The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 13

The theoremes of Geometry brieflye declared by shorte examples.

_The firste Theoreme._

When .ij. triangles be so drawen, that the one of th? 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._

[Ill.u.s.tration]

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 secd trigle & y^e hole trigle. 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 groud line, are equal togither. And if the sides that be equal, be draw? out in l?gth th? wil the corners that are vnder the ground line, be equal also togither.

_Example_

[Ill.u.s.tration]

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.

[Ill.u.s.tration]

_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 fr 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 begi at the same pointes, and end in anie other pointe then the twoo first did.

_Example._

[Ill.u.s.tration]

The first line is A.B, on which I haue erected two other lines A.C, and B.C, that meete in the p.r.i.c.ke 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 th? 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 trigles haue there ij. sides equal one to an other, and their groud lines equal also, then shall their corners, whiche are contained betwene like sides, be equall one to the other.

_Example._

[Ill.u.s.tration]

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 groud 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._

[Ill.u.s.tration]

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 p.r.i.c.ke 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._

[Ill.u.s.tration]

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 p.r.i.c.ke 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.

[Ill.u.s.tration]

_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._