other sides
340to containe, and to that length open your compasse, and

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341then worke as you did in the threleke triangle, remembryng
342this, that in a nouelike triangle you must take ij. lengthes besyde
343the fyrste lyne, and draw an arche lyne with one of the~
344at the one ende, and with the other at
345the other end, the exa~ple is as in the other
346before.
347
348

THE III. CONCL.

349

TO DIUIDE AN ANGLE OF RIGHT
350LINES INTO IJ. EQUAL PARTES.

351    First open your compasse as largely as you can, so that it do
352not excede the length of the shortest line y^t incloseth the angle.
353Then set one foote of the compasse in the verye point of
354the angle and with the other fote draw a compassed arch fro~
355the one lyne of the angle to the other,
356that arch shall you deuide in halfe, and
357the~ draw a line fro~ the a~gle to y^e middle
358of y^e arch, and so y^e angle is diuided
359into ij. equall partes. Example.
360Let the tria~gle be A.B.C, the~ set I one
361foot of y^e co~passe in B, and with the other
362I draw y^e arch D.E, which I part
363into ij. equall parts in F, and the~ draw
364a line fro~ B, to F, & so I haue mine inte~t
365
366

THE IIII. CONCL.

367

TO DEUIDE ANY MEASURABLE
368LINE INTO IJ. EQUALL PARTES.

369
370    Open your compasse to the iust le~gth of
371y^e line. And the~ set one foote steddely at
372the one ende of the line, & w^t the other
373fote draw an arch of a circle against y^e
374midle of the line, both ouer it, and also
375vnder it, then doo lykewaise

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376at the other ende of the line. And marke where those arche
377lines do meet crosse waies, and betwene those ij. pricks draw
378a line, and it shall cut the first line in two equall portions.
379Example.
380    The lyne is A.B. accordyng to which I open the compasse
381and make .iiij. arche lines, whiche meete in C. and D, then
382drawe I a lyne from C, so haue I my purpose.
383    This conclusion serueth for makyng of quadrates and squires,
384beside many other commodities, howebeit it maye bee
385don more readylye by this conclusion that foloweth nexte.
386
387

THE FIFT CONCLVSION.

388

TO MAKE A PLUMME LINE OR ANY PRICKE THAT
389YOU WILL IN ANY RIGHT LYNE APPOINTED.

390
391    Open youre compas so that it be not wyder then from the
392pricke appoynted in the line to the shortest ende of the line,
393but rather shorter. Then sette the one foote of the compasse
394in the firste pricke appointed, and with the other fote marke
395ij. other prickes, one of eche syde of that fyrste, afterwarde
396open your compasse to the wydenes of those ij. new prickes,
397and draw from them ij. arch
398lynes, as you did in the fyrst
399conclusion, for making of a
400threlyke tria~gle. then if you
401do mark their crossing, and
402from it drawe a line to your
403fyrste pricke, it shall bee a
404iust plum lyne on that place.
405Example.
406The lyne is A.B. the prick on
407whiche I shoulde make the plumme lyne, is C. then open I
408the compasse as wyde as A, C, and sette one foote in C.
409and with the other doo I marke out C.A. and C.B, then open
410I the compasse as wide as A.B, and make ij. arch lines which
411do crosse in D, and so haue I doone.
412    Howebeeit, it happeneth so sommetymes, that the

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413pricke on whiche you would make the perpendicular or plum
414line, is so nere the eand of your line, that you can not extende
415any notable length from it to thone end of the line, and if so be
416it then that you maie not drawe your line lenger fro~ that end,
417then doth this conclusion require a newe ayde, for the last deuise
418will not serue. In suche case therfore shall you dooe thus:
419If your line be of any notable length, deuide it into fiue partes.
420And if it be not so long that it maie yelde fiue notable partes,
421then make an other line at will, and parte it into fiue equall
422portio~s: so that thre of those partes maie be found in your line.
423Then open your compas as wide as thre of these fiue measures
424be, and sette the one foote of the compas in the pricke, where
425you would haue the plumme line to lighte (whiche I call the
426first pricke,) and with the other foote drawe an arche line
427righte ouer the pricke, as you can ayme it: then open youre
428compas as wide as all fiue measures be, and set the one foote in
429the fourth pricke, and with the other foote draw an other arch
430line crosse the first, and where thei two do crosse, thense draw
431a line to the poinct where you woulde haue the perpendicular
432line to light, and you haue doone.
433Example.
434    The line is A.B. and
435A. is the prick, on whiche
436the perpendicular
437line must light. Therfore
438I deuide A.B. into fiue
439partes equall, then do I
440open the compas to the
441widenesse of three partes
442(that is A.D.) and
443let one foote staie in A.
444and with the other I
445make an arche line in C.
446Afterwarde I open the
447compas as wide as A.B.

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448(that is as wide as all fiue partes) and set one foote in the .iiij.
449pricke, which is E, drawyng an arch line with the other foote
450in C. also. Then do I draw thence a line vnto A, and so haue
451I doone. But and if the line be to shorte to be parted into fiue
452partes, I shall deuide it into iij. partes only, as you see the line
453F.G, and then make D. andan other line (as is K.L.) whiche I
454deuide into .v. suche diuisions, as F.G. containeth .iij, then open
455I the compaas as wide as .iiij. partes (whiche is K.M.)
456and so set I one foote of the compas in F, and with the other I
457drawe an arch lyne toward H, then open I the co~pas as wide
458as K.L. (that is all .v. partes) and set one foote in G, (that is the
459iij. pricke) and with the other I draw an arch line toward H.
460also: and where those .ij. arch lines do crosse (whiche is by H.)
461thence draw I a line vnto F, and that maketh a very plumbe
462line to F.G, as my desire was. The maner of workyng of this
463conclusion, is like to the second conclusion, but the reason of it
464doth depe~d of the .xlvi. prorosicio~ of y^e first boke of Euclide.
465An other waie yet. set one foote of the compas in the prick, on
466whiche you would haue the plumbe line to light, and stretche
467forth thother foote toward the longest end of the line, as wide
468as you can for the length of the line, and so draw a quarter of a
469compas or more, then without stirring of the compas, set one
470foote of it in the same line, where as the circular line did begin,
471and extend thother in the circular line, settyng a marke where
472it doth light, then take half that quantitie
473more there vnto, and by that
474prick that endeth the last part, draw
475a line to the pricke assigned, and it
476shall be a perpendicular.
477Example.
478 A.B. is the line appointed, to whiche
479I must make a perpendicular line
480to light in the pricke assigned, which
481is A. Therfore doo I set one foote of
482the compas in A, and extend the other
483vnto D. makyng a part of a circle,

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484more then a quarter, that is D.E. Then do I set one foote
485of the compas vnaltered in D, and stretch the other in the circular
486line, and it doth light in F, this space betwene D. and F.
487I deuide into halfe in the pricke G, whiche halfe I take with
488the compas, and set it beyond F. vnto H, and therfore is H. the
489point, by whiche the perpendicular line must be drawen, so say
490I that the line H.A, is a plumbe line to A.B, as the conclusion
491would.
492
493

THE .VI. CONCLVSION.

494

TO DRAWE A STREIGHT LINE FROM ANY PRICKE
495THAT IS NOT IN A LINE, AND TO MAKE IT PERPENDICULAR
496TO AN OTHER LINE.

497
498    Open your compas so wide that it may extend somewhat farther,
499the~ from the prick to the
500line, then sette the one foote of
501the compas in the pricke, and
502with the other shall you draw
503a co~passed line, that shall crosse
504that other first line in .ij. places
505Now if you deuide that arch
506line into .ij. equall partes, and
507from the middell pricke therof
508vnto the prick without the
509line you drawe a streight line,
510it shalbe a plumbe line to that
511firste lyne, accordyng to the
512conclusion. Example.
513    C. is the appointed pricke, from whiche vnto the line A.B. I
514must draw a perpe~dicular. Therfore I open the co~pas so wide,
515that it may haue one foote in C, and thother to reach ouer the
516line, and with y^t foote I draw an arch line as you see, betwene
517A. and B, which arch line I deuide in the middell in the point
518D. Then drawe I a line from C. to D, and it is perpendicular
519to the line A.B, accordyng as my desire was.

page E4R

520

THE XXXIIJ. THEOREME.

521

IN ALL RIGHT ANGULED TRIANGLES, THE SQUARE OF
522THAT SIDE WHICHE LIETH AGAINST THE RIGHT ANGLE, IS
523EQUALL TO THE .IJ. SQUARES OF BOTH THE OTHER SIDES

524
525

EXAMPLE.

526
527    A.B.C. is a triangle, hauing
528a ryght angle in B. Wherfore
529it foloweth, that the square of
530A.C, (whiche is the side that
531lyeth agaynst the right angle)
532shall be as muche as the two
533squares of A.B. and B.C.
534which are the other .ij. sides.
535    By the square of any lyne,
536you muste vnderstande a figure
537made iuste square, hauyng
538all his iiij. sydes equall
539to that line, whereof it is the square, so is A.C.F, the square
540of A.C. Lykewais A.B.D. is the square of A.B. And B.C.E.
541is the square of B.C. Now by the numbre of the diuisions in
542eche of these squares, may you perceaue not onely what the
543square of any line is called, but also that the theoreme is true,
544and expressed playnly bothe by lines and numbre. For as you
545see, the greatter square (that is A.C.F.) hath fiue diuisions on
546eche syde, all equall togyther, and those in the whole square
547are twenty and fiue. Nowe in the left square, whiche is
548A.B.D. there are but .iij. of those diuisions in one syde, and
549that yeldeth nyne in the whole. So lykeways you see in the
550meane square A.C.E. in euery syde .iiij. partes, whiche in the
551whole amount vnto sixtene. Nowe adde togyther all the
552partes of the two lesser squares, that is to saye, sixtene and
553nyne, and you perceyue that they make twenty and fiue, whyche
554is an equall numbre to the summe of the greatter square.

page E4V

555    By this theoreme you may vnderstand a redy way to know
556the syde of any ryght anguled triangle that is vnknowen, so
557that you knowe the lengthe of any two sydes of it. For by
558tournynge the two sydes certayne into theyr squares, and so
559addynge them togyther, other subtractynge the one from the
560other (accordyng as in the vse of these theoremes I haue sette
561foorthe) and then fyndynge the roote of the square that remayneth,
562which roote (I meane the syde of the square) is the
563iuste length of the unknowen syde, whyche is sought for. But
564this appertaineth to the thyrde booke, and therefore I wyll
565speake no more of it at this tyme.
566
567

THE XXXIIIJ. THEOREME.

568

IF SO BE IT, THAT IN ANY TRIANGLE, THE SQUARE
569OF THE ONE SYDE BE EQUALL TO THE .IJ. SQUARES OF
570THE OTHER IJ. SIDES, THAN MUST NEDES THAT CORNER
571BE A RIGHT CORNER, WHICH IS CONTEINED BETWENE
572THOSE TWO LESSER SYDES.

573
574

EXAMPLE.

575
576    As in the figure of the laste Theoreme, bicause A.C, made
577in square, is as much as the square of A.B, and also as the square
578of B.C. ioyned bothe togyther, therefore the angle that is inclosed
579betwene those .ij. lesser lynes, A.B. and B.C. (that is
580to say) the angle B. whiche lieth against the line A.C, must nedes
581be a ryght angle. This teoremetheoreme dothe so depende of the
582truthe of the laste, that whan you perceaue the truthe of the
583one, you can not iustly doubt of the others truthe, for they
584conteine one sentence, contrary waies pronounced.
585
586

THE .XXXV. THEOREME.

587

IF THERE BE SET FORTH .IJ. RIGHT LINES, AND ONE
588OF THEM PARTED INTO SUNDRY PARTES, HOW MANY

page F1R

589OR FEW SO EUER THEY BE, THE SQUARE THAT IS MADE
590OF THOSE IJ. RIGHT LINES PROPOSED, IS EQUALL TO ALL
591THE SQUARES, THAT ARE MADE OF THE VNDIUIDED
592LINE, AND EUERY PARTE OF THE DIUIDED LINE.
593
594

EXAMPLE.

595
596    The ij. lines proposed ar A
597B. and C.D, and the lyne A.B.
598is deuided into thre partes by
599E. and F. Now saith this theoreme,
600that the square that is
601made of those two whole lines
602A.B. and C.D, so that the
603line A.B. sta~deth for the le~gth
604of the square, and the other
605line C.D. for the bredth of the same. That square (I say) willwil be
606equall to all the squares that be made, of the vndiueded lyne
607(which is C.D.) and euery portion of the diuededdiuided line. And to
608declare that particularly, Fyrst I make an other line G.K, equall
609to the line C.D, and the line G.H. to be equal to the line
610A.B, and to bee diuided into iij. like partes, so that G.M. is equall
611to A.E, and M.N. equal to E.F, and then muste N.H.
612nedes remaine equall to F.B. Then of those ij. lines G.K, vndeuided,
613and G.H. which is deuided, I make a square, that is
614G.H.K.L, In which square if I drawe crosse lines frome one
615side to the other, according to the diuisions of the line G.H,
616then will it appear plaine, that the theoreme doth affirme. For
617the first square G.M.O.K, must needes be equal to the square
618of the line C.D, and the first portio~ of the diuided line, which
619is A.E, for bicause their sides are equall. And so the seconde

page F1V

620square that is M.N.P.O, shall be equall to the square of C.D,
621and the second part of A.B, that is E.F. Also the third square
622which is N.H.L.P, must of necessitee be equal to the square
623of C.D, and F.B, bicause those lines be so coupeled that euery
624couple are equall in the seuerall figures. And so shal you not
625only in this example, but in all other finde it true, that if one
626line be deuided into sondry partes, and another line whole
627and vndiuided, matched with him in a square, that square
628which is made of these two whole lines, is as muche iuste and
629equally, as all the seuerall squares, whiche bee made of the
630whole line vndiuided, and euery part seuerally of the diuided
631line.
632
633

THE XXXVI. THEOREME.

634
635

IF A RIGHT LINE BE PARTED INTO IJ. PARTES, AS
636CHAUNCE MAY HAPPE, THE SQUARE THAT IS MADE OF
637THAT WHOLE LINE, IS EQUALL TO BOTHE THE SQUARES
638THAT ARE MADE OF THE SAME LINE, AND THE TWOO
639PARTES OF IT SEUERALLY.

640
641

EXAMPLE.

642
643    The line propouned beyng A.B. and deuided, as chaunce happeneth,
644in C. into ij. unequall partes,
645I say that the square made of the hole
646line A.B, is equal to the two squares
647made of the same line with the twoo
648partes of it selfe, as with A.C, and
649with C.B, for the square D,E.F.G.
650is equal to the two other partial squares
651of D.H.K.G and H.E.F.K, but
652that the greater square is equall to the
653square of the whole line A.B, and the

page F2R

654partiall squares equall to the squares of the second partes of
655the same line ioyned with the whole line, your eye may iudg
656without muche declaracion, so that I shall not neede to make
657more exposition therof, but that you may examine it, as you
658did in the laste Theoreme.
659
660

THE XXXVIJ THEOREME.

661
662

IF A RIGHT LINE BE DEUIDED BY CHAUNCE, AS IT
663MAYE HAPPEN, THE SQUARE THATEdition: HAT IS MADE OF THE
664WHOLE LINE, AND ONE OF THE PARTES OF IT WHICH
665SOEUER IT BE, SHAL BE EQUALL TO THAT SQUARE THAT IS
666MADE OF THE IJ. PARTES IOYNED TOGITHER, AND TO
667AN OTHER SQUARE MADE OF THAT PART, WHICH WAS
668BEFORE IOYNED WITH THE WHOLE LINE.

669
670

EXAMPLE.

671
672    The line A.B. is deuided
673in C. into twoo
674partes, though not equally, of which two
675partes for an example
676I take the first, that is
677A.C, and of it I make
678one side of a square,
679as for example D.G.
680accomptinge those two lines to be equall, the other side of the
681square is D.E, whiche is equall to the whole line A.B.
682Now may it appeare, to your eye, that the great square made
683of the whole line A.B, and of one of his partes that is A.C,

page F2V

684(which is equall with D.G.) is equal to two partiall squares,
685wherof the one is made of the saide greatter portion A.C, in
686as muche as not only D.G, beynge one of his sides, but also D.
687H. beinge the other side, are eche of them equall to A.C. The
688second square is H.E.F.K, in which the one side H.E, is equal
689to C.B, being the lesser parte of the line, A.B, and E.F. is equall
690to A.C. which is the greater parte of the same line. So
691that those two squares D.H.K.G, and H,E,F,K, bee bothe of
692them no more then the greate square D.E,F,G, accordinge to
693the wordes of the Theoreme afore saide.
694
695

THE XXXVIIJ. THEOREME.

696
697

IF A RIGHTE LINE BE DEUIDED BY CHAUNCE, INTO
698PARTES, THE SQUARE THAT IS MADE OF THAT WHOLE
699LINE, IS EQUALL TO BOTH THE SQUARES THAT AR MADE
700OF ECHE PARTE OF THE LINE, AND MOREOUER TO TWO
701SQUARES MADE OF THE ONE PORTION OF THE DIUIDED
702LINE IOYNED WITH THE OTHER IN SQUARE.

703
704

EXAMPLE.

705
706    Lette the diuided line bee A,B,
707and parted in C, into twoo partes:
708Nowe saithe the Theoreme, that
709the square of the whole lyne A,B,
710is as mouche iuste as the square
711of A.C, and the square of C.B., eche
712by it selfe, and more ouer as
713muche twise, as A.C. and C.B.

page F3R

714ioyned in one square will make. For as you se, the great square
715D.E.F.G, conteyneth in hym foure lesser squares, of whiche
716the first and the greatest is N.M.F.K, and is equall to the
717square of the lyne A.C. The second square is the lest of them
718all, that is D.H.L.N, and it is equall to the square of the
719line B.CC.B. Then are there two other longe squares both of one
720bygnes, that is H.E.N.M. and L.N.G.K, eche of them both
721hauyng .ij. sides equall to A.C, the longer parte of the diuided
722line, and there other two sides equall to C.B, beeyng the
723shorter parte of the said line A.B.
724    So is that greatest square beeyng made of the hole lyne A.
725B, equal to the ij. squares of eche of his partes seuerally, and
726more by as muche iust as .ij. longe squares, made of the longer
727portion of the diuided lyne ioyned in square with the
728shorter parte of the same diuided line as the theoreme wold.
729And as here I haue put an example of a lyne diuided into .ij.
730partes, so the theoreme is true of all diuided lines, of what
731number so euer the partes be, foure, fyue, or syxe. etc.
732    This theoreme hath great vse not only in geometrie, but also
733in arithmetike, as herafter I will declare in conuenient place
734
735

THE .XXXIX. THEOREMETHEOREME.

736
737

IF A RIGHT LINE BE DEUIDED INTO TWO EQUALL PARTES,
738AND ONE OF THESE .IJ. PARTES DIUIDED AGAYN
739INTO TWO OTHER PARTES, AS HAPPENETH THE LONGE
740SQUARE THAT IS MADE OF THE THYRD OR LATER PART
741OF THAT DIUIDED LINE, WITH THE RESIDUE OF THE
742SAME LINE, AND THE SQUARE OF THE MYDLEMOSTE
743PARTE, ARE BOTHE TOGITHER EQUALL TO THE SQUARE
744OF HALFE THE FIRSTE LINE.

page F3V

745

EXAMPLE.

746
747    The line A.B. is diuided into
748ij. equal partes in C, and
749that parte C.B. is diuided agayne as hapneth
750in D. WherfereWherfore saith the
751Theorem that the long
752square made of D.B.
753and A.D, with the square
754of C.D. (which is the
755mydle portion) shall bothe be equall to the square of half the
756lyne A.B, that is to saye, to the square of A.C, or els of C.
757D, which make all one. The long square F.G.N.O. whiche is
758the longe square that the theoreme speaketh of, is made of .ij.
759long squares, wherof the fyrst is F.G.M.K, and the seconde
760is K.N.O.M. The square of the myddle portion is L.M.
761O.P. And the square of the halfe of the fyrste lyne is E.K.
762Q.L. Nowe by the theoreme, that longe square F.G.M.
763O, with the iuste square L.M.O.P, muste bee equall to the
764greate square E.K.Q.L, whyche thynge bycause it seemeth
765somewhat difficult to vnderstande, althoughe I intende not
766here to make demonstrations of the Theoremes, bycause it
767is appoynted to be done in the newe edition of Euclide, yet I
768wyll shew you brefely how the equalitee of the partes doth
769stande. And fyrst I say, that where the comparyson of equalitee
770is made betweene the greate square (whiche is made of
771halfe the line A.B.) and two other, where of the fyrst is the
772longe square F.G.N.O, and the seconde is the full square L.
773M.O.P, which is one portion of the great square allredye,
774and so is that longe square K.N.M.O, beynge a parcell also
775of the longe square F.G.N.O, Wherfore as those two partes
776are common to bothe partes compared in equalitee, and
777therfore beynge bothe abated from eche parte, if the reste of
778bothe the other partes bee equall, than were those whole partes
779equall before: Nowe the reste of the great square, those

page F4R

780two lesser squares beyng taken away,Original has a closing bracket ')' with no corresponding opening bracket is that longe square E.
781N.P.Q, whyche is equall to the long square F.G.K.M, beyng
782the rest of the other parte. And that they two be equall,
783theyr sydes doo declare. For the longest lynes that is F.K and
784E.Q are equall, and so are the shorter lynes, F.G, and E.N,
785and so appereth the truthe of the Theoreme.
786
787

THE .XL. THEOREME.

788
789

IF A RIGHT LINE BE DIUIDED INTO .IJ. EUEN PARTES,
790AND AN OTHER RIGHT LINE ANNEXED TO ONE ENDE
791OF THAT LINE, SO THAT IT MAKE ONE RIGHTE LINE
792WITH THE FIRSTE. THE LONGE SQUARE THAT IS MADE
793OF THIS WHOLE LINE SO AUGMENTED, AND THE PORTION
794THAT IS ADDED WITH THE SQUARE OF HALFE THE
795RIGHT LINE, SHALL BE EQUALL TO THE SQUARE OF THAT
796LINE, WHICHE IS CONPOUNDEDCOMPOUNDED OF HALFE THE FIRSTE
797LINE, AND THE PARTE NEWLY ADDED.

798
799

EXAMPLE.

800
801    The fyrst lyne propouned is
802A.B, and it is diuided into
803ij. equall partes in C, and an
804other ryght lyne, I meane
805B.D. annexed to one ende
806of the fyrste lyne.
807Nowe say I, that the long
808square A.D.M.K, is made
809of the whole lyne so augme~ted,
810that is A.D, and the portio~ annexed, y^t is D.M. for D.M
811is equall to B.D, wherfore y^t long square A.D.M.K, with the

page F4V

812square of halfe the first line, that is E.G.H.L, is equall to the
813great square E.F.D.C. whiche square is made of the line C.
814D. that is to saie, of a line compounded of halfe the first line,
815beyng C.B, and the portion annexed, that is B.D. And it is
816easyly perceaued, if you consyder that the longe square A.C.
817L.K. (whiche onely is lefte out of the great square) hath another
818longe square equall to hym, and to supply his steede
819in the great square, and that is G,F.M.H. For theytheir sydes be
820of lyke lines in length.
821
822

THE XLI. THEOREME.

823
824

IF A RIGHT LINE BEBEE DIUIDED BY CHAUNCE, THE
825SQUARE OF THE SAME WHOLE LINE, AND THE SQUARE
826OF ONE OF HIS PARTES ARE IUSTE EQUALL TO THE LO~G
827SQUARE OF THE WHOLE LINE, AND THE SAYDE PARTE
828TWISE TAKEN, AND MORE OUER TO THE SQUARE OF
829THE OTHER PARTE OF THE SAYD LINE.

830
831

EXAMPLE.

832
833    A.B. is the line diuided in C. And
834D.E.F.G, is the square of the whole
835line, D.H.K.M. is the square of
836the lesser portion (whyche I take
837for an example) and therfore must bee
838twise reckened. Nowe I saye that
839those ij. squares are equall to two
840longe squares of the whole line A.
841B, and his sayd portion A.C, and also
842to the square of the other portion
843of the sayd first line, whiche portion
844is C.B, and his square K.N.F.L In this theoreme there is
845no difficultie, if you co~syder that the litle square D.H.K.M.
846is iiij. tymes reckened, that is to say, fyrst of all as a parte of
847the greatest square, whiche is D.E.F.G. Secondly he is rekned

page G1R

848by him selfe. Thirdely he is accompted as parcell of the long
849square D.E.N.M, And fourthly he is taken as a part of the other
850long square D.H.L.G, so that in as muche as he is twise
851reckened in one part of the compariso~ of equalitee, and twise
852also in the second parte, there can rise none occasion of errour
853or doubtfulnes therby.