A History of the Mechanical Universe II

Posted in history of science
Based on Stephen Gaukroger, Descartes: An Intellectual Biography (Oxford, 1995), Richard Westfall, Force in Newton’s Physics (Neale Watson Academic Publications, 1971), and J. B. Shank, The Newton Wars and the Beginning of the French Enlightenment (University of Chicago Press, 2008)

The publication of Galileo’s Dialogue heralded a broadening of astronomical research. Kepler, Galileo, and Descartes were contemporaries and read each other’s work. Descartes corresponded with Christiaan Huygens’ father, and Christiaan Huygens, Gottfried Leibniz, and Isaac Newton were simultaneously active. Voltaire’s Lettres philosophiques, published shortly after Newton’s death, popularized Newtonian ideas on the Continent. Descartes’ followers resisted the new system, but once Newton’s ideas had won out, their development was driven by a new generation of mathematicians whose main tool was Leibniz’s calculus.

The Beginnings of Mechanical Philosophy

Descartes’ theory of the planetary system displaced Aristotle’s “nested spheres” cosmology. This theory was the product of Descartes’ own reasoning, but the underlying methodology was not unique to him. He embraced mechanical philosophy, a new way of thinking about the physical world that would ultimately dominate the physical sciences.

The guiding principle of mechanical philosophy was that all phenomena were reduced to matter and motion, to bits of inert matter striking other bits of inert matter. This principle was first proposed by the Greek atomists, but their ideas had been swept aside by those of Aristotle and his followers. Europeans rediscovered atomism when Lucretius’s On the Nature of Things was found and translated in the early fifteenth century. Lucretius inspired Europe’s scholars, but he didn’t lay out a road map. The Europeans had to find their own way.

Galileo accepted the premise of mechanical philosophy:

I do not believe that for exciting in us tastes, odours, and sounds there are required in external bodies anything but sizes, shapes, numbers, and slow or fast movements; and I think that if ears, tongues, and noses were taken away, shapes and numbers and motions would remain but not odours or tastes or sounds.1

His own research, though, dealt only with aggregated matter and did not contribute to mechanical philosophy.

Descartes was directly influenced by Isaac Beeckman, with whom he first studied and then collaborated. For Beeckman, a phenomenon is explained by “a picturable or imaginable structure of parts” whose workings are mechanical.2 The parts are corpuscles — irreducible bits of matter. Beeckman’s corpuscular theory differed from classical atomism in significant ways:

The atomism of Epicurus had been one that regarded the size and shape of atoms, and to a lesser extent the direction of their motion, as the features that carried explanatory weight…For Beeckman, and for seventeenth-century corpuscularians generally, it was above all speed and direction of motion that did the explanatory work. As often as not, atoms were thought of as invariant in shape, as spheres…A few degrees of size were generally acknowledged.

Atoms have the ultimate explanatory role for the atomists,…[but] it is conglomerations of atoms that provide the explanations for Beeckman…Atoms are perfectly hard, since they are by definition simply regions of space fully occupied by matter. Perfectly hard bodies cannot rebound on impact, yet if any mechanical account of macroscopic phenomena is to be given purely in terms of microscopic parcels of matter in motion, then impact and elastic rebound must play a very significant role…In an attempt to reconcile this approach with the inelasticity of atoms, he took elastic congeries of atoms and empty space as his fundamental mechanical entities.3

Descartes’ corpuscular philosophy differed from his mentor’s. Descartes assumed that space was filled with corpuscles, while Beeckman imagined corpuscles colliding in a void. Descartes abandoned Beeckman’s elastic conglomerates of corpuscles: everything was to be explained in terms of the motion of individual corpuscles.

Descartes was also influenced by the Catholic friar and philosopher Marin Mersenne. Renaissance culture had given rise to beliefs that, in Mersenne’s view, eroded the critical distinction between the natural and the supernatural. Matter was imagined to have sympathies and antipathies; herbal concoctions were believed to have occult powers; sorcery, astrology, and cabbalism thrived. Neoplatonic philosophers imagined a “world soul,” and Kepler, in his early writings, claimed that the planets were kept in their paths by their souls (animae motrices). Even the church’s favoured philosophy, Aristotelianism, was suspect: it claimed that material objects strive toward some preferred state. In its place Mersenne advocated a strictly mechanical view of nature in which matter was entirely inert.

Beeckman’s application of mechanical philosophy was piecemeal, and Mersenne was an advocate of mechanical philosophy but not a practitioner. It was left to Descartes to set out a comprehensive vision of the mechanical universe.

Descartes’ Cosmology

Descartes believed unreservedly in the power of the human intellect:

The deduction or pure inference of one thing from another can never be performed wrongly by an intellect which is in the least degree rational, though we may fail to make the inference if we do not see it…In fact, none of the errors to which men are liable is ever due to faulty inference. They are due only to the fact that men take for granted certain poorly understood experiences, or lay down rash or groundless judgements.4

Deduction was one of Descartes’ two tools. The other was intuition, by which he meant the instantaneous, “clear and distinct” apprehension of a concept. No other tools were necessary: “Nothing can be added to the clear light of reason which does not in some way dim it.”5

Descartes’ confidence in his own intellect was misplaced: his planetary theory was mere fantasy. Voltaire wrote that Descartes was “born to bring to light the errors of antiquity and to put his own in their place.”6 Nevertheless, the idea of a strictly mechanical universe was attractive to natural philosophers. Leibniz and Huygens were among the prominent thinkers who saw more promise in Descartes’ mechanics than in the “occult power” — universal gravitation — posited by Newton.

Descartes assumed that the earth and the other planets are contained in a sun-centered vortex of matter, and that the stars are the suns at the center of other vortices. The vortices press against each other; together, they fill all of space. The swirling of the vortex explains the rotation of the planets, their orbital movements, and the variation in orbital speeds.

Let us assume that the matter of the heavens, in which the planets are situated, revolves unceasingly, like a vortex having the Sun as its centre, and that those of its parts that are close to the Sun move more quickly than those further away, and that all planets always remain suspended among the same parts of this celestial matter. For by that alone, and without any other devices, all their phenomena are easily understood. Thus if some straws are floating in the eddy of a river, where the water doubles back on itself and forms a vortex as it swirls, we can see that it carries them along and make them move in circles with it. Further, we can often see that some of these straws rotate about their own centres, and that those that are closer to the centre of the vortex that contains them complete their circle more rapidly than those that are further away from it. Finally, we see that, although these whirlpools always attempt a circular motion, they practically never describe perfect circles, but sometimes become too great in width or in length. Thus we can easily imagine that all the same things happen to the planets; and this is all we need to explain all their remaining phenomena.7

The swirling of the vortex erodes the matter within it, resulting in three elements that differ only in their sizes. The third element is the largest, and is the building block for earthy objects, including planets and comets. The second element is much smaller and is formed into airy things. It fills the space between the planets, and this orbiting stream of matter carries the planets along with it. The first element is indefinitely small. It has the character of fire and is the source of light.

The collisions yield very small parts of matter, which accommodate themselves to the space available so that a void is not formed, and these move at great speed because, “having to go off to the side through very narrow passages and out of the small spaces left between the parts of the second element as they proceeded to collide head-on with one another, [the first element] had a longer route to traverse than the second in the same time.” But the first element is formed in a greater quantity than is needed simply to fill in the spaces between pieces of second and third element, and the excess naturally moves towards the centre because the second element has a greater centrifugal tendency to move to the periphery, leaving the centre the only place for the first element to settle. There it “composes perfectly liquid and subtle round bodies which, incessantly turning much faster than and in the same direction as the parts of surrounding second element, have the force to increase the agitation of those parts to which they are are closest and even, in moving from the centre towards the circumference, to push the parts in all directions, just as they push one another.” These concentrations of first element in the form of fluid, round bodies at the centre of each system are suns, and the pushing action that Descartes describes is “what we shall take to be light.”8

Descartes claimed that the only fundamental property of matter is that it occupies space — it is defined by “extension alone.” Every other property is a secondary property requiring a mechanical explanation. Weight, or equivalently, the propensity to fall to earth, is one of these secondary properties and is explained by the vortex. An object released above the earth would be immersed in a stream of the second element. This stream, having great speed and great centrifugal force, would surge over the object and push it downwards. Huygens, in 1669, argued that the same explanation

… shows the correctness of the principle that Galileo took to demonstrate the proportion of the acceleration of falling bodies, which is that their velocity increases equally over equal times. For these bodies being pushed successively by the parts of the neighboring matter, which tries to occupy their place,…the result is that the action of the matter that presses upon them can be considered always as strong as when it finds them at rest, from which one concludes rather easily that the increase of velocities is proportionate to that of the times.9

Huygens was not the only philosopher to find explanatory power in vortex theory. In 1730 — decades after Newton’s Principia was published — the mathematician Johann Bernoulli used it to derive Kepler’s third law.

The Puzzle of Circular Motion

Galileo, like the philosophers who preceded him, believed that planetary orbits were either circles or composites of circles. Aristotle explained this motion by declaring that the planets were composed of ether, and that ether’s natural motion was circular. Galileo, believing that the earth and the heavens were governed by the same physics, rejected this idea and proposed in its place a form of inertial movement. Imagine a frictionless horizontal plane in a vacuum. An object on that plane, once in motion, would forever remain in motion. The essential characteristic of this plane is that all parts of it are equidistant from the center of gravitational attraction. The plane appears horizontal from our limited human perspective, but zooming out shows it to be a circular path around the gravitational center. The planets, Galileo argued, follow this kind of inertial path around the sun.

Circular motion was an essential part of Descartes’ planetary model. Each vortex pressed against other vortices and was therefore bounded. Matter pressed outward but could not escape. Since matter entirely filled the vortex, it could only move in roughly circular bands, the forward movement of each corpuscle creating space for the forward movement of the corpuscle behind it. (And matter had to move: Descartes declared that the “quantity of motion” was fixed, a reflection of the immutability of God.)

Descartes also discussed circular movement on a smaller scale. He used a stone in a sling (as in David and Goliath) as an example of such motion:

To move in a circle, it [the stone] must be externally constrained; in resisting the constraint, it strives to move away from the centre. Whirl a stone in a sling, and you can feel it pull on the string as it strives to move away from its centre of revolution.10

Christiaan Huygens would later give a name to this tension: centrifugal (center-fleeing) force.

Descartes’ discussion provided the template for later studies of circular motion: it was to be understood as a state of equilibrium between two opposing forces.11 Isaac Newton took this approach in his early studies. A letter from Robert Hooke, a brilliant experimentalist but a poor mathematician, induced him to “think outside the box.”

Hooke presented his own ideas about the heavens in a lecture that was published as An Attempt to Prove the Motion of the Earth (1674).

Hooke put forward three powerful and suggestive suppositions about celestial motions. According to the first, “All celestial bodies whatsoever, have an attraction or gravitating power toward their own centers, whereby they attract not only their own parts…[but also] all the other celestial bodies that are within the sphere of their activity.”…The second supposition states that once a body is set in motion, it will continue its motion in a straight line until deflected by some external power so as to describe a circle, an ellipse, or a more complex curve. The third argues, “These attractive powers are so much more powerful in operating, by how much the nearer the body wrought upon is to their own centers…”12

Hooke discussed his ideas in a letter written to Newton in 1679. Newton, unable to overcome problems inherent in the equilibrium approach, soon began to explore Hooke’s approach. This exploration would lead, only a few years later, to the Principia. Hooke’s influence on Newton can be overstated — as Hooke himself had — but it cannot be denied.

Hooke was the one who in fact set upright the crucial problem of orbital motion, which had been conceptualised, as it were, upside down. Hooke rather than Newton stated the mechanical elements of orbital motion in terms adequate to the concept of inertia. There is no good reason to doubt that in this matter Hooke was Newton’s mentor.13

Newton

By 1684, the astronomer Edmond Halley, the mathematician/architect Christopher Wren, and Robert Hooke all suspected that the sun’s gravitational pull on the planets was inversely proportional to the square of their distances from the sun.14 They wondered what implications this rule would have for the shape of planetary orbits. Halley, on a visit to Cambridge, posed the question to Newton, who immediately answered that the orbits would be elliptical. He had already worked out the answer, he said, and would send to Halley a copy of his calculations. He eventually sent a nine-page document titled De motu corporum in gyrum (On the motion of bodies in orbit). But Newton had become fascinated with celestial mechanics while preparing this document, and he continued to study it exclusively and obsessively. His research culminated in the book Philosophiae naturalis principia mathematica (Mathematical principles of natural philosophy, 1687). De motu was innovative; Principia was revolutionary.

De motu

Newton imagined a single body orbiting a fixed center of attraction. The body’s innate tendency was to move in a straight line, but it was continually turned inward by the attractive force, creating a closed orbit. Newton gave this force a new name: centripetal (center-seeking) force, in conscious opposition to Huygens’ centrifugal force. Newton used this model to show that “an elliptical orbit entails an inverse-square force to one focus,” and that “an inverse-square force entails a conic orbit, which is an ellipse for velocities below a certain limit.”15 He also demonstrated Kepler’s area and harmonic (period of revolution) laws.

It might seem that Newton’s accomplishment was to provide the underpinnings for Kepler’s three laws, but such a claim would be historically inaccurate. The idea that Kepler set out three “laws” postdates the publication of the Principia; before that time, the laws had been regarded as no more than useful approximations.16 The evidence supporting elliptical orbits was thin:

The minor axis of Mercury is only 2 percent shorter than the major axis, the minor axis of Mars, only 0.4 percent shorter, and in all other cases the difference between an ellipse and an eccentric circle was beyond detection.17

Newton claimed that Kepler, finding that the orbits were not circular, had simply “guessed” that they were elliptical. The harmonic law was likewise mistrusted. The area law was well known, but was so intractable that it was almost always replaced with a simpler rule.18

Pulling so many important results out of a single model was a significant achievement, but Newton very quickly became dissatisfied with his work. De motu was revised within months of its issuance and then converted into the foundations for Principia. Looking backwards from Principia reveals the weaknesses of De motu.

The concept of inertial motion does not appear in De motu. Instead, Newton proposed a “force of a body” or “force inherent in a body” that maintained the body in uniform rectilinear motion. The body’s motion would change only if the body encountered an outside force such as centripetal force. In this scheme, an orbit resulted from the interaction of the inherent and centripetal forces. The problem that Newton could not solve to his own satisfaction was how to analyze this interaction, because the forces were not directly comparable.

Whereas “force” as inherent force causes a uniform motion, “force” as centripetal force causes a uniform acceleration…The two forces…have utterly different relations to motion.19

Newton was already shifting away from the idea of inherent force in the revisions to De motu. He would replace it with the concept of inertia, thereby reinventing the field of mechanics.

The idea of universal gravitation also does not appear in De motu. Gravity had originally been imagined to be a property of certain bodies: it caused them to fall towards the center of the earth, their natural place. Other objects possessed the opposite property, levity, which caused them to pull away from the earth. Galileo abandoned this division, instead arguing that there is only gravity and that objects differ in their heaviness. Gravity became a property of the attracting body — the earth — rather than of the attracted body. The nature and scope of this attraction remained a mystery. Perhaps it was to be explained by Descartes’ vortices, but if so, its domain was determined by the shape and extent of the unseen vortices. Newton, in postulating universal gravitation, claimed that gravity is a property of matter itself, and that its strength is proportional to “mass,” a measure of the quantity of matter. Every body has this property, and every body attracts every other body. Newton had not yet formulated this concept when he wrote De motu but it dominated Principia, which was completed less than two years later.

Something that was tacitly accepted in De motu is denied in Principia: ether, the substance believed to fill the seemingly empty space between the planets. As well as describing celestial mechanics, De Motu had described the motion of projectiles. Confronting theory with evidence gave rise to a conundrum:

The motion of projectiles in the terrestrial atmosphere deviated from the idealized parabolic paths postulated under the assumption of no resistance because the atmospheric air did provide resistance to motion. Yet the motion of celestial bodies did not deviate from the idealized elliptical paths predicted under similar assumptions of no resistance due to the celestial ether. This observation posed a problem. If Newton accounted for this celestial behavior by assuming that the ether was so diffuse that it caused no resistance, then he could no longer assume that the ether was dense enough to provide the mechanical collisions needed for the gravitational interaction.20

Newton chose to abandon ether entirely. He could not argue that gravity was mechanical and he could propose no other compelling explanation for it, so he simply accepted its existence.

Principia

The Principia consists of three books. Book 1 sets out Newton’s theory of motion and applies it to abstract orbital bodies. Book 2 extends the analysis of orbital bodies by introducing a resisting medium, and then uses the results to sharply criticize Cartesian vortex theory. Book 3 examines the implications of Book 1 for a range of “real world” issues, including the layout of the solar system, the behaviour of comets, the precession of the equinoxes, and tides. Geometry is the main analytical tool throughout.

Newton immediately introduces the idea of inertia: a continuing force isn’t needed to keep a body in motion, but a force is needed to change either its speed or its direction. Inertial motion had previously been discussed by Galileo, Huygens, and Descartes, but none of them had integrated it into a general theory of motion. Newton was able to do so, possibly because he thoroughly understood De motu’s technical shortcomings. He had a concrete problem to solve, and recognized that the concept of inertia was the key to its solution.

Newton’s characterization of inertia is contained in his first and second laws of motion:

Every body perseveres in its state of being at rest or moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed [upon it].

A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.21

Impressed force exists only during a change in motion.

Impressed force is the action exerted on a body to change its state either of resting or of moving uniformly straight forward.

This force consists solely in the action and does not remain in a body after the action has ceased…There are various sources of impressed force, such as percussion [impact], pressure, or centripetal force.22

Impressed force is resisted by what Newton called “inherent force,” a term carried over from De motu but given a new meaning.

Inherent force of matter is the power of resisting by which every body, as far as it is able, perseveres in its state either of resting or of moving uniformly straight forward…

Because of the inertia of matter, every body is only with difficulty put out of its state either of resting or of moving. Consequently, inherent force may also be called by the very significant name of force of inertia. Moreover, a body exerts this force only during a change of its state, caused by another force impressed upon it.23

Inherent force is determined by the mass and velocity of the body: the modern term for it is momentum. Inherent force and impressed force are mathematically comparable because they are both exerted only during a change in motion.

Orbital motion is caused by the interaction of momentum and centripetal force. Momentum tries to keep the planet moving in a straight line, centripetal force tries to pull it towards the source of the attraction. Neither force fully succeeds in its endeavour.

Newton begins his analysis of orbital motion with a generalization of the area law.

First, he considers the inertial (linear) motion of a body in the absence of any external forces; and he shows that with regard to any point in space not in the line of motion, a line drawn from the body to that point will sweep out equal areas in any equal times…Newton then “compounds” this inertial motion with a blow, a sudden impact, an “impulsive” force (as it later came to be known), directed toward that point, and he proves geometrically that area is still conserved. In the new inertial motion following the first blow, a second blow is given — again in the direction of the point (or center) — and again area is conserved. These blows follow one another at regular intervals and produce a polygonal path, whose sides, together with the lines from the central point to the extremities of the sides, determine a set of equal-area triangles. Then, in the limit, as the time between successive blows becomes indefinitely small,…the “ultimate perimeter” will be a curve and “the centripetal force by which the body is perpetually drawn back from the tangent of this curve will act continually.” [He then] proves the converse, that the area law implies inertial motion in a central force field. Thus the transformed area law provides…a necessary and sufficient condition for a centripetal force.24

If an orbiting planet sweeps out equal areas in equal times, a centripetal force of some kind is operating. What kind? If the harmonic law is satisfied, the strength of the centripetal force is inversely related to the square of distance.25

Newton initially assumes that a single body is in motion near a fixed center of attraction, and shows that the body’s motion will take the form of a conic section if and only if the centripetal force varies inversely with the square of the distance. The body sometimes follows an elliptical orbit with the center of attraction at one focus (as the planets do), and sometimes follows a hyperbolic orbit, skating past the center but not being captured by it (as comets do).

Newton then presents the two-body problem: two bodies are in motion, and each attracts the other in proportion to its mass. He shows that this system has a fixed center of mass. The bodies follow elliptical orbits around the center of mass, and also around each other.

This system was still not an analogue of the actual planetary system, in which there were (in Newton’s time) six known planets, but Newton was unable to solve a system involving three bodies, let alone seven. He was nevertheless confident that the dynamics of the planetary system were not much different from those of the two-body problem. In a work only published after his death, he wrote:

Because the fixed stars are quiescent one in respect of another, we may consider the Sun, Earth, and Planets as one system of bodies carried hither and thither by various motions among themselves…[Their] common center of gravity will be quiescent: And from it the Sun is never far removed…The sun, according to the various situation of the Planets, is variously agitated and always wandering to and fro with a slow motion of libration, yet is never recedes one entire diameter of its own body from the quiescent center of the whole system…

About the Sun thus librated the other Planets are revolved in elliptic orbits, and, by radii drawn to the Sun, describe areas nearly proportional to the times. If the Sun was quiescent, and the other Planets did not act mutually one upon another, their orbits would be [exactly] elliptic, and the areas exactly proportional to the times. But the actions of the Planets among themselves, compared with the actions of the Sun on the Planets, are of no moment, and produce no sensible errors.26

But such reflections were not part of Book 1, which is structured as a mathematical exercise from beginning to end.

It is only in Book 3 that Newton deals with the physical universe. “Nature,” Newton explains, “is always simple and ever consonant with itself,” and physical hypotheses should reflect this simplicity:

No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena…Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same.

Those qualities…that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally.27

After setting out these rules, Newton previews their use:

If it is universally established by experiments and astronomical observations that all bodies on or near the earth gravitate toward the earth, and do so in proportion to the quantity of matter in each body, and that the moon gravitates toward the earth in proportion to the quantity of its matter, and that our sea in turn gravitates toward the moon, and that all planets gravitate towards one another, and that there is a similar gravity of comets toward the sun, it will have to be concluded…that all bodies gravitate toward one another.28

These phenomena are, of course, precisely the ones that he would examine in Book 3, compelling (in his view) the conclusion of universal gravitation.

Newton interprets the observational evidence in the light of his theoretical results. The moons of Jupiter obey both the area law and the harmonic law, so they are subject to a centripetal force that varies inversely with the square of distance. The same claim is made for the satellites of Saturn and for the known planets. Moreover, Earth’s moon obeys the area law, and the centripetal force required to hold it in its orbit matches Huygens’ measurement of the Earth’s gravity. Newton’s rules for forming hypotheses invite the hypothesis that this centripetal force is the Earth’s gravity, and also the hypothesis that the same identification holds for the rest of the solar system:

Hitherto we have called “centripetal” that force by which celestial bodies are kept in their orbits. It is now established that this force is gravity, and therefore we shall call it gravity from now on. For the cause of the centripetal force by which the moon is kept in its orbit ought to be extended to all the planets.29

Book 3 presents other findings, but universal gravitational attraction is the showstopper.

Newton cannot explain gravitation. He sums up what he has done and what he has been unable to do in this fashion:

I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. Indeed, this force arises from some cause that penetrates as far as the centers of the sun and planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do) but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the squares of the distances…I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses.30

Newton believes that he has shown (in Book 2) vortex theory to be incompatible with the evidence, so to adopt a mechanical explanation of gravity would certainly be to “feign” a hypothesis. But mechanical philosophy, for its adherents, is a methodology that had delivered natural philosophy from the occult beliefs that had dogged it earlier in the century. Acceptance of a mysterious action-at-a-distance power that pervades the entire universe would be, for them, a retreat from sound principles.

Newton does not believe that God plays an active role in maintaining the universe, but he is certain of his existence.

This most elegant system of the sun, planets, and comets could not have arisen without the design and dominion of an intelligent and powerful being…He rules all things, not as the world soul but as the lord of all.31

In this one thing, the mechanical philosophers would agree with him.

The Response to Principia

The Principia wasn’t intended for a wide audience, and in Britain, it didn’t get one. It was well-received by astronomers such as John Flamsteed and Edmond Halley, and by natural philosophers with a strong mathematical bent. Some of these “mathematical Newtonians” contributed to a new mathematical mechanics, but

… their interpretation of Newtonian mechanics was decidedly in the minority in eighteenth-century Britain, and their program increasingly disappeared over the course of the century even as Newton’s reputation as the supreme authority in the physical sciences reached heroic ascendance. Instead it was the natural theological approach to the Principia adopted by Samuel Clarke, Richard Bentley and William Whiston along with the experiment-centered and application-oriented approach personified by J. T. Desaguliers and Francis Hauksbee that more fully represented British appropriations of Newton during his lifetime.32

Newton was not alone in believing that the majesty of the heavens constituted a clear demonstration of God’s existence. Roger Cotes — mathematician and astronomer, fellow of Trinity College, and the editor of the second edition of the Principia — wrote,

He must be blind who does not at once see, from the best and wisest structures of things, the infinite wisdom and goodness of their almighty creator; and he must be mad who refuses to acknowledge them.33

Similar sentiments were put forward by other leading scholars, and yet there was also an ever-present fear that atheists and deists might find support in the Principia. Physical laws diminished the role of God, inviting the thought, “If God is absent now, perhaps He was absent always.”34

Newton identified himself with experimental philosophy, which sought knowledge in observation and experiment, and distrusted imaginative and speculative hypotheses. He was not alone in this: Hooke and Robert Boyle were committed to the same approach, designing experiments in order to draw out specific results. Boyle wrote,

I am very sensible of my being far from having such a stock of experiments and observations, as I judge requisite to write systematically; and I am apt to impute many of the deficiencies to be met with in the theories and reasonings of such great wits as Aristotle, Campanella, and some other celebrated Philosophers, chiefly to this very thing, that they have too hastily, and either upon a few observations, or at least without a competent number of experiments, presumed to establish principles and deliver axioms.35

Desaguliers and Hauksbee were among the Newtonians who adopted the experimental approach. British science soon became known for its emphasis on experiments.

Britain’s middle and upper classes acknowledged Newton as a national hero. They could not read the Principia but they were keen to know more about its contents. To this end, they could purchase a nontechnical exposition such as Henry Pemberton’s View of Sir Isaac Newton’s Philosophy (1728) or Colin Maclaurin’s Account of Sir Isaac Newton’s Philosophical Discoveries (1748). Or they could subscribe to a lecture series that included demonstrations of “mechanical, hydrostatical, pneumatical, and optical experiments.” In 1718 the residents of London had three such series to choose from. One series was presented by Hauksbee and William Whiston (a former Cambridge fellow), another by Desaguiliers.36 They could also attend the Boyle Lectures, which had been endowed by Robert Boyle in his will. The lectures used natural philosophy to demonstrate the superiority of Christianity over other beliefs, and especially over atheism, deism, and materialism. In their early years, the lectures were invariably presented by followers of Newton. Newton endorsed them. He wrote to Richard Bentley, the first of the lecturers, “When I wrote my treatise upon our system I had an eye upon such principles as might work with considering men for the belief of a Deity, and nothing can rejoice me more than to find it useful for that purpose.”37

Voltaire lived in London from 1726 to 1728, having been exiled from France for affronting a nobleman. He was among those who were intrigued by Newton’s ideas. Other aspects of English society, such as religious toleration and open debate, also appealed to him. After his return to France, he published a collection of essays in two editions, Letters concerning the English Nation (1733) and Lettres philosophiques (1734). These essays praised aspects of English society that were not to be found in France, so that their praise of England was implicitly criticism of France. This alone would have intrigued the French public, but there was more. For reasons having little to do with Voltaire, Lettres philosophiques was published without the approval of the royal censors. Voltaire faced prosecution and imprisonment for this act, but escaped to the countryside where he remained as a fugitive for many years. A copy of his book was “publicly burned by the hangman on the steps of the judicial chambers” and the remaining copies were confiscated. Voltaire, previously a struggling author, was now not just famous but notorious.38

Newton, though well-known among French scholars, was little known among the French public. Voltaire, in two provocative essays in Lettres philosophiques, stimulated the public’s interest in him and his work. Voltaire began,

A Frenchman arriving in London finds changes in philosophy as in other matters. He left a universe that was filled; he discovers the void; in Paris, they imagine a universe composed of vortices of subtle matter; in London none of this; we think it is the pressure of the moon that causes the fluctuation of the tides; the English believe that the ocean gravitates toward the moon…Light, for a Cartesian, exists in the air; for a Newtonian, it comes from the sun in six minutes and a half.39

In short, Descartes’ ideas ruled in France and Newton’s in England.

Voltaire praised Descartes’ geometry, but then continued,

Geometry…would have guided him faithfully in his study of physics; yet finally he abandoned this guide and surrendered to the desire for system. At this point his philosophy became no more than an ingenious tale, at most merely plausible for ignorant people. He mistook the nature of the soul, the proofs of the existence of God, the nature of matter, the laws of movement, and the nature of light…Yet it is not too much to say that even in his mistakes he is admirable; he erred, but it was at least systematically and with consistency. He demolished the absurd fantasies with which children have been entertained these past two thousand years; he taught the men of his time to reason, and even to use his own weapons against him. If he did not use true coinage, he at least denounced the counterfeit.40

Having dismissed Descartes’ theories entirely, Voltaire discussed major elements of Newton’s Principia, sometimes in surprising detail. He concluded with an imagined exchange between French scholars and Newton. The scholars haughtily ask, “What then have you taught us?” Newton replies, “I have taught you that the mechanics of this central force gives weight to objects in proportion to their mass, that this central force alone moves the planets and the comets in their evident relations. I have demonstrated that there cannot be any other cause of weight and movement in celestial bodies, for…if there were yet another force that works on all these bodies, it would increase their speed or change their direction. Now, not one of these bodies has a single degree of movement, of speed, of direction that cannot be proved to be the effect of this central force; thus it is impossible that there be any other.”41

The slighting reference to French scholars follows from their response to Principia: they readily absorbed its mathematical components, and summarily dismissed its physical claims.

Continental scholars were at the forefront of mathematical innovation. The first steps in the development of analytic geometry were taken, independently and with different purposes, by Descartes and Pierre de Fermat in the middle of the seventeenth century. Leibniz, the co-discoverer of calculus (with Newton), published his papers on differential and integral calculus in 1684 and 1686. His methods were soon adopted by Jakob and Johann Bernoulli, and then by others, including the Marquis de l’Hôpital and Pierre Varignon. Differential equations were introduced by Leibniz and the Bernoullis in the 1680s.

The Continental mathematicians applied their skills to the development of mathematical mechanics. When Principia was published, two things were apparent to them: the value of the theorems, and the old-fashioned way in which they had been proved. As an example, the proof of the generalized area law (described above) used calculus-like arguments but no calculus. The mathematicians “translated” Newton’s proofs into state-of-the-art mathematics (Varignon, l’Hôpital, and Johann Bernoulli were key contributors) and made Newton’s theorems a part of the broader discipline of mathematical mechanics. The discipline that had floundered in Britain, flourished in France.

Book 3 was a different matter. Mechanical philosophy was deeply entrenched on the Continent. Phenomena had to be explained by matter and motion, by physical contact: any other premise would allow occult causes to creep back into natural philosophy. Newton’s claim that a mysterious force governs all celestial motion could not be accepted.

Christiaan Huygens was the most respected natural philosopher on the Continent. He was willing to accept that the sun attracts the planets with a force that diminishes with the square of distance, because it was conceivable that such a force could be explained by vortices. Moreover, since this force “counterbalances so well the centrifugal forces of the planets,” it was possible to fit it into the old idea of orbital motion as an equilibrium between two opposing forces. But he could not accept any sort of mutual attraction because it could not be made consistent with vortices.42

The French were equally unwilling to accept the concept of universal gravitation. They set aside Newton’s explanation of celestial motions, preferring in its place the explanation of their own national champion, Descartes. The extensive empirical evidence that Newton had presented in Book 3 was ignored. French scholars maintained this position into the 1730s.

Cartesian and Newtonian ideas clashed in the 1730s, and the Newtonian ideas gradually prevailed. There was no single turning point in the clash, but two influential persons can be identified.

One of them was Voltaire. He wrote, with the assistance of the Marquise du Châtelet, a book that introduced the Newtonian system to the public, Eléments de la philosophie de Newton (1738). Such a book would have been unexceptional in England, but in France, it challenged the establishment. France’s Royal Academy was a hierarchical institution, able to influence the discourse of its members, and by extension, the opinions of the public. The Academy was strongly Cartesian. Voltaire presented himself to the public as “an independent, critical thinker beholden only to universal reason,”43 and he left little doubt that the modern thinker was a Newtonian.

By contrast, Pierre Louis Maupertuis worked within the establishment, eventually becoming a senior member of the Royal Academy. He proposed, organized, and participated in a pair of expeditions that would determine the shape of the earth. Cartesian vortex theory (as developed by Huygens) and Newtonian theory gave different predictions, so the expeditions were effectively a head-to-head test of the two theories.

Newton’s and Huygens’ theories both implied that the Earth would have an oblate shape, flattened at the poles to different degrees…The controversy could be settled by pendulum measurements far apart, as close as possible to the north pole and the equator.44

Maupertuis led an expedition to Lapland in 1736, while the mathematician Charles Marie de La Condamine led an expedition to Peru. The results supported the Newtonian theory but did not convince committed Cartesians, who claimed that untested instruments and inexperienced observers had introduced errors into the measurements.

In 1732, before the expeditions took place, Maupertuis had written Discourse on the Different Figures of the Planets, which has been described as “not so much a defense of the Newtonian theory of universal gravitation as a defense of the possibility of such a defense.”45 He argued that gravitational attraction is not metaphysically impossible and not obviously contradicted by evidence, so the a priori dismissal of gravitation was not an intellectually acceptable position. He concluded that

….attraction is now only a question of fact. It is in the system of the universe that one must look in order to determine whether it is a principle which holds in nature, and if so how far it goes in explaining the phenomena, or whether one introduces it uselessly to explain facts that can better be explained without it.46

Gravitation’s existence was an empirical question, but for Maupertuis, the evidence was not to be found in experiments. Instead, it lay in the ability of mathematics to model the heavens.

Maupertuis’ Newton believed that God was a mathematically minded creator, free to establish whatever natural order he saw fit, but one who was then constrained ever after by the mathematical laws that guided his decision….He [Maupertuis] built his defense of Newtonianism upon the regularities of the heavens that only Newton’s simple mathematical principles could explain. 47

Arguments of this kind appealed to young scholars who were now well-trained in mathematical mechanics, and they increasingly applied themselves to Newtonian questions. Among them were Alexis Clairaut, Jean le Rond d’Alembert, Leonhard Euler, Joseph-Louis Lagrange, and Pierre-Simon Laplace.

God fell out of the discussion of planetary systems — it’s not clear when or why. One answer is implied by the story (possibly apocryphal) of Laplace’s meeting with Napoleon. Napoleon observed that Laplace had explained the solar system without mentioning God; Laplace replied, “I had no need of that hypothesis.” So God disappeared because clear-sighted and cool-headed scientists had no use for him. This explanation is unpersuasive. God was not a necessary hypothesis for Descartes and Newton, either, although they did make room for him in remote corners of their theories.48 They were nevertheless deeply religious men for whom natural philosophy and theology formed a coherent whole. Laplace, by contrast, was not conventionally religious: he is thought to have been either a deist or an atheist. The difference between Newton and Descartes on the one hand and Laplace on the other is the nature of the man, not the nature of the theory. The simplest explanation for this difference is that they lived on opposite sides of the Enlightenment, the time during which Europeans shifted their focus from religious matters to secular matters. They were all men of their times.

  1. Galileo, quoted by Westfall, Force in Newton’s Physics, pp. 6-7. ↩
  2. Gaukroger, Descartes: An Intellectual Biography, p. 71, ↩
  3. Gaukroger, Descartes: An Intellectual Biography, p 71 and p. 72. ↩
  4. Descartes, quoted by Gaukroger, Descartes: An Intellectual Biography, p. 116. ↩
  5. Descartes, quoted by Gaukroger, Descartes: An Intellectual Biography, p. 117. ↩
  6. Voltaire, quoted by J. B. Shank, The Newton Wars and the Beginning of the French Enlightenment (University of Chicago, 2008), p. 313. ↩
  7. Descartes in Principia Philosophiae. Quoted in Gaukroger, Descartes’ System of Natural Philosophy, pp. 144-5. ↩
  8. Gaukroger, Descartes: An Intellectual Biography, p. 250. The included quotes are from Descartes’ Le Monde, a manuscript that Descartes held back from publication in response to the Inquisition’s condemnation of Galileo. ↩
  9. Michel Blay, Reasoning with the Infinite (University of Chicago Press, 1998), p. 51. ↩
  10. Westfall, Force in Newton’s Physics, p. 78. ↩
  11. Westfall, Force in Newton’s Physics, p. 82. ↩
  12. Domenico Bertoloni Meli, Thinking with Objects, (Johns Hopkins University Press, 2006), pp. 219-20. ↩
  13. Westfall, Force in Newton’s Physics, p. 427. “Upside down” means, of course, in terms of repulsion from the center instead of attraction to it. ↩
  14. Newton discovered the inverse square rule in the late 1660s. He first found the formula for centrifugal force, and then combined it with Kepler’s third law to obtain the rule (Westfall, Never at Rest, pp. 149-50 and 152). He withheld from publication both the formula and the rule. Christiaan Huygens had discovered the formula earlier, in 1659, but did not publish it until 1673. After its publication, Halley combined the formula and Kepler’s third law (exactly as Newton had) to obtain the inverse square rule. Wren seems to have arrived at the rule intuitively — perhaps by analogy with light, whose intensity was known to follow an inverse square rule. ↩
  15. Westfall, Never at Rest, p. 404. ↩
  16. Leibniz appears to have been the first to “elevate” Kepler’s rules into laws. See Chris Smeenk and Eric Schliesser, “Newton’s Principia” in Jed Buchwald and Robert Fox, eds., The Oxford Handbook of the History of Physics (2013), p. 114. ↩
  17. George Smith, “Newton’s Philosophiae Naturalis Principia Mathematica“, Stanford Encyclopedia of Philosophy (online resource, 2008). ↩
  18. Smith, “Newton’s Philosophiae Naturalis Principia Mathematica”. ↩
  19. Westfall, Force in Newton’s Physics, p. 435. ↩
  20. Bruce Brackenridge, The Key to Newton’s Dynamics (University of California Press, 1995), p. 23. ↩
  21. Principia, p. 62. All Principia references are to Bernard Cohen and Anne Whitman, The ‘Principia’: The Authoritative Translation (University of California Press, 1999). ↩
  22. Principia, p. 51, Definition 4 and subsequent text. ↩
  23. Principia, p. 50, Definition 3 and subsequent text. ↩
  24. Bernard Cohen, The Newtonian Revolution (Cambridge, 1980), p. 251. ↩
  25. Principia, p. 97, Corollary 6 of Proposition 4. ↩
  26. Isaac Newton, The System of the World (Motte and Bathurst, London, 1787), p. 49. System is the first draft of Book 3 of Principia. It is nontechnical, intended to be widely read. Newton set it aside, replacing it with a technical exposition intended for a highly specialized audience. System was not published until 1728, more than forty years after Principia. ↩
  27. Principia, pp. 440-1, Book 3, rules 1-3. ↩
  28. Principia, p. 442. ↩
  29. Principia, p. 452. ↩
  30. Principia, p. 589. ↩
  31. Principia, p. 586. ↩
  32. J. B. Shank, “There was No Such Thing as the ‘Newtonian Revolution’ and the French Initiated It,” Early Science and Medicine (2004), p. 260. ↩
  33. Principia, p. 44 (editor’s preface). ↩
  34. Edward Dolnick, The Clockwork Universe (Harper, 2011), p. 309. ↩
  35. Robert Boyle. Quoted by Peter Anstey and Alberto Vanzo, Experimental Philosophy and the Origins of Empiricism (Cambridge, 2023), p. 51. ↩
  36. Jeffrey Wigelsworth, Deism in Enlightenment England (Manchester University Press, 2009), p. 142. ↩
  37. Newton, quoted by Margaret Jacob, The Newtonians and the English Revolution, 1689-1720, p. 156. ↩
  38. Shank, The Newton Wars and the Beginning of the French Enlightenment, pp. 301-3. ↩
  39. Voltaire, Philosophical Letters (1634; Hackett Publishing, 2007), p. 47. ↩
  40. Voltaire, Philosophical Letters, p. 50. The mathematician d’Alembert offered a parallel but more respectful appraisal: “Let us recognize that Descartes, who was forced to create a completely new physics, could not have created it better; that it was necessary, so to speak, to pass by the way of the vortices in order to arrive at the true system of the world; and that if he was mistaken concerning the laws of movement, he was the first, at least, to see that they must exist.” (Quoted in Shank, The Newton Wars, p. 16. )↩
  41. Voltaire, Philosophical Letters, p. 57. ↩
  42. Cohen, The Newtonian Revolution, pp. 81-2. ↩
  43. Shank, The Newton Wars, p. 243. ↩
  44. Smeenk and Schliesser, “Newton’s Principia”, pp. 154-5. ↩
  45. Shank, The Newton Wars, pp. 238-9. ↩
  46. Maupertuis, quoted by Shank, The Newton Wars, p. 288. ↩
  47. Shank, The Newton Wars, p. 290. ↩
  48. Descartes made reference to God in his laws of motion, arguing that the “quantity of motion” and the “quantity of rest” were fixed by the immutability of God. Newton found perturbations in the planetary orbits that seemed to make the orbits unstable, and argued that God must periodically intervene to set things right. Leibniz, himself deeply religious, mocked this claim: couldn’t the creator of the universe get it right the first time? ↩