Green Lion PressDensmore will not allow the students to take anything for granted. For example, she devotes no fewer than seventy-four pages to a penetrating scrutiny of Section 1, the method of first and ultimate ratios. As the title states, the book confines itself to what it calls the central argument, that is, Sections 1-3 of Book I and the early propositions of Book 3, the argument that concludes in the law of universal gravitation- and Densmore pursues that argument with all the intensity that the St. John's program encourages. Densmore's husband, William H. Donahue, has translated afresh all the propositions and other material from the Principia that the volume includes.
This is a wonderful book. Taking Newton in his own terms, it insists on the full rigor of the demonstrations and does not hesitate to point out where full rigor appears to be lacking. The flavor of the book can be sampled in its treatment of the phenomena cited at the beginning of Book 3, where Densmore pauses to explain in what sense generalizations not directly observable can be called phenomena (for example, Kepler's third law applied to the satellites of Jupiter) and how the data for them were collected in the late seventeenth century. She devotes no fewer than forty pages, virtually a tenth of the book, to the careful examination of a passage essential to the argument that occupies five pages of the Principia. As she says in the "Preliminaries," "we understand Newton only in understanding why he proved things as he did" (p. xxiv). Students are not the only ones who can profit from the exercise.
Richard S. Westfall
Indiana University
The first three sections of Book I deal with the motion of a point mass in a central force field. This study culminates with the demonstration that if the orbit is a conic section and the area law is valid for a focus, then the acceleration is inverse-square. The opening propositions of Book 3 deal with the main phenomena of the planetary system and contain a demonstration of the law of universal gravitation. It is a pleasure to follow Densmore's reconstruction of this momentous discovery in science, since the argument supporting it requires on the one hand very elementary mathematical tools, and on the other a profound understanding of the relationships between mathematical models and astronomical data.
Densmore's book is interesting not only for teaching purposes. Historians of science have a great deal to learn from it. The Principia is always difficult to read since Newton is often quite brief and leaves the reader to reconstruct the steps of the complete argument. This guidebook provides such an analysis. Every proposition, lemma or corollary from the above-mentioned sections of the Principia is first quoted (in a reliable translation provided by William H. Donahue) and then explained in a series of notes and in a carefully expanded proof. Of course, experts might disagree on specific points: the way in which the Principia's cryptic proofs can be expanded is not uniquely determined by Newton's text. For instance, there is no agreement on how the brief Corollary I to Proposition 13, Book 1-in which Newton presents his proof that conic sections are necessary orbits in an inverse-square force field-should be expanded. The author's statement that "if we want a mathematical proof of this corollary, it seems that we will need to look elsewhere" (p. 207) might be challenged. Furthermore, some scholars, such as Eric Aiton and Tom Whiteside, consider Newton's proof of Propositions 1 and 2, Book 1, to be problematic. Densmore does not tackle these complex interpretative issues. There is another criticism that might be raised: it concerns the contiguity between Newton's geometric methods and the calculus. The author tries to adhere to a geometric presentation faithful to Newton's text. However, in some cases, Newton seems to rely on calculus, rather than geometry. For instance, in the demonstration of Proposition 9, Book 1, concerning motion along an equiangular spiral (in a central force field it implies an inverse-cube force), Newton employs properties of this curve which he had studied in his fluxional writings. In many cases, even in these more elementary parts of the Principia, Newton's knowledge of the calculus is evident and might have been noted. In the commentary to Proposition 9, I would have liked a reference to Corollary 3 to Proposition 41, Book 1, where Newton faces the inverse problem of determining the orbit of a point mass accelerated by an inverse-cube central force via reduction to a polar "fluxional" (differential) equation. One might continue with such kind of observations. But this is inevitable when a text as complex as the Principia is commented upon and interpreted. I hope that the reader of this short review will understand that Densmore's book is a first-class work: it is a detailed, useful and enjoyable commentary on those mathematical demonstrations in which the theory of universal gravitation was first established.
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The intent of Densmore's guidebook is to involve the student actively in Newton's analysis; Newton's tendency to omit intermediate steps in the analysis offers ample opportunity for such involvement. To that end, the guidebook is designed on three levels: The first consists of the translation, by W. H. Donahue, from Newton's Latin text itself, and it is distinct from the author's notes and expanded proofs. The student can thus attempt to follow Newton without intervention. The second level offers minimal help in the form of notes that alert the student to possible omissions and potential pitfalls in Newton's presentation and then challenge the student to fill in any steps that are missing. The third level provides an expansion of Newton's sketch of the demonstration and offers a step-by-step demonstration of what Densmore thinks "Newton would have given as a complete proof."
Throughout the guidebook, the student is urged to attempt the demonstration before reading these extended notes, but the notes are always there as a safety net when needed. The challenge to understand Newton's analysis excites the author, and she has written the guidebook to communicate that excitement to the student.
On what level and in what time frame is such a communication possible? Densmore appears to gear the guidebook toward an upper-division undergraduate course, when she notes that "the Muses of this guidebook have been the students in my junior mathematics tutorials [at St John's]." Moreover, it is evident that some knowledge of Euclid's Elements is assumed, in both technique and substance. Specific references to the Elements are given, however, for those less familiar with Euclid. The time span for the course is a semester, although the author notes that "those who have more than a semester to spend on Newton can profitably work...out some of the intriguing side paths [not covered in the guidebook]." I can only look with envy at an institution such as St. John's that is willing to offer a semester to the Principia, and with absolute admiration at those that offer more.
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J. BRUCE BRACKENRIDGE
Lawrence University
Appleton, Wisconsin
Green Lion Press