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that the area A S B is less than the area BSa; andas the body moves more sluggishly from B to a, soalso it is more sluggishly retarded by the orb at S.But it is precisely because of these opposite differencesthat the course of the body from B to a resemblesin shape the course from A to B. To show that thisis in accordance with the facts,—that is, to explainthe relation without undertaking to prove that it musthold,—let us consider the state of the body at p, apoint symmetrically placed with respect to P (that is,as far from B C, a C, as P is from B C, C A). At p thecourse of the body is for the moment in directionp t, and by the properties of the ellipse the angle S ptis equal to the angle S P T ; so that as far as directionis concerned the retarding influence of the orb at S isas effective on the body when at p as when at P. Inmagnitude, however, the pull is less in the proportionof the square of S P to the square of S p. But to makeup for this,— exactly ,—the velocity at p is less than thevelocity at P, in the same proportion that S P is lessthan S p.* This deficiency acts doubly (or, as it were,squares itself): for the reduced velocity causes the
* The velocities in an elliptic orhit are not generally proportionalinversely to the distances from the attracting centre, but they are inthe case of two such positions as p and P. In reality, because ofthe eqnal description of areas the velocities at different pointsare inversely proportional to the perpendiculars from the point S tothe tangent through the respective points. But if tangents weredrawn from S to P T and p t, these perpendiculars would clearlybe proportional to S P and S p, simply because S P and S p areequally inclined to P T and p t.