PROPERTIES OF THE

TRACT 13.

248

Demons .—Let p be the given point, andAb the two points in which the line pabmeets the surface of the sphere; throughPAB and the centre let a plane cut thesphere in the great circle tab, to whichdraw the tangent pt. Then the rectanglepa . pb is equal to the square of pt (Eucl.iii. 36); but pt, and consequently itssquare, is constant by Prop. 1; thereforepb, which is always equal to this square,the same constant magnitude.

,r

Ti

A

the rectangle pa .is every where of

prop. 5.

If any two lines intersect each other within asphere, and beterminated at the surface on both sides; the rectangle of theparts of the one line, will be equal to the rectangle of theparts of the other. And, universally, the rectangles of thetwo parts of all lines passing through the point of intersec-tion, are all of the same magnitude.

Demons .—Through any one-of thelines, as ab, conceive a plane to bedrawn through the centre c of thesphere, cutting the sphere in the greatcircle abd ; and draw its diameterdcpf through the points of intersectionp of all the lines. Then the rectangle AP . pb is equal to therectangle dp . pl (Eucl. iii. 35).

Again, through any other of the intersecting lines gi-i, andthe centre, conceive another plane to pass, cutting the spherein another great circle dgfh. Then, because the points cand p are in this latter plane, the line cp, and consequentlythe whole diameter dcpf, is in the same plane; and thereforeit is a diameter of the circle dgfh, of -which gpii is a chord.Therefore, again, the rectangle gp . ph is equal to the rect-angle dp . pf (Eucl. iii. 35).

Consequently all the rectangles ap . pb, gp . ph, &c, areequal, being each equal to the constant rectangle dp . pf,