and Proposition 10

Following the solution to Proposition 9, Newton inserted Lemma 12, a relationship that appeared as Lemma 1 in On Motion and that is required in the analysis of the direct problem that appears in Proposition 10.

Principia : Lemma 12. On Motion : Lemma 1.

All parallelograms described around a given ellipse are equal to each other.

Understand the same for parallelograms described in a hyperbola around its diameters .

All parallelograms described around a given ellipse are equal to each other .

This is evident from Conics. This is established from the Conics.

This relationship is demonstrated in Book 7, Proposition 31 in the Conics of Apollonius.^[25]

Newton has added the reference to a hyperbola because in the 1687 Principia he discussed hyperbolic orbits as well as elliptical orbits. See the discussion and diagram for Lemma 1 in chapter 4.

Principia : Proposition 10. Problem 5. On Motion : Problem 2.

Let a body be orbited on an ellipse; there is required the law of centripetal force being directed to the center of the ellipse .

A body orbits on a classical ellipse; there is required the law of centripetal force being directed to the center of the ellipse .

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The opening statements of this proposition in On Motion and in the Principia are almost identical, except for the change from the active, "a body orbits," to the passive, "let a body be orbited." The bodies of the texts differ only by the addition of three parenthetical expressions:

"(from the Conics )," "(by Lemma 12)," and "(by the Corollary to Theorem 5)," and by the addition of a line of qualification concerning a set of propositions. Except for a slight

rephrasing of one other line, the texts are identical, word for word. The reader should refer to the detailed discussion of Problem 2 in chapter 5 if there are questions concerning Proposition 10 of the 1687 Principia .

Section 3

Section 3. Of the motion of bodies in eccentric Conic Sections

The preceding section contains the solutions to the direct problems of a circular path with a center of force on the circumference, a spiral path with the center of force at its pole, and an elliptical path with the center of force at the center of the ellipse. These are preliminary examples of the application of the paradigm of Proposition 6. The direct Kepler problem commands much more respect than do these preliminary examples, however, and it is with Proposition 11 and the solution to that problem that Newton opens this new section. It

provides the answer to the question raised by Halley on his visit to Newton, a question that set into motion the activity that eventually resulted in the publication of the Principia . As he concluded Proposition 11 Newton referred to "the dignity of the problem" and its place of honor at the beginning of a new section. He also gave the solutions to the other conic sections, the hyperbola and the parabola, as separate propositions rather than in a scholium to the proposition on elliptical motion, as he had done in On Motion .

Principia : Proposition 11. Problem 6. On Motion : Problem 3.

Let a body be revolved on an ellipse; there is required the law of centripetal force being directed to a focus of the ellipse .

A body orbits on an ellipse; there is required the law of centripetal force directed to a focus of the ellipse .

The opening statements are essentially identical, as are the bodies of the text, except that the active statement "a body orbits" in On Motion is now the passive statement "let a body be revolved."^[26] Newton has added to the demonstration in the Principia the following items: a description of a parallelogram, four parenthetical expressions, a qualification, and references to Lemma 8, Lemma 12, and Theorem 5. He has removed the auxiliary ratio of M to N that appeared in the original solution, and he has replaced the scholium with a closing statement.

With these minor exceptions, the

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texts are identical, word for word, and the reader should refer to the detailed discussion of Problem 3 in chapter 5 if there are questions concerning Proposition 11 of the 1687 Principia as it is given in the Appendix.

Principia : Proposition 12. Problem 7. Let a body be moved on a hyperbola; there is required the law of centripetal force being directed to a focus of the figure .

Newton did not present a separate demonstration for the solution to the direct problem of hyperbolic orbits in On Motion . In Proposition 12 of the Principia , however, he has very carefully constructed a demonstration of hyperbolic orbits that follows in detail the demonstration of elliptical orbits in Proposition 11. After the opening lines describing the figure, the construction follows the previous solution line for line with only the few necessary accommodations to the new figure. The reader should refer to the detailed discussion of Problem 3 in chapter 5 if there are questions concerning Proposition 12 of the 1687 Principia .

Principia : Lemma 13. The latus rectum of a parabola pertaining to any vertex is quadruple the distance of that vertex from the focus of the figure. This is evident from the Conics.^[27]

Principia : Lemma 14. A perpendicular dropped from the focus of a parabola to its tangent is a mean proportional between the distance of the focus from the point of contact and the distance from the principal vertex of the figure .

These demonstrations of the properties of a parabola are required in the analysis of parabolic motion to follow in Proposition 13.^[28]

Principia : Proposition 13. Problem 8. Let a body be moved on the perimeter of a parabola;

there is required the law of centripetal force being directed to the focus of this figure . Newton did not present a separate demonstration for the solution to the direct problem of parabolic orbits in On Motion , but he does in the Principia . Of particular interest is the first corollary to this proposition, in which Newton claims that solutions to the three direct problems also provide solutions to the inverse problem.

Principia : Proposition 13. Corollary 1. From the last three propositions it follows that if any body P should depart from position P along any straight line PR, with any velocity, and is at

the same time acted upon by a centripetal force that is reciprocally proportional to the square of the distance from the center, this body will be moved in one of the sections of conics having a focus at the center of forces; and conversely .

Newton was criticized for failing to defend this assumption and he provided an outline of a defense in an expanded version of this corollary in the 1713 edition of the Principia . (The discussion of this point will be continued in chapter 10.) Whether Newton succeeded or failed in providing a satisfactory solution for the inverse problem has been the subject of

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considerable scholarly debate from the late seventeenth century until the present.^[29]

Principia : Proposition 14. Theorem 6. If several bodies should be revolved around a common center, and the centripetal force should decrease in the doubled ratio of the distances from the center, I say that the latera recta of orbits are in the doubled ratio of the areas that bodies describe by radii constructed to the center in the same time .

This relationship is required in the demonstration of Proposition 15 to follow.

Principia : Proposition 15. Theorem 7. On Motion : Theorem 4.

With the same suppositions, I say that the periodic times in ellipses are in the three-halved ratio of the transverse axes .

Supposing that the centripetal force is reciprocally proportional to the square of the distance from the center, the squares of the periodic times in ellipses are as the cubes of their transverse axes .

The revised demonstration of Kepler's "three-halves power law" in the 1687 Principia , which is given in the Appendix, is much simpler than the demonstration in On Motion , which is discussed in detail in chapter 6.

Principia : Proposition 16. Theorem 8. With the same suppositions, and with straight lines drawn to bodies that touch the orbits in the same places, and with perpendiculars dropped to these tangents from a common focus, I say that the velocities of the bodies are in a ratio compounded of the ratio of perpendiculars inversely, and the half ratio of the latera recta directly .

The demonstration of this proposition is followed by nine corollaries in which the relationship of the speeds and latera recta of conic sections is explored. Specifically, Corollaries 1 and 3 are employed in the proposition to follow in which the nature of a particular type of conic (elliptical, hyperbolic, or parabolic) is given by the relative magnitude of its initial projection speed.

Principia : Proposition 17. Problem 9. On Motion : Problem 4.

Supposing that the centripetal force be

made reciprocally proportional to the Supposing that the centripetal force be made reciprocally proportional to the

square of the distance from its center, and that the absolute quantity of that force is known; there is required a line which a body will describe, when released from a given position with a given velocity along a given straight line .

square of the distance from its center, and that the absolute quantity of that force is known; there is required an ellipse which a body will describe, when released from a given position with a given speed along a given straight line .

Let the centripetal force directed to point S be that which makes the body p orbit in any given orbit pq. . . . [emphasis added]

Let the centripetal force directed to point S be that which makes body p orbit in a circle pq. . . . [emphasis added]

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The opening statements in Problem 4 from On Motion and Proposition 17 from the Principia are identical except for a description of the general path as "a line" rather than "an ellipse" and the replacement of the word for "speed" (celeritate ) with the word for "velocity" (velocitate ).

In the detailed statement of the text, however, Newton has made many changes, including changing the reference "circle" to "any orbit." He has not changed the proof in any substantive way, but clearly he was not satisfied with the presentation in On Motion . The interesting changes in the text are from the specific required "ellipse" for body P in On Motion to the more general required "line" in the Principia , and from the "a circle pq " for the reference body p in On Motion to the more general "any given orbit pq " in the Principia . Neither of these changes are more than cosmetic. In the revision for the Principia , Newton soon lets the body P "deflect under the compulsion of the centripetal force into the conic section PQ ." The

"conic section" is still more general than a specific "ellipse," but it is still not a general "line."

The shift from the reference "a circle pq " to "any given orbit pq " is simply the recognition that the reference circle is sufficient but not necessary: any orbit will do.^[30]

Conclusion

Thus, the first three theorems and the first three problems of On Motion have provided the basis for the first seventeen propositions of the Principia . In Section 1 of Book One, Newton has added detailed definitions, set forth the laws of motion in an axiomatic fashion, and provided a number of lemmas designed to provide a formal background to the limiting procedures that he used in the earlier tract without a defense. Section 2 opens with a

demonstration in Proposition 1 of the law of equal areas, which was the first theorem in On Motion , and Section 3 closes with the demonstration of Proposition 17, which was also the final problem on planetary motion in On Motion . Independent of the additional material, the core of the dynamics of the two works remains essentially unchanged. The basic paradigm for solving problems remains the linear dynamics ratio, which now appears as Proposition 6.

Newton follows it with the same two preliminary problems given in On Motion plus the addition of a problem on spiral motion. The single method of Proposition 6 is applied to the three preliminary problems and then Newton presents the solution to the distinguished Kepler problem of elliptical/focal motion. In the revisions to follow, however, Newton introduces two other methods to solve the same problems.

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Eight—