Part 7

THE arte of makyng of Dials, bothe for the daie and the nyght, with certayn new formes of fixed dialles for the moon and other for the sterres, whiche may bee sette in glasse windowes to serue by daie and by night. And howe you may by those dialles knowe in what degree of the Zodiake not only the sonne, but also the moone is. And how many howrs old she is. And also by the same dial to know whether any eclipse shall be that moneth, of the sonne or of the moone.

The makyng and vse of an instrument, wherby you maye not onely measure the distance at ones of all places that you can see togyther, howe muche eche one is from you, and euery one from other, but also therby to drawe the plotte of any countreie that you shall come in, as iustely as maie be, by mannes diligence and labour.

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.

[Sidenote: I.]

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.

[Sidenote: II.]

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.

[Sidenote: III.]

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._

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_

A.B.C. is a twileke triangle, for the one side A.C, is equal to the other side B.C. And therfore I saye that the inner corners A. and B, which are about the ground lines, (that is A.B.) be equall togither. And farther if C.A. and C.B. bee drawen forthe vnto D. and E. as you se that I haue drawen them, then saye I that the two vtter angles vnder A. and B, are equal also togither: as the theorem said. The profe wherof, as of al the rest, shal apeare in Euclide, whome I intende to set foorth in english with sondry new additions, if I may perceaue that it wilbe thankfully taken.

_The thirde Theoreme._

If in annye triangle there bee twoo angles equall togither, then shall the sides, that lie against those angles, be equal also.

_Example._

This triangle A.B.C. hath two corners equal eche to other, that is A. and B, as I do by supposition limite, wherfore it foloweth that the side A.C, is equal to that other side B.C, for the side A.C, lieth againste the angle B, and the side B.C, lieth against the angle A.

_The fourth Theoreme._

When two lines are drawen from the endes of anie one line, and meet in anie pointe, it is not possible to draw two other lines of like lengthe ech to his match that shal begin at the same pointes, and end in anie other pointe then the twoo first did.

_Example._

The first line is A.B, on which I haue erected two other lines A.C, and B.C, that meete in the pricke C, wherefore I say, it is not possible to draw ij. other lines from A. and B. which shal mete in one point (as you se A.D. and B.D. mete in D.) but that the match lines shalbe vnequal, I mean by _match lines_, the two lines on one side, that is the ij. on the right hand, or the ij. on the lefte hand, for as you se in this example A.D. is longer then A.C, and B.C. is longer then B.D. And it is not possible, that A.C. and A.D. shall bee of one lengthe, if B.D. and B.C. bee like longe. For if one couple of matche lines be equall (as the same example A.E. is equall to A.C. in length) then must B.E. needes be vnequall to B.C. as you see, it is here shorter.

_The fifte Theoreme._

If two triangles haue there ij. sides equal one to an other, and their ground lines equal also, then shall their corners, whiche are contained betwene like sides, be equall one to the other.

_Example._

Because these two triangles A.B.C, and D.E.F. haue two sides equall one to an other. For A.C. is equall to D.F, and B.C. is equall to E.F, and again their ground lines A.B. and D.E. are lyke in length, therfore is eche angle of the one triangle equall to ech angle of the other, comparyng together those angles that are contained within lyke sides, so is A. equall to D, B. to E, and C. to F, for they are contayned within like sides, as before is said.

_The sixt Theoreme._

When any right line standeth on an other, the ij. angles that thei make, other are both right angles, or els equall to .ij. righte angles.

_Example._

A.B. is a right line, and on it there doth light another right line, drawen from C. perpendicularly on it, therefore saie I, that the .ij. angles that thei do make, are .ij. right angles as maie be iudged by the definition of a right angle. But in the second part of the example, where A.B. beyng still the right line, on which D. standeth in slope wayes, the two angles that be made of them are not righte angles, but yet they are equall to two righte angles, for so muche as the one is to greate, more then a righte angle, so muche iuste is the other to little, so that bothe togither are equall to two right angles, as you maye perceiue.

_The seuenth Theoreme._

If .ij. lines be drawen to any one pricke in an other lyne, and those .ij. lines do make with the fyrst lyne, two right angles, other suche as be equall to two right angles, and that towarde one hande, than those two lines doo make one streyght lyne.

_Example._

A.B. is a streyght lyne, on which there doth lyght two other lines one frome D, and the other frome C, but considerynge that they meete in one pricke E, and that the angles on one hand be equal to two right corners (as the laste theoreme dothe declare) therfore maye D.E. and E.C. be counted for one ryght lyne.

_The eight Theoreme._

When two lines do cut one an other crosseways they do make their matche angles equall.

_Example._

What matche angles are, I haue tolde you in the definitions of the termes. And here A, and B. are matche corners in this example, as are also C. and D, so that the corner A, is equall to B, and the angle C, is equall to D.

_The nynth Theoreme._

Whan so euer in any triangle the line of one side is drawen forthe in lengthe, that vtter angle is greater than any of the two inner corners, that ioyne not with it.

_Example._

The triangle A.D.C hathe hys grounde lyne A.C. drawen forthe in lengthe vnto B, so that the vtter corner that it maketh at C, is greater then any of the two inner corners that lye againste it, and ioyne not wyth it, whyche are A. and D, for they both are lesser then a ryght angle, and be sharpe angles, but C. is a blonte angle, and therfore greater then a ryght angle.

_The tenth Theoreme._

In euery triangle any .ij. corners, how so euer you take them, ar lesse then ij. right corners.

_Example._

In the firste triangle E, whiche is a threlyke, and therfore hath all his angles sharpe, take anie twoo corners that you will, and you shall perceiue that they be lesser then ij. right corners, for in euery triangle that hath all sharpe corners (as you see it to be in this example) euery corner is lesse then a right corner. And therfore also euery two corners must nedes be lesse then two right corners. Furthermore in that other triangle marked with M, whiche hath .ij. sharpe corners and one right, any .ij. of them also are lesse then two right angles. For though you take the right corner for one, yet the other whiche is a sharpe corner, is lesse then a right corner. And so it is true in all kindes of triangles, as you maie perceiue more plainly by the .xxij. Theoreme.

_The .xi. Theoreme._

In euery triangle, the greattest side lieth against the greattest angle.

_Example._

As in this triangle A.B.C, the greattest angle is C. And A.B. (whiche is the side that lieth against it) is the greatest and longest side. And contrary waies, as A.C. is the shortest side, so B. (whiche is the angle liyng against it) is the smallest and sharpest angle, for this doth folow also, that is the longest side lyeth against the greatest angle, so it that foloweth

_The twelft Theoreme._

In euery triangle the greattest angle lieth against the longest side.

For these ij. theoremes are one in truthe.

_The thirtenth theoreme._

In euerie triangle anie ij. sides togither how so euer you take them, are longer then the thirde.

For example you shal take this triangle A.B.C. which hath a very blunt corner, and therfore one of his sides greater a good deale then any of the other, and yet the ij. lesser sides togither ar greater then it. And if it bee so in a blunte angeled triangle, it must nedes be true in all other, for there is no other kinde of triangles that hathe the one side so greate aboue the other sids, as thei y^t haue blunt corners.

_The fourtenth theoreme._

If there be drawen from the endes of anie side of a triangle .ij. lines metinge within the triangle, those two lines shall be lesse then the other twoo sides of the triangle, but yet the corner that thei make, shall bee greater then that corner of the triangle, whiche standeth ouer it.

_Example._

A.B.C. is a triangle. on whose ground line A.B. there is drawen ij. lines, from the ij. endes of it, I say from A. and B, and they meete within the triangle in the pointe D, wherfore I say, that as those two lynes A.D. and B.D, are lesser then A.C. and B.C, so the angle D, is greatter then the angle C, which is the angle against it.

_The fiftenth Theoreme._

If a triangle haue two sides equall to the two sides of an other triangle, but yet the angle that is contained betwene those sides, greater then the like angle in the other triangle, then is his grounde line greater then the grounde line of the other triangle.

_Example._

A.B.C. is a triangle, whose sides A.C. and B.C, are equall to E.D. and D.F, the two sides of the triangle D.E.F, but bicause the angle in D, is greatter then the angle C. (whiche are the ij. angles contayned betwene the equal lynes) therfore muste the ground line E.F. nedes bee greatter thenne the grounde line A.B, as you se plainely.

_The xvi. Theoreme._

If a triangle haue twoo sides equalle to the two sides of an other triangle, but yet hathe a longer ground line then that other triangle, then is his angle that lieth betwene the equall sides, greater then the like corner in the other triangle.

_Example._

This Theoreme is nothing els, but the sentence of the last Theoreme turned backward, and therfore nedeth none other profe nother declaration, then the other example.

_The seuententh Theoreme._

If two triangles be such sort, that two angles of the one be equal to ij. angles of the other, and that one side of the one be equal to on side of the other, whether that side do adioyne to one of the equall corners, or els lye againste one of them, then shall the other twoo sides of those triangles bee equalle togither, and the thirde corner also shall be equall in those two triangles.

_Example._