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 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.