Purpose of this chapter

At the time that Newton's Principia was published, Christiaan Huygens was universally recognized as the greatest authority in natural philosophy and a leading geometer of his age. His work in mathematics, mechanics, technology, astronomy and optics was outstanding. One can enumerate his main contributions: the mathematical treatment of probability, the study of impact laws, the observations of the planetary system, the mathematical treatment of simple and compound pendula, the discovery of the tautochronism of the cycloid, the understanding of conservation laws in dynamics nowadays seen as ‘equivalent’ to energy conservation, the mathematical study of centrifugal force, the wave theory of light and the study of double refraction in a crystal of Iceland spar. This is indeed an impressive list: both Newton and Leibniz declared their indebtedness to the Dutch natural philosopher.

What is perhaps more relevant with Huygens is that he showed how far-reaching could be the use of mathematics in the understanding of natural phenomena. In particular, Huygens' work in mechanics and optics was framed in a highly sophisticated mathematical language. While Galileo's ‘two new sciences’ required relatively simple mathematical tools, while Descartes' ‘natural philosophy’ was mainly qualitative and divorced from mathematics, Huygens' work was heavily mathematical. One of his masterpieces, significantly entitled Horologium oscillatorium, sive de motu pendulorum ad horologia aptato demonstrationes geometricae, published in Paris in 1673, required very complex mathematical techniques. Newton's Principia and Leibniz's dynamical works were written after the example of mathematization of mechanics given in the Horologium, and, despite the numerous differences, both men owed a great deal to this work.

The Principia was to remain a classic fossilized, on the wrong side of the frontier between past and future in the application of mathematics to physics.

The problem

Principia as a plural

Newton's magnum opus bears a title which is both imposing and perplexing. Undoubtedly, the great achievement referred to in the title consists in the application of ‘mathematical principles’ to the physical world, or better to ‘natural philosophy’. However, when we ask ourselves which, and how many, are Newton's mathematical principles, the answer does not come so easily. We know much more about the natural philosophy. Many scholars have taught us about the laws of motion, absolute time and space, the law of universal gravitation, the cosmology of void and matter. The names of I. Bernard Cohen, Richard Westfall, Rupert Hall and John Herivel come immediately to the mind of any historian of science. But, from the point of view considered in my research, Tom Whiteside comes first.

In his many papers, but especially in the critical apparatus of the sixth volume of ‘his’ Mathematical Papers, Whiteside has given a profound and detailed analysis of Newton's mathematical natural philosophy. As a study of Newton's mathematical achievements in the Principia Whiteside's studies will endure and this book has not been written to replace them. Actually, I began the research which led to this book by reading and following Whiteside's studies.

Purpose of this chapter

In this chapter we will be concerned with the small group of enthusiasts who extended and promoted Leibniz's calculus at the turn of the century. A handful of mathematicians, mainly based in Basel and Paris, invested their intellectual strengths in the new and controversial algorithm first published in the Acta eruditorum for 1684. Their reaction to the Principia resulted in a clearly stated programme: Newton's demonstrations had to be translated into Leibnizian language. I am interested in understanding not only what these mathematicians achieved, but especially how they justified the relevance of their research programme.

In §8.2 and §8.3 the reader will find some introductory material concerning the Leibnizian school. Then we will move on to consider some results related to the Principia achieved by Varignon (§8.4), Hermann (§8.5) and Johann Bernoulli (§8.6). This rather selective choice of material will allow us to show the following.

• Varignon did not prove new results. Nevertheless, his work is important since he insisted publicly on the advantages of adopting Leibniz's calculus in dealing with some of the main themes of the Principia. Varignon stressed the generality of calculus methods.

• […]

A common heritage

Despite the chauvinistic divisions originated by the Newton–Leibniz controversy, the Newtonian and the Leibnizian schools shared a common mathematical method. They adopted two algorithms, the analytical method of fluxions and the differential and integral calculus, which were translatable one into the other. As far as foundations are concerned, the situation was much more fragmented than is usually thought. The Channel did not divide the friends of infinitesimals from the adherents to limits. One finds British who refer to the infinitesimalist side of the Newtonian method (i.e. to ‘moments’) and equate moments with differentials; one finds Continentals who support a theory of limits.

Rather than seeing the Newtonians and the Leibnizians as two groups practising different mathematical methods, it is more fruitful, and more adherent to historical evidence, to focus on the amount of shared knowledge between the two schools. The British and the Continentals shared a vast quantity of common algorithmic techniques, notational devices and foundational ideas. Newton and Leibniz, however, tried, with partial success, to orient their disciples along different research lines. The difference between the Newtonians and the Leibnizians consisted in the manner in which they oriented themselves in this common mathematical heritage: it consisted in the values that they adopted, in the purposes they had in mind, and in the ways in which they placed this heritage in historical perspective.

Equivalence

Passing from the fluxional to the differential notation was a triviality.

Purpose of this chapter

Leibniz's response to the Principia is one of the most complex intellectual events in the history of science. The homo universalis took a critical attitude towards a number of issues, spanning absolute time and space, the gravitational force, the basic laws of motion, and the role of God in the Universe. In this chapter I will confine myself to Leibniz's reaction to Newton's mathematical methods. Notwithstanding the priority dispute, Leibniz's evaluation of Newton's mathematics was a positive one. As a matter of fact, in their mathematical works Newton and Leibniz shared many techniques and concepts. The algorithms of their calculi (the analytical method of fluxions and the differential and integral calculus) were translatable, and translated, one into the other. Furthermore, contrary to what is generally believed, their ideas on the interpretation of these algorithms were strikingly similar. However, the equivalence between the two breaks down when we move from the abstract level of algorithmic techniques and foundational matters, and we consider the mathematical practices. In fact, as I will show, Newton and Leibniz oriented their research along different lines, since they held different values and different expectations for future research. The idea that I would like to convey is that the two mathematicians shared a common mathematical tool, but used it for different purposes.

Purpose of this chapter

In this chapter I will try to chart the failures and successes that the British Newtonians encountered in reading Newton's magnum opus. I will abandon technical matters and I will adopt a descriptive level of presentation, while in the next chapter the reader will find some information on the mathematical approach to natural philosophy developed in the British school. Even though my analysis is far from complete, I hope that the material collected allows some generalizations on the Newtonian school. I am particularly interested in identifying the validation criteria that were shared among Newton's closest associates. After the publication of the Principia, and particularly during the priority dispute with Leibniz, Newton surrounded himself with a small group of mathematicians with whom he shared his discoveries. They adhered to many of Newton's methodological values. As a bonus they were often helped by Newton in their academic careers: e.g. David Gregory was appointed to the Savilian Chair of Astronomy in Oxford through Newton's recommendation. The same holds true for the appointment of Edmond Halley to the Savilian Chair of Geometry, Colin Maclaurin to the Chair of Mathematics in Edinburgh, or William Whiston to the Lucasian Chair in Cambridge. We will have to pay some attention to the unpublished, private, side of their scientific production. As will become clearer in the next chapter, the values and expectations that the Newtonians shared oriented their research along lines different from those pursued by the Leibnizians.

Purpose of this chapter

In this chapter I will try to give the reader some information about the mathematical methods employed by Newton in the Principia. We will have to consider with some patience a variety of lemmas, propositions, corollaries and scholia. I have found it convenient to postpone a more detailed analysis of some propositions to later chapters.

I have tried to render the structure of Newton's demonstrations without wasting too much space on trivial details. Therefore, I have skipped the lines of demonstration in which appeal is made to simple geometrical properties (e.g. similarity of triangles). We would adopt the same strategy in presenting Laplace's or Poincaré's demonstrations. When dealing with proofs given in a more familiar symbolic language we do not expect that every substitution of variable, or that every elementary integration, should be made explicit and commented on. The Principia has a reputation for being a very difficult, if not tedious, work. This is mainly due to the fact that nowadays we are not used to geometrical techniques. However, when the trivialities are skipped, the structure of most demonstrations emerges as remarkably simple. I hope that the reader will thus be able to follow the essential steps.

In the quotations from the Principia I am using Cohen and Whitman's forthcoming translation. I deemed it necessary not to follow their policy of altering some of Newton's notations and technical terms (see §1.3).

Purpose of this chapter

Section 2.2 is concerned with a mathematical method which Newton developed from the year 1664. He labelled it the ‘analytical method of fluxions’. It can be defined as the Newtonian equivalent, or counterpart, of the Leibnizian differential and integral calculus, and briefly – but somewhat improperly – called the ‘fluxional calculus’. From 1664 to the 1690s Newton elaborated several versions of it. Furthermore, Newton distinguished between an analytical and a synthetic method of fluxions (§2.3). In this chapter I will attempt a periodization of these versions, paying attention to concepts, rather than to results. What follows cannot be taken as an exposition or introduction to Newton's mathematical discoveries. This would take us too much space, and indeed, after Whiteside's works, would also be useless. What I will do rather is to sketch the nature of the mathematical method for drawing tangents and ‘squaring curves’ that Newton had developed before the composition of the Principia.

The analytical method

Mathematical background

Newton's interest in mathematics began around 1664 when he read François Viète's works (1646), the second Latin edition of René Descartes' Geometria (1659–61), William Oughtred's Clavis mathematicae (1631), and John Wallis's Arithmetica infinitorum (1656). It was in reading this selected group of works that Newton learned about the most exciting discoveries in analytical geometry, algebra, tangents, maxima and minima, quadratures, infinite series and infinite products.

Purpose of this chapter

The second and third editions of the Principia appeared in 1713 and 1726 respectively. The emendation and variations between these editions are remarkable and have been studied in detail by scholars such as Hall and Cohen. However, in broad outline, the structure of the first edition remained unaltered. The number and order of the propositions, as well as the methods of proof, remained almost unchanged.

Thanks to the recent edition of Newton's Mathematical Papers, we now know that more radical restructurings were considered from the early 1690s up to the late 1710s. Despite the fact that nothing of these projects appeared in print during Newton's lifetime, it is interesting to consider them since they reveal Newton's evaluations of his own mathematical methods for natural philosophy. For instance we know of projects of gathering all the mathematical Lemmas in a separate introductory section. From David Gregory's retrospective memorandum of a visit he paid to Newton in May 1694 we learn about projects of expanding the geometrical Sections 4 and 5, Book 1, into a separate appendix on the ‘Geometria Veterum’, and of adding a treatise on the quadrature of curves as a second mathematical appendix in order to show the method whereby ‘curves can be squared’. In 1712 Newton was still thinking of adding a treatise on series and quadratures as an appendix to the Principia.

Frontispiece from Isaac Newton's The method of fluxions and infinite series, London 1736. This work is an English translation of a Latin tract written by Newton in 1670–71. Colson, who held the Lucasian Chair of Mathematics in Cambridge from 1739 to 1760, added a long commentary in which he deals inter alia with foundational problems. This image expresses values that Newton was able to communicate to some of his followers. The figure refers to a problem solved on pages 267–76. Actually the bucolic scene is superimposed on a geometric diagram. Two points 1 (top) and 2 (lower) ‘flow’ from left to right along two straight trajectories. The motion of 1 is retarded, the speed of 2 is constant. The motion of a point 3 along the trajectory LMN, to be found, is such that at each instant the two points 1 and 2 must lie on the tangent at 3. This is a typical ‘inverse tangent problem’: a curve, to be found, is defined by the properties of its tangent. These problems lead to fluxional (or ‘differential’ equations). One of Colson's preferred ideas was that Newton's fluents and fluxions exist in nature (in Newton's words ‘Hae Geneses in rerum natura locum vere habent’ (Mathematical Papers, 8: 122–3)), while Leibniz's differentials are just fictions. According to Colson, the superiority of Newton's method over Leibniz's calculus – a point that Berkeley's Analyst (1734) would have missed – is that the fluxional symbols always refer to finite quantities which have an existence.

FOUR DIFFERENT METHODS FOR INVESTIGATING PROBABILITIES

The classical conceptions of probability emerged in the seventeenth century as byproducts of the discussion on issues related to games of chance. The question that was raised concerned the betting rates appropriate to different gambling situations. At first blush, the question seemed to be solvable by simple calculations. However, when different writers tried to give a comprehensive answer to the question, it became clear that such a solution could be attempted in different ways and that the relation between them was not self-evident.

1. a. The first type of answer was the following: If you want to inquire into the future results of, for example, the throw of a pair of dice, you should inquire into the physical aspects of the dice-throwing process.

2. b. Second, it was maintained that a prudent gambler should try to gather information before gambling. For example, before gambling on the future results of tossing a coin, the gambler should toss the same coin a large number of times and use the frequencies of the types of results as empirical data that should enable him to determine his betting rates.

3. c. Next it was suggested that questions having to do with betting rates could be answered only by introducing special symmetry or invariance assumptions. For example, it was suggested that if we are ignorant with respect to the results of throwing a pair of dice, and if by observing the dice we do not detect any asymmetry, we should be led to believe that all of the possible results are equally likely. Therefore, if there are n possibilities and the prize is 1 dollar, we should be willing to gamble 1/n dollars on any one of the possibilities.

4. […]

ELEMENTARY MEASURE THEORY

Let S be a set. A collection B0 of subsets of S is called an algebra iff:

1. If A, A' ∈ B0, then A ∩ A' ∈ B0 and A ∪ A' ∈ B0.

2. If A ∈ B0, then S\A ∈ B0.

3. S ∈ B0.

A collection B is called a σ-algebra if it is an algebra and, in addition,

1. if {An} is a countable collection of subsets of S such that, for every n, An ∈ B, then ∪nAn ∈ B.

A measurable space is a structure 〈S,B〉, where S is a set and B is a σ-algebra of subsets of S.

Let 〈S,B〉 be a measurable space. A finitely additive measure is a function m: B→[0,∞) such that:

1. If A,A' ∈ B and A ∩ A' = Ø, then m(A ∪ A') = m(A) + m(A').

2. m(Ø) = 0.

If m(S) < ∞, m is a finite measure. If m(S) = 1 m is a normalized probability measure.

m is a measure if it is a finitely additive measure and, in addition,

1. If {An} ⊆ B is a countable collection of mutually exclusive measurable sets, then m(∪nAn) = Σnm(An) (σ-additivity).

〈S,B,m〉 is called a measure space.

Theorem (Caratheodory). If m is a σ-additive measure on an algebra B0, it can be extended, uniquely, to a measure on the σ-algebra that extends B0.

GROUP-INVARIANT MEASURES

In 1933, Alfred Haar wrote a paper, entitled “The Concept of Measure in the Theory of Continuous Groups,” whose main objective was to construct a measure invariant to the action of a topological group. The importance of the paper must have been noticed immediately, and further studies followed shortly on its heels. Von Neumann proved the uniqueness of this measure in 1934. In 1937, Stefen Banach wrote an appendix about the Haar measure for a textbook about integration theory, and in 1940 both Weil and Cartan offered new uniqueness proofs. What is interesting and important about Haar's construction is that it leads (in most important cases) to a measure that is different from the Lebesgue measure by only a factor of proportionality. However, because the Haar measure is more general and abstract, it can illuminate and even justify the choice of the Lebesgue measure as the natural measure. In particular, the construction makes it easier to explain why we use the Lebesgue measure as a probability measure when it is defined as a volume measure.

The Haar measure is hardly unknown to mathematicians. In fact, for many mathematicians the Haar construction constitutes the foundation of their views on probabilities. It contains existence and uniqueness proofs in a fairly general setting, and it coincides with our pretheoretical notions of how probability measures ought to behave. However, the importance of the Haar measure is not due solely to its mathematical character.