Page:Newton's Principia (1846).djvu/232

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[Book I.
the mathematical principles

cases to revolve in conic sections, and composing sphaerical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do; which is very remarkable. It would be tedious to run over the other cases, whose conclusions are less elegant and important, so particularly as I have done these. I choose rather to comprehend and determine them all by one general method as follows.

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 evanscent 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 aequo, 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.

Suppose a superficies as EFfe to have its breadth infinitely diminished, and to be just vanishing and that the same superficies by its revolution round the axis PS describes a sphaerical concavo-convex solid, to the several equal particles of which there tend equal centripetal forces; I say, that the force with which that solid attracts a corpuscle situate in P is in a ratio compounded of the ratio of the solid DE² Ff and the ratio of the force with which the given particle in the place Ff would, attract the same corpuscle.

For if we consider, first, the force of the sphaerical superficies FE which