Newton’s Principia is one of the few scientific books which has sensibly affected the methods of scientific research and the ideas of men about the universe. It is on this aspect of the subject I propose, in this paper, to make a few remarks. The work itself is a classic in the history of mathematics: the exposition of the subject, the enunciation of the principle of prime and ultimate ratios, the creation of mechanics as a science resting on experiments, and the theory of universal gravitation with concrete applications to the solar system, make it a masterpiece. Here I avoid all technicalities, and confine myself to a general description of its genesis and contents and the reason why its publication affected scientific thought and methods.

Newton’s exposition arose from an investigation of the cause of the motion of the planets round the sun, and this in due course led to the enunciation and establishment of the Newtonian theory of attraction. The origin of this theory has been often told, but will bear repetition. The fundamental idea occurred to Newton in 1665 or 1666, shortly after he had taken his degree at Cambridge, when, as he [226] wrote later, “I was in the prime, of my age for invention, and minded Mathematicks and Philosophy more than at any time since.” His reasoning was as follows. He knew that gravity extended to the highest hills, he saw no reason why it should cease to act at greater heights, accordingly he believed that it would be found in operation as far as the moon, and he suspected that it might be the force which retained that body in its path round the earth.

This hypothesis he verified thus. If a stone is allowed to fall near the surface of the earth, the attraction of the earth causes it to move through sixteen feet in one second: also Kepler’s Laws, if accurate and applicable, involve the conclusion that the attraction of the earth on a distant body varies inversely as the square of its distance from the earth. Now the radius of the earth and the distance of the moon were known to Newton, and therefore, on this hypothesis, he could find the magnitude of the earth’s attraction on the moon. Further, assuming that the moon moved in a circle, he could calculate the force required to retain it in its orbit. At this time his estimate of the radius of the earth was inaccurate, and, when he made the calculation, he found that this force was rather greater than the earth’s attraction on the moon. The discrepancy did not shake his faith in his theory, but he conjectured that the moon’s motion was also [227] affected by other influences, such for example, as the effect of a resisting medium which might itself be in motion as supposed by Descartes in his hypothetical vortices.

In 1679 Newton knew with approximate correctness the value of the radius of the earth. He repeated his calculations, and found the results to be in accordance with his former hypothesis. He then proceeded to the general theory of the motion of a particle under a force directed to a fixed point, and showed that the vector to the particle would sweep over equal areas in equal times. He also proved that, if a particle describes an ellipse under a force directed to a focus, the law must be that of the inverse square of the distance from the focus, and conversely, that the orbit of a particle projected in free space under the influence of such a force must be a conic. The application to the solar system was obvious, since Kepler had shown that the planets describe ellipses with the sun in one focus, and that the vectors from the sun to them sweep over equal areas in equal times. This investigation was made for his own satisfaction and was not published at the time. In it he treated the solar bodies as if they were particles, and he must have realized that the results could be taken as being only approximately correct.

In 1684 the subject of the planetary orbits was [228] discussed in London by Halley, Hooke, and Wren. They were aware that, if Kepler’s conclusions were correct, the attraction of the sun or earth on a distant external particle must vary inversely as the square of the distance, but they could not determine the orbit of a particle subjected to the action of a central force of this kind. It was suggested that Newton might be able to assist them. Accordingly in August, Halley went to Cambridge for a talk on the subject, and then found that Newton had solved the problem some five years previously, and that the path was necessarily a conic. At Halley’s request Newton wrote out the substance of his argument, and sent it to London.

Halley at once realized the importance of the communication, and later in the autumn returned to Cambridge to urge Newton to prosecute the theory further. He found that Newton had already done something in the matter, the results being contained in a manuscript which he saw. Probably this reference is to the holograph manuscript, still preserved in the University Library at Cambridge, of Newton’s lectures in the Michaelmas Term, which served as the basis of his memoir sent to the Royal Society a few months later. The great value of these investigations was recognized, and Newton was persuaded to attack the more general problem. His results are given in the Principia.

[229]
As yet Newton had dealt with the problem as if the sun and the planets might be regarded as heavy masses concentrated at their centres. Clearly at the best this was only an approximation, though considering the enormous distances involved it was not unreasonable. In January or February, 1685, he considered the question of the attraction of bodies of finite size, and found, to his surprise and gratification, that a sphere or spherical shell attracts an external particle as if condensed into a heavy mass at its centre. Hence the results he had already proved for the relative motion of particles were true for the solar system, save for small errors due partly to the fact that the bodies were not perfectly spherical and partly to disturbances caused by the planets attracting one another. It was no longer a question of rough approximation: the problem was reducible to mathematical analysis, subject to the introduction of minute corrections, which, given the necessary observations, could be calculated very closely. This was a new discovery of first-rate importance, and initiated the modern theory of attractions.

The first book of the Principia was finished before the summer of 1685. It deals with the motion of particles or bodies in free space either in known orbits or under the action of known forces. In it the law of attraction is generalized into the [230] statement that every particle of matter attracts every other particle with a force which varies directly as the product of their masses and inversely as the square of the distance between them. Thus gravitation was brought into the domain of Science.

The second book was completed by the summer of 1686. It treats of motion in a resisting medium and of various problems connected with waves. At the end of it, it is shown that the Cartesian theory of vortices is inconsistent with the laws of motion, and necessarily leads to incorrect results. This book opened another world to the application of mathematics and, in effect, created the science of hydrodynamics.

The third book was finished in March 1687. In this, the theorems previously established are applied to the chief phenomena of the universe, and briefly we may say that all the facts then known about the solar system and, in particular, the motion of the moon with its various inequalities, the figure of the earth, and the phenomena of the tides, were shown to be in accord with the theory. Much of the material for these calculations was collected by Flamsteed and Halley.

The Principia, as I have said, is a classic. Like other books to which that compliment is paid, it is rarely read: indeed, I doubt whether there are a [231] dozen men in Cambridge who have glanced all through it, even in a cursory manner. When I was an undergraduate the course for the Tripos involved five sections (1, 2, 3, 9, and 11) of the first book, but now, probably with good reason, even this slight acquaintance with the work is no longer required, and to-day the character of these investigations is unfamiliar to most mathematicians, while the fact that it is written in Latin tends to diminish the number of its readers. I will, then, with your permission, describe briefly its frame-work.

First, however, let me remark on how different was the knowledge of mathematics, even among experts, at the time it was written from that current to-day. In the geometry of the circle and conics mathematicians were familiar with the methods of Greek science, and in their application Newton was unrivalled among his contemporaries, but outside geometry methods of investigation were far to seek. Analysis had been but little developed; algebraic notation had only recently taken definite form; trigonometry was still used mainly as an adjunct to astronomy; analytical geometry had been invented by Descartes, but no text-books on it of modern type were available; while nothing about the calculus had been published. Mechanics, however, had recently been treated as a science—statics by Stevinus and dynamics by Galileo—and this paved the way for [232] Newton’s investigations. In particular, Galileo had established principles which foreshadowed the first two laws of motion, and had deduced formulae in linear motion like v²=2fs, s=½ft², and in circular motion like f=v²/r.

Newton prefaced the Principia by explaining that the earliest problems in natural philosophy which attract attention are connected with the phenomena of motion, and it was with motion). that the book dealt. To discuss motion effectively, it was necessary to give precision to the language used, and accordingly he propounded definitions of mass, momentum, inertia, and so on, which have settled the language of the subject. He next enunciated his three well-known laws of motion, and described the experiments on which he based them. He followed this up by deducing rules for the composition and resolution of forces, and discussed relative motion.

This preliminary matter is followed by the first book, concerned with the motion of bodies in an unresisting medium. It is divided into fourteen sections containing ninety-eight propositions with various interpolated lemmas, corollaries, and scholia.

The first section is on the method of prime and ultimate ratios, by the use of which Newton was able, in effect, to integrate. He applied this to the curvature and the areas of curves, and proved that, [233] at the very beginning of the motion of a body from rest under any force, the space described is proportional to the force and the square of the time.

The second section is concerned with the motion of a particle under a central force. It contains the well-known propositions that if the force is central the area swept out by the vector to the centre is proportional to the time, and conversely that if such area is proportional to the time the particle is acted on by a central force. Newton further discussed particular cases of circular, elliptic, and spiral motion. In the third section he dealt with motion in a conic under a central force to the focus, showed that in this case the force must vary inversely as the square of the distance, and conversely that if a particle be projected from any point in any direction with any velocity under such a force it must describe a conic about the centre of force as a focus, and that in such elliptic orbits the periodic times are in the sesquiplicate ratio of the major axes of the ellipses. He also explained how to treat the problem if disturbing forces are introduced. These two sections solved the problem of planetary motion if the planets could be treated as particles and did not disturb one another’s motions.

The fourth and fifth sections are given up to the proof of certain geometrical propositions in conics required for subsequent discussions: in particular [234] the construction of a conic when a focus and three other conditions or when five points on it or five tangents to it are given.

In the sixth section Newton returned to the problem of the motion of a particle in an ellipse under a central force to a focus, and discussed how to determine the position of the particle at any given time. (Kepler’s Problem.)

The seventh and eighth sections are devoted to the motion of a particle under a central force which is any function of the distance. The geometrical treatment of these problems is ingenious, but necessarily more involved than when modern analysis is used.

In the ninth section Newton dealt with the motion of particles in orbits which are revolving about the centre of force, and on the motion of the apses of such orbits: this introduced the theory of disturbing forces. The tenth section is concerned with constrained motion, and particularly with the motion of pendulums. The eleventh section deals with the motion of particles under their mutual attractions and incidentally with the problem of three bodies. These three sections afford a notable illustration of Newton’s analytical powers.

The twelfth and thirteenth sections deal with the attraction under various laws of force of spherical bodies, circular laminae, and solids of revolution. [235] These sections brought the problem of the solar system, consisting of solid bodies of finite size and approximately spherical in form, into the domain of mathematics, and led up to the generalization that all particles of matter attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them, from which law it would seem that all the known phenomena of the motions of the solar system can be deduced.

The fourteenth section is concerned with the motion of minute corpuscles, with applications to the corpuscular theory of light.

The second book is devoted to the discussion of the motion of bodies in resisting mediums: there are fifty-three propositions besides lemmas, scholia, etc.

In the first section, Newton considered the motion of a particle or sphere moving in a medium whose resistance varies as the velocity of the particle: in the second section the resistance is assumed to vary as the square of the velocity: and in the third section the resistance is supposed to consist of two terms, one varying as the velocity and the other as the square of the velocity. The fourth section is on spiral motion caused by resistance of the medium.

The fifth section deals with the density and pressure of liquids and gases at rest (Hydrostatics).

[236]
The sixth section treats of the motion of pendulums in a resisting medium; and the seventh section is concerned with the motion of fluids, and the resistance they offer to the motion of projectiles. The latter section contains the celebrated statement of the form of the solid of least resistance, whose demonstration proved a puzzle to mathematicians until the invention of the calculus of variations. Newton’s solution is in the Portsmouth papers, and elsewhere I have published it: it involves the use of fluxions, and it is probable that it was his failure to translate this demonstration into geometrical language that led him to give the result without a proof.

The eighth section deals with the motion of waves with applications to the theory of sound and the undulatory theory of light; and the ninth section deals with vortices; it is here shown that the theory of vortices suggested by Descartes to explain the motion of the solar system is untenable.

This book created the theory of hydrodynamics. Much of it is incomplete, but it is astonishing that Newton proved as much as he did; of course to-day no one would suggest that the best way of attacking these problems is by Newtonian geometrical methods.

The third book contains the practical application of the propositions in the two earlier books to the solar system. I need not describe this in detail. In order to justify this application, Newton commenced [237] by laying down four rules which have since been accepted as binding in scientific investigations. These, as given in the third edition, are to the following effect: (1)We are not to assume more causes than are sufficient and necessary for the explanation of observed facts. (2)Hence, as far as possible, similar effects must be assigned to the same cause; for instance, the fall of stones in Europe and America. (3)Properties common to all bodies within reach of our experiments are to be assumed as pertaining to all bodies; for instance, extension. (4)Propositions in science obtained by a wide induction are to be regarded as exactly or approximately true, until phenomena or experiments show that they may be corrected or are liable to exceptions. The substance of these rules is now accepted as the basis of scientific investigation. Their formal enunciation here serves as a landmark in the history of thought.

As soon as the Copernican view of the solar system was accepted, it was natural for men to seek to explain the reason why the planets moved as they did. Descartes, in 1644, had suggested that the explanation might be found in the existence of vortices in space. This conjecture, although based on arbitrary assumptions, and in fact untenable, played an important part in the history of the subject, for it accustomed men to think that planetary phenomena might be explicable by the same laws [238] as are found to be true on the earth. That this was so was established by Newton in his Principia, where all the motions of the solar system were made to depend on one assumption as to the law of attraction. The question whether this law could itself be deduced from some more fundamental assumption was raised by Newton, but he could not devise a satisfactory hypothesis. It has been discussed again and again since his time, and the problem is still unsolved.

Newton’s conclusions were immediately accepted in Britain, and very rapidly by the leading mathematicians in Europe: indeed Huygens came expressly from Holland in order to make the personal acquaintance of a writer whose work promised to revolutionize the history of science. The refutation of the Cartesian hypothesis ran, however, counter to the sentiments and wishes of a certain number of philosophers, and some few years elapsed before the truth of the gravitation theory was universally admitted, but it would be ungracious to dwell further on this. In Britain the work exercised a profound influence in philosophy as well as in science, and educated men of all schools of thought acquainted themselves with the general line of Newton’s reasoning and his deductions.

That men of science and philosophers should have approved Newton’s theory is not surprising, but it is somewhat curious that it excited so little [239] opposition among theologians. Galileo’s discoveries of hills, vales, and (supposed) seas on the moon and planets had already suggested that life might exist there, and in the popular (but illogical) view this involved the idea of the existence of beings with spiritual and intellectual faculties not unlike those of men. Newton’s results seemed to show that there was nothing in the nature of things to differentiate the earth from the other planets, and therefore considerably strengthened the view that life might be found on them. It might well be asked whether such life, and indeed whether the mechanism of the solar system as expounded by Newton, was in accordance with Scripture. That these difficulties were not pressed against Newton’s conclusions is, I think, attributable to the fact that his theory was explicitly concerned only with non-organic matter. His own opinion was that the extension of the reign of law was an additional argument in favour of a divine creation: this view, set out at the end of the Principia and in his five letters to Bentley in 1692–93, was generally accepted by the leaders of religious thought in Britain.

Lagrange more than once remarked that Newton was not only the greatest mathematician of former days, but the most fortunate, since, as there is but one universe, it can happen to but one man in the [240] world’s history to be the interpreter of its laws. It is true that Newton applied his theory only to the solar system for which alone he had the necessary data, but after the publication of the Principia, no one doubted that gravity extended to the most distant regions of space. The work of Sir William Herschel and that of all later astronomers on binary and other systems rests on this hypothesis.

The influence of the Principia on dynamical astronomy has been permanent. It is not too much to say that when it was published, the theory, as there set out, had outstripped observation, but during the succeeding century large numbers of new facts were collected, and applications of the theory to new problems were made, notably by Clairaut, Euler, and Lagrange. All these researches tended to confirm it.

The demonstrations in the Principia are expressed in the language of classical geometry, and, though unnecessarily concise and difficult, their correctness is unimpeachable. That Newton could carry his calculations so far with the limited mathematics then at his command is not the least wonderful part of the performance, but it is the prerogative of genius to get great results with but scanty equipment.

Newton’s methods, which even in the seventeenth century were archaic, became in time quite out of [241] date. This reason, the growth of the subject, and the development of analysis made it desirable to expound dynamical astronomy afresh. Towards the end of the eighteenth century the task was undertaken by Laplace in his MÉcanique CÉleste. This is far more than the translation of the Principia into the language of modern analysis, for it greatly extends the theory of some branches of the subject which had been left incomplete by Newton, either on account of his not having the requisite analysis at his command or because the necessary facts were not available. Laplace acknowledged his debt to Newton, and expressed his deliberate opinion that the Principia was pre-eminent over every previous production of human genius—“so near the gods, man cannot nearer go.” A century later a fresh exposition of the subject embodying the discoveries of the nineteenth century was given by F.F.Tisserand in his MÉcanique CÉleste; this presents the subject in its modern form.

Newton had applied his theory to the solar system as it existed, and had not investigated its origin. We owe to Laplace the enunciation of a hypothesis as to its evolution. According to this conjecture, the solar system originated in a quantity of incandescent gas rotating round an axis through its centre of mass. Laplace assumed that as this gas cooled, it would contract, and that successive rings [242] would break off from its outer edge; these rings in their turn would cool, and finally condense into the planets with their satellites; while the sun represents the central core which would be left. Recent investigations show that this cannot be taken as correct without numerous modifications. Moreover every extension of our knowledge requires the introduction of alterations in the hypothesis, and this clearly suggests that the conjecture is untenable. It played, however, a useful part in its day, as suggesting a common origin for all members of the system. Perhaps I ought to add that a nebular origin had been previously outlined by Kant, who had also suggested meteoric aggregations and tidal friction as agents concerned, but these were little more than vague conjectures.

The Principia convinced its readers that the laws of mechanics, discovered by experiment on the earth, were operative throughout the solar system. It was reserved for the nineteenth century to extend the reign of law to other celestial phenomena. Newton and his successors had proved that the law of gravity extends through all parts of space where observations are possible. That the sun, stars, and planets are constituted of similar materials was generally believed; and this has now been confirmed by the use of the spectroscope which has enabled us to calculate the temperature of gaseous stars, and [243] specify the chemical elements comprised in them. Thus the composition of far-distant suns has been reduced to problems to be settled in our laboratories. The scientific world, however, in admitting the validity of the theory of universal gravity had implicitly accepted the principle that the reign of law, as investigated on the earth, extends throughout the universe. Thus the daring which permits us, living on a medium-sized planet attached to one of the smaller suns, to analyse the universe is, I venture to say, the direct outcome of the genius of Newton as displayed in his Principia.