The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 12

Two right lines make no platte forme.

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A platte forme, as you harde before, hath bothe length and bredthe, and is inclosed with lines as with his boundes, but ij.

right lines cannot inclose al the bondes of any platte forme.

Take for an example firste these two right lines A.B. and A.C.

whiche meete togither in A, but yet cannot be called a platte forme, bicause there is no bond from B. to C, but if you will drawe a line betwene them twoo, that is frome B. to C, then will it be a platte forme, that is to say, a triangle, but then are there iij. lines, and not only ij. Likewise may you say of D.E.

and F.G, whiche doo make a platte forme, nother yet can they make any without helpe of two lines more, whereof the one must be drawen from D. to F, and the other frome E. to G, and then will it be a longe rquare. So then of two right lines can bee made no platte forme. But of ij. croked lines be made a platte forme, as you se in the eye form. And also of one right line, & one croked line, maye a platte fourme bee made, as the semicircle F. doothe sette forth.

Certayn common sentences manifest to sence, and acknowledged of all men.

_The firste common sentence._

What so euer things be equal to one other thinge, those same bee equall betwene them selues.

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Examples therof you may take both in greatnes and also in numbre. First (though it pertaine not proprely to geometry, but to helpe the vnderstandinge of the rules, whiche may bee wrought by bothe artes) thus may you perceaue. If the summe of monnye in my purse, and the mony in your purse be equall eche of them to the mony that any other man hathe, then must needes your mony and mine be equall togyther. Likewise, if anye ij. quant.i.ties, as A. and B, be equal to an other, as vnto C, then muste nedes A.

and B. be equall eche to other, as A. equall to B, and B. equall to A, whiche thinge the better to perceaue, tourne these quant.i.ties into numbre, so shall A. and B. make sixteene, and C.

as many. As you may perceaue by multipliyng the numbre of their sides togither.

_The seconde common sentence._

And if you adde equall portions to thinges that be equall, what so amounteth of them shall be equall.

Example, Yf you and I haue like summes of mony, and then receaue eche of vs like summes more, then our summes wil be like styll.

Also if A. and B. (as in the former example) bee equall, then by adding an equal portion to them both, as to ech of them, the quarter of A. (that is foure) they will be equall still.

_The thirde common sentence._

And if you abate euen portions from things that are equal, those partes that remain shall be equall also.

This you may perceaue by the last example. For that that was added there, is subtracted heere. and so the one doothe approue the other.

_The fourth common sentence._

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 quant.i.ties 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.

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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 quant.i.tee be laide on an other, and thei agree, so that the one excedeth not the other, then are they equall togither.

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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.

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It shall be mete to expresse both w^t one example, for of thys last sentence many m? at the first hearing do make a doubt.

Therfore as in this example of the circle deuided into sudry 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 seas take for an exple 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 groudes 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 csequently of the other artes mathematical, which haue the grounds (as Arithmeticke, musike and astronomy) aboue all other artes and sciences, that be vsed amgest 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.

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