Chapter IV: Part 4

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 w^t one example, for of thys last sentence many men at the first hearing do make a doubt. Therfore as in this example of the circle deuided into sundry partes it doeth appere that no parte can be so great as the whole 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 season take for an example this square figure following A.B.C.D, w^{ch} 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, y^t hath for his groundes 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 consequently of the other artes mathematical, which haue the grounds (as Arithmeticke, musike and astronomy) aboue all other artes and sciences, that be vsed amongest 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 them 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, w^{ch} is the third line in y^e second triangle & y^e hole triangle. A.B.C. must nedes be equal to y^e 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 ground
line, are equal togither. And if the sides that be equal, be
drawen out in length then 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 from 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 begin
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 then 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 triangles haue there ij. sides equal one to an other,
and their ground 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 ground 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
them, ar lesse then 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 then 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 y^t 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 angle 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 then that
other triangle, then is his angle that lieth betwene the
equall sides, greater then 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.

[Sec.] 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.

_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 ground or ground 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 twenty 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 y^e 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
drawen 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, y^t their ground 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 vnderstandyng of this theorem. [Sidenote: _Bias lyne._] 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, those doo I call _Fyll squares_, [Sidenote: _Fyll squares._] accordyng to the greke name, which is _anapleromata_, [Sidenote: #anaple:ro:mata#] and called in latin _supplementa_, bycause that they make one generall square, includyng and enclosyng the other diuers squares, as in this example 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.

[Sec.] 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. standeth for the length 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 portion 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._

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.M.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 augmented, that is A.D, and the portion annexed, y^t is D.M, for D.M is equall to B.D, wherfore y^t 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 long 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 consyder 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 comparison 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 comprehend 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 chaunce
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 theoreme
is verified in the
figures of this
example.

_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._