The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 15

[Ill.u.s.tration]

The .ij. fyrst lynes are A.B. and C.D, the thyrd lyne that crosseth them is E.F. And bycause that E.F. maketh ij. matche angles with A.B, equall to .ij. other lyke matche angles on C.D, (that is to say E.G, equall to K.F, and M.N. equall also to H.L.) therfore are those ij. lynes A.B. and C.D. gemow lynes, vnderstand here by _lyke matche corners_, those that go one way as doth E.G, and K.F, lyke ways N.M, and H.L, for as E.G. and H.L, other N.M. and K.F. go not one waie, so be not they lyke match corners.

_The nyntenth Theoreme._

When on two right lines there is drawen a thirde right line crossewaies, and maketh the ij. ouer corners towarde one hande equall togither, then ar those .ij. lines paralleles.

And in like maner if two inner corners toward one hande, be equall to .ii. right angles.

_Example._

As the Theoreme dothe speake of .ij. ouer angles, so muste you vnderstande also of .ij. nether angles, for the iudgement is lyke in bothe. Take for example the figure of the last theoreme, where A.B, and C.D, be called paralleles also, bicause E. and K, (whiche are .ij. ouer corners) are equall, and lykewaies L.

and M. And so are in lyke maner the nether corners N. and H, and G. and F. Nowe to the seconde parte of the theoreme, those .ij.

lynes A.B. and C.D, shall be called paralleles, because the ij.

inner corners. As for example those two that bee toward the right hande (that is G. and L.) are equall (by the fyrst parte of this nyntenth theoreme) therfore muste G. and L. be equall to two ryght angles.

_The xx. Theoreme._

When a right line is drawen crosse ouer .ij. right gemow lines, it maketh .ij. matche corners of the one line, equall to two matche corners of the other line, and also bothe ouer corners of one hande equall togither, and bothe nether corners like waies, and more ouer two inner corners, and two vtter corners also towarde one hande, equall to two right angles.

_Example._

Bycause A.B. and C.D, (in the laste figure) are paralleles, therefore the two matche corners of the one lyne, as E.G. be equall vnto the .ij. matche corners of the other line, that is K.F, and lykewaies M.N, equall to H.L. And also E. and K. bothe ouer corners of the lefte hande equall togyther, and so are M.

and L, the two ouer corners on the ryghte hande, in lyke maner N. and H, the two nether corners on the lefte hande, equall eche to other, and G. and F. the two nether angles on the right hande equall togither.

-- Farthermore yet G. and L. the .ij. inner angles on the right hande bee equall to two right angles, and so are M. and F. the .ij. vtter angles on the same hande, in lyke manner shall you say of N. and K. the two inner corners on the left hand. and of E. and H. the two vtter corners on the same hande. And thus you see the agreable sentence of these .iii. theoremes to tende to this purpose, to declare by the angles how to iudge paralleles, and contrary waies howe you may by paralleles iudge the proportion of the angles.

_The xxi. Theoreme._

What so euer lines be paralleles to any other line, those same be paralleles togither.

_Example._

[Ill.u.s.tration]

A.B. is a gemow line, or a parallele vnto C.D. And E.F, lykewaies is a parallele vnto C.D. Wherfore it foloweth, that A.B. must nedes bee a parallele vnto E.F.

_The .xxij. theoreme._

In euery triangle, when any side is drawen forth in length, the vtter angle is equall to the ij. inner angles that lie againste it. And all iij. inner angles of any triangle are equall to ij. right angles.

[Ill.u.s.tration]

_Example._

The triangle beeyng A.D.E. and the syde A.E. drawen foorthe vnto B, there is made an vtter corner, whiche is C, and this vtter corner C, is equall to bother the inner corners that lye agaynst it, whyche are A. and D. And all thre inner corners, that is to say, A.D. and E, are equall to two ryght corners, whereof it foloweth, _that all the three corners of any one triangle are equall to all the three corners of euerye other triangle_. For what so euer thynges are equalle to anny one thyrde thynge, those same are equalle togitther, by the fyrste common sentence, so that bycause all the .iij. angles of euery triangle are equall to two ryghte angles, and all ryghte angles bee equall togyther (by the fourth request) therfore must it nedes folow, that all the thre corners of euery triangle (accomptyng them togyther) are equall to iij. corners of any triangle, taken all togyther.

_The .xxiii. theoreme._

When any ij. right lines doth touche and couple .ij. other righte lines, whiche are equall in length and paralleles, and if those .ij. lines bee drawen towarde one hande, then are thei also equall together, and paralleles.

_Example._

[Ill.u.s.tration]

A.B. and C.D. are ij. ryght lynes and paralleles and equall in length, and they ar touched and ioyned togither by ij. other lynes A.C. and B.D, this beyng so, and A.C. and B.D. beyng drawen towarde one syde (that is to saye, bothe towarde the lefte hande) therefore are A.C. and B.D. bothe equall and also paralleles.

_The .xxiiij. theoreme._

In any likeiamme the two contrary sides ar equall togither, and so are eche .ij. contrary angles, and the bias line that is drawen in it, dothe diuide it into two equall portions.

_Example._

[Ill.u.s.tration]

Here ar two likeiammes ioyned togither, the one is a longe square A.B.E, and the other is a losengelike D.C.E.F. which ij.

likeiammes ar proued equall togither, bycause they haue one ground line, that is, F.E, And are made betwene one payre of gemow lines, I meane A.D. and E.H. By this Theoreme may you know the arte of the righte measuringe of likeiammes, as in my booke of measuring I wil more plainly declare.

_The xxvi. Theoreme._

All likeiammes that haue equal grounde lines and are drawen betwene one paire of paralleles, are equal togither.

_Example._

Fyrste you muste marke the difference betwene this Theoreme and the laste, for the laste Theoreme presupposed to the diuers likeiammes one ground line common to them, but this theoreme doth presuppose a diuers ground line for euery likeiamme, only meaning them to be equal in length, though they be diuers in numbre. As for example. In the last figure ther are two parallels, A.D. and E.H, and betwene them are drawen thre likeiammes, the firste is, A.B.E.F, the second is E.C.D.F, and the thirde is C.G.H.D. The firste and the seconde haue one ground line, (that is E.F.) and therfore in so muche as they are betwene one paire of paralleles, they are equall accordinge to the fiue and twentye Theoreme, but the thirde likeiamme that is C.G.H.D. hathe his grounde line G.H, seuerall frome the other, but yet equall vnto it. wherefore the third likeiam is equall to the other two firste likeiammes. And for a proofe that G.H.

being the ground or groud line of the third likeiamme, is equal to E.F, whiche is the ground line to both the other likeiams, that may be thus declared, G.H. is equall to C.D, seynge they are the contrary sides of one likeiamme (by the foure and tw?ty theoreme) and so are C.D. and E.F. by the same theoreme.

Therfore seynge both those ground lines E.F. and G.H, are equall to one thirde line (that is C.D.) they must nedes bee equall togyther by the firste common sentence.

_The xxvii. Theoreme._

All triangles hauinge one grounde lyne, and standing betwene one paire of parallels, ar equall togither.

_Example._

[Ill.u.s.tration]

A.B. and C.F. are twoo gemowe lines, betweene which there be made two triangles, A.D.E. and D.E.B, so that D.E, is the common ground line to them bothe. wherfore it doth folow, that those two triangles A.D.E. and D.E.B. are equall eche to other.

_The xxviij. Theoreme._

All triangles that haue like long ground lines, and bee made betweene one paire of gemow lines, are equall togither.