History and Philosophy of Mechanics: Newton's Principia Mathematica

Readings

Required Text

[Densmore] = Densmore, Dana. Newton’s Principia: The Central Argument, Translation, Notes, and Expanded Proofs. 3rd ed. Green Lion Press, 2003. ISBN: 9781888009231.

SES #	TOPICS	READINGS	DISCUSSION QUESTIONS
1	The problem of motion in Greek Antiquity: Zeno and Aristotle	Aristotle’s Physics.	Do Zeno’s paradoxes pose genuine problems for the intelligibility of motion? What is gained, and what is lost, by defining “motion” the way Aristotle does?
2	Motion according to Galileo	Galileo’s Discourses Concerning Two New Sciences (1638). Galilei, Galileo. Galileo: Two New Sciences. Translated by Stillman Drake. University of Wisconsin Press, 1974. ISBN: 9780299064044: -–. Third Day, On Local Motion. [Preview with Google Books] -–. On Equable Motion. [Preview with Google books] -–. On Naturally Accelerated Motion through Proposition II. [Preview with Google Books]	How does Galileo’s mathematical science of motion differ from later “equations of motion”? How does Galileo deal with the problem of instantaneous velocity?
3	Introduction to Newton’s Principia	[Densmore] Foreword by: Curtis Wilson, p. xiii [Densmore] Preliminaries, pp. xxiii–lv [Densmore] Newton’s Preface to the Readers, pp. 3–4 Philosophical Reading: “Aristotle on Place, Time, Void, and Projectile Motion.” In Aristotle’s Physics.	What does Newton mean by “philosophy”?
4	The Definitions: Newton’s understanding of force	[Densmore] Definitions, pp. 5–21 [Densmore] Newton’s Commentary to Definitions 6-8, p. 21 [Densmore] Scholium to Definitions, pp. 22–7	How else might you define “quantity of motion”? What is motion and how might one measure it? How does Newton’s understanding of motion differ from Aristotle’s? What do we mean by “force”? Do we have a direct experience of it or is its meaning based on a definition? How are we to understand “forces propagated out into surrounding spaces”?
5	Newton’s Axioms or Laws of Motion	[Densmore] Laws 1, 2 and 3, pp. 28–31 [Densmore] Corollaries to the Laws of Motion, pp. 31–9 [Densmore] Scholium after the Laws of Motion, pp. 39–46	In Law 1, what is meant by a “state of resting or of moving uniformly”? Is Law 1 explained by (tacit) reference to an inherent force of inertia? What is the relation between Newton’s verbal statement of Law 2 and the equation now known as Newton’s Second Law of Motion? What are “attractions”? Has Newton proved that Law 3 applies to attractions?
6	Book I Section 1—The Lemmas: Newton’s application of mathematics to motion—his “method of first and ultimate ratios”	[Densmore] Lemma 1, pp. 47–50 [Densmore] Lemma 2, pp. 50–5 Archimedes and Euclid’s Elements. Philosophical Reading: Buridan, John. “The Medieval Theory of Impetus.” In Questions on the Eight Books of [Aristotle’s] Physics (1509).	What does Newton mean by “ultimately equal”? How does Newton’s treatment of vanishingly small quantities differ from that of Archimedes?
7	The Lemmas (cont.)	[Densmore] Lemma 3 and Corollaries 1–4, pp. 55–62 [Densmore] Lemma 4 and Corollary, pp. 62–70	How can a figure that is always rectilinear have a curvilinear limit?
8	The Lemmas (cont.)	[Densmore] Lemma 5, pp. 70–1 [Densmore] Lemma 6, pp. 72–7 [Densmore] Notes on ratios between vanishing quantities, pp. 77–83	What is “continuous curvature”?
9	The Lemmas (cont.)	[Densmore] Lemma 7 and Corollaries 1–3, pp. 83–91 [Densmore] Lemma 8 and Corollary, pp. 91–4 [Densmore] Lemma 9, pp. 94–9 Philosophical Reading: Galilei, Galileo. Discourses Concerning Two New Sciences (1634).
10	The Lemmas (cont.)	[Densmore] Lemma 10 and Corollaries 1–5, pp. 99–107 [Densmore] Scholium after Lemma 10, pp. 107–8 [Densmore] Lemma 11 and Corollaries 1–3, pp. 108–17 [Densmore] Scholium to the Lemmas, pp. 117–21	What are “nascent” and “evanescent” quantities and how do they differ? How does Newton’s method differ from the “method of indivisibles”?
11	Book I Section 2—Newton’s Propositions on Centripetal Forces	[Densmore] Reading: Proposition 1 and Corollaries 1–6, pp. 123–41	Does Proposition 1 depend in any way on the assumption of an inverse-square force? Is Newton assuming that there is a body at the center of forces? Is it significant that Newton speaks of a center of forces rather than a center of force? Do the impulsive forces acting at each point become ultimately a continuously varying force? What is the relationship between Proposition 1 and Kepler’s Second Law?
12	Newton’s Propositions (cont.)	[Densmore] Proposition 2, pp. 142–7 [Densmore] Scholium, p. 147 [Densmore] Proposition 3 and Corollaries 1–3, pp. 148–53 [Densmore] Scholium, pp. 153–4 Philosophical Reading: Descartes, René. Principia Philosophiae, Part II (1644). [Preview with Google Books] A more recent edition of the text: Descartes, René. Principia Philosophiae (1677). Kessinger Publishing, 2010. ISBN: 9781166267605.	Proposition 2 proves the converse of Proposition 1. Is there an easier way to prove it? Is Newton’s use of “vanishing quantities” (“least triangles”) mathematically satisfactory? Why does Proposition 3 introduce a body as the center of forces?
13	Newton’s Propositions (cont.)	[Densmore] Proposition 4 and Corollaries 1–9, pp. 154–77 [Densmore] Scholium, pp. 177–8
14	Newton’s Propositions (cont.)	[Densmore] Proposition 6 and Corollaries 1 and 5, pp. 178–88	How can there be “least times” that are unequal? In Corollary 1, why does Newton say, “inversely as the solid”?
15	Newton’s Propositions (cont.)	[Densmore] Proposition 7 and Corollaries 2 and 3, pp. 188–200 [Densmore] Proposition 9, pp. 201–8 Philosophical Reading: Boyle, Robert. “Of the Excellency and Grounds of the Corpuscular or Mechanical Philosophy” (1674). A more recent edition of the text: Ariew, Roger, ed. Modern Philosophy: An Anthology of Primary Sources. Edited by Eric Watkins. Hackett Publishing Company, 2009. ISBN: 9780872209794.	Does Proposition 7 show that eccentric circular orbits are somehow “unnatural”?
16	Excursus on Apollonius’ On Conic Sections	Apollonius’ On Conic Sections: -–Book I, Definitions 4–8, Propositions 11–3, 15; Definitions 9 and 11; Propositions 17, 21, 32, 35, 37, 46, 47, 49, and 60. -–Book III Propositions 45, 48, 49, 51, and 52. Philosophical Reading: Huygens, Christian. “Huygens on the Hypothetico-Deductive Method.” In Treatise on Light (1687). [Preview with Google Books] A more recent edition of the text: Huygens, Christian. Treatise on Light. Tredition, 2011. ISBN: 9783842476653.
17	Excursus on Apollonius’ On Conic Sections (cont.)	Apollonius’ On Conic Sections: Book I, Definitions 4–8, Propositions 11-3, 15; Definitions 9 and 11; Propositions 17, 21, 32, 35, 37, 46, 47, 49, and 60. Book III Propositions 45, 48, 49, 51, and 52. Philosophical Reading: Huygens, Christian. “Huygens on the Hypothetico-Deductive Method.” In Treatise on Light (1687). [Preview with Google Books] A more recent edition of the text: Huygens, Christian. Treatise on Light. Tredition, 2011. ISBN: 9783842476653.
18	Excursus on Apollonius’ On Conic Sections (cont.)	Apollonius’ On Conic Sections: -–Book I, Definitions 4–8, Propositions 11–13, 15; Definitions 9 and 11; Propositions 17, 21, 32, 35, 37, 46, 47, 49, and 60. -–Book III Propositions 45, 48, 49, 51, and 52. Philosophical Reading: Huygens, Christian. “Huygens on the Hypothetico-Deductive Method.” In Treatise on Light (1687). [Preview with Google Books] A more recent edition of the text: Huygens, Christian. Treatise on Light. Tredition, 2011. ISBN: 9783842476653.
19	Newton’s Propositions (cont.)	[Densmore] Lemma 12, pp. 208–9 [Densmore] Notes on Lemma 12, Appendix A, pp. 491–6 [Densmore] Proposition 10, pp. 209–15	What is the connection between the elliptical orbits of Proposition 10 and simple harmonic motion?
20	Newton’s Propositions (cont.)	[Densmore] Corollaries 1 and 2 to Proposition 10, pp. 215–24 [Densmore] Scholium, pp. 224–6	Why, in Corollary 1 to Proposition 10, does Newton present the converse of Proposition 10 without proof? How might he prove it? Is it surprising, in Corollary 2, that the periods of these elliptical orbits are independent of the size and shape of the ellipses?
21	Book I Section 3—Newton’s Propositions on the motion of bodies in eccentric conic sections	[Densmore] Proposition 11, pp. 227–36 [Densmore] Proposition 12, pp. 236–47 Philosophical Reading: Newton on absolute space and time: Isaac Newton, Scholium (to the Definitions, Principia, Book 1) on Space and Time (1687).
22	Newton’s Propositions on conic sections (cont.)	[Densmore] Lemmas 13 and 14, pp. 247–8 [Densmore] Notes on Lemma 13, Appendix A, pp. 496-500 [Densmore] Notes on Lemma 14, Appendix A, pp. 500–3 [Densmore] Proposition 13 and Corollaries 1 and 2, pp. 248–58	Does Corollary 1 to Proposition 13 succeed in proving the converse of Propositions 11–13? Is the argument entirely mathematical, or does it depend on assumptions about how things work in the real world? Is it mathematically or merely physically impossible to “describe two mutually tangent orbits with the same centripetal force and the same velocity”?
23	Newton’s Propositions on conic sections (cont.)	[Densmore] Proposition 14 and Corollary, pp. 258–63 [Densmore] Proposition 15 and Corollary, pp. 263–7	Is it intuitively plausible that the period time of elliptical orbits would not depend on the size of the minor axis?
24	Newton’s Propositions on conic sections (cont.)	[Densmore] Proposition 16 and Corollaries 1–3, pp. 267–73 [Densmore] Proposition 17 and Corollaries 1 and 2, pp. 273–92 Philosophical Reading: Newton’s exchange with Bentley on action at a distance: Isaac Newton, letter to Bentley (1693).	How can the “absolute quantity of force” be “known”?
25	Book III—Of the System of the World	[Densmore] Newton’s Preface to Book III, p. 301 [Densmore] Rules of Philosophizing, pp. 303–5	What does Newton mean by the phrase, “System of the World”? What does Newton mean, in the Preface, by distinguishing “mathematical principles of philosophy” from “philosophical principles of philosophy”? In what sense have Newton’s scholia been “philosophical”? What does Newton mean by “true causes”? How do philosophers know that “nature does nothing in vain” and “does not indulge herself in superfluous causes”? Why does Newton say, “I do not at all assert that gravity is essential to bodies”? What is an “argument from induction”?
26	The Phenomena	[Densmore] Phenomenon 1, pp. 308–21 [Densmore] Phenomenon 2, pp. 321–5	Are the “Phenomena” observations, or conclusions based on observations? In the first edition of the Principia, Newton called them “Hypotheses.” What is the significance of the change of terminology?
27	The Phenomena (cont.)	[Densmore] Phenomenon 3, pp. 325–35 [Densmore] Phenomenon 4, pp. 335–47 Philosophical Reading: Descartes, René. “Descartes v. Newton on the Rules of Philosophy.” In Rules for the Direction of the Mind (composed ca. 1628). A more recent edition of the text: Descartes, René. Rules for the Direction of the Mind. Bobbs-Merrill Co, 2000. ISBN: 9780672603341.	In Phenomenon 3, why does Newton leave out Earth from the list of planets orbiting the sun? Has Newton proved that the planets revolve around the sun?
28	The Phenomena (cont.)	[Densmore] Phenomenon 5, pp. 347–8 [Densmore] Phenomenon 6, pp. 349–50
29	Book III Propositions on the planets and their moons	[Densmore] Proposition III.1, pp. 351–3 [Densmore] Proposition III.2, pp. 353–5
30	Book III Propositions (cont.)	[Densmore] Proposition III.3, pp. 356–60 [Densmore] Proposition III.4, pp. 360–76 [Densmore] Scholium, pp. 376–80 Philosophical Reading: “Newton v. Leibniz on Space and Time.” In the Leibniz-Clarke Correspondence (1716). A more recent edition of the text: Alexander, H. G., ed. The Leibniz-Clarke Correspondence: Together with Extracts from Newton’s Principia and Opticks. Manchester University Press, 1998. ISBN: 9780719006692.	What does Newton mean by “gravitates” and “gravity” in Proposition III.4? Why does Newton feel the need to “display the demonstration more amply” in the Scholium?
31	Book III Propositions (cont.)	[Densmore] Proposition III.5 and Corollaries 1–3, pp. 380–7 [Densmore] Scholium, pp. 385–8	What does Newton mean, in the Scholium, when he says “the cause of the centripetal force by which the moon is confined to its orbit is obligated to extend to all planets, by Rules 1, 2 and 4”? Why is Newton entitled, after Proposition III.5, to call centripetal force gravity? Why isn’t this enough to establish universal gravitation?
32	Flashback to Proposition II.24	[Densmore] Proposition II.24 and Corollaries 1, 6 and 7, pp. 387–402
33	Book III Propositions (cont.)	[Densmore] Proposition III.6 and Corollaries 1–5, pp. 402–22 Philosophical Reading: Berkeley, George. “Berkeley’s Philosophical Critique of the Calculus.” In The Analyst (1754). [Preview with Google Books] A more recent edition of the text: Berkeley, George. The Analyst A Discourse Addressed To An Infidel Mathematician. Kessinger Publishing, 2010. ISBN: 9781161456448.
34	Flashback to Proposition I.69; Proposition III.7	[Densmore] Proposition I.69 and Corollary, pp. 422–6 [Densmore] Scholium, pp. 426–7 [Densmore] Proposition III.7, pp. 427–34	When Newton speaks of bodies “pulling” one another, is he assuming a particular model of how gravitational attraction? How do we know that Newton’s Third Law applies to gravitation? How can inverse-square particle-to-particle attraction explain the center-to-center gravitation of the heavenly bodies?
35	Flashback to Propositions I.71-I.76	[Densmore] Proposition I.71, pp. 435–50 [Densmore] Scholium, p. 450	Intuitively, what would you say is the reason that spherical masses attract as if all the mass were concentrated at the center? Are your intuitions indicated by Proposition 71?
36	Flashback to Propositions I.71-I.76 (cont.)	[Densmore] Proposition I.74, pp. 450–2 [Densmore] Proposition I.75 and Corollaries 1 and 2, pp. 452–9 [Densmore] Proposition I.76, pp. 459–60 Philosophical Reading: Kant, Immanuel. “Kant’s Newtonian Philosophy of Nature.” In Metaphysical Foundations of Natural Science (1786). A more recent edition of the text: Kant, Immanuel. Metaphysical Foundations of Natural Science. Bobbs-Merrill Co, 1970. ISBN: 9780672513800.
37	Book III Propositions (cont.)	[Densmore] Proposition III.8 and Corollaries 1 and 2, pp. 460–77	Has Newton now proved the principle of universal gravitation? What remains to be done?
38	Book III Propositions (cont.)	[Densmore] Proposition III.13, pp. 477–85	Does Newton’s celestial mechanics (elliptical orbits sweeping out areas proportional to the times) depend on assuming the truth of “Kepler’s Laws”?
39	Conclusion of the Course: Newton’s General Scholium	[Densmore] The General Scholium, pp. 485–9	What are the theological implications of Newton’s “System of the World”? Regarding the cause of gravity, what does Newton mean by saying, “I do not contrive hypotheses”? What is “experimental philosophy”?