Part 14

THE vse bothe of the Globe and the Sphere, and therin also of the arte of Nauigation, and what instrumentes serue beste thervnto, and of the trew latitude and longitude of regions and townes.

Euclides woorkes in foore partes, with diuers demonstrations Arithmeticall and Geometricall or Linearie. The fyrst parte of platte formes. The second of numbres and quantitees surde or irrationall. The third of bodies and solide formes. The fourthe of perspectiue, and other thynges thereto annexed.

BESIDE these I haue other sundrye woorkes partely ended, and partely to bee ended, Of the peregrination of man, and the originall of al nations, The state of tymes, and mutations of realmes, The image of a perfect common welth, with diuers other woorkes in naturall sciences, Of the wonderfull workes and effectes in beastes, plantes, and minerals, of whiche at this tyme, I will omitte the argumentes, beecause thei doo appertaine littel to this arte, and handle other matters in an other sorte.

To haue, or leaue, Nowe maie you chuse, No paine to please, Will I refuse.

The Theoremes of Geometry, before WHICHE ARE SET FORTHE certaine grauntable requestes which serue for demonstrations Mathematicall.

That from any pricke to one other, there may be drawen a right line.

As for example A--------B. A. being the one pricke, and B. the other, you maye drawe betwene them from the one to the other, that is to say, frome A. vnto B, and from B. to A.

That any right line of measurable length may be drawen forth longer, and straight.

Example of A.B, which as it is a line of measurable lengthe, so may it be drawen forth farther, as for example vnto C, and that in true streightenes without crokinge.

That vpon any centre, there may be made a circle of anye quantitee that a man wyll.

Let the centre be set to be A, what shal hinder a man to drawe a circle aboute it, of what quantitee that he lusteth, as you se the forme here: other bygger or lesse, as it shall lyke him to doo:

That all right angles be equall eche to other.

Set for an example A. and B, of which two though A. seme the greatter angle to some men of small experience, it happeneth only bicause that the lines aboute A, are longer then the lines about B, as you may proue by drawing them longer, for so that B. seme the greater angle yf you make his lines longer then the lines that make the angle A. And to proue it by demonstration, I say thus. If any ij. right corners be not equal, then one right corner is greater then an other, but that corner which is greatter then a right angle, is a blunt corner (by his definition) so must one corner be both a right corner and a blunt corner also, which is not possible: And againe: the lesser right corner must be a sharpe corner, by his definition, bicause it is lesse then a right angle. which thing is impossible. Therefore I conclude that all right angles be equall.

Yf one right line do crosse two other right lines, and make ij. inner corners of one side lesser then ij. righte corners, it is certaine, that if those two lines be drawen forth right on that side that the sharpe inner corners be, they wil at length mete togither, and crosse on an other.

The ij. lines beinge as A.B. and C.D, and the third line crossing them as dooth heere E.F, making ij inner cornes (as ar G.H.) lesser then two right corners, sith ech of them is lesse then a right corner, as your eyes maye iudge, then say I, if those ij. lines A.B. and C.D. be drawen in lengthe on that side that G. and H. are, the will at length meet and crosse one an other.

Two right lines make no platte forme.

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.

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. quantities, 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 quantities 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.