The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 16

_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 draw? 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 groud 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 x.x.xi. 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._

[Ill.u.s.tration]

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 .x.x.xij. 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.

[Ill.u.s.tration]

_Example._

Fyrst before I declare the examples, it shal be mete to shew the true vnderstdyng 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: ??ap????ata] and called in latin _supplementa_, bycause that they make one generall square, includyng and enclosyng the other diuers squares, as in this exple 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 x.x.xiij. 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._

[Ill.u.s.tration]

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

-- By the square of any lyne, you muste vnderstande a figure made iuste square, hauyng all his iiij. sydes equall to that line, whereof it is the square, so is A.C.F, the square of A.C.

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

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

_The x.x.xiiij. 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 p.r.o.nounced.

_The .x.x.xv. 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.

[Ill.u.s.tration]

_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. stdeth for the l?gth 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 porti 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 x.x.xvi. 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._

[Ill.u.s.tration]