**Newton's ** *
Principia*

Dear ARITHMOSticians,

Here are my suggestions for our readings on 19-20 September 2009:

- Newton's three laws, pp. 416-417 of the Cohen/Whitman edition of
Newton's
*Principia*.- Historical question: When were these laws expressed in the language of the calculus? Who is responsible?

- First and ultimate ratios, Book I, Section I, Lemmas 1, 2
and 7, pp. 433-434 and 436.
- Historical question: Why did it take so long to get the epsilon-delta definition of the limit?

- Differential calculus, Book II, Section II, Lemma 2, pp. 646-650.
- Integral calculus, Book I, Section XIII, Proposition 91 and its Lemmas 1 and 2, pp. 615-617.
- Centripetal force: Book I, Section 2, Propositions 10 and 11, pp. 459-463.
- Curve fitting: Book III, Lemma 5, pp. 896-897.

You will observe that I have not taken Newton's advice (see below) on reading
the *Principia*. But I have made these choices influenced by what George
Smith had to say about the *Principia *
at the January 2009 joint
meetings. The information below is a more careful enumeration of what Smith
discussed together with a few books dealing with the *
Principia*; of
course, this is not comprehensive.

At the beginning of Book 3 of the *
Principia*, Newton provides the
following advice to the reader:

It will be sufficient to read with care the Definitions [pp. 403-415], the Laws of Motion [pp. 416-430], and the first three sections of book 1 [pp. 433-472], and then turn to this book 3 on the system of the world, consulting at will the other propositions of Books 1 and 2 which are referred to here. [p. 793 of the Cohen/Whitman translation; the parenthetical page numbers also refer to this edition.]

Newton also noted that he had presented the "propositions in a mathematical style, so that they may be read only by those who have first mastered the principles" [p. 793 of C/W].

At the joint AMS/MAA meetings in January 2009 in Washington, DC, George E. Smith
of the Philosophy Department at Tufts made a presentation on "The Mathematics of
Newton's *Principia Mathematica*". Since he will be one of the speakers
at the MAA Short Course in Berkeley in January 2010, I plan to read carefully
the portions of the *Principia* that he mentioned in his talk:

- The method of first and last ratios:
- Lemma 1. Cohen/Whitman, p. 433; Desnmore pp. 17-19. Question: Why did it take so long to get the epsilon-delta definition of the limit?
- Lemma 2. C/W, p. 433; Densmore, pp. 20-24.
- Lemma 7. C/W, p. 436; Densmore, pp. 54-61. These page numbers indicate that Densmore has an abundance of commentary.

- Differential Calculus:
- Book 2, Lemma 2. C/W pp. 646-647 and 50.
- Leibniz. C/W p. 649.

- Integral Calculus:
- Book 1, Prop. 81. C/W pp. 603-606.
- Book 1, Prop. 91. C/W pp. 615-618.

- Centripetal Force:
- Book 1, Prop. 6. C/W p. 453.
- Book 1, Prop. 10. C/W p. 459.
- Book 1, Prop. 11. C/W p. 462.

- Motion of apsides:
- Book 1, Prop.43. C/W p. 543.
- Book 1, Prop. 44. C/W p. 535.
- Book 1, Prop. 44, Cor. 2. C/W p. 537.
- Book 1, Prop. 44, Cor. 3. C/W p. 438.
- Book 1, Prop. 45. C/W p. 539, and 51.
- Book 1, Prop. 45, Prob. 31. C/W p. 539.

And now, a very selective bibliography:

Newton, Isaac,
*The Principia: mathematical
principles of natural philosophy.* *A new translation by I. Bernard Cohen and Anne Whitman, assisted by Julia Budenz. Preceded by ``A guide to Newton's Principia'' by Cohen*.
*
University of California Press, Berkeley, CA,* 1999. xviii+974 pp. ISBN:
0-520-08816-6. MR1965137 (2004a:01042)
only lists the contents. [Have book]

This is the translation that I am reading. I am reading it in conjunction with the Densmore book listed below.

Densmore, Dana, *Newton's Principia: The Central Argument. Translation,
Notes, and Expanded Proofs. *Green Lion Press, Third edition. [Have book]

Prepared as a text for students at St. John's College Great Books Program in Santa Fe.

Newton, Isaac, 1642-1727, Philosophiae naturalis principia mathematica, Harvard University Press, 1972. MR0485110 (58 #4966a). 3d ed. (1726), with variant readings, assembled and edited by Alexandre Koyré and I. Bernard Cohen, with the assistance of Anne Whitman. Latin text, with editorial matter in English.

"The history and purpose of the edition are explained at length in the companion volume,Introduction to Newton's Principia[by I. B. Cohen] published simultaneously with these two text volumes." This is the definitive Latin edition.

Cohen, I Bernard, 1914-2003, *
Introduction to Newton's "Principia", *
Harvard University Press, 1971. Reviewed by M. A. Hoskin MR0465755 (57 #5647).
[Have book]

Cohen, I. Bernard, "The review of the first edition of
Newton's *Principia* in the Acta Eruditorum, with notes on the other
reviews," pp 323-353*
*in* The investigation of difficult
things *(Cambridge Univ. Press, Cambridge, 1992). Reviewed by Emily Grosholz,
MR1200060 (94e:01010). [Have volume]

Newton, Isaac, 1642-1727, *
The mathematical papers of Isaac Newton, edited
by D. T. Whiteside with the assistance in publication of M. A. Hoskin***,
**Cambridge, London, Cambridge U.P., 1967-1981. USMA QC16.N7 C35
2002.

Volume 6, dealing with the years 1684 to 1691, deals with thePrincipia. Volume 8 deals with the priority dispute and has much information on thePrincipia.

Guicciardini, Niccolò, "Isaac Newton, *
Philosophiae
naturalis principia mathematicaI, *first edition," (1687)," pp. 59-87
in* Landmark writings in Western mathematics 1640-1940* (Elsevier, 2005),
edited by Ivor Grattan-Guinness.

Guicciardini, Niccolò, "Conceptualism and contextualism
in the recent historiography of Newton's *Principia,*"
*Historia
Mathematica*,
Volume 30, Issue 4, November 2003, Pages 407-431. [Online with
ScienceDirect]

Recently the *Principia* has been the object of renewed
interest among mathematicians and physicists. This technical interpretative work
has remained somewhat detached from the busy and fruitful Newtonian industry run
by historians of science. In this paper will advocate an approach to the study
of the mathematical methods of Newton's *Principia* in which both
conceptual and contextual aspects are taken into consideration.

Cohen, I Bernard and Smith, George E., The Cambridge Companion to Newton. Cambridge: Cambridge University Press, 2002. [USMA: QC16.N7 C35 2002]

For controversy, some papers of Robert Weinstock should be included.

**Papers by G. E. Smith:**

"How Did Newton Discover Universal Gravity?" in Beyond Hypothesis: Newton's Experimental Philosophy, The St. John's Review, vol. xlv, no. 2 (1999), pp. 32-63.

"The Vis Viva Dispute: A Controversy at the Dawn of Dynamics,"
*Physics Today*, October 2006, pp. 31-36.

Comments on Ernan McMullin's "The Impact of Newton's Principia on the Philosophy of Science." In JSTOR.

This list must be very incomplete.

It seems that no authenticated portrait of Hooke survives
today. Newton instigated the removal of Hooke's portrait in the Royal Society.
In 2003, the historian
Lisa Jardine
claimed a recently discovered portrait represents Robert Hooke. However,
Jardine's hypothesis was soon disproved by William Jensen (University of
Cincinnati) and independently by the German researcher Andreas Pechtl (Johannes
Gutenberg University, Mainz). Actually, the portrait depicts
Jan Baptist van
Helmont. A seal used by Hooke displays an unusual profile portrait of a man's head,
that some have argued portrays Hooke. Both these claims remain in dispute,
however. Moreover, the engraved frontispiece to the 1728 edition of Chambers'
*Cyclopedia* shows as an interesting detail a bust of Robert Hooke.

A paper describing Jensen’s re-identification of the portrait
appeared in the November 2004 issue of the British journal *Ambix*, and the
January 2005 issue of *Nature* featured an article about the portrait mystery.