1. Whiteside 1974, Vol. 6: 568 n. 1.

2. Whiteside 1974, Vol. 6: 568 n. 1.

3. Newton [corres. 1688-1694] 1961, Vol. 3: 384.

4. Newton [math. 1684-1691] 1974, Vol. 6: 573. Proposed Proposition 6 . Where bodies describe all similar parts of similar figures in proportional times, their centripetal forces tend to centers similarly positioned in those figures and are to one another in a ratio compounded of the ratio of the heights directly and the doubled ratio of the times inversely.

5. Newton [math. 1684-1691] 1974, Vol. 6: 575-577. Proposed Proposition 7 . If in two orbits proportional ordinates stand at any given angles on proportional abscissas, and the centers of forces are similarly located in the abscissas, bodies shall describe corresponding parts of the orbits in proportional times and the centripetal forces will be as the heights of the bodies directly and the squares of the times inversely. . . .

Corollary . Therefore if one of the orbits APB be a circle and the other orbit AQB any ellipse, and the point S be the center of both, since the force whereby a body revolves with uniform motion in a circle is given . . . the force whereby another body might simultaneously revolve in the ellipse will be as the height SQ of that body.

6. Newton [math. 1684-1691] 1974, Vol. 6: 579-581. Proposed Proposition 8 . The force whereby any body P can revolve in any orbit APB whatever round the center S of force is to the force whereby another body P can revolve in the same orbit and in the same periodic time round any other center R of force as the product of the height of the first body and the square of the height of the second body, SP × RP ² , to the cube of the straight line PT which the straight line ST parallel to the orbit's tangent cuts off from the height of the second body in the direction of that body.

Corollary 1 . . . .

Corollary 2 . . . . the centripetal force will be reciprocally as the square of the height PR .

Corollary 3 . . . . will also in this case—where, that is, the ellipse has passed into a parabola—be reciprocally as the square of the height.

Corollary 4 . . . . the force whereby a body can revolve in a hyperbola about its focus as center will be reciprocally as the square of the height.

7. Newton [math. 1684-1691] 1974, Vol. 6: 581. Proposed Proposition 9 . If a body should, in a nonresisting space, revolve round a stationary center in any orbit whatever and describe any just barely nascent arc in a minimal time, and an "arrow" of the arc be drawn to bisect its chord and pass, when produced, through the center of force, then the centripetal force at the mid-point of the arc will be as that sagitta directly and the square of the time inversely.

Corollary 1 . . . . the centripetal force will then be reciprocally as the "solid" SP ² × QT ² / QR  . . .

Corollary 2 . . . .

Corollary 3 . . . . the centripetal force will be reciprocally as the "solid" SY ² × P .

Corollary 4 . . . .

Corollary 5 . . . .

8. Newton [math. 1684-1691] 1974, Vol. 6: 583. Proposed Lemma 12 . If in any diameter PG of a conic there be taken, on its concave side, PM equal to the latus rectum pertaining to that diameter, and through the points P and M a circle be described to touch the conic at P , then this circle will have the same curvature as that conic at P .

9. Newton [math. 1684-1691] 1974, Vol. 6: 585-589. Proposed Proposition 10 . Let a body move in the perimeter of the conic PQ : there is required the centripetal force tending to any given point S .

Corollary 1 . If the force tends toward the center of the conic . . .

Corollary 2 . If the force tends toward the conic's focus . . .

Corollary 3 . If the force tends toward an infinitely distant point . . .

Corollary 4 . If the conic passes into a circle and the centripetal force should tend to a point given in its circumference . . .

10. Whiteside 1974, Vol. 6: 581 n. 29.

11. Newton [math. 1664-1666] 1967, Vol. 1: 456. In Whiteside's original transcription of Newton's English manuscript, the word given here as may appeared as will . The change in meaning is quite dramatic: may indicates that Newton is giving one possible method for solving the problem while will indicates that something specific must follow. The editor, D. T. Whiteside, has informed me that he now believes that the word he first transcribed as "will" is in fact "may." In Herivel's transcription the word is given in brackets with a question mark: [will?] (Herivel 1965, 130). Clearly, Herivel also had difficulties in deciphering Newton's English script. Herivel, however, mistakenly transcribes Newton's "y ⁿ " as "that" rather than as "then" (Herivel 1965, 130), which further obscure's Newton's intent and makes more difficult the choice of "will" or "may" by context (i.e., " then the force . . . may be found" versus " that the force . . . will be found").

12. Newton [math. 1684-1691] 1974, Vol. 6: 578.

13. PT is equal to CA , where CA is half the major axis of the ellipse. Newton had demonstrated this relationship in the proof to Proposition 11 in the 1687 edition and intended it to appear in an introductory lemma in the unpublished radical revision. See Whiteside 1974, Vol. 6: 580 n. 25.

14. Newton [math. 1684-1691] 1974, Vol. 6: 585.

15. For the central ellipse, Proposition 10, the line PE = PS = PC and, thus, the force is directly as PC . For the focal ellipse, Proposition 11, the line PE is a constant of the ellipse (equal to the semi-major axis AC ) and thus the force is inversely proportional to the square of the distance SP . The remaining two published problems concerning circular paths are both dispatched in a similarly simple fashion as Corollaries 3 and 4 of this new Problem 2.

16. Whiteside 1974, Vol. 6: 589 n. 48.

17. Hall and Hall 1962, 293.

18. Ball 1893, 116.

19. Two copies of Newton's tract, both in English, survive in manuscript: one, at Cambridge, is an autograph and is not dated; the other, at Oxford, is a copy in the hand of Locke's amanuensis Brownover and is dated "March 1689," which is continue

presumably March 1690 (new style). The major portion of the manuscript is found in Ball 1893, 116-120. A complete copy appears in Hall and Hall 1962, 293-301, and in Herivel 1965, 246-254.

20. Between 1961 and 1969, four related articles appeared in Archives internationales d'histoire des sciences that gave rise to a debate which has continued intermittently ever since: Herivel 1961 and 1963, Hall and Hall 1963, and Westfall 1969. The discussion in this section of chapter 8 is taken from Brackenridge 1993. See Erlichson 1993 for an alternate argument concerning the "missing solution of 1679." For a suggestion that the curvature method may have been employed by Newton before 1679, see Nauenberg 1994 and my discussion in chapter 10 to follow.

21. Whiteside 1989, xv.

22. This special case is present in the copy of the manuscript at Cambridge but missing from the copy of the manuscript at Oxford.

23. Herivel 1965, 249.

24. It will be demonstrated in chapter 10 that the circular ratio 1 / ( YS ² × PV ) = 1 / ( SP ² × r × sin ³ a ) where SP is the radius, r the radius of curvature, and a the sine of the angle between the tangent and the radius. The radius of curvature is equal at aphelion and perihelion and, for any point on the major axis of the ellipse, the angle a is the same. Thus, the force is as the inverse square of the distance SP to any point on the major axis. This relationship was called to my attention by D. T. Whiteside.