If about the centre S there be described any circle as AEB, and about the centre P there be also described two circles EF, ef, cutting the first in E and e, and the line PS in F and f; and there be let fall to PS the perpendiculars ED, ed; I say, that if the distance of the arcs EF, ef be supposed to be infinitely diminished, the last ratio of the evanescent line Dd to the evanescent line Ff is the same as that of the line PE to the line PS.
For if the line Pe cut the arc EF in q; and the right line Ee, which coincides with the evanescent arc Ee, be produced, and meet the right line PS in T; and there be let fall from S to PE the perpendicular SG; then, because of the like triangles DTE, dTe, DES, it will be as Dd to Ee so DT to TE, or DE to ES: and because the triangles, Eeq, ESG (by Lem. VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or Ff so ES to SG; and, ex æquo, as Dd to Ff so DE to SG; that is (because of the similar triangles PDE, PGS), so is PE to PS. Q.E.D.