The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men.

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.

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 mo∣ny

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and mine be e∣quall togyther. Likewise, if anye ij. quantities, as A and B, be equal to an other, as vn¦to C, then muste nedes A. and B. be equall eche to o∣ther, as A. equall to B, and B. equall to A, whiche thinge the better to peceaue, tourne these quantities into numbre, so shall A. and B. make fixteene, 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 thin∣ges that be equall, what so amounteth of them

shallbe 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 e∣quall, 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 laste 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 thin¦ges, 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.

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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 e∣quall to forty.

The seuenth common sentence. If any two thinges be the halfes of one other thing, than 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 num¦bre declareth.

The eyght common sentence. If any one quantitee be laide on an o∣ther, and thei agree, so that the one

excedeth not the other, then are they e∣quall togither.

As if this figure A. B. C, be layed on

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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 e∣quall to D. E, and the third lyne C. A, is equal to F. D so that euery side in the one is equall to some one side of the other. wherfore it is playne, that the two triangles are equall to∣gither.

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 com∣men 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 sētence many mē at the first hearing do make a doubt. Ther¦fore as in this example of the circle deuided into sūdry partes

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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 par∣tes 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

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A. B, whose partes A. and B, are trian∣gles, 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 thin¦ges (as some do) then is it moste false, for e∣uery compound thyng is made of partes of diuerse matter and substance. Take for example a man, a house, a boke, and all o∣ther compound thinges. Some vnderstand it thus, that the par∣tes 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 seasō take for an exāple this square figure folowing A. B. C. D, w^ch is deuided but into two parts, and yet (as youse) 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

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hath for his gro¦ūdes soche pla∣yne truthes, & as I may say, suche vndow∣btfull and sensi¦ble principles, And this is the cause why all learned menne dooth approue the certenty of geometry, and cōsequently of the other artes mathematical, which haue the grounds (as A∣rithmetike, mu¦sike and astro∣nomy) aboue all other artes and sciences, that be vsed amōgest men. Thus muche haue I sayd of the first principles, and now will I go on with the theoremes, whiche I do only by exam∣ples declae, minding to reserue the proofes to a peculiar boke which I will then set forth, when I perceaue this to be thank∣fully taken of the readers of it.