The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men.

The theoremes of Geometry brieflye declared by shorte examples.

The firste Theoreme. When .ij. triangles be so drawen, that the one of thē hath ij. sides equal to ij. sides of the

other triangle, and that the angles enclosed with those sides, bee equal also in bothe trian∣gles, 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

[illustration] diagram

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 bee∣tweene F.G, and G.H, for both of them answere to the

[illustration] diagram

eight parte of a circle. Ther¦fore doth it remayne that A. B. whiche is the thirde lyne in the firste triangle, doth a∣gre in lengthe with F.H, w^ch is the third line in y^t secōd tri¦āgle & y^e hole triāgle. A.B.C. must nedes be equal to y^e hole tri¦angle 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 groud line, are equal togither. And if the sides that be equal, be drawē out in lēgth thē wil the corners that are vnder the ground line, be equal also togither.

Example

A.B.C. is a twileke triangle, for

[illustration] diagram

the one side A.C, is equal to the o∣ther 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 to gither. 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 an∣gles 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 a∣gainst those angles, be equal also.

Example

This triangle A.B.C. hath two corners e∣qual

[illustration] diagram

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 frō 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 begī at the same poin¦tes, and end in anie other pointe then the twoo first did.

Example.

The first line is A.B, on which I haue

[illustration] diagram

erected two other lines A.C, and B. C, that meete in the pricke C, where∣fore 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 li¦nes shalbe vnequa••, 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 youse in this ex¦ample A. D. is longer thē 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 triāgles haue there ij. sides equal one to another, and their groūd lines equal also, then

shall their corners, whiche are contained be∣twene 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.

[illustration] diagram

For A. C. is equall to D. F, and B.C. is equall to E. F, and again their groūd lines A.B. and D.E. are lyke in length, therfore is eche angle of the one triangle equall to ech an¦gle 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.

[illustration] diagram

A.B. is a right line, and on it there doth light another right line, drawen from C. perpen∣dicularly on it, therefore saie J, that the .ij. angles that thei do make, are .ij. right angles as maie be iudged by the defi∣nition of a right angle. But in the second part of the exam∣ple, where A.B. beyng still the right line, on whiche D. stan∣deth

inslope 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,

[illustration] diagram

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 cor∣ners (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 crosse ways they do make their matche angles equall.

Example.

What matche angles are, I

[illustration] diagram

haue tolde you in the defini∣tions 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 an∣gle is greater than any of the two inner cor∣ners, that ioyne not with it.

Example.

The triangle A. D. C

[illustration] diagram

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 in∣ner corners that lye a∣gainste 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 e∣uer you take thē, ar lesse thē ij. right corners.

Example.

In the firste triangle E, whiche is a

[illustration] diagram

threlyke, and therfore hath all his an∣gles sharpe, take anie twoo corners that you will, and you shall perceiue that they be lesser then .ij. right cor∣ners, for in euery triangle that hath all sharpe corners (as you see it to be

[illustration] diagram

in this example) euery corner is lesse then a right corner. And therfore al∣so euery two corners must nedes be lesse then two right corners. Fur∣thermore in that other triangle mar∣ked 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 tri∣angles, as you maie perceiue more plainly by the .xxij. Theo∣reme.

The .xi. Theoreme. In euery triangle, the greattest side lieth against the greattest angle.

Example.

As in this triangle A. B. C,

[illustration] diagram

the greattest angle is C. And A. B. (whiche is the side that lieth against it) is the greatest and longest side. And contra∣ry waies, as A. C. is the shor∣test side, so B. (whiche is the angle liyng against it) is the

smallest and sharpest angle, for this doth folow also, that as 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 thē the thirde.

For example you shal take this triangle A.B.C. which hath a vee¦ry

[illustration] diagram

blunt corner, and therfore one of his sides greater a good deale then any of the other, and yet thr ij. lesser sides togither ar greate 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.

[illustration] diagram

A.B.C. is a triangle. on whose ground line A.B. there is dra¦wen ij. lines, from the ij. endes of it, I say from A. and B, and they meete within the trian∣gle 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 āgle 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.

[illustration] diagram

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. an∣gles contayned betwene the equal ly∣nes)

therfore muste the

[illustration] diagram

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 thē that other triangle, then is his angle that lieth betwene the equall sides, greater thē 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 of such sort, that two angles of the one be equal to ij. angles of the o∣ther, 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.

[illustration] diagram

Bicause that A. B. C, the one triangle hath two cor¦ners 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 e∣quall, that is A. B, which is equa••l 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 shalbe equall, that is, the angle C. shalbe equall to t 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.

[illustration] diagram

And bycause that E. F. ma∣keth 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 mat∣che corners, those that go one way as doth E. G, and K. F, lykeways 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 crosse waies, and maketh the ij. ouer corners towarde one hande equall to∣gither, 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 an example the figure of the last theo∣reme, where A. B, and C. D, be called paralleles also, bicause E. and K, (whiche are .ij. ouer corners) are equall, and lyke waies 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 pa∣ralleles, because the ij. inner corners. As for example those two that bee toward the right hande (that is G. and L.) are e∣quall

(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 cor∣ners of the one line, equall to two matche cor∣ners of the other line, and also bothe ouer cor¦ners of one hande equall togither, and bothe nether corners likewaies, and more ouer two inner corners, and two vtter corners also to∣warde 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 .iij. theore∣mes to tende to this purpose, to declare by the angles how to iudge paralleles, and contrary waies howe you may by pa∣ralleles 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 parel∣lele

[illustration] diagram

vnto C. D. and E. F, lyke waies 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

[illustration] diagram

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 bothe the in∣ner corners that lye a∣gaynst 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 a∣ny 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 togy∣ther (by the fourth request) therfore must it nedes folow, that all the thre corners of euery triangle (accomptyng them to∣gyther) are equall to iij. corners of any other triangle, taken all togyther.

The .xxiii. theoreme. when any ij. right lines doth touche and cou∣ple .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 e∣quall

[illustration] diagram

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 to∣warde 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.

[illustration] diagram

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, by∣cause 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 pa∣ralleles, are equal togither.

Example.

Fyrste you muste marke the difference betwene this Theo∣reme 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 like iamme, only meaning them to be equal in length, though they be diuers in numbbe. 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 groud 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 twē¦ty 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 e∣quall to one thirde line (that is C.D.) they must nedes bee e∣quall togyther by the firste common sentence.

The xxvii. Theoreme. All triangles hauinge one grounde lyne; an standing betwene one paire of parallels, ar equall togither.

Example.

[illustration] diagram

A.B. and C. F. are twoo gemowe lines, betweene which there be made two tri¦angles, 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 ge∣mow 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 li∣nes 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 fixe 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 com¦men 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 e∣quall, 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. A∣gaine A.D.E, is equal to eche of them both, for his ground line D.E, is equall to A. B, in so muche as they are the contrary si∣des 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. Alequal 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 on ground line D.E. And ag aine they rise towarde one side, that is to say, vp warde toward the line A. B, wher¦fore they must needes be inclosed, betweene one paire of pa∣rallels, which are heere in this example A.B. and D.E.

The thirty Theoreme. Equal triangles that haue the irground lines equal, and be drawē 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 groūd lines are equall togither, that is C.D, and E. F, now this bee∣ynge true, and considering that they are made towarde one side, it foloweth, that they are made betwene one paire of pa¦rallels 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 be∣twene two paire of parallels, presupposinge one to bee dra∣wen by their ground line, and then do they ryse toward con¦trary 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. ge∣mow

[illustration] diagram

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 lykeiamme A.B.G.C. is double to the triangle B. C. G. For euery halfe of that lyke∣iamme 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 like¦iamme 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 dou∣ble to the triangle, so the halfe of it, must needes be equall to the triangle. Compare the .xv. 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 ex¦amples,

[illustration] diagram

it shalbe mete to shew the true vnderstādyngof this heorem. Therfore by the Bi∣as line,* 1.1 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, ac∣cordyng to the greke name,* 1.2 whiche is anapleromata, and called in latin supplementa, bycause that they make one ge∣nerall square, includyng and enclosyng the other diuers squa∣res, as in this exāple 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. syll squares, for they doo syll vp the sydes of the .ij. fyrste square sykeiammes, in suche sorte, that of 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 squres. 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 squres of two likeiammes made aboute one bias line, as the exaumple sheweth. Conferre the twelfthe con∣clusion with this the oreme.

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

[illustration] diagram

a ryght angle in B. Wherfore it soloweth, 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 syne, you muste vnderstande a fi∣gure made iuste square, ha∣uyng 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, why∣che 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 theorem es I haue sette foorthe) and then fyndynge the roote of the square that re∣mayneth, 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 quare of B.C. ioyned bothe togyther, therefore the angle that is in∣closed 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 ne∣des 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.

[illustration] diagram

The ij. lines proposed ar A. B. and C.D, and the lyne A.B.

[illustration] diagram

is deuided into thre partes by E. and F. Now saith this theo∣reme,

[illustration] diagram

that the square that is made of those two whole li∣nes A.B. and C.D, so that the line A.B. stādeth for the lēgth 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, e∣quall 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 e∣quall 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, vn∣deuided, 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 portiō 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 vndiuided, 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 diui∣ded line.

The xxxvi. Theoreme. If a right line be parted into ij. partes, as chaunce may happe, the square that is made of that 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 propouned beyng A.B. and deuided, as chaunce hap∣peneth, in C. into ij. vnequall partes,

[illustration] diagram

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 it selfe, as with A. C, and with C. B, for the square D, E.F.G. is equal to the two other partial squa¦res 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: hat is made of the whole line, and one of the partes of it which so euer it be, shal be equall to that square that is made of the ij. partes ioyned togither, and to another square made of that part, which was before ioyned with the whole line.

Example.

The line A.B. is de¦uided

[illustration] diagram

in C. into twoo partes, though not e∣qually, 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, wherof 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 e∣quall 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,

[illustration] diagram

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 youse, 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 A. 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 diui∣ded line, and there other two sides equall to C.B, beeyng the 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 lon∣ger 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 ingeometrie, but also in arithmetike, as herafter I will declare in conuenient place

The .xxxix. theoreme. If a right line be deuided into two equall par∣tes, 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

[illustration] diagram

ij. equal partes in C, and that parte C.B. is diui∣ded agayne as hapneth in D. Wherfore saith the Theorem that the long square made of D. B. and A.D, with the squa¦re of C.D. (which is the L 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 halft 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 equa∣litee 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 seconde 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.O.O, Wherfore as those two par∣tes are common to bothe partes compared in equalitee, and therfire ••eynge bothe abated from eche parte, if the reste of bothe the other partes bee equall, than were those whole par¦tes equall before: Nowe the resle 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, be∣yng 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 par∣tes, 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 por∣tion 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 propouned is A.B, and it is diuided into

[illustration] diagram

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 aug∣mēted) that is A.D, and the portiō 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 a nother 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 rightline bee diuided by chaunce, the square of the same whole line, and the square of one of his partes are iuste equall to the lōg 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

[illustration] diagram

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 al∣so to the square of the other portion of the sayd first line, whiche porti∣on is C. B, and his square K. N. F. L. In this theoreme there is no difficultie, if you cōsyder 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 o∣ther long square D.H.L.G, so that in as muche as he is twise reckened in one part of the comparisō 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 happe∣neth 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 vn∣equall

[illustration] diagram

partes as happeneth. the longsquare of yt and his lesser portion A. C, is foure

[illustration] diagram

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 appea∣reth that one of the little squares (I meane K.L.P O) is reckened twise, ones as par cell of the second longsquare and agayne as parte of the

thirde longsquare, 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 accompethy as per∣cell of any one of them, and then haue you iiij. long squares di¦stinctly 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 comprehēd in it all the other fiue squares, that is to say, foure longsquares (as is before mencioned) and one full square. which is the in∣tent of the Theoreme.

The xliij. Theoreme. If a right line be deuided into ij. equal par∣tes first, and one of those parts again into o∣ther ij. parts, as chaūce hapeneth, the square that is made of the last part of the line so di∣uided, 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 por∣tion 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 par¦tes 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

[illustration] diagram

square of A.D, which is the residue of the whole line. Those two squares, I say, ar double to the square of the 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 F. 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 big∣nes, by reason that I did diuide C.B. into two equall partes, so that you may 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 eayser is to be vnderstande, bicause that the greate square hath in his side iij. quarters of the firste line, whiche multiplied by it selfe 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 e∣qually, 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 double 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.

[illustration] diagram

The line is A.B, and is di¦uided firste tnto twoo e∣qual partes in C, and thē is there annexed to it an other portion whiche is B.D. Now saith the The¦oreme, 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. cōtaining four par¦tes, then must needes his halfe containe ij. partes of such partes I suppose B.D. (which is the ānex¦ed line) to containe thre, so shall the hole line cōprehend vij. parts, and his square xlix. parts, whervnto if you ad y^e 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, wher∣by 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 (in∣closyng the blunt corner) and that 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 dra∣wen from one of the sharpe angles of the fore∣sayd triangle.

Example.

For the declaration of this theoreme and the next also, whose vse are wonderfull in the practise of Geometrie, and in mea¦suryng especially, it shall be nedefull to declare that euery tri¦angle that hath no ryght angle, as those be whyche are called (as in the boke of practise is declared) sharp cornered trian∣gles, and blunt couered triangles, yet may they be brought to haue a ryght angle, eyther by partyng them into two lesser tri¦angles,

or els by addyng an other triangle vnto them, whiche may be a great helpe for the ayde of measuryng, as more large∣ly shall be sette foorthe in the boke of measuryng. But for this present place, this forme wyll I vse, (whiche Theon also v∣seth) 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 blunte cor∣nered triangle may the better bee knowen.

Fyrst ther∣fore

[illustration] diagram

I sette foorth the tri¦angle A.B.C, whose cor∣ner by C. is a blunt corner as you maye well iudge, than to make an other tri∣angle of yt with a ryght angle, I must drawe forth the side B.C. vnto D, and frō the sharp corner by A. I brynge a plumbe lyne or perpēdi∣cular 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 be∣twene 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 Theo∣reme, 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.

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 perpen∣dicular line falleth, and the portion of that same line, liyng betweene the perpendicular, and the foresaid sharpe corner.

Example.

Fyrst I sette

[illustration] diagram

foorth the tri¦angle A. B.C, and in yt I draw a plūbe line from the angle C. vnto the line A. B, and it lighteth in D. Nowe by the theore∣me 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 mu¦che as is twise conteyned in the lōg 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 C. 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. Wher¦by you may consyder agayn, an other proportion of equalitee,

that is to saye, that the square E. with the twoo long squares 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 ly∣eth againste it, the square of anye one side, with the ij. longesquares made of 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 per¦pendiculare falseth, 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 theo∣reme 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 bene∣fite, no manne that hathe not well wayde their wonderfull commoditee, woulde credite the posibilitie of their wonder∣full vse, and large ayde in knowledge. But all this wyll I re∣mitte to a place conuenient.

The xlvij Theoreme. If ij. points be marked in the circumferēce of a circle, and a right line drawen frome the one to the other, that line must needes fal with in 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

[illustration] diagram

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, inany of these ca¦ses you see, that the line that is dra∣wen from the one pricke to the other dothe euermore run within the edge of the circle, els canne it be no right line. Howbeit, that a croked line, especially being more cro¦ked 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 be 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

[illustration] diagram

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, acordinge 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 o∣ther into two equall partions.

Example.

[illustration] diagram

The circle is A. B. C. D, and the cen¦tre is E, the one line A. C, and the o∣ther 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 cutte the other into equall portions. For as you may easily iudge, A C. hath one portiō lō¦ger and an other shorter, and so like wise B. D. Howbeit, it is not so to be vnderstād, but one of them may be diuided into ij. euē parts,

but bothe to bee cutte equally in the middle, is not possible, onles both passe through the cētre, therfore much rather whē 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

[illustration] diagram

so manifest, that it neadeth nother demonstration nother declaraciō. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be drawē, 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 a sondre, as the edges of the circles, where they bee moste 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

[illustration] diagram

B. D. E, and his centre is by F, so that it is easy enough to perceiue that their centres doe dyffer as muche a sonder, as the halfe dia∣meter of the greater circle is lō∣ger then the half diameter of the lesser circle. And so must it nee∣des be thought and said of all o∣ther circles in lyke kinde.

The .lij. theoreme. If a certaine pointe be assigned in the dia∣meter 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 pas∣seth by the centre, and the shortest is the resi∣dew 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 cōtra∣ry waies, that is shortest, that is farthest from it. And amongest thē all there can be but one∣ly .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.

Example.

[illustration] diagram

The circle is A. B. C. D. E. H, and his centre is F, the diame∣ter is A. E, in whiche diameter I haue taken a certain point di∣staunt from the centre, and that pointe is G, from whiche I haue drawen .iiij. lines to the circum∣ference, 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 reside we 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 longèr 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.B, betweene whiche the lyne G. E, (as the shortest in all the circle) doothe stande euen distaunte from eche of them, and that 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 dia∣meter, so that the saide poynte be (as the Theoreme doeth suppose) somewhat distaunt from the centre of the said cir∣cle. For if they be drawen from the centre, then must they of necessitee be all equall, howe many so euer they bee, is 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 The∣oreme, by neerenesse and distaunce is vnderstand the nere∣nesse 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 longest of all them that crosse the circle. And of thother 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 thē, 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 lēgth▪ and they two muste be on the two contrarie si∣des of the shortest line.

Example.

[illustration] diagram

Take the circle to be A. B. C, and the point assigned without it to be D. Now say J, that if there be drawen sundrie lines from D, and crosse the circle, endyng in the circumference on the cōtrary 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 a∣mongest the rest. And contrarie waies, D. B, is the shorteste, be∣cause 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 like∣waies 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 whi∣che 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 therfore also in shortnes, so D.G, is farthest from it in distance, and thērfore is the lon∣gest of them. Now D.H, beyng nerer then D. G, is also shor∣ter

then it, and beynge farther of, then D.K, is longer then it▪ So that for this parte of the theoreme (as J think) you do plain ly perceaue the truthe thereof, so the residue hathe no dif∣ficulte. For seing that the nearer any line is to D.C, (which ioy¦neth 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 ap∣peareth the meaning of the whole Theoreme.

And of this Theoreme dothe there folo we an other lyke, whiche you maye calle other a theoreme by it selfe, or else a Corollary vnto this laste theoreme, J passe not so muche for the name. But his sentence is this: when so euer any ly∣nes be drawen frome any pointe, withoute a circle, whether they crosse the circle, or cande in the utter edge of his circumference, those two lines that bee e∣qually 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 casyly perceaue both the meanyng and also the truch of this proposition.

The L iiij. Theoreme. If a point be set forthe in a circle, and frō that pointe vnto the circumference many li∣nes drawen, of which more then two are equal togither, then is that point the centre of that circle.

Example.

[illustration] diagram

The circle is A.B.C, and with init J haue sette fourth for an example three prickes, which are D.E. and F, and from eue∣ry one of them J haue drawē (at the leaste) iiij. lines vnto the circumference of the circle but frome D, J haue drawen more, yet maye it appear rea∣dily vnto your eye, that of all the lines whiche be drawen from E. and F, vnto the circum∣ference, 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 fame 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 cou∣ples, 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 circumfe∣rence, 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 poittes you maye drawe ij. equall lines to the circumfe∣rence of the cirle, whether that pointe be within the circle or without it.

The l v. Theoreme. No circle canne cut an other circle in more

pointes then two.

Example.

[illustration] diagram

The first circle is A.B.F.E, the second circle is B. C. D, F, and they crosse one an o∣ther 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 exā∣ple. where I haue set forthe one tunne forme, and one eye forme, and eche of them cutteth euery of their two cir∣cles 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 scarsely 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.

[illustration] diagram

The one circle, which is the greattest and vttermost is A. B. C, the other circle that is y^e lesser, and is drawen within the firste, is A. D. E. The cētre 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 frome 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 and other, that their edges doo touche and a right line bee drawenne frome the centre of the oneto 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 secōd cir¦cle is D, B.C, and his cētre is H, the point wher they do touch is B. Nowe doo you se that the line K.H, whiche is drawen

[illustration] diagram

from K, that is cen∣tre of the firste cir∣cle, 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 tu∣che together.

The .lviij. theoreme. One circle can not touche an other in more pointes then one, whether they touche within or without.

Example.

[illustration] diagram

For the declaration of this Theoreme, J 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 cen¦tre K. the .iiij. is D.G.L.M, and his centre E. Nowe as you perceiue the second cir∣cle A.D.G, toucheth the first in the inner side, inso much as it is drawen within the o∣ther, and yet it toucheth him but in one point, that is to say in A, so lyke waies the third

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.

[illustration] diagram

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 then distaunce from the centre is all one.

The .lx. Theoreme. In euerie circle the longest line is the diame∣ter, and of all the other lines, thei are still lon∣gest

that be nexte vnto the centre, and they be the shortest, that be farthest distaunt from it.

Example.

[illustration] diagram

In this circle A.B.C.D, I haue drawen first the diame∣ter, whiche is A.D, whiche passeth (as it must) by the cen¦tre E, Then haue I drawen ij. other lines as M.N, whi∣che is neerer the centre, and F. G, that is farther from the centre, The fourth line also on the other side of the dia∣meter, that is B. C, is neerer to the centre then the line F. G, for it is of lyke distance as is the line M.N. Nowe saie I, that A. D, beyng the diame∣ter, 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, because it is nerer to the centre of the cir∣cle 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 pro∣portion 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 exaunple, If two squares be sette foorthe, whereof the one containeth in it fiue square seete, 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 mea∣sure

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 con∣taineth .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 con∣uenient 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 di∣ameter in perpendicular forme, and do make a right angle with the diameter, that right line shall light without the circle, and yet so ioint∣ly knitte to it, that it is not possible to draw a∣ny other right line betwene that saide line and the circumferēce of the circle And the angle that is made in the semicircle is greater then any sharpe angle that may be made of right li∣nes, 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 perpendicu∣lar line, which maketh a right angle with the diameter, is E.A, whiche line falleth without the circle, and yet ioyneth so ex∣actly vnto it, that it is not possible to draw an other right line betwene the circumference of the circle and it, whiche thyng

[illustration] diagram

is so plainly seene of the eye, that it needeth no farther de¦claracion. For euery man wil easily consent, that betwene the croked line A.F, (whiche is a parte of the circumferēce of the circle) and A.E (which is the said perpēdicular line) there can none other line bee drawen in that place where they make the angle. Nowe for the residue of the theo∣reme. The angle D. A. B, which is made in the semicircle, is greater then anye sharpe angle that maye 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 greatter then the other, for they twoo doo make the whole right angle, so that if anye corner coulde bee 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 tge circle to the point where they touch, that

line whiche is drawenne frome the centre, shall be a perpendicular line to the touch line.

Example.

[illustration] diagram

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 cen¦tre of the circle bee in that same line, so dra∣wen.

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

[illustration] diagram

it appeareth mani∣fest, 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 cētre 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 ••heoreme is supposed.

The lxiiij. Theoreme. If an angle be made on the centre of a cir¦cle, and an other angle made on the circumfe¦rence of the same circle, and their grounde line be one common portion of the circumfe∣rence, then is the angle on the centre twise so great as the other angle on the circūferēce

Example.

[illustration] diagram

The cirle 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 one the centre, is twise so greate as the angle C.A.D, which is on the cir¦cumference.

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 an example, it shall bee needefull to declare, what is to be vnderstande by the wor∣des 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 ang••es 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 ma∣keth the cantle of the circle. So that if any angle do not touch the circumference, or if the lines that in close that angle, doo not ende in the extremities of the corde line, but ende other in some other part of the faid corde, or in the circumference, or that any one of them do so eande, then is not that angle ac∣compted to be drawen in the faid 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

[illustration] diagram

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. F. C, and the gea∣ter is D.A.C.C. In this greater can¦tle 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 ligh∣teth 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 circum∣ference 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.

[illustration] diagram

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 con∣trary, those two angles are like waies equall to two right an∣gles. 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 contrary to the angle by B, or yet to the angle by D, for they onely be accompted contrary an∣gles, 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 ly∣nes be distinct, where as the angle by A, and the angle by D, haue one common line A. D, and therfore can not be accomp∣ted contrary angles, So the angle by D, and the angle by C,

haue D.C, as a common line, and therfore be not contrary an∣gles. 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 drawent 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 whi∣che

[illustration] diagram

I draw a cantle of a cir¦cle, whiche is A.B.C. Now saieth the Theoreme, that it is not possible to draw an o∣ther cantle of a circle, whi∣che 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 an∣gle were made in it, that angle would be greatter then the an∣gle made in the cantle A. B. C, and therfore ban not they be cal∣led lyke cantess, but and if any other cantle were made great∣ter then the first, then would the angle it 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 equall

righte lynes, are equall together.

Example.

What is mentby like cantles you haue heard before. and it is easie to vnderstand, that suche figures are 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 a∣gree, that if the one be layed on the other, they shall exactly a∣gree in all their boundes, so that nother shall excede other.

[illustration] diagram

Nowe for the ex∣ample of the Theo∣reme, I haue set for∣the diuers varieties of cantles of circles, amongest which the first and seconde are made vpō equall li∣nes, and ar also both equall and like. The third couple ar ioy∣ned in one, and be no¦ther 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 e∣quall cantles, therfore can not they be like cantles, for necessa∣rily it goeth together, that all cantles of circles made vpon e∣quall 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 circum∣ference, whether the angle light on the cir∣cumference, or on the centre.

Example.

Firste I haue sette for an exaumple twoo equall circles, that

[illustration] diagram

is A. B. C. D, whose centre is K, and the second cir∣cle E. F. G. H, and his centre L, and in eche of thē is there made two angles, one on the circum¦ference, and the o∣ther 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 equall 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 to∣gether, whether they be made on the centre, or on the circumference.

Example.

This Theoreme doth but conuert the sentence of the last The∣oreme

before, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archesynes, 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 af∣firming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste the∣oreme 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. e∣quall to the angle F.L.H, consideringe that they are made on the centres of equall circles. And so of the other angles, bi∣cause those two arche lines aforesaid ar equal, therfore the an¦gle 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 circum∣ference 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 circumference, so that the grea∣ter arche line of the one is equall to the grea∣ter arche line of the other, and the lesser to the lesser.

Example.

The circle A.

[illustration] diagram

B. C. D, is made equall to the cir∣cle E.F.G.H, and the right line B. D. is equal to the righte line F.H, wherfore it folo∣weth, 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 to∣gither.

Example.

This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other ex∣ample. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, so do the this Theoreme

declare the equalnes of the right lines to ensue of the equal∣nes 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 estemed 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 an∣gle, 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.

Jn 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 y^e angle made in a semicircle. Also it is meet to note y^t al angles that be made in y^e part of a circle, ar made other in a se¦micircle (which is the iuste half circle) or els in a cantle of the circle, which cantle is other greater or lesser then the semi∣circle is, as in this figure annexed you maye perceaue euerye one of the thinges seuerallye.

[illustration] diagram

Firste the circle is, as you see, A.B.C.D, and his cen∣tre E, his diameter is A.D, Then is ther a line drawē 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 can∣tles of the whole circle, The greater cantle is D.A B, and the lesser cantle is B. C. D, Jn 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 can∣tle, and an angle of a cantle, firste for an example J take the greater cātle B.A.D, in which is but one angle made, and that is the angle by A, which is made of the 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 cal∣led the angles of that cantle, so the twoo angles made of the righte line D.B, and the arche line D.••.B, are the twoo an∣gles of this cantle, whereof the one is by D, and the other is by B. Wher you must remēbre, that the āgle by D. is made of the right line B.D, and the arche line D.A. And this angle is diui∣ded by an other right line A.E.D, which in this case must be omitted as no line. Also the āgle 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 w^ch the one is the right line B. A, & tho∣ther the right line B.E, yet in this case they ar not to be cōside¦red. And by this may you perceaue also which be the angles of the lesser cantle, the first of thē is made of y^t right line B. D, & of y^e arch line B.C, the secōd is made of the right line. D.B, & of the arch line D.C. Then ar ther ij. other lines, w^ch deuide those ij. corners, y^t is the line B.C, & the line C.D, w^ch 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 an∣gle in a cantle, and an angle of a cantle. And ••n lyke sorte shall you iudg of the āgle made in a semicircle, whiche is distinct frō 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 the angle made in the semicir∣cle 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 cir∣cumference, and make an angle, so they ende with their other extremities at the endes of the diameter. These thynges pre∣mised, now saie J touchyng the Theoreme, that euerye angle that is made in a semicircle, is a right angle, and if it be made in any cātle of a circle, thē must it neds be other a blūt āgle, 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 cātle 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 of 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 con∣trary 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, J saie accordyng to the Theoreme, that the .ij. an∣gles 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 an∣gles 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

[illustration] diagram

the touche line is E. F. The pointe of the touchyng is D, from which point J suppose the line D. B, to be drawen crosse the circle, and to de∣uide it into .ij. cantles, wher∣of the greater is B.A.D and the lesser is B.C.D, and in ech of them an angle drawen, for in the greater cantle the an∣gle 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 y^e 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 an∣gle B.D.F, is on the one side of the line B.D, (whiche is accor∣dyng

to the supposition of the Theoreme drawen crosse the cir¦cle) and the angle B.A.D, is in the cātle on the other side. Like∣waies 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 āgle by C,) whi∣che is made in the cātle on the other side of the right line B.D. The profe of all these J do reserue, as J 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 por¦tions of the one line, shall be equall to the lyke∣iamme made of the partes of the other lyne.

[illustration] diagram

Because this The∣oreme doth serue to many vses, and wold be wel vn∣derstande, J haue set forth .ij. exam∣ples of it. Jn the firste, the lines by their crossyng do make their porti∣ons somewhat to∣ward an equalitie Jn the second the portiōs of the ly∣nes be very far frō an equalitie, and yet in bothe these and in all other y^e Theoreme is true. Jn the first exāple the circle is A.B.C.D, in which thone line A.C, doth crosse thother line B. D, in y^e point E. Now if you do make one likeiāme or lōgsquare of D.E, & E. B, being y^e .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 o∣ther line G.K, in the pointe L. Wherfore if you make a lyke∣iamme 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 J 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 longe square that is made of that whole lyne whiche crosseth the circle, and the portion of it, that lyeth betwene the vtter cir∣cumference 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,

[illustration] diagram

to lie without the circle, betweene the vtter cir∣cumference of it, and the pointe assigned, whiche was A. No we concer∣nyng the meanyng of the Theoreme, if you make a longsquare of the whole line A. C, and of that parte of it that lyeth be∣twene the circumference and the point, (whiche is A. D,) that longe square shall be equall to the full square of the touche line A. B, accordyng not one∣ly as this figure she weth, but also the saied nyne∣teenth conclusion dooeth proue, if you lyste to ex∣amyne the one by the o∣ther.

The .lxxvij. 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 cros∣seth

the circle made with the portiō of the same line beyng without the circle betweene the vt∣ter 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.

Jn 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 point assigned, and cros∣seth the circle, and for his other shorter sides is made of the por¦tion 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 por∣tion 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 J make an ende of the Theoremes.

Notes

• * 1.1

  Bias lyne.
• * 1.2

  Fyll squa∣res. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉