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

by Robert Record

The circle is A.B.C.D, and the touche line is E.F. The pointe of the touchyng is D, from which point I suppose the line D.B, to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater is B.A.D, and the lesser is B.C.D, and in ech of them an angle is drawen, for in the greater cantle the angle is by A, and is made of the right lines B.A, and A.D, in the lesser cantle the angle is by C, and is made of y^e right lines B.C, and C.D. Now saith the Theoreme that the angle B.D.F, is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantle B.A.D, so that the angle B.D.F, is equall to the angle B.A.D, because the angle B.D.F, is on the one side of the line B.D, (whiche is according

to the supposition of the Theoreme drawen crosse the circle) and the angle B.A.D, is in the cãtle on the other side. Likewaies the angle B.D.E, beyng on the one side of the line B.D, must be equall to the angle B.C.D, (that is the ãgle by C,) whiche is made in the cãtle on the other side of the right line B.D. The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.

[ The .lxxv. Theoreme.]

In any circle when .ij. right lines do crosse one an other, the likeiamme that is made of the portions of the one line, shall be equall to the lykeiamme made of the partes of the other lyne.

Because this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie. In the second the portiõs of the lynes be very far frõ an equalitie, and yet in bothe these and in all other y^e Theoreme is true. In the first exãple the circle is A.B.C.D, in which thone line A.C, doth crosse thother line B.D, in y^e point E. Now if you do make one likeiãme or lõgsquare of D.E, & E.B, being y^e .ij.

portions of the line D.B, that longsquare shall be equall to the other longsquare made of A.E, and E.C, beyng the portions of the other line A.C. Lykewaies in the second example, the circle is F.G.H.K, in whiche the line F.H, doth crosse the other line G.K, in the pointe L. Wherfore if you make a lykeiamme or longsquare of the two partes of the line F.H, that is to saye, of F.L, and L.H, that longsquare will be equall to an other longsquare made of the two partes of the line G.K. which partes are G.L, and L.K. Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.

[ The .lxxvi. Theoreme.]

If a pointe be marked without a circle, and from that pointe two right lines drawen to the circle, so that the one of them doe runne crosse the circle, and the other doe touche the circle onely, the long square that is made of that whole lyne which crosseth the circle, and the portion of it, that lyeth betwene the vtter circumference of the circle and the pointe, shall be equall to the full square of the other lyne, that onely toucheth the circle.

Example.