Part 15

If you abate equalle partes from vnequal thinges, the remainers shall be vnequall.

As bicause that a hundreth and eight and forty be vnequal if I take tenne from them both, there will remaine nynetye and eight and thirty, which are also vnequall. and likewise in quantities it is to be iudged.

The fifte common sentence.

When euen portions are added to vnequalle thinges, those that amounte shalbe vnequall.

So if you adde twenty to fifty, and lyke ways to nynty, you shall make seuenty and a hundred and ten whiche are no lesse vnequall, than were fifty and nynty.

The syxt common sentence.

If two thinges be double to any other, those same two thinges are equal togither.

Bicause A. and B. are eche of them double to C, therefore must A. and B. nedes be equall togither. For as v. times viij. maketh xl. which is double to iiij. times v, that is xx so iiij. times x, likewise is double to xx. (for it maketh fortie) and therefore muste neades be equall to forty.

The seuenth common sentence.

If any two thinges be the halfes of one other thing, then are thei .ij. equall togither.

So are D. and C. in the laste example equal togyther, bicause they are eche of them the halfe of A. other of B, as their numbre declareth.

The eyght common sentence.

If any one quantitee be laide on an other, and thei agree, so that the one excedeth not the other, then are they equall togither.

As if this figure A.B.C, be layed on that other D.E.F, so that A. be layed to D, B. to E, and C. to F, you shall see them agre in sides exactlye and the one not to excede the other, for the line A.B. is equall to D.E, and the third lyne C.A, is equall to F.D so that eueryside in the one is equall to some one side of the other. Wherfore it is playne, that the two triangles are equall togither.

The nynth common sentence.

Euery whole thing is greater than any of his partes.

This sentence nedeth none example. For the thyng is more playner then any declaration, yet considering that other common sentence that foloweth nexte that.

The tenthe common sentence.

Euery whole thinge is equall to all his partes taken togither.

It shall be mete to expresse both w^t one example, for of thys last sentence many men at the first hearing do make a doubt. Therfore as in this example of the circle deuided into sundry partes it doeth appere that no parte can be so great as the whole circle, (accordyng to the meanyng of the eight sentence) so yet it is certain, that all those eight partes together be equall vnto the whole circle. And this is the meanyng of that common sentence (whiche many vse, and fewe do rightly vnderstand) that is, that All the partes of any thing are nothing els, but the whole. And contrary waies: The whole is nothing els, but all his partes taken togither. whiche saiynges some haue vnderstand to meane thus: that all the partes are of the same kind that the whole thyng is: but that that meanyng is false, it doth plainly appere by this figure A.B, whose partes A. and B, are triangles, and the whole figure is a square, and so are they not of one kind. But and if they applie it to the matter or substance of thinges (as some do) then it is most false, for euery compound thyng is made of partes of diuerse matter and substance. Take for example a man, a house, a boke, and all other compound thinges. Some vnderstand it thus, that the partes all together can make none other forme, but that that the whole doth shewe, whiche is also false, for I maie make fiue hundred diuerse figures of the partes of some one figure, as you shall better perceiue in the third boke. And in the meane season take for an example this square figure following A.B.C.D, w^{ch} is deuided but in two parts, and yet (as you se) I haue made fiue figures more beside the firste, with onely diuerse ioynyng of those two partes. But of this shall I speake more largely in an other place. In the mean season content your self with these principles, whiche are certain of the chiefe groundes wheron all demonstrations mathematical are fourmed, of which though the moste parte seeme so plaine, that no childe doth doubte of them, thinke not therfore that the art vnto whiche they serue, is simple, other childishe, but rather consider, howe certayne the profes of that arte is, y^t hath for his groundes soche playne truthes, & as I may say, suche vndowbtfull and sensible principles, And this is the cause why all learned menne dooth approue the certenty of geometry, and consequently of the other artes mathematical, which haue the grounds (as Arithmeticke, musike and astronomy) aboue all other artes and sciences, that be vsed amongest men. Thus muche haue I sayd of the first principles, and now will I go on with the theoremes, whiche I do only by examples declare, minding to reserue the proofes to a peculiar boke which I will then set forth, when I perceaue this to be thankfully taken of the readers of it.

The theoremes of Geometry brieflye declared by shorte examples.

The firste Theoreme.

When .ij. triangles be so drawen, that the one of them hath ij. sides equal to ij sides of the other triangle, and that the angles enclosed with those sides, bee equal also in bothe triangles, then is the thirde side likewise equall in them. And the whole triangles be of one greatnes, and euery angle in the one equall to his matche angle in the other, I meane those angles that be inclosed with like sides.

Example.

This triangle A.B.C. hath ij. sides (that is to say) C.A. and C.B, equal to ij. sides of the other triangle F.G.H, for A.C. is equall to F.G, and B.C. is equall to G.H. And also the angle C. contayned beetweene F.G, and G.H, for both of them answere to the eight parte of a circle. Therfore doth it remayne that A.B. whiche is the thirde lyne in the firste triangle, doth agre in lengthe with F.H, w^{ch} is the third line in y^e second triangle & y^e hole triangle. A.B.C. must nedes be equal to y^e hole triangle F.G.H. And euery corner equall to his match, that is to say, A. equall to F, B. to H, and C. to G, for those bee called match corners, which are inclosed with like sides, other els do lye against like sides.

The second Theoreme.

In twileke triangles the ij. corners that be about the ground line, are equal togither. And if the sides that be equal, be drawen out in length then wil the corners that are vnder the ground line, be equal also togither.

Example