Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/365

Rh evanecent line Dd to the evanecent line Ff is the ame 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 evanecent 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 becaue of the like triangles DTE, dTe, DES; it will be as Dd to Ee o DT to TE, or DE to ES; and becaue the triangles Eeq, ESG (by lem. 8. and cor. 3. lem. 7.) are imilar, it will be as Ee to eq or Ff o ES to SG; and ex æquo, as Dd to Ff o DE to SG; that is (becaue is the imilar triangles PDE, PGS) o is PE to PS. Q. E. D.

Supoe a uperficies as EFfe (Pl. 22 Fig. 5.) to have its breadth infinitely diminished, and to be jut vanihing; and that the ame uperficies by its revolution round the axis PS decribes a phærical concavo-convex olid to the everal equal particles of which there tend equal centripetal forces; I ay that the force with which that olid attracts a corpucle ituate in P, is in a ratio compunded of the ratio of the olid $$\scriptstyle DE^2 \times Ff$$ and the ratio of the force with