Mechanics

Aristotle's physics stands at the head of the Western science of motion, and the developments which follow it are in large part corrections of the account he gives. His governing principle is that whatever is in motion is moved by something, so that no motion occurs without a mover. A stone falls because earth, as an element, moves naturally toward the center of the cosmos; fire rises because it is the nature of fire to move upward. Such natural motions call for no external push, since they express the nature of the element itself. Violent or forced motions, as when a stone is thrown upward, do require an external cause, and they cease when that cause is withdrawn.

Aristotle argues further that the speed of a moving body varies with the force applied and varies inversely with the resistance of the medium. A heavier stone, on this view, falls faster than a lighter one, and motion through water proceeds more slowly than motion through air, the denser medium offering the greater resistance. Motion through a void would on these terms be instantaneous, which Aristotle takes as a demonstration that a void cannot exist. His account of motion is thus inseparable from his doctrine of the elements, of place, and of the plenum, matters treated more fully under the ideas of Element and Space.

Two features of this physics may be noted, since the modern criticism of it turns on them. It is qualitative rather than mathematical, treating hot and cold, heavy and light, as real attributes of bodies; and it explains by four kinds of cause rather than one, admitting formal and final as well as efficient and material causes. Both features are denied by the mechanical philosophy which Galileo and Descartes restate from the ancient atomists, and it may be argued that the doctrine of natural places, by dividing motions according to the natures of bodies, discouraged the search for laws of motion applicable to all bodies whatsoever.

Aristotelian mechanics held the field for some eighteen centuries, during which, as Fourier observes, little was added to the science that Archimedes had begun. The reasons appear to be philosophical rather than technical. So long as motion required a mover throughout its course, and so long as qualities were taken to have physical reality equally with quantities, the program of Galileo and Newton could not be conceived. Whether the rejection of final causes in physics represents a gain in truth or only a restriction of the questions physics permits itself to ask is considered further under the idea of Cause.

With Archimedes, according to Fourier, the science of rational mechanics begins, for he is the first to explain the mathematical principles of the equilibrium of solids and of fluids. Where Aristotle reasons qualitatively about motion and cause, Archimedes demonstrates theorems. His treatise On the Equilibrium of Planes derives the law of the lever from a small number of postulates, showing that two weights balance at distances inversely proportional to their magnitudes. His work On Floating Bodies establishes that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces.

The method may be considered as significant as the results. Archimedes begins with axioms, proceeds by deduction, and reaches conclusions which measurement can test. He does not inquire why the lever works, nor what the natural place of water may be. He asks instead what must follow, given the conditions assumed. This abstraction from physical causes to mathematical relations is the mark of his procedure, and Galileo takes it as the model for the science he undertakes to found. Upon a single proposition in the book on Equilibrium, Galileo remarks, depend not only the law of the lever but the laws of most other mechanical devices as well.

The part of mechanics which Archimedes treats is statics, the science of bodies at rest or in equilibrium. Statics may be regarded as a limiting case of dynamics, the rest of a body being the condition to which the principles of motion apply when the forces upon it are balanced. Pascal later enlarges this branch by showing, in his treatise on the equilibrium of liquids, that a vessel full of water constitutes a new mechanical principle capable of multiplying force to any degree desired. Archimedes also devised methods for measuring the areas and volumes of curved figures which bear on the questions treated under the idea of Mathematics.

It may be wondered why, with the start Archimedes made, no application of his principles and his method followed for eighteen centuries. The answer is sometimes sought in the physics of Aristotle, whose doctrine of natural motions to natural places, and whose treatment of qualities alongside quantities, discouraged the mathematical study even of local motion. Whatever the cause of the delay, Galileo's self-conscious adoption of the Archimedean manner of demonstration marks the point at which the science resumes.

With Galileo, dynamics is founded as a science, some eighteen centuries after Archimedes had established statics. His aim, as he states it, is to describe with precision the motions found in a child's play, stones dropped and stones thrown, the one the natural motion of free fall and the other the violent motion of a projectile. That a falling body accelerates is clear to observation. Galileo wishes to know the properties of that acceleration: how the rate of increase in velocity is related to the times and the distances of the fall, and what velocity the body has at any given point. Rolling balls down inclined planes and measuring the distances traversed in successive equal times, he finds that the distances increase as the squares of the times, and that the velocity increases in direct proportion to the time elapsed. All bodies, he holds, fall with the same acceleration whatever their weight, contrary to the Aristotelian doctrine that the heavier falls the faster.

Galileo resolves the curvilinear path of a projectile into an imparted rectilinear motion and the deflecting pull of gravity, and shows the path to be a parabola. In the course of this analysis he expresses the insight which Newton later formulates as the first law of motion, that a body continues in rest or in uniform rectilinear motion unless compelled by impressed force to change that state. The innovation is considerable, for Aristotle and his followers had looked for the cause which keeps a body moving, whereas on this view uniform motion continues without cause and only a change of velocity or direction calls for one.

Of the causes themselves Galileo declines to speak. Of the several opinions philosophers have advanced about the cause of the acceleration of natural motion, attraction to the center, repulsion among the small parts of the body, a stress in the surrounding medium, he says that all ought perhaps to be examined, but that it is not worthwhile to debate them in sciences where mathematical demonstrations are applied to natural phenomena. When Simplicio appeals to gravity as a manifest cause, Salviati replies that everyone knows it is called gravity, but that the essence of the thing cannot be defined. The doctrine of primary and secondary qualities which Galileo states in Il Saggiatore, and which is treated more fully under the ideas of Quality and Sense, belongs to the same restriction of physics to measurable quantities.

Newton's laws of motion and of universal gravitation rest on the ground Galileo prepared, for without the law of uniform acceleration and the composition of motions in projectile flight there would have been little for the Principia to generalize. It may be argued, however, that Galileo's pure program, which initiated mathematical physics, was not by itself sufficient to advance it, and that the worrying about causes which he set aside, and which Huygens and Newton could not escape, provided the pivot for the discoveries that followed.

Descartes states the mechanical philosophy in the boldest form it takes in the seventeenth century, and states it as a program for physical research. The laws of mechanics, he writes, are the laws of nature. In Part II of the Principles of Philosophy he holds that matter is nothing but extension, that all physical change consists in the motion of extended parts, and that the universe is a plenum admitting no void. God created matter with a certain total quantity of motion and conserves it, so that motion is redistributed among bodies by contact and collision while the sum remains constant. Descartes states an early version of the law of inertia, according to which a body in motion continues in a straight line unless deflected by another body, together with three rules of impact whose particulars were later found to be mistaken.

The true properties of bodies, in Descartes' opinion, consist solely in motion or its absence and in the configuration and situation of their parts. Colors, odors, savors, and the rest he takes to be merely sensations existing in thought, differing from the real properties of bodies as much as pain differs from the shape and motion of the instrument which inflicts it. What is excluded from physics by this doctrine are the occult qualities and sympathies of Aristotelian and Renaissance natural philosophy. Gravity, hardness, and the power of heating are to be accounted for by the impact of parts in motion, and by nothing else.

From these principles follows a cosmology in which the planets are carried about the sun by vortices of subtle matter, and in which magnetism, the tides, and even the sensation of animals are to be explained by the shapes, sizes, and motions of invisible particles. Descartes opposes atomism as plainly as Aristotle does, so that the mechanical view is shown not to depend on indivisible particles moving in a void. The relation between his physics and his account of mind, which is what makes the beast-machine possible, is treated under the ideas of Matter and Soul.

Newton accepts the ideal of a mathematical physics and the law of inertia, but rejects the vortices as the cause of gravitation, and appears to class Descartes with Aristotle when he says that hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. Yet the ether which Newton himself proposes in the Queries to the Optics is judged by later physicists to be no less mechanical than the Cartesian vortices, and no philosophical ground has been found for preferring one hypothesis to the other. Newton's victory over Descartes may therefore be described as a victory in mathematical and experimental physics rather than in philosophy.

Huygens opens his Treatise on Light by describing optics as the kind of science in which geometry is applied to matter, and at once expresses his wish to advance it by investigating the origin and the causes of the truths already known. Such explanations, he thinks, will be found only if the causes of all natural effects are conceived in terms of mechanical motions, and he declares that we must do this or else renounce all hope of comprehending anything in physics. Unlike Galileo and Newton, then, he holds that the mathematical physicist may properly inquire after causes. His own inquiry proposes that light consists in waves propagated through an all-pervading medium, each point of a wave front serving as the source of a new spherical wavelet whose envelope forms the succeeding front, a construction from which the laws of reflection and refraction may be derived.

The analogy on which the theory rests is that of sound, which spreads through the air by a movement passed on from part to part, forming spherical surfaces that enlarge until they strike the ear. Huygens is aware that the analogy is imperfect, since a sounding body shut in an evacuated vessel makes no sound while the vessel remains transparent. Waves must be waves of something, and he therefore posits an ethereal matter, a transparent medium permeating the universe. This ether, posited to explain light mechanically, then calls in turn for a mechanical account of its own properties. Huygens does not avoid the new problem, but neither does he undertake to settle it, saying that it is not necessary to examine here the causes of the hardness or the springiness of the ethereal particles.

Huygens was also a master of practical mechanics. He constructed the first reliable pendulum clock, working out the mathematics of cycloidal motion so that the period should be independent of the amplitude. He formulated correct laws for the collision of elastic bodies, in correction of Descartes' rules, and his treatment of centripetal force supplied one of the instruments Newton required for the theory of gravitation. Of Newton's attraction he wrote to Leibnitz that he was not at all pleased with theories built on that principle, which seemed to him absurd, an objection turning on the scandal of action at a distance discussed under the idea of Cause.

The rivalry between Huygens' waves and Newton's corpuscles continues for something like two centuries, the two theories proving equally competent to account for reflection, refraction, and dispersion. The discoveries of Young and Fresnel tend to favor the wave theory, though the requirement that the vibrations be transverse converts the ether from an air-like into a jelly-like substance, and so aggravates the very difficulty Huygens had postponed. Einstein calls the ether the enfant terrible among hypothetical physical substances, and observes that physicists abandoned the whole program of mechanical explanation rather than continue to make the artificial assumptions its construction demanded. Eddington's coinage of the word "wavicle" suggests how far the controversy has been transformed rather than concluded.

Newton's achievement in the Mathematical Principles of Natural Philosophy is a synthesis of what had been accomplished before him and a criticism of much that had stood in the way. Three laws of motion are laid down, and from these, together with the law of universal gravitation, there follow the motions of projectiles and of planets alike. Kepler had shown the orbits to be ellipses but, lacking the first law of motion, could theorize about their cause only by looking for a force projected outward from the sun to sweep the planets along their paths. A follower of Galileo would rather seek a force deflecting the planet inward from its own rectilinear course, and this is what Newton found. With that discovery the terrestrial dynamics of Galileo becomes celestial as well, and the traditional separation of the heavens from the earth is overcome.

The mathematical instrument was of a piece with the physics. Newton devised the calculus to handle the continuously varying quantities which mechanics required, and though the proofs of the Principia are given geometrically, the reasoning beneath them turns on computing instantaneous rates of change and summing infinitely small contributions. From the laws of motion and the inverse-square law of force he was able to account by deduction for the perturbations of the moon, the oblateness of the earth, the precession of the equinoxes, the tides, and the paths of comets. The relation between mathematical demonstration and physical measurement which this work exhibits is treated further under the idea of Mathematics.

Newton's stated policy is to argue from phenomena without feigning hypotheses, and in mathematics, he says, we are to investigate the quantities of force with their proportions before we inquire into the physical species and causes of those forces. The policy postpones rather than excludes the question of cause, and in the Queries appended to the Optics he does propose an ethereal medium, rarer within dense bodies than in the empty spaces between them, whose differences of pressure might produce what we call gravity. To Bentley he wrote that the notion of one body acting on another at a distance through a vacuum, without any mediating thing, was so great an absurdity that no competent thinker could fall into it. Whether such a medium can be reconciled with the unretarded motion of the planets is the objection Newton himself raises against Huygens' ether.

Newton's picture of the world governs the science of the following century, and it is his name that later physicists attach to classical mechanics, although he disavowed the framing of hypotheses. The reason may be that his concepts of attraction and force are left dark unless they are given some mechanical interpretation, and that the crises to which his work and Galileo's led could not be escaped without discussion of the causes of gravity and of the propagation of light. Faraday will convert that difficulty into a question about fields rather than forces, and the question whether mechanical conceptions belong in physics at all is the issue that occupies the science down to Einstein.

In the opening pages of the Analytical Theory of Heat, where he reviews the triumphs of explanation achieved by Newton and his successors, Fourier declares that whatever the range of mechanical theories may be, they do not apply to the effects of heat, which make up a special order of phenomena not to be explained by the principles of motion and equilibrium. His aim with respect to this class of phenomena, produced not by mechanical forces but by the presence and accumulation of heat, is to reduce the physical questions to problems of pure analysis and to express the most general conditions of the propagation of heat in differential equations. The method of solving those equations by decomposing complex distributions of temperature into sums of simple periodic functions has since been applied throughout physics.

What Fourier proposes is thus a mathematical physics which dispenses with mechanical interpretation altogether. Of the nature of heat, he writes, only uncertain hypotheses could be formed, but knowledge of the mathematical laws to which its effects are subject is independent of all hypothesis; primary causes are unknown to us, yet they are subject to simple and constant laws which observation may discover. He does not decide between the caloric theory, which treats heat as a subtle fluid, and the kinetic theory, which identifies it with molecular motion. Lavoisier had already granted that we are not obliged to suppose caloric a real substance, it being sufficient to consider it as the repulsive cause, whatever that may be, so that we remain at liberty to investigate its effects in an abstract and mathematical manner.

Fourier acknowledges a debt to Descartes for the analytical equations first introduced into the study of curves and surfaces, but insists that these equations are not restricted to the properties of figures or to the objects of rational mechanics. They extend, he says, to all general phenomena. The claim asserts the mathematical character of nature as the ground of a purely mathematical physics, and in this it resembles Galileo's declaration that the book of nature is written in mathematical language, a matter treated further under the ideas of Mathematics and Nature.

Where Newton returns to the question of causes after disavowing it, Fourier never deviates from his indifference to causes and never softens his judgment that mechanics is irrelevant to the subject he investigates. His trust in mathematical analysis, able of itself to yield and to organize physical discoveries, appears to have encouraged Clerk Maxwell to turn from a mechanical to a mathematical theory of electricity, and certain of Fourier's results, such as his theory of dimensions, proved directly useful to Maxwell. The larger question raised by his example, whether physics should content itself with description or strive for explanation, is the issue on which the rise and decline of the mechanical point of view depends.

Faraday comes between Fourier and Maxwell, and he discovers by experiment the phenomena whose mathematical structure Maxwell afterward develops. His researches establish electromagnetic induction, by which a changing magnetic condition produces a current; the laws of electrolysis; and the rotation of polarized light by magnetism. Gilbert had compared magnetism and electricity without converting either into the other, and Oersted had shown that an electric current has a magnetic effect. Faraday shows the reverse, that magnetism has electrical power, and his interest in such reversibilities extends to the relation between electrical and gravitational force, a speculation which points beyond Maxwell toward a field theory unifying all physical phenomena under a single set of mathematical laws.

Faraday sees no incompatibility between experimentation and speculation. As an experimentalist, he says, he feels bound to let experiment guide him into any train of thought it may justify, being satisfied that experiment must in the end declare the truth. His speculative instrument is the conception of lines of force. Electric and magnetic influence, on this view, is not exerted instantaneously across empty space, as the Newtonian model of attraction would have it, but is carried by lines stretching continuously from one body to another, whose arrangement and density determine the strength and direction of the force at each point. The space between bodies is thus not empty but occupied by a physically real condition which mediates their action.

The bearing of this on mechanics may be stated in terms of the long controversy over action at a distance, which had troubled Newton in the case of gravity and Huygens in the case of light. If force is mediated by a field, then action at a distance is avoided without recourse to the ether that both had found unsatisfactory, but the field must itself be admitted as a physical thing possessing energy and obeying laws of its own. Maxwell gives Faraday's lines of force a mathematical expression, and his equations, verified experimentally by Hertz, show light to be an electromagnetic wave. Content to let the field theory state the mathematical structure of the phenomena, Maxwell gives up the attempt to supply a mechanics for his equations.

The point to which Faraday carries the story suggests its sequel in Maxwell and in Einstein. Once the field is admitted, the mechanical picture of particles acting on one another across empty space no longer suffices for the whole of physics, and Einstein's judgment is that science did not succeed in carrying out the mechanical program convincingly. Whether the abandonment of that program represents the discovery of a better way of explaining nature, or rather the relinquishing of explanation in favor of description at its most general, remains a question on which physicists and philosophers have differed.

Kant adds no new mechanical laws, but he raises a question which the practitioners of mechanics had left unexamined, namely how mechanical knowledge is possible at all. Newton's laws claim universal and necessary validity, yet Hume had argued that experience alone can justify no universal claim. That bodies have hitherto attracted one another according to the inverse-square law may be granted; that they must always do so no number of observations can establish. The difficulty is the same one that arises over the first law of motion, which, as James remarks, is never a matter of experience but has to be disengaged from under experience by ignoring conditions that are always present.

Kant's answer is that the fundamental principles of mechanics are not read off from nature but are conditions the mind imposes upon experience in order that experience should be intelligible. The three Analogies of Experience correspond to the categories of relation: that something permanent persists through all change, that every event has a cause determining it according to a rule, and that substances in space stand in mutual interaction. These are not empirical generalizations but the framework within which empirical generalization becomes possible. The question whether the mind supplies the order it finds in nature is treated more fully under the ideas of Experience and Knowledge.

On this view Newton's laws, so far as they express substance, causality, and reciprocity, have a transcendental ground which no empirical science can furnish or overturn, while the particular form of the law of gravitation remains empirical, since reason cannot determine why the force should vary as the inverse square rather than the inverse cube. Kant takes Newtonian physics as the model of a rational science of nature, and in the Metaphysical Foundations he undertakes to derive its principles from the concept of matter as the movable in space. He is nevertheless among those who find attraction without contact a chimerical fancy which we have no right to assume.

Kant's transcendental philosophy does not compete with Newtonian mechanics but proposes to justify it. Whether that justification stands after the revolutions of twentieth-century physics has been much disputed, since relativity revises the conceptions of space and time which Kant took to be forms of intuition, and quantum theory unsettles the universality of causal determination. The dispute belongs to the larger question whether the principles of mechanics are findings of experimental research or philosophical assumptions brought to it.