During the past few decades, starting with I. B. Cohen’s and D. Whiteside’s groundbreaking studies of the historical background, development, and content of Newton’s Principia, and A. Shapiro’s edition of Newton’s Optical Papers, there have appeared a large number of articles and books enlarging our understanding on this subject. This scholarly work is attested by the voluminous list of references that appear in the new books by Ducheyne and Harper. These two authors focus their attention primarily on Newton’s scientific method and its successful applications to astronomical phenomena, while Ducheyne also considered his less-successful application to optics.

Already in the preface of the first edition of his Principia (1687), Newton explained with great clarity the fundamental features of his methodology for astronomy: “For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions, then to demonstrate the other phenomena from these forces.” He continued by outlining succinctly the content of the Principia: “It is to these ends that the general propositions in books 1 and 2 are directed, while in book 3 our explanation of the system of the world illustrates these propositions. For in book 3, by means of propositions demonstrated mathematically in books 1 and 2, we derive from celestial phenomena the gravitational forces by which bodies tend towards the sun and towards the individual planets. Then the motions of the planets, the comets, the moon, and the sea are deduced from these forces by propositions that are also mathematical.”

The full title of Newton’s magisterial book, Mathematical Principles of Natural Philosophy, emphasized the fact that the calculations of celestial phenomena were based on mathematical principles that were deduced from observations. These principles have become known as Newton’s laws of motion, although he did not credit himself with their discovery, stating in an early Scholium of his book, that “The principles that I have set forth are accepted by mathematicians and confirmed by experiments of many kinds.”

During the preparation of the second edition of the Principia (1713), its editor, Roger Cotes, gave Newton some excellent advice: “I think it will be proper besides the account of the Book & its Improvements to add something more particular concerning the manner of Philosophing made use of & wherein it differs from that of De Carters & and others. I mean in first demonstrating the Principles it imploys. This I would not only assert but make evident by a short deduction of the Principle of Gravity from the Phenomena of Nature, in a popular way, that may be understood by ordinary Readers & may serve at the same time as Specimen to them of the Method of the whole Book.”

Unfortunately, Newton failed to heed Cotes’s advice, and neither did Ducheyne or Harper. But I will follow it by discussing Newton’s moon test, because it gives an explicit demonstration of Newton’s methodology, which is the main subject of the two books under review.

Already eight years before the publication of the Principia, Newton deduced that the radial dependence of the gravitational attractive force between the sun and the planets varies inversely as the square of their distance. Like Christiaan Huygens, Newton had demonstrated that a body revolving with uniform velocity v on a circle with radius r, has a radial acceleration a = v ²/r directed toward the center of the circle. Newton argued that the force that confined a body in such an orbit was an attractive centripetal force proportional to a. Later on, this condition appeared as Prop. 4 of the Principia. Accepting the Copernican view that each planet revolves approximately on such an orbit around the sun with period t, and substituting for v the kinematical relation v = 2πr/t, Newton found that a = 4π² r/t ². Then, applying Kepler’s harmonic law, based on astronomical observations, that t ² is proportional to r ³, led Newton to his seminal discovery that a is proportional to 1/r ². Furthermore, Newton assumed that this radial dependence is also a terrestrial property, and tested this additional hypothesis by comparing the radial acceleration of the moon on its orbit around the earth with the Galilean acceleration g of falling bodies on the surface of the earth. In this case r corresponds to the radius of the earth, but at the time Newton used an incorrect value for it, and his comparison failed. But by 1684, when he began to write his Principia, he found the correct value of this radius, and verified his hypothesis. Moreover, a year later he found the remarkable mathematical proof, given in Prop. 71, that the 1/r ² dependence remained valid at the earth’s surface. Later on, in book 3 of the Principia, Newton also included the additional observational evidence that the four known satellites of Jupiter satisfy Kepler’s harmonic law, as required by his gravitational hypothesis.

In his book, Harper devotes an entire chapter and an appendix to discussing Newton’s detailed numerical comparison in the Principia (corollary 7 of Prop. 37, book 3) between the observed value of g, that he and Huygens had obtained from pendulum experiments, with the values of the lunar radial acceleration a obtained from more accurate observations of its period and of the mean distance of the moon from the earth. Harper concludes that this is an example among several others contained in his book demonstrating in Newton’s scientific method that “a theory succeeds by having its theoretical parameters receive convergent accurate measurements from the phenomena it purports to explain.”

Ducheyne also discusses in great detail Newton’s moon test in the form presented in the Principia to support the main contention in his book that Newton’s methodology “was more demanding and rich than a standard hypothetico-deductive methodology, and to highlight that it encompassed procedures to minimize inductive risk.” He stresses that one of the important innovations of Newton’s approach was the implicit requirement that a theory had to be not only sufficient, but also necessary to explain the phenomena. For example, Newton’s assumption that the gravitational force is a central force is necessary in order to maintain uniform velocity of a planet in a circular orbit, but in Prop. 1 book 1, Newton gave a proof that such forces are both necessary and sufficient for general orbits that satisfy the requirement that equal areas are swept by the radial line in equal times. For the special case of an inverse square force law, this requirement corresponds to Kepler’s area law. Kepler proposed this law empirically, based on his fit to Tycho Brahe’s accurate observations of the orbit of Mars. Moreover, Kepler also found the remarkable result that the orbit of Mars could be fitted by an ellipse, and in Prop. 11, book 1 Newton gave a mathematical proof to one of the seminal results in the history of science, namely, that elliptical motion satisfying Kepler’s area law also requires that the force varied inversely as the square of the distance. Additionally, in Prop. 45, book 1, Newton showed that even a small deviation from this inverse square dependence implied that the axis of the ellipse precesses and a planet’s orbit is not closed. But in fitting astronomical observations of the planets Newton found that to a very good approximation the orbits are ellipses with the major axis fixed in space. Such a result, which could not have been anticipated, illustrates what Cohen calls the Newtonian style that Ducheyne discusses in his book at some length. Unfortunately, for the general reader he mars some of his presentation with technical expressions like “counterfactual-nomological,” and jargon like “inference tickets.” Ducheyne also admits that his claims about Newton’s methodology are entirely based on the presentation in the Principia (the method of justification), and “do not pertain to the chronological sequence of discovery of the theory of universal gravitation (the method of discovery).” But it is insufficient to present Newton’s methodology without taking into account his method of discovery.

The main emphasis in Harper’s book is on how Newton “employs theory mediated measurements to turn data into far more informative evidence that can be achieved by hypothetico-deductive evidence alone” and on how “Newton’s inferences from phenomena realize an ideal of empirical success that is richer than prediction.” The Principia established a new quantitative paradigm of physical science. “Newton enlarged the definition of science to include those very approximations by which material phenomena diverge from the ideal patterns that had represented the objects of science to an earlier age. The Principia submitted the perturbations themselves to quantitative analysis and it proposed the exact correlations of theory and material event as the ultimate criterion of scientific truth.”

Both Ducheyne and Harper emphasize that Newton went beyond the hypothetico-deductive methodology that was prevalent in his time. Responding to criticisms that he had failed to explain the physical origin of gravitational force, in a “General Scholium” appended to the second edition of the Principia (1713), Newton stated that “I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy.”

Under the title “Facing the Limits of Deductions from Phenomena: Newton’s Quest for a Mathematical-Demonstrative Optics,” a chapter of Duchenye’s book is devoted to Newton’s Opticks. In agreement with I. B. Cohen, Duchenye concludes that Newton’s methodological ideal envisioned in the Principia “was not equally attainable in the study of optical phenomena.” In his book, Harper ignores this topic, and neither he nor Ducheyne discussed Newton’s failures in describing fluid flow and motion in a resisting medium in book 3 of the Principia.