2.1 Foreword
We should not be fooled by Newton’s triumph in the Age of Enlightenment. Despite what we might call Newton’s seeming apotheosis—as brought about by the public of the Italian Republic of Letters in the mid-eighteenth century—Newtonianism struck those mathematicians, astronomers and physicists who adhered to it as an open system: a field of research filled with unsolved questions. The historical accuracy of this claim may be appreciated by considering the case of Bologna and that of Laura Bassi in particular, where—as Cavazza,1 Ceranski 2 and Findlen 3 have shown—Newton’s work was approached more as a repository of open problems than as a closed doctrine.
2.2 Mathematics
Ce géomètre, dont tous les ouvrages portent un caractère singulier de sublimité, paroît en particulier dans celui-ci s’être élevé à une hauteur immense, à laquelle tout autre génie moins pénétrant, et moins fort que le sien, auroit tenté vainement d’atteindre: mais la route qu’il a tenuë dans une entreprise si difficile, se dérobe aux yeux de ceux qui apperçoivent avec étonnement le degré d’élévation auquel il est parvenu. On doit cependant en excepter quelques legeres traces qu’il a eu soin de laisser sur son passage, aux endroits qui avoient mérité qu’il s’y arrêtât plus long-tems. Ces endroits, au reste, sont presque toujours assez distants les uns des autres. Si l’on se propose donc de suivre la même carrière, on est obligé se de guider soi-même dans de longs intervales.4
Isaacus Newtonus enumerationem linearum tertii gradus in lucem protulit, licet nulla edita demonstratione, regulisque quibus usus erat, minime attactis, quippe qui magis sibi ipsi admirationem comparare, quam alios edocere cupiebat.5
For a variety of reasons which cannot be explored in the present article, in his mathematical works Newton left the—often far from straightforward—task of completing his demonstrations up to his readers.
This is evidently the case in the Principia. Here readers must often refer to properties of conics which require knowledge that is not readily available. Other propositions take for granted an acquaintance with techniques for the squaring of curves which left even mathematicians of the likes of Gottfried Wilhelm Leibniz and Christiaan Huygens at a loss. Newton, for instance, starts many of the propositions in the Principia (such as proposition 41, Book 1) with the premise that the demonstration implies a method for the squaring of curves (concessis curvilinearum figurarum quadraturis); he reduces the problem examined to the squaring of a curve and then—usually in a corollary—presents a solution which depends on this squaring, yet without ever providing any details on how to perform the operation itself.6
It was precisely the piecemeal quality of the demonstrations of the Principia and the mathematical appendices of the Opticks, Enumeratio linearum tertii ordinis and De quadratura curvarum which prodded early seventeenth-century mathematicians to comment upon, complete and often bitterly criticise Newton’s mathematical work. In the context of the polemic between Newton and Leibniz, the supporters of the German mathematician attributed these gaps to Newton’s incompetence in the field of infinitesimal calculus: his elliptic style—so they argued—betrays the inferiority of his method compared to that of Leibniz. For these and other reasons, the Principia long remained a puzzling text for those seeking to embark on a mathematically informed reading of it. As late as 1716, Eustachio Manfredi wrote that the language of Newton’s magnum opus was no more comprehensible to him than Arabic, and wished that Giuseppe Verzaglia might complete the commentary of the Principia he had promised. Nor should we attribute the difficulty faced by Manfredi to his provincialism as a Bolognese: for much the same bafflement was expressed in the Hague, Paris and Basel.7
It should further be noted that the very mathematical methods of Newton considerably vary from work to work. Take the collection of mathematical writings edited by William Jones in 1711 (a copy is now to be found in the University Library of Bologna).8 We here find early works such as De analysi per aequationes numero terminorum infinitas, which makes use of infinitesimals (or momenta, according to Newton’s terminology), alongside more mature works such as De quadratura curvarum, where Newton instead claims he only wishes to employ finite magnitudes and passages to the limit. Finally, in his Principia Newton adopts a geometrical language and seldom makes use of infinite series or symbolic procedures. The work begins with a section on the “method of first and last ratios”, which presents a geometrical theory for the passages to the limit necessary to define the area of a curvilinear surface, and the tangent and radius of curvature of a plane curve. Anyone approaching Newton’s mathematics was bound to ask himself what method should be used and what course followed to carry on his mathematical legacy. Was one to employ geometry or algebra? Infinitesimals or limits? Such questions were not raised on a purely technical-mathematical level, since they were also bound to affect one’s stance in heated debates such as the one focusing on the contrast between the geometry of the Veteres and the algebra of the Recentes, between Leibniz’s infinitesimals and Newton’s “evanescent quantities”. As Luigi Pepe has shown in his studies on Italian mathematical treatises, Bolognese, and more generally Italian, mathematicians chose to adopt a Newtonian approach based on limits without thereby renouncing Leibniz’s more flexible differential notation.9,10 Bassi presents this as quite a natural choice in De problemate quodam mechanico, where she employs Newtonian terminology for variable magnitudes—termed fluentes—while adopting Leibniz’s differential notation for her simple calculations.11 In this regard, mention must also be made of Gabriele Manfredi’s research on differential equations, and of the support of Leibnizian methods voiced by Verzaglia, who had studied under Johann Bernoulli in Basel.
2.3 The Principia
The Principia is no doubt best regarded as a work which mathematises natural philosophy to an astoundingly sweeping and detailed degree. Before 1687, mathematics had been successfully applied to statics, collisions, parabolic projectile motion and the motion of the cycloid pendulum (Huygens’ Horologium dates from 1673); Huygens had also made some progress in the study of the motion of projectiles through resisting media (his underlying hypothesis being that resistance is proportional to speed). In the Principia, Newton ventured to examine phenomena of a level of complexity quite unforeseen by his predecessors: take his (qualitative) study of the motion of three gravitationally interacting bodies in Sect. 11 of Book 1, of the attraction of extended bodies in Sects. 12 and 13 of Book 1, or of the motion of projectiles through resisting media—under the hypothesis that resistance to motion varies as the linear combination between a term proportional to the speed and one proportional to the square of the speed—in Sects. 3 and 4 of Book 2. In Book 3, planetary astronomy is mathematised, ultimately lending physical-mathematical confirmation to the Copernican system.12
As is widely known, the whole Copernican issue was still being made the object of censures which could hardly be overlooked in Bologna. Experimental confirmation of the motion of the Earth was later found in the phenomenon of the aberration of light, which was recorded and interpreted by James Bradley; in 1729 Eustachio Manfredi published his own observations on the phenomenon, although when doing so he was first forced to state his own personal adherence to geocentricism.13 In the third book of the Principia, Newton presents some striking results concerning the ebb and flow of tides, the procession of the equinoxes, the orbit of comets, the shape of the Earth and the perturbations of planetary orbits, while also broaching the difficult problem of the motion of the Moon. These are all undeniable achievements. Still, the Principia initially struck competent readers as a rather puzzling text. The secondary literature on the reception of Newton’s work has often focused on the issue of the nature of gravitation. Many readers with a Cartesian background were perplexed by this force acting from a distance, for which no mechanical explanation was provided. The debate which ensued is well known, but it is not on this great riddle which I wish to focus. Rather, I would like to stress how many astronomers, mathematicians and natural philosophers found an array of unsolved problems in the Principia, and how the reception of Newtonianism also came about through an acceptance of Newton’s challenge to solve what he himself had only solved in nuce—or approached from a wrong angle. I will here list some of the questions Newton left open. Each of these matters provided a stimulus for eighteenth-century mathematicians to develop an approach that might prove more satisfying from a physical as well as mathematical point of view.
First of all, it must be observed that Newton’s treatment of the motion of fluids in the second book of the Principia could hardly be used by researchers interested in either ballistics or water management. Engineering water management studies were indeed made by mathematicians associated with Bologna. However, only with the completely new approach to hydrodynamics developed by Johann Bernoulli, his son Daniel and Leonhard Euler, among others, were initial steps taken in the right direction. New mathematical tools, such as partial differential equations, unknown to Newton, enabled this new chapter in the history of analytical mechanics to be written. In this regard, mention must be made of Bassi’s De problemate quodam hydrometrico, a work devoted to the outflow of liquid from a container with an opening made at its bottom.14 Newton had already examined this problem in the second book of the Principia. In fact, he had been forced to completely rewrite this section for his second edition of the work (1713). The idea, which had first been suggested by Evangelista Torricelli and refined by Edme Mariotte, Domenico Guglielmini and Huygens, among others, consisted in working out the speed of the outpouring fluid in function of its height.15
In Newton’s day, the shape of the Earth was not quite clear. According to the Cartesians, the Earth was shaped like a “melon”: they believed that the distance between the North and South Pole had to be greater than the diameter of the equator. Newton, by contrast, believed that the Earth was flattened at its poles: a theoretical estimate which was confirmed in the eighteenth century through the expeditions that measured meridian arcs in Lapland and Peru. Newton reasoned as follows. He envisaged the Earth as a homogeneous and rotating fluid mass. The shape of the Earth, he argued, must be the equilibrium shape of this mass. If the mass is in equilibrium, the following must hold true: if we imagine solidifying the entire mass with the exception of two mutually communicating rectilinear channels—the first linking the North Pole to the centre of the Earth, the second connecting a point at the equator with the centre of the Earth—the fluid of the two channels will be found to be in equilibrium. Their length must therefore be such that the fluid they contain will remain still. Because of the centrifugal force generated by the rotation of the Earth, it will only be possible to achieve this equilibrium if the equatorial channel is longer than the polar one. By taking account of the speed of rotation of the Earth, of the mathematical results (proposition 91, Book 1) for the attraction exercised by an ellipsoid of revolution on a mass point situated on the extension of its axis, and of empirical data regarding the variation of the oscillation period of a pendulum as a function of latitude, Newton obtained an approximate measurement for the Earth’s flattening at its poles. Note that Newton did not demonstrate what the surface shape of a rotating fluid mass must necessarily be. Yet, his principle of “solidification” has played an important role in the study of the equilibrium of fluids. Newton also carefully examined the variation of gravity according to latitude. It must be observed, as George Smith has often emphasised, that this is the only result in the Principia crucially dependent upon the universality of the Law of Gravitation, which is to say upon the fact that gravity must apply to each constitutive particle of the Earth—and not just to heavenly bodies, at a macroscopic level. These geodesic results are obtained by assuming that the density of the Earth is homogeneous; hence, they are only valid in a very approximate way.16
Newton reached some interesting results concerning the motion of the Moon. He succeeded in lending satisfying mathematical expression to some anomalies in the planet’s motion: its divergence from the Law of Areas, the motion of its line of nodes, the fluctuation of the inclination of the plane of its orbit. As previously noted, the Sun generates forces which affect the Earth-Moon system, making the motion of our satellite irregular. One of these anomalies, the precession of the lunar apogee, escaped Newton’s analysis and became a celebrated problem addressed by all the leading mathematicians of the eighteenth century, from Euler to Laplace.17
The Copernican System attributes the procession of the equinoxes to a gradual shift in the rotation axis of the Earth. This axis is not fixed in relation to the stars: while keeping a fixed tilt in relation to the terrestrial orbit, it traces a cone. The complete precession period is of about 26,000 years. Newton’s explanation is as follows. Since the Moon’s period of revolution around the Earth is very small compared to the precession of the terrestrial axis, we can envisage the mass of the Moon as a ring distributed around the Earth. The rotation axis of the Earth is tilted in relation to the plane of this ring. The Earth, moreover, is flattened at its poles: it may be conceived of as a sphere with an “equatorial bulge”. The result is the subjecting of a non-spherical rotating body to a torque. While Newton never explored the dynamics of rigid bodies, he realised that the effect exercised by the lunar ring on the equatorial bulge of the Earth generates a conical motion of precession of the rotation axis. The same kind of reasoning made for the Moon may be applied to the Sun: for the gravitational action of the Sun must be regarded as one of the causes of the precession of the equinoxes. Finally, it is easy to realise that if the Earth were perfectly spherical, there would be no such thing as the precession of the equinoxes. This was the first physical explanation ever provided for an astronomic phenomenon which had been known since Antiquity. It must nonetheless be observed that Newton did not know the mass of the Moon and hence did not know how to gauge the intensity of the gravitational pull exercised by our satellite upon the equatorial bulge of the Earth. Ultimately, what he did was to assign the lunar mass a value that would enable him to reach the result he expected. Besides, Newton did not possess a theory for the dynamics of rigid bodies and his discussion of the precession of the equinoxes had to be completely revised in the light of Euler’s results on the dynamics of extended bodies.18
The way Newton inferred the phenomenon of tides from the theory of gravitation was by showing that the pull of the Moon (and, to a lesser extent, the Sun) upon the ocean masses generates two swellings. Consider, for instance, the gravitational pull of the Moon: this will be stronger on the water particles closer to our satellite and weaker on those situated at the antipodes. It is this discrepancy which causes two swellings of the ocean surface. Since the Earth spins on its axis, the two swellings will cover the whole terrestrial surface, producing two high tides and two low ones in 24 h. This theory had to be radically revised in the eighteenth century, when scientists discovered that the force at work is not the one perpendicular to the surface of the water, but the one tangent to it and responsible for the horizontal flow of the water particles. Newton’s theory, moreover, is a static one, which fails to take account of the fact that tides are a dynamic phenomenon. To put it briefly, according to the theory developed by Laplace, tides are wave-motions produced by periodical gravitational disturbances from the Sun and Moon. These motions are influenced by a wide range of factors, including the Coriolis effect and the geometry of ocean basins, which favours given frequencies in the oscillation of ocean masses.19
As noted above, the great value of Newton’s results is self-evident. A single force, the familiar gravitational force responsible for the falling of bodies towards the centre of the Earth, also explains a large number of terrestrial and celestial phenomena. This force may be subjected to mathematics, which will then become the language enabling us to grasp the causes behind the most varied phenomena. And it was precisely the methamaticisation of the force of gravity which enabled solutions to be found at the time for the greatest puzzles surrounding the “World System”: is there any good reason to choose the Copernican system over the geocentric? And if the planets revolve around the Sun, what is it that keeps them within their orbits?
The solution to these puzzles provided in the Principia, while valuable in itself, is beset by difficulties of both a technical and foundational sort. The former I have already referred to above by emphasising how in the second and third book of the Principia Newton explores many topics (the motion of fluids, that of the lunar apogee, the shape of the Earth, the tides, the procession of the equinoxes) through physical and mathematical tools which retrospectively strike us as inadequate. It was up to mathematicians such as Clairaut, Daniel Bernoulli, Euler, d’Alembert, Lagrange and Laplace—to mention but a few names—to develop the dynamics of fluids and rigid bodies, the calculus of variations and the theory of partial differential equations, the least action principle, elliptical integrals, the series expansion of trigonometric functions, and many other techniques which enabled the solving of those problems inadequately addressed in the Principia. The reception of Newtonianism among mathematicians, astronomers and physicists in the eighteenth century must be envisaged not as the embracing of a world system on their part, but rather as an expression of their desire to take part in a research programme that Newton had left open—and this precisely because many of the problems tackled in the Principia required new tools in order to be solved.
2.4 Opticks
The open character of Newton’s legacy is particularly evident in the case of Opticks, which ends with a series of queries which Newton couches in a hypothetical and speculative language. These queries ultimately outline a research plan still awaiting to be implemented for the study of chemical, electrical and magnetic phenomena. Their role in shaping enquiries into the so-called “Baconian sciences”—to adopt Thomas Kuhn’s terminology—in the eighteenth century can hardly be overemphasised.20 Laura Bassi, who also made contributions in the fields of mechanics and hydrometry (see notes 11, 14), was especially fascinated by this aspect of Newton’s legacy. After all, Opticks was a significant point of reference for the Bolognese—at least from 1728, when Newton’s experiments on refraction were confirmed by Francesco Maria Zanotti and Francesco Algarotti at the Institute, proving Giovanni Rizzetti’s arguments wrong.21 This successful outcome of the experimentum crucis in Bologna, while highly meaningful for the reception of Newton’s theories on light, ought not be seen as the straightforward embracing of a theory devoid of any gaps and leaving no questions open.22
It is worth noting how Newton’s queries regarding chemical and electric phenomena were also grounded in his interest for the phenomena of perception and volition, which is to say vital phenomena that fell within the sphere of interests of the Veratti-Bassi couple. To further appreciate this aspect of the queries, one might turn to the Newtonian manuscript which ultimately laid their foundation, An Hypothesis Explaining the Properties of Light. Prodded by the criticism he had received from Robert Hooke and Huygens, Newton presented this text at the Royal Society in December 1675.23
In his Hypothesis, Newton argues that space is pervaded by very fine aether, a kind of fluid possessing great elasticity. This fluid may be found not just in empty space, but also in bodies, for it permeates the pores of crystals, glass and water. In empty space, however, it occurs in a denser form than in solids. Through this aether vibrations travel that are similar to acoustic vibrations, only much faster and more minute (nowadays one would say they have a much shorter wavelength). Light, according to Newton, consists in a flow of corpuscles of various form which interact with the aether: aether refracts light and light warms aether. When these corpuscles meet the surface of a reflecting or refracting medium, they generate aether waves, like stones falling into a pond. According to this hypothesis, light corpuscles all move at the same speed. Aether refracts light: the corpuscles, that is, tend to deflect towards regions where the aether is less dense; and since aether is less dense in glass than in air, the luminous corpuscles are deflected when they pass through the surface of separation between air and glass. When they pass from air to glass—Newton explains—they are accelerated in the normal direction of the surface. Newton, therefore, used this model to explain the mechanism behind refraction.
As has just been mentioned, when the corpuscles meet the surface separating two media with different refractive indexes, they set the aether in vibration, and this in turn lends periodic properties to light. The phe-nomena Hooke observed on thin surfaces (soap bubbles, mica, etc.) and recorded in his Micrographia may be explained, according to Newton, by attributing periodic properties to light—something which supporters of the wave theory would have found much easier to do.
Confirmation of the fact that the aether invoked by Newton is a vital element comes from many passages of the Hypothesis, and particularly those concerning the relations between volition and muscular movement. Newton refers to a “puzzling” problem: how are muscles contracted and dilated in such a way as to generate movement in animals? The most popular answer was that provided by the Cartesians, who believed the nervous system to consist of thin channels through which the animal spirit flows, this being a gaseous or possibly fiery substance filling the nerves of the body and the pores of the brain. Volition and perceptions were thus explained by the Cartesians by positing a hydraulic exchange between the brain, from which the animal fluid was believed to flow, and the limbs of the body, which were held to be driven by this fluid when it steeped into the muscles. The hypothesis suggested by Newton is not far from the Cartesian: the aether permeating muscles he regarded as an animal spirit capable of dilation and contraction. According to his view, the soul has the power to fill muscles with this spirit or “wind” through the nerves. It is not necessary—Newton adds—to posit a very large variation in the density of the aether in muscles, since thanks to its considerable elasticity all it would take would be a small variation in density to engender a big variation in pressure. Newton’s language, however, is more reminiscent of Henry More and Thomas Willis than it is of Descartes.
In the closing queries of Opticks, Newton reframes his 1675 hypothesis on aether according to the principles of attraction and repulsion. It is nonetheless clear that he is still approaching themes such as those of the elasticity of air, the short-range forces behind chemical affinities and electric forces starting from an idea of matter that cannot be reduced to the “inert and brute” matter of Cartesian mechanics. Newton’s research on elastic fluids—such as air and the electric fluid—presupposes a conception of matter as shaped by active principles; this, in turn, led many Newtonians who carried on the programme outlined in Newton’s Opticks to cultivate an interest in the world of life.24 Indeed, Bologna was destined to become an important centre in the debate on the therapeutic applications of electricity, a debate which peaked with Luigi Galvani’s work in the late eighteenth century.
2.5 Closing Remarks
Laura Bassi’s physics would appear to have been influenced chiefly by Opticks, and especially its queries, although she also made a number of contributions to mathematical physics inspired by the Principia. Newton’s queries, as is well known, are literally open questions which the mathematician posed his followers, as if seeking to suggest the possible lines of research they ought to follow. What I wished to draw attention to in this paper was the fact that the open character of Newton’s heritage also extends to his mathematical work and the Principia. Italian Newtonians of the eighteenth century, in whose ranks Laura Bassi may certainly be counted, often regarded Newton’s work not as a system one might convert to, but rather as a repository of open problems on which to focus one’s research.
Isaac Newton (1642–1727), Roman school, oil on canvas, 18th century, Biblioteca Universitaria di Bologna, inv. 466
Giambattista Pittoni the younger, Domenico and Giuseppe Valeriani, An allegorical Monument to Sir Isaac Newton, 1727–29, oil on canvas, Cambridge, The Fitzwilliam Museum; Museum accession number PD.52-1973. The beam of light refracted by a prism showed the spectrum of colours comprised by sunlight
In this famous picture Newton shows the universality of the gravitational force (I. Newton, A Treatise of the System of the World, 2nd ed., Fayram, London 1731
This research was funded by the Department of Philosophy “Piero Martinetti” of the University of Milan under the Project “Departments of Excellence 2018-2022” awarded by the Ministry of Education, University and Research (MIUR).