Part 12 (1/2)

Two right lines make no platte forme.

[Ill.u.s.tration]

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.

[Ill.u.s.tration]

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._