Part 8

Bicause that A.B.C, the one triangle hath two corners A. and B, equal to D.E, that are twoo corners of the other triangle. D.E.F. and that they haue one side in theym bothe equall, that is A.B, which is equall to D.E, therefore shall both the other ij. sides be equall one to an other, as A.C. and B.C. equall to D.F. and E.F, and also the thirde angle in them both shal be equall, that is, the angle C. shal be equall to the angle F.

_The eightenth Theoreme._

When on ij. right lines ther is drawen a third right line crosse waies, and maketh .ij. matche corners of the one line equall to the like twoo matche corners of the other line, then ar those two lines gemmow lines, or paralleles.

_Example._

The .ij. fyrst lynes are A.B. and C.D, the thyrd lyne that crosseth them is E.F. And bycause that E.F. maketh ij. matche angles with A.B, equall to .ij. other lyke matche angles on C.D, (that is to say E.G, equall to K.F, and M.N. equall also to H.L.) therfore are those ij. lynes A.B. and C.D. gemow lynes, vnderstand here by _lyke matche corners_, those that go one way as doth E.G, and K.F, lyke ways N.M, and H.L, for as E.G. and H.L, other N.M. and K.F. go not one waie, so be not they lyke match corners.

_The nyntenth Theoreme._

When on two right lines there is drawen a thirde right line crossewaies, and maketh the ij. ouer corners towarde one hande equall togither, then ar those .ij. lines paralleles. And in like maner if two inner corners toward one hande, be equall to .ii. right angles.

_Example._

As the Theoreme dothe speake of .ij. ouer angles, so muste you vnderstande also of .ij. nether angles, for the iudgement is lyke in bothe. Take for example the figure of the last theoreme, where A.B, and C.D, be called paralleles also, bicause E. and K, (whiche are .ij. ouer corners) are equall, and lykewaies L. and M. And so are in lyke maner the nether corners N. and H, and G. and F. Nowe to the seconde parte of the theoreme, those .ij. lynes A.B. and C.D, shall be called paralleles, because the ij. inner corners. As for example those two that bee toward the right hande (that is G. and L.) are equall (by the fyrst parte of this nyntenth theoreme) therfore muste G. and L. be equall to two ryght angles.

_The xx. Theoreme._

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

_Example._

Bycause A.B. and C.D, (in the laste figure) are paralleles, therefore the two matche corners of the one lyne, as E.G. be equall vnto the .ij. matche corners of the other line, that is K.F, and lykewaies M.N, equall to H.L. And also E. and K. bothe ouer corners of the lefte hande equall togyther, and so are M. and L, the two ouer corners on the ryghte hande, in lyke maner N. and H, the two nether corners on the lefte hande, equall eche to other, and G. and F. the two nether angles on the right hande equall togither.

[Sec.] Farthermore yet G. and L. the .ij. inner angles on the right hande bee equall to two right angles, and so are M. and F. the .ij. vtter angles on the same hande, in lyke manner shall you say of N. and K. the two inner corners on the left hand. and of E. and H. the two vtter corners on the same hande. And thus you see the agreable sentence of these .iii. theoremes to tende to this purpose, to declare by the angles how to iudge paralleles, and contrary waies howe you may by paralleles iudge the proportion of the angles.

_The xxi. Theoreme._

What so euer lines be paralleles to any other line, those same be paralleles togither.

_Example._

A.B. is a gemow line, or a parallele vnto C.D. And E.F, lykewaies is a parallele vnto C.D. Wherfore it foloweth, that A.B. must nedes bee a parallele vnto E.F.

_The .xxij. theoreme._

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

_Example._

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

_The .xxiii. theoreme._

When any ij. right lines doth touche and couple .ij. other righte lines, whiche are equall in length and paralleles, and if those .ij. lines bee drawen towarde one hande, then are thei also equall together, and paralleles.

_Example._

A.B. and C.D. are ij. ryght lynes and paralleles and equall in length, and they ar touched and ioyned togither by ij. other lynes A.C. and B.D, this beyng so, and A.C. and B.D. beyng drawen towarde one syde (that is to saye, bothe towarde the lefte hande) therefore are A.C. and B.D. bothe equall and also paralleles.

_The .xxiiij. theoreme._

In any likeiamme the two contrary sides ar equall togither, and so are eche .ij. contrary angles, and the bias line that is drawen in it, dothe diuide it into two equall portions.

_Example._

Here ar two likeiammes ioyned togither, the one is a longe square A.B.E, and the other is a losengelike D.C.E.F. which ij. likeiammes ar proued equall togither, bycause they haue one ground line, that is, F.E, And are made betwene one payre of gemow lines, I meane A.D. and E.H. By this Theoreme may you know the arte of the righte measuringe of likeiammes, as in my booke of measuring I wil more plainly declare.

_The xxvi. Theoreme._

All likeiammes that haue equal grounde lines and are drawen betwene one paire of paralleles, are equal togither.

_Example._

Fyrste you muste marke the difference betwene this Theoreme and the laste, for the laste Theoreme presupposed to the diuers likeiammes one ground line common to them, but this theoreme doth presuppose a diuers ground line for euery likeiamme, only meaning them to be equal in length, though they be diuers in numbre. As for example. In the last figure ther are two parallels, A.D. and E.H, and betwene them are drawen thre likeiammes, the firste is, A.B.E.F, the second is E.C.D.F, and the thirde is C.G.H.D. The firste and the seconde haue one ground line, (that is E.F.) and therfore in so muche as they are betwene one paire of paralleles, they are equall accordinge to the fiue and twentye Theoreme, but the thirde likeiamme that is C.G.H.D. hathe his grounde line G.H, seuerall frome the other, but yet equall vnto it. wherefore the third likeiam is equall to the other two firste likeiammes. And for a proofe that G.H. being the ground or ground line of the third likeiamme, is equal to E.F, whiche is the ground line to both the other likeiams, that may be thus declared, G.H. is equall to C.D, seynge they are the contrary sides of one likeiamme (by the foure and twenty theoreme) and so are C.D. and E.F. by the same theoreme. Therfore seynge both those ground lines E.F. and G.H, are equall to one thirde line (that is C.D.) they must nedes bee equall togyther by the firste common sentence.

_The xxvii. Theoreme._

All triangles hauinge one grounde lyne, and standing betwene one paire of parallels, ar equall togither.

_Example._

A.B. and C.F. are twoo gemowe lines, betweene which there be made two triangles, A.D.E. and D.E.B, so that D.E, is the common ground line to them bothe. wherfore it doth folow, that those two triangles A.D.E. and D.E.B. are equall eche to other.

_The xxviij. Theoreme._

All triangles that haue like long ground lines, and bee made betweene one paire of gemow lines, are equall togither.

_Example._

Example of this Theoreme you may see in the last figure, where as sixe triangles made betwene those two gemowe lines A.B. and C.F, the first triangle is A.C.D, the seconde is A.D.E, the thirde is A.D.B, the fourth is A.B.E, the fifte is D.E.B, and the sixte is B.E.F, of which sixe triangles, A.D.E. and D.E.B. are equall, bicause they haue one common grounde line. And so likewise A.B.E. and A.B.D, whose commen grounde line is A.B, but A.C.D. is equal to B.E.F, being both betwene one couple of parallels, not bicause thei haue one ground line, but bicause they haue their ground lines equall, for C.D. is equall to E.F, as you may declare thus. C.D, is equall to A.B. (by the foure and twenty Theoreme) for thei are two contrary sides of one lykeiamme. A.C.D.B, and E.F by the same theoreme, is equall to A.B, for thei ar the two y^e contrary sides of the likeiamme, A.E.F.B, wherfore C.D. must needes be equall to E.F. like wise the triangle A.C.D, is equal to A.B.E, bicause they ar made betwene one paire of parallels and haue their groundlines like, I meane C.D. and A.B. Againe A.D.E, is equal to eche of them both, for his ground line D.E, is equall to A.B, inso muche as they are the contrary sides of one likeiamme, that is the long square A.B.D.E. And thus may you proue the equalnes of all the reste.

_The xxix. Theoreme._

Al equal triangles that are made on one grounde line, and rise one waye, must needes be betwene one paire of parallels.

_Example._

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

_The thirty Theoreme._

Equal triangles that haue their ground lines equal, and be drawen toward one side, ar made betwene one paire of paralleles.

_Example._

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

_The xxxi. theoreme._

If a likeiamme haue one ground line with a triangle, and be drawen betwene one paire of paralleles, then shall the likeiamme be double to the triangle.

_Example._

A.H. and B.G. are .ij. gemow lines, betwene which there is made a triangle B.C.G, and a lykeiamme, A.B.G.C, whiche haue a grounde lyne, that is to saye, B.G. Therfore doth it folow that the lyke iamme A.B.G.C. is double to the triangle B.C.G. For euery halfe of that lykeiamme is equall to the triangle, I meane A.B.F.E. other F.E.C.G. as you may coniecture by the .xi. conclusion geometrical.

And as this Theoreme dothe speake of a triangle and likeiamme that haue one groundelyne, so is it true also, yf theyr groundelynes bee equall, though they bee dyuers, so that thei be made betwene one payre of paralleles. And hereof may you perceaue the reason, why in measuryng the platte of a triangle, you must multiply the perpendicular lyne by halfe the grounde lyne, or els the hole grounde lyne by halfe the perpendicular, for by any of these bothe waies is there made a lykeiamme equall to halfe suche a one as shulde be made on the same hole grounde lyne with the triangle, and betweene one payre of paralleles. Therfore as that lykeiamme is double to the triangle, so the halfe of it, must needes be equall to the triangle. Compare the .xi. conclusion with this theoreme.

_The .xxxij. Theoreme._

In all likeiammes where there are more than one made aboute one bias line, the fill squares of euery of them must nedes be equall.

_Example._

Fyrst before I declare the examples, it shal be mete to shew the true vnderstandyng of this theorem. [Sidenote: _Bias lyne._] Therfore by the _Bias line_, I meane that lyne, whiche in any square figure dooth runne from corner to corner. And euery square which is diuided by that bias line into equall halues from corner to corner (that is to say, into .ij. equall triangles) those be counted _to stande aboute one bias line_, and the other squares, whiche touche that bias line, with one of their corners onely, those doo I call _Fyll squares_, [Sidenote: _Fyll squares._] accordyng to the greke name, which is _anapleromata_, [Sidenote: #anaple:ro:mata#] and called in latin _supplementa_, bycause that they make one generall square, includyng and enclosyng the other diuers squares, as in this example H.C.E.N. is one square likeiamme, and L.M.G.C. is an other, whiche bothe are made aboute one bias line, that is N.M, than K.L.H.C. and C.E.F.G. are .ij. fyll squares, for they doo fyll vp the sydes of the .ij. fyrste square lykeiammes, in suche sorte, that all them foure is made one greate generall square K.M.F.N.

Nowe to the sentence of the theoreme, I say, that the .ij. fill squares, H.K.L.C. and C.E.F.G. are both equall togither, (as it shall bee declared in the booke of proofes) bicause they are the fill squares of two likeiammes made aboute one bias line, as the exaumple sheweth. Conferre the twelfthe conclusion with this theoreme.

_The xxxiij. Theoreme._

In all right anguled triangles, the square of that side whiche lieth against the right angle, is equall to the .ij. squares of both the other sides.

_Example._

A.B.C. is a triangle, hauing a ryght angle in B. Wherfore it foloweth, that the square of A.C, (whiche is the side that lyeth agaynst the right angle) shall be as muche as the two squares of A.B. and B.C. which are the other .ij. sides.

[Sec.] By the square of any lyne, you muste vnderstande a figure made iuste square, hauyng all his iiij. sydes equall to that line, whereof it is the square, so is A.C.F, the square of A.C. Lykewais A.B.D. is the square of A.B. And B.C.E. is the square of B.C. Now by the numbre of the diuisions in eche of these squares, may you perceaue not onely what the square of any line is called, but also that the theoreme is true, and expressed playnly bothe by lines and numbre. For as you see, the greatter square (that is A.C.F.) hath fiue diuisions on eche syde, all equall togyther, and those in the whole square are twenty and fiue. Nowe in the left square, whiche is A.B.D. there are but .iij. of those diuisions in one syde, and that yeldeth nyne in the whole. So lykeways you see in the meane square A.C.E. in euery syde .iiij. partes, whiche in the whole amount vnto sixtene. Nowe adde togyther all the partes of the two lesser squares, that is to saye, sixtene and nyne, and you perceyue that they make twenty and fiue, whyche is an equall numbre to the summe of the greatter square.

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

_The xxxiiij. Theoreme._

If so be it, that in any triangle, the square of the one syde be equall to the .ij. squares of the other .ij. sides, than must nedes that corner be a right corner, which is conteined betwene those two lesser sydes.

_Example._

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

_The .xxxv. theoreme._

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

_Example._

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

_The xxxvi. Theoreme._

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

_Example._

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

_The xxxvij. Theoreme._

If a right line be deuided by chaunce, as it maye happen, the square that is made of the whole line, and one of the partes of it which soeuer it be, shal be equall to that square that is made of the ij. partes ioyned togither, and to an other square made of that part, which was before ioyned with the whole line.

_Example._

The line A.B. is deuided in C. into twoo partes, though not equally, of which two partes for an example I take the first, that is A.C, and of it I make one side of a square, as for example D.G. accomptinge those two lines to be equall, the other side of the square is D.E, whiche is equall to the whole line A.B.

Now may it appeare, to your eye, that the great square made of the whole line A.B, and of one of his partes that is A.C, (which is equall with D.G.) is equal to two partiall squares, whereof the one is made of the saide greatter portion A.C, in as muche as not only D.G, beynge one of his sides, but also D.H. beinge the other side, are eche of them equall to A.C. The second square is H.E.F.K, in which the one side H.E, is equal to C.B, being the lesser parte of the line, A.B, and E.F. is equall to A.C. which is the greater parte of the same line. So that those two squares D.H.K.G and H.E.F.K, bee bothe of them no more then the greate square D.E.F.G, accordinge to the wordes of the Theoreme afore saide.

_The xxxviij. Theoreme._

If a righte line be deuided by chaunce, into partes, the square that is made of that whole line, is equall to both the squares that ar made of eche parte of the line, and moreouer to two squares made of the one portion of the diuided line ioyned with the other in square.

_Example._

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

So is that greatest square, beeyng made of the hole lyne A.B, equal to the ij. squares of eche of his partes seuerally, and more by as muche iust as .ij. longe squares, made of the longer portion of the diuided lyne ioyned in square with the shorter parte of the same diuided line, as the theoreme wold. And as here I haue put an example of a lyne diuided into .ij. partes, so the theoreme is true of all diuided lines, of what number so euer the partes be, foure, fyue, or syxe. etc.

This theoreme hath great vse, not only in geometrie, but also in arithmetike, as herafter I will declare in conuenient place.

_The .xxxix. theoreme._