A Short History of Astronomy (1898)/Chapter 9

by Arthur Berry

CHAPTER IX.

 

UNIVERSAL GRAVITATION.

 

"Nature and Nature's laws lay hid in night;
God said 'Let Newton be!' and all was light."
Pope.

164. Newton's life may be conveniently divided into three portions. First came 22 years (1643–1665) of boyhood and undergraduate life; then followed his great productive period, of almost exactly the same length, culminating in the publication of the Principia in 1687; while the rest of his life (1687–1727), which lasted nearly as long as the other two periods put together, was largely occupied with official work and studies of a non-scientific character, and was marked by no discoveries ranking with those made in his middle period, though some of his earlier work received important developments and several new results of decided interest were obtained.

165. Isaac Newton was born at Woolsthorpe, near Grantham, in Lincolnshire, on January 4th, 1643;^[1] this was very nearly a year after the death of Galilei, and a few months after the beginning of our Civil Wars. His taste for study does not appear to have developed very early in life, but ultimately became so marked that, after some unsuccessful attempts to turn him into a farmer, he was entered at Trinity College, Cambridge, in 1661.

Although probably at first rather more backward than most undergraduates, he made extremely rapid progress in mathematics and allied subjects, and evidently gave his teachers some trouble by the rapidity with which he absorbed what little they knew. He met with Euclid's Elements of Geometry for the first time while an undergraduate, but is reported to have soon abandoned it as being "a trifling book," in favour of more advanced reading. In January 1665 he graduated in the ordinary course as Bachelor of Arts.

166. The external events of Newton's life during the next 22 years may be very briefly dismissed. He was elected a Fellow in 1667, became M.A. in due course in the following year, and was appointed Lucasian Professor of Mathematics, in succession to his friend Isaac Barrow, in 1669. Three years later he was elected a Fellow of the recently founded Royal Society. With the exception of some visits to his Lincolnshire home, he appears to have spent almost the whole period in quiet study at Cambridge, and the history of his life is almost exclusively the history of his successive discoveries.

167. His scientific work falls into three main groups, astronomy (including dynamics), optics, and pure mathematics. He also spent a good deal of time on experimental work in chemistry, as well as on heat and other branches of physics, and in the latter half of his life devoted much attention to questions of chronology and theology; in none of these subjects, however, did he produce results of much importance.

168. In forming an estimate of Newton's genius it is of course important to bear in mind the range of subjects with which he dealt; from our present point of view, however, his mathematics only presents itself as a tool to be used in astronomical work; and only those of his optical discoveries which are of astronomical importance need be mentioned here. In 1668 he constructed a reflecting telescope, that is, a telescope in which the rays of light from the object viewed are concentrated by means of a curved mirror instead of by a lens, as in the refracting telescopes of Galilei and Kepler. Telescopes on this principle, differing however in some important particulars from Newton's, had already been described in 1663 by James Gregory (1638–1675), with whose ideas Newton was acquainted, but it does not appear that Gregory had actually made an instrument. Owing to mechanical difficulties in construction, half a century elapsed before reflecting telescopes were made which could compete with the best refractors of the time, and no important astronomical discoveries were made with them before the time of William Herschel (chapter xii.), more than a century after the original invention.

Newton's discovery of the effect of a prism in resolving a beam of white light into different colours is in a sense the basis of the method of spectrum analysis (chapter xiii., § 299), to which so many astronomical discoveries of the last 40 years are due.

169. The ideas by which Newton is best known in each of his three great subjects gravitation, his theory of colours, and fluxions seem to have occurred to him and to have been partly thought out within less than two years after he took his degree, that is before he was 24. His own account—written many years afterwards—gives a vivid picture of his extraordinary mental activity at this time:—

"In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their orbs I deduced that the forces which keep the Planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since."^[2]

170. He spent a considerable part of this time (1665-1666) at Woolsthorpe, on account of the prevalence of the plague.

The well-known story, that he was set meditating on gravity by the fall of an apple in the orchard, is based on good authority, and is perfectly credible in the sense that the apple may have reminded him at that particular time of certain problems connected with gravity. That the apple seriously suggested to him the existence of the problems or any key to their solution is wildly improbable.

Several astronomers had already speculated on the "cause" of the known motions of the planets and satellites; that is they had attempted to exhibit these motions as consequences of some more fundamental and more general laws. Kepler, as we have seen (chapter vii., § 150), had pointed out that the motions in question should not be considered as due to the influence of mere geometrical points, such as the centres of the old epicycles, but to that of other bodies; and in particular made some attempt to explain the motion of the planets as due to a special kind of influence emanating from the sun. He went, however, entirely wrong by looking for a force to keep up the motion of the planets and as it were push them along. Galilei's discovery that the motion of a body goes on indefinitely unless there is some cause at work to alter or stop it, at once put a new aspect on this as on other mechanical problems; but he himself did not develop his idea in this particular direction. Giovanni Alfonso Borelli (1608-1679), in a book on Jupiter's satellites published in 1666, and therefore about the time of Newton's first work on the subject, pointed out that a body revolving in a circle (or similar curve) had a tendency to recede from the centre, and that in the case of the planets this might be supposed to be counteracted by some kind of attraction towards the sun. We have then here the idea— in a very indistinct form certainly—that the motion of a planet is to be explained, not by a force acting in the direction in which it is moving, but by a force directed towards the sun, that is about at right angles to the direction of the planet's motion. Huygens carried this idea much further—though without special reference to astronomy—and obtained (chapter viii., § 158) a numerical measure for the tendency of a body moving in a circle to recede from the centre, a tendency which had in some way to be counteracted if the body was not to fly away. Huygens published his work in 1673, some years after Newton had obtained his corresponding result, but before he had published anything; and there can be no doubt that the two men worked quite independently.

171. Viewed as a purely general question, apart from its astronomical applications, the problem may be said to be to examine under what conditions a body can revolve with uniform speed in a circle.

Fig. 70.—Motion in a circle.

Let a represent the position at a certain instant of a body which is revolving with uniform speed in a circle of centre o. Then at this instant the body is moving in the direction of the tangent a a to the circle. Consequently by Galilei's First Law (chapter vi., §§ 130, 133), if left to Itself and uninfluenced by any other body, it would continue to move with the same speed and in the same direction, i.e. along the line a a, and consequently would be found after some time at such a point as a. But actually it is found to be at b on the circle. Hence some influence must have been at work to bring it to b instead of to a. But b is nearer to the centre of the circle than a is; hence some influence must be at work tending constantly to draw the body towards o, or counteracting the tendency which it has, in virtue of the First Law of Motion, to get farther and farther away from o. To express either of these tendencies numerically we want a more complex idea than that of velocity or rate of motion, namely acceleration or rate of change of velocity, an idea which Galilei added to science in his discussion of the law of falling bodies (chapter vi., §§ 116, § 133). A falling body, for example, is moving after one second with the velocity of about 32 feet per second, after two seconds with the velocity of 64, after three seconds with the velocity of 96, and so on; thus in every second it gains a downward velocity of 32 feet per second; and this may be expressed otherwise by saying that the body has a downward acceleration of 32 feet per second per second. A further investigation of the motion in a circle shews that the motion is completely explained if the moving body has, in addition to its original velocity, an acceleration of a certain magnitude directed towards the centre of the circle. It can be shewn further that the acceleration may be numerically expressed by taking the square of the velocity of the moving body (expressed, say, in feet per second), and dividing this by the radius of the circle in feet. If, for example, the body is moving in a circle having a radius of four feet, at the rate of ten feet a second, then the acceleration towards the centre is $\left({\frac {10\times 10}{4}}=\right)$ 25 feet per second per second.

These results, with others of a similar character, were first published by Huygens—not of course precisely in this form—in his book on the Pendulum Clock (chapter viii., § 158); and discovered independently by Newton in 1666.

If then a body is seen to move in a circle, its motion becomes intelligible if some other body can be discovered which produces this acceleration. In a common case, such as when a stone is tied to a string and whirled round, this acceleration is produced by the string which pulls the stone; in a spinning-top the acceleration of the outer parts is produced by the forces binding them on to the inner part, and so on.

172. In the most important cases of this kind which occur in astronomy, a planet is known to revolve round the sun in a path which does not differ much from a circle. If we assume for the present that the path is actually a circle, the planet must have an acceleration towards the centre, and it is possible to attribute this to the influence of the central body, the sun. In this way arises the idea of attributing to the sun the power of influencing in some way a planet which revolves round it, so as to give it an acceleration towards the sun; and the question at once arises of how this "influence" differs at different distances. To answer this question Newton made use of Kepler's Third Law (chapter vii., § 144). We have seen that, according to this law, the squares of the times of revolution of any two planets are proportional to the cubes of their distances from the sun; but the velocity of the planet may be found by dividing the length of the path it travels in its revolution round the sun by the time of the revolution, and this length is again proportional to the distance of the planet from the sun. Hence the velocities of the two planets are proportional to their distances from the sun, divided by the times of revolution, and consequently the squares of the velocities are proportional to the squares of the distances from the sun divided by the squares of the times of revolution. Hence, by Kepler's law, the squares of the velocities are proportional to the squares of the distances divided by the cubes of the distances, that is the squares of the velocities are inversely proportional to the distances, the more distant planet having the less velocity and vice versa. Now by the formula of Huygens the acceleration is measured by the square of the velocity divided by the radius of the circle (which in this case is the distance of the planet from the sun). The accelerations of the two planets towards the sun are therefore inversely proportional to the distances each multiplied by itself, that is are inversely proportional to the squares of the distances. Newton's first result therefore is: that the motions of the planets—regarded as moving in circles, and in strict accordance with Kepler's Third Law—can be explained as due to the action of the sun, if the sun is supposed capable of producing on a planet an acceleration towards the sun itself which is proportional to the inverse square of its distance from the sun; i.e. at twice the distance it is 1/4 as great, at three times the distance 1/9 as great, at ten times the distance 1/100 as great, and so on.

The argument may perhaps be made clearer by a numerical example. In round numbers Jupiter's distance from the sun is five times as great as that of the earth, and Jupiter takes 12 years to perform a revolution round the sun, whereas the earth takes one. Hence Jupiter goes in 12 years five times as far as the earth goes in one, and Jupiter's velocity is therefore about 5/12 that of the earth's, or the two velocities are in the ratio of 5 to 12; the squares of the velocities are therefore as 5 × 5 to 12 × 12, or as 25 to 144. The accelerations of Jupiter and of the earth towards the sun are therefore as 25 ÷ 5 to 144, or as 5 to 144; hence Jupiter's acceleration towards the sun is about 1/28 that of the earth, and if we had taken more accurate figures this fraction would have come out more nearly 1/25. Hence at five times the distance the acceleration is 25 times less.

This law of the inverse square, as it may be called, is also the law according to which the light emitted from the sun or any other bright body varies, and would on this account also be not unlikely to suggest itself in connection with any kind of influence emitted from the sun.

173. The next step in Newton's investigation was to see whether the motion of the moon round the earth could be explained in some similar way. By the same argument as before, the moon could be shewn to have an acceleration towards the earth. Now a stone if let drop falls downwards, that is in the direction of the centre of the earth, and, as Galilei had shewn (chapter vi., § 133), this motion is one of uniform acceleration; if, in accordance with the opinion generally held at that time, the motion is regarded as being due to the earth, we may say that the earth has the power of giving an acceleration towards its own centre to bodies near its surface. Newton noticed that this power extended at any rate to the tops of mountains, and it occurred to him that it might possibly extend as far as the moon and so give rise to the required acceleration. Although, however, the acceleration of falling bodies, as far as was known at the time, was the same for terrestrial bodies wherever situated, it was probable that at such a distance as that of the moon the acceleration caused by the earth would be much less. Newton assumed as a working hypothesis that the acceleration diminished according to the same law which he had previously arrived at in the case of the sun's action on the planets, that is that the acceleration produced by the earth on any body is inversely proportional to the square of the distance of the body from the centre of the earth.

It may be noticed that a difficulty arises here which did not present itself in the corresponding case of the planets. The distances of the planets from the sun being large compared with the size of the sun, it makes little difference whether the planetary distances are measured from the centre of the sun or from any other point in it. The same is true of the moon and earth; but when we are comparing the action of the earth on the moon with that on a stone situated on or near the ground, it is clearly of the utmost importance to decide whether the distance of the stone is to be measured from the nearest point of the earth, a few feet off, from the centre of the earth, 4000 miles off, or from some other point. Provisionally at any rate Newton decided on measuring from the centre of the earth.

It remained to verify his conjecture in the case of the moon by a numerical calculation; this could easily be done if certain things were known, viz. the acceleration of a falling body on the earth, the distance of the surface of the earth from its centre, the distance of the moon, and the time taken by the moon to perform, a revolution round the earth. The first of these was possibly known with fair accuracy; the last was well known; and it was also known that the moon's distance was about 60 times the radius of the earth. How accurately Newton at this time knew the size of the earth is uncertain. Taking moderately accurate figures, the calculation is easily performed. In a month of about 27 days the moon moves about 60 times as far as the distance round the earth; that is she moves about 60 × 24,000 miles in 27 days, which is equivalent to about 3,300 feet per second. The acceleration of the moon is therefore measured by the square of this, divided by the distance of the moon (which is 60 times the radius of the earth, or 20,000,000 feet); that is, it is 3,300 × 3,300/60 × 20,000,000, which reduces to about 1/110. Consequently, if the law of the inverse square holds, the acceleration of a falling body at the surface of the earth, which is 60 times nearer to the centre than the moon is, should be 60 × 60/110, or between 32 and 33; but the actual acceleration of falling bodies is rather more than 32. The argument is therefore satisfactory, and Newton's hypothesis is so far verified.

The analogy thus indicated between the motion of the moon round the earth and the motion of a falling stone may be illustrated by a comparison, due to Newton, of the moon to a bullet shot horizontally out of a gun from a high place on the earth. Let the bullet start from b in fig. 71, then moving at first horizontally it will describe a curved path and reach the ground at a point such as c, at some distance from the point a, vertically underneath its starting-point. If it were shot out with a greater velocity, its path at first would be flatter and it would reach the ground at a point c' beyond c; if the velocity were greater still, it would reach the ground at c" or at c'"; and it requires only a slight effort of the imagination to conceive that, with a still greater velocity to begin with, it would miss the earth altogether and describe a circuit round it, such as b d e. This is exactly what the moon does, the only difference being that the moon is at a much greater distance than we have supposed the bullet to be, and that her motion has not been produced by anything analogous to the gun; but the motion being once there it is immaterial how it was produced or whether it was ever produced in the past. We may in fact say of the moon "that she is a falling body, only she is going so fast and is so far off that she falls quite round to the other side of the earth, instead of hitting it; and so goes on for ever."^[3]

In the memorandum already quoted (§ 169) Newton speaks of the hypothesis as fitting the facts "pretty nearly"; but in a letter of earlier date (June 20th, 1686) he refers to the calculation as not having been made accurately enough. It is probable that he used a seriously inaccurate value of the size of the earth, having overlooked the measurements of Snell and Norwood (chapter viii., § 159); it is known that even at a later stage he was unable

Fig. 71.—The moon as a projectile.

to deal satisfactorily with the difficulty above mentioned, as to whether the earth might for the purposes of the problem be identified with its centre; and he was of course aware that the moon's path differed considerably from a circle. The view, said to have been derived from Newton's conversation many years afterwards, that he was so dissatisfied with his results, as to regard his hypothesis as substantially defective, is possible, but by no means certain; whatever the cause may have been, he laid the subject aside for some years without publishing anything on it, and devoted himself chiefly to optics and mathematics.

174. Meanwhile the problem of the planetary motions was one of the numerous subjects of discussion among the remarkable group of men who were the leading spirits of the Royal Society, founded in 1662. Robert Hooke (1635–1703), who claimed credit for most of the scientific discoveries of the time, suggested with some distinctness, not later than 1674, that the motions of the planets might be accounted for by attraction between them and the sun, and referred also to the possibility of the earth's attraction on bodies varying according to the law of the inverse square. Christopher Wren (1632–1723), better known as an architect than as a man of science, discussed some questions of this sort with Newton in 1677, and appears also to have thought of a law of attraction of this kind. A letter of Hooke's to Newton, written at the end of 1679, dealing amongst other things with the curve which a falling body would describe, the rotation of the earth being taken into account, stimulated Newton, who professed that at this time his "affection to philosophy" was "worn out," to go on with his study of the celestial motions. Picard's more accurate measurement of the earth (chapter viii., § 159) was now well known, and Newton repeated his former calculation of the moon's motion, using Picard's improved measurement, and found the result more satisfactory than before.

175. At the same time (1679) Newton made a further discovery of the utmost importance by overcoming some of the difficulties connected with motion in a path other than a circle.

He shewed that if a body moved round a central body, in such a way that the line joining the two bodies sweeps out equal areas in equal times, as in Kepler's Second Law of planetary motion (chapter vii., § 141), then the moving body is acted on by an attraction directed exactly towards the central body; and further that if the path is an ellipse, with the central body in one focus, as in Kepler's First Law of planetary motion, then this attraction must vary in different parts of the path as the inverse square of the distance between the two bodies. Kepler's laws of planetary motion were in fact shewn to lead necessarily to the conclusions that the sun exerts on a planet an attraction inversely proportional to the square of the distance of the planet from the sun, and that such an attraction affords a sufficient explanation of the motion of the planet.

Once more, however, Newton published nothing and "threw his calculations by, being upon other studies."

176. Nearly five years later the matter was again brought to his notice, on this occasion by Edmund Halley (chapter x., §§ 199–205), whose friendship played henceforward an important part in Newton's life, and whose unselfish devotion to the great astronomer forms a pleasant contrast to the quarrels and jealousies prevalent at that time between so many scientific men. Halley, not knowing of Newton's work in 1666, rediscovered, early in 1684, the law of the inverse square, as a consequence of Kepler's Third Law, and shortly afterwards discussed with Wren and Hooke what was the curve in which a body would move if acted on by an attraction varying according to this law; but none of them could answer the question.^[4] Later in the year Halley visited Newton at Cambridge and learnt from him the answer. Newton had, characteristically enough, lost his previous calculation, but was able to work it out again and sent it to Halley a few months afterwards. This time fortunately his attention was not diverted to other topics; he worked out at once a number of other problems of motion, and devoted his usual autumn course of University lectures to the subject. Perhaps the most interesting of the new results was that Kepler's Third Law, from which the law of the inverse square had been deduced in 1666, only on the supposition that the planets moved in circles, was equally consistent with Newton's law when the paths of the planets were taken to be ellipses.

177. At the end of the year 1684 Halley went to Cambridge again and urged Newton to publish his results. In accordance with this request Newton wrote out, and sent to the Royal Society, a tract called Propositiones de Motu, the 11 propositions of which contained the results already mentioned and some others relating to the motion of bodies under attraction to a centre. Although the pro- positions were given in an abstract form, it was pointed out that certain of them applied to the case of the planets. Further pressure from Halley persuaded Newton to give his results a more permanent form by embodying them in a larger book. As might have been expected, the subject grew under his hands, and the great treatise which resulted contained an immense quantity of material not contained in the De Motu. By the middle of 1686 the rough draft was finished, and some of it was ready for press. Halley not only undertook to pay the expenses, but superintended the printing and helped Newton to collect the astronomical data which were necessary. After some delay in the press, the book finally appeared early in July 1687, under the title Philosophiae Naturalis Principia Mathematica.

178. The Principia, as it is commonly called, consists of three books in addition to introductory matter: the first book deals generally with problems of the motion of bodies, solved for the most part in an abstract form without special reference to astronomy; the second book deals with the motion of bodies through media which resist their motion, such as ordinary fluids, and is of comparatively small astronomical importance, except that in it some glaring inconsistencies in the Cartesian theory of vortices are pointed out; the third book applies to the circumstances of the actual solar system the results already obtained, and is in fact an explanation of the motions of the celestial bodies on Newton's mechanical principles.

179. The introductory portion, consisting of "Definitions" and "Axioms, or Laws of Motion," forms a very notable contribution to dynamics, being in fact the first coherent statement of the fundamental laws according to which the motions of bodies are produced or changed. Newton himself does not appear to have regarded this part of his book as of very great importance, and the chief results embodied in it, being overshadowed as it were by the more striking discoveries in other parts of the book, attracted comparatively little attention. Much of it must be passed over here, but certain results of special astronomical importance require to be mentioned.

Galilei, as we have seen (chapter vi., §§ 130, 133), was the first to enunciate the law that a body when once in motion continues to move in the same direction and at the same speed unless some cause is at work to make it change its motion. This law is given by Newton in the form already quoted in § 130, as the first of three fundamental laws, and is now commonly known as the First Law of Motion.

Galilei also discovered that a falling body moves with continually changing velocity, but with a uniform acceleration (chapter vi., § 133), and that this acceleration is the same for all bodies (chapter vi., § 116). The tendency of a body to fall having been generally recognised as due to the earth, Galilei's discovery involved the recognition that One effect of one body on another may be an acceleration produced in its motion. Newton extended this idea by shewing that the earth produced an acceleration in the motion of the moon, and the sun in the motion of the planets, and was led to the general idea of acceleration in a body's motion, which might be due in a variety of ways to the action of other bodies, and which could conveniently be taken as a measure of the effect produced by one body on another.

180. To these ideas Newton added the very important and difficult conception of mass.

If we are comparing two different bodies of the same material but of different sizes, we are accustomed to think of the larger one as heavier than the other. In the same way we readily think of a ball of lead as being heavier than a ball of wood of the same size. The most prominent idea connected with "heaviness" and "lightness" is that of the muscular effort required to support or to lift the body in question; a greater effort, for example, is required to hold the leaden ball than the wooden one. Again, the leaden ball if supported by an elastic string stretches it farther than does the wooden ball; or again, if they are placed in the scales of a balance, the lead sinks and the wood rises. All these effects we attribute to the "weight" of the two bodies, and the weight we are mostly accustomed to attribute in some way to the action of the earth on the bodies. The ordinary process of weighing a body in a balance shews, further, that we are accustomed to think of weight as a measurable quantity. On the other hand, we know from Galilei's result, which Newton tested very carefully by a series of pendulum experiments, that the leaden and the wooden ball, if allowed to drop, fall with the same acceleration. If therefore we measure the effect which the earth produces on the two balls by their acceleration, then the earth affects them equally; but if we measure it by the power which they have of stretching strings, or by the power which one has of supporting the other in a balance, then the effect which the earth produces on the leaden ball is greater than that produced on the wooden ball. Taken in this way, the action of the earth on either ball may be spoken of as weight, and the weight of a body can be measured by comparing it in a balance with standard bodies.

The difference between two such bodies as the leaden and wooden ball may, however, be recognised in quite a different way. We can easily see, for example, that a greater effort is needed to set the one in motion than the other; or that if each is tied to the end of a string of given kind and whirled round at a given rate, the one string is more tightly stretched than the other. In these cases the attraction of the earth is of no importance, and we recognise a distinction between the two bodies which is independent of the attraction of the earth. This distinction Newton regarded as due to a difference in the quantity of matter or material in the two bodies, and to this quantity he gave the name of mass. It may fairly be doubted whether anything is gained by this particular definition of mass, but the really important step was the distinct recognition of mass as a property of bodies, of fundamental importance in dynamical questions, and capable of measurement.

Newton, developing Galilei's idea, gave as one measurement of the action exerted by one body on another the product of the mass by the acceleration produced—a quantity for which he used different names, now replaced by force. The weight of a body was thus identified with the force exerted on it by the earth. Since the earth produces the same acceleration in all bodies at the same place, it follows that the masses of bodies at the same place are proportional to their weights; thus if two bodies are compared at the same place, and the weight of one (as shewn, for example, by a pair of scales) is found to be ten times that of the other, then its mass is also ten times as great. But such experiments as those of Richer at Cayenne (chapter viii., § 161) shewed that the acceleration of falling bodies was less at the equator than in higher latitudes; so that if a body is carried from London or Paris to Cayenne, its weight is altered but its mass remains the same as before. Newton's conception of the earth's gravitation as extending as far as the moon gave further importance to the distinction between mass and weight; for if a body were removed from the earth to the moon, then its mass would be unchanged, but the acceleration due to the earth's attraction would be 60 × 60 times less, and its weight diminished in the same proportion.

Rules are also given for the effect produced on a body's motion by the simultaneous action of two or more forces.^[5]

A further principle of great importance, of which only very indistinct traces are to be found before Newton's time, was given by him as the Third Law of Motion in the form: "To every action there is always an equal and contrary reaction; or, the mutual actions of any two bodies are always equal and oppositely directed." Here action and reaction are to be interpreted primarily in the sense of force. If a stone rests on the hand, the force with which the stone presses the hand downwards is equal to that with which the hand presses the stone upwards; if the earth attracts a stone downwards with a certain force, then the stone attracts the earth upwards with the same force, and so on. It is to be carefully noted that if, as in the last example, two bodies are acting on one another, the accelerations produced are not the same, but since force is measured by the product of mass and acceleration, the body with the larger mass receives the lesser acceleration. In the case of a stone and the earth, the mass of the latter being enormously greater,^[6] its acceleration is enormously less than that of the stone, and is therefore (In accordance with our experience) quite insensible.

181. When Newton began to write the Principia he had probably satisfied himself (§ 173) that the attracting power of the earth extended as far as the moon, and that the acceleration thereby produced in any body—whether the moon, or whether a body close to the earth—is inversely proportional to the square of the distance from the centre of the earth. With the ideas of force and mass this result may be stated in the form: the earth attracts any body with a force inversely proportional to the square of the distance from the earths centre, and also proportional to the mass of the body.

In the same way Newton had established that the motions of the planets could be explained by an attraction towards the sun producing an acceleration inversely proportional to the square of the distance from the sun's centre, not only in the same planet in different parts of its path, but also in different planets. Again, it follows from this that the sun attracts any planet with a force inversely proportional to the square of the distance of the planet from the sun's centre, and also proportional to the mass of the planet.

But by the Third Law of Motion a body experiencing an attraction towards the earth must in turn exert an equal attraction on the earth; similarly a body experiencing an attraction towards the sun must exert an equal attraction on the sun. If, for example, the mass of Venus is seven times that of Mars, then the force with which the sun attracts Venus is seven times as great as that with which it would attract Mars if placed at the same distance; and therefore also the force with which Venus attracts the sun is seven times as great as that with which Mars would attract the sun if at an equal distance from it. Hence, in all the cases of attraction hitherto considered and in which the comparison is possible, the force is proportional not only to the mass of the attracted body, but also to that of the attracting body, as well as being inversely proportional to the square of the distance. Gravitation thus appears no longer as a property peculiar to the central body of a revolving system, but as belonging to a planet in just the same way as to the sun, and to the moon or to a stone in just the same way as to the earth.

Moreover, the fact that separate bodies on the surface of the earth are attracted by the earth, and therefore in turn attract it, suggests that this power of attracting other bodies which the celestial bodies are shewn to possess does not belong to each celestial body as a whole, but to the separate particles making it up, so that, for example, the force with which Jupiter and the sun mutually attract one another is the result of compounding the forces with which the separate particles making up Jupiter attract the separate particles making up the sun. Thus is suggested finally the law of gravitation in its most general form: every particle of matter attracts every other particle with a force proportional to the mass of each, and inversely proportional to the square of the distance between them.^[7]

182. In all the astronomical cases already referred to the attractions between the various celestial bodies have been treated as if they were accurately directed towards their centres, and the distance between the bodies has been taken to be the distance between their centres. Newton's doubts on this point, in the case of the earth's attraction of bodies, have been already referred to (§ 173); but early in 1685 he succeeded in justifying this assumption. By a singularly beautiful and simple course of reasoning he shewed (Principia, Book I., propositions 70, 71) that, if a body is spherical in form and equally dense throughout, it attracts any particle external to it exactly as if its whole mass were concentrated at its centre. He shewed, further, that the same is true for a sphere of variable density, provided it can be regarded as made up of a series of spherical shells, having a common centre, each of uniform density throughout, different shells being, however, of different densities. For example, the result is true for a hollow indiarubber ball as well as for a solid one, but is not true for a sphere made up of a hemisphere of wood and a hemisphere of iron fastened together.

183. The law of gravitation being thus provisionally established, the great task which lay before Newton, and to which he devotes the greater part of the first and third books of the Principia, was that of deducing from it and the "laws of motion" the motions of the various members of the solar system, and of shewing, if possible, that the motions so calculated agreed with those observed. If this were successfully done, it would afford a verification of the most delicate and rigorous character of Newton's principles.

The conception of the solar system as a mechanism, each member of which influences the motion of every other member in accordance with one universal law of attraction, although extremely simple in itself, is easily seen to give rise to very serious difficulties when it is proposed actually to calculate the various motions. If in dealing with the motion of a planet such as Mars it were possible to regard Mars as acted on only by the attraction of the sun, and to ignore the effects of the other planets, then the problem would be completely solved by the propositions which Newton established in 1679 (§ 175) and by their means the position of Mars at any time could be calculated with any required degree of accuracy. But in the case which actually exists the motion of Mars is affected by the forces with which all the other planets (as well as the satellites) attract it, and these forces in turn depend on the position of Mars (as well as upon that of the other planets) and hence upon the motion of Mars. A problem of this kind in all its generality is quite beyond the powers of any existing mathematical methods. Fortunately, however, the mass of even the largest of the planets is so very much less than that of the sun, that the motion of any one planet is only slightly affected by the others; and it may be regarded as moving very nearly as it would move if the other planets did not exist, the effect of these being afterwards allowed for as producing disturbances or perturbations in its path. Although even in this simplified form the problem of the motion of the planets is one of extreme difficulty (cf. chapter xi., § 228), and Newton was unable to solve it with anything like completeness, yet he was able to point out certain general effects which must result from the mutual action of the planets, the most interesting being the slow forward motion of the apses of the earth's orbit, which had long ago been noticed by observing astronomers (chapter iii., § 59). Newton also pointed out that Jupiter, on account of its great mass, must produce a considerable perturbation in the motion of its neighbour Saturn, and thus gave some explanation of an irregularity first noted by Horrocks (chapter viii., § 156).

184. The motion of the moon presents special difficulties, but Newton, who was evidently much interested in the problems of lunar theory, succeeded in overcoming them much more completely than the corresponding ones connected with the planets.

The moon's motion round the earth is primarily due to the attraction of the earth; the perturbations due to the other planets are insignificant; but the sun, which though at a very great distance has an enormously greater mass than the earth, produces a very sensible disturbing effect on the moon's motion. Certain irregularities, as we have seen (chapter ii., §§ 40, 48; chapter v., § 111), had already been discovered by observation. Newton was able to shew that the disturbing action of the sun would necessarily produce perturbations of the same general character as those thus recognised, and in the case of the motion of the moon's nodes and of her apogee he was able to get a very fairly accurate numerical result;^[8] and he also discovered a number of other irregularities, for the most part very small, which had not hitherto been noticed. He indicated also the existence of certain irregularities in the motions of Jupiter's and Saturn's moons analogous to those which occur in the case of our moon.

185. One group of results of an entirely novel character resulted from Newton's theory of gravitation. It became for the first time possible to estimate the masses of some of the celestial bodies, by comparing the attractions exerted by them on other bodies with that exerted by the earth.

The case of Jupiter may be given as an illustration. The time of revolution of Jupiter's outermost satellite is known to be about 16 days 16 hours, and its distance from, Jupiter was estimated by Newton (not very correctly) at about four times the distance of the moon from the earth. A calculation exactly like that of § 172 or § 173 shews that the acceleration of the satellite due to Jupiter's attraction is about ten times as great as the acceleration of the moon towards the earth, and that therefore, the distance being four times as great, Jupiter attracts a body with a force 10 × 4 × 4 times as great as that with which the earth attracts a body at the same distance; consequently Jupiter's mass is 160 times that of the earth. This process of reasoning applies also to Saturn, and in a very similar way a comparison of the motion of a planet, Venus for example, round the sun with the motion of the moon round the earth gives a relation between the masses of the sun and earth. In this way Newton found the mass of the sun to be 1067, 3021, and 169282 times greater than that of Jupiter, Saturn, and the earth, respectively. The corresponding figures now accepted are not far from 1047, 3530, 324439. The large error in the last number is due to the use of an erroneous value of the distance of the sun—then not at all accurately known—upon which depend the other distances in the solar system, except those connected with the earth-moon system. As it was necessary for the employment of this method to be able to observe the motion of some other body attracted by the planet in question, it could not be applied to the other three planets (Mars, Venus, and Mercury), of which no satellites were known.

186. From the equality of action and reaction it follows that, since the sun attracts the planets, they also attract the sun, and the sun consequently is in motion, though—owing to the comparative smallness of the planets—only to a very small extent. It follows that Kepler's Third Law is not strictly accurate, deviations from it becoming sensible in the case of the large planets Jupiter and Saturn (cf. chapter vii., § 144). It was, however, proved by Newton that in any system of bodies, such as the solar system, moving about in any way under the influence of their mutual attractions, there is a particular point, called the centre of gravity, which can always be treated as at rest; the sun moves relatively to this point, but so little that the distance between the centre of the sun and the centre of gravity can never be much more than the diameter of the sun.

It is perhaps rather curious that this result was not seized upon by some of the supporters of the Church in the condemnation of Galilei, now rather more than half a century old; for if it was far from supporting the view that the earth is at the centre of the world, it at any rate negatived that part of the doctrine of Coppernicus and Galilei which asserted the sun to be at rest in the centre of the world. Probably no one who was capable of understanding Newton's book was a serious supporter of any anti-Coppernican system, though some still professed themselves obedient to the papal decrees on the subject.^[9]

187. The variation of the time of oscillation of a pendulum in different parts of the earth, discovered by Richer in 1672 (chapter viii., § 161), indicated that the earth was probably not a sphere. Newton pointed out that this departure from the spherical form was a consequence of the mutual gravitation of the particles making up the earth and of the earth's rotation. He supposed a canal of water to pass from the pole to the centre of the earth, and then from the centre to a point on the equator (b o a a in fig. 72), and then found the condition that these two columns of water o b, o a, each being attracted towards the centre of the earth, should balance. This method involved certain assumptions as to the inside of the earth, of which little can be said to be known even now, and consequently, though Newton's general result, that the earth is flattened at the poles and bulges out at the equator, was right, the actual numerical expression which he found was not very accurate. If, in the figure, the dotted line is a circle the radius of which is equal to the distance of the

pole b from the centre of the earth o, then the actual surface of the earth extends at the equator beyond this circle as far as a, where, according to Newton, a a is about 1/230 of o b or o a, and according to modern estimates, based on actual measurement of the earth as well as upon theory (chapter x., § 221), it is about 1/293 of o a. Both Newton's fraction and the modern one are so small that the resulting flattening cannot be made sensible in a figure; in fig. 72

Fig. 72.—The spheroidal form of the earth.

the length a a is made, for the sake of distinctness, nearly 30 times as great as it should be.

Newton discovered also in a similar way the flattening of Jupiter, which, owing to its more rapid rotation, is considerably more flattened than the earth; this was also detected telescopically by Domenico Cassini four years after the publication of the Principia.

188. The discovery of the form of the earth led to an explanation of the precession of the equinoxes, a phenomenon which had been discovered 1,800 years before (chapter ii., § 42), but had remained a complete mystery ever since.

If the earth is a perfect sphere, then its attraction on any other body is exactly the same as if its mass were all concentrated at its centre (§ 182), and so also the attraction on it of any other body such as the sun or moon is equivalent to a single force passing through the centre o of the earth; but this is no longer true if the earth is not spherical. In fact the action of the sun or moon on the spherical part of the earth, inside the dotted circle in fig. 72, is equivalent to a force through o, and has no tendency to turn the earth in any way about its centre; but the attraction on the remaining portion is of a different character, and Newton shewed that from it resulted a motion of the axis of the earth of the same general character as precession. The amount of the precession as calculated by Newton did as a matter of fact agree pretty closely with the observed amount, but this was due to the accidental compensation of two errors, arising from his imperfect knowledge of the form and construction of the earth, as well as from erroneous estimates of the distance of the sun and of the mass of the moon, neither of which quantities Newton was able to measure with any accuracy.^[10] It was further pointed out that the motion in question was necessarily not quite uniform, but that, owing to the unequal effects of the sun in different positions, the earth's axis would oscillate to and fro every six months, though to a very minute extent.

189. Newton also gave a general explanation of the tides as due to the disturbing action of the moon and sun, the former being the more important. If the earth be regarded as made of a solid spherical nucleus, covered by the ocean, then the moon attracts different parts unequally, and in particular the attraction, measured by the acceleration produced, on the water nearest to the moon is greater than that on the solid earth, and that on the water farthest from the moon is less. Consequently the water moves on the surface of the earth, the general character of the motion being the same as if the portion of the ocean on the side towards the moon were attracted and that on the opposite side repelled. Owing to the rotation of the earth and the moon's motion, the moon returns to nearly the same position with respect to any place on the earth in a period which exceeds a day by (on the average) about 50 minutes, and consequently Newton's argument shewed that low tides (or high tides) due to the moon would follow one another at any given place at intervals equal to about half this period; or, in other words, that two tides would in general occur daily, but that on each day any particular phase of the tides would occur on the average about 50 minutes later than on the preceding day, a result agreeing with observation. Similar but smaller tides were shewn by the same argument to arise from the action of the sun, and the actual tide to be due to the combination of the two. It was shewn that at new and full moon the lunar and solar tides would be added together, whereas at the half moon they would tend to counteract one another, so that the observed fact of greater tides every fortnight received an explanation. A number of other peculiarities of the tides were also shewn to result from the same principles.

Newton ingeniously used observations of the height of the tide when the sun and moon acted together and when they acted in opposite ways to compare the tide-raising powers of the sun and moon, and hence to estimate the mass of the moon in terms of that of the sun, and consequently in terms of that of the earth (§ 185). The resulting mass of the moon was about twice what it ought to be according to modern knowledge, but as before Newton's time no one knew of any method of measuring the moon's mass even in the roughest way, and this result had to be disentangled from the innumerable complications connected with both the theory and with observation of the tides, it cannot but be regarded as a remarkable achievement. Newton's theory of the tides was based on certain hypotheses which had to be made in order to render the problem at all manageable, but which were, certainly not true, and consequently, as he was well aware, important modifications would necessarily have to be made, in order to bring his results into agreement with actual facts. The mere presence of land not covered by water is, for example, sufficient by itself to produce important alterations in tidal effects at different places. Thus Newton's theory was by no means equal to such a task as that of predicting the times of high tide at any required place, or the height of any required tide, though it gave a satisfactory explanation of many of the general characteristics of tides.

190. As we have seen (chapter v., § 103; chapter vii., § 146), comets until quite recently had been commonly regarded as terrestrial objects produced in the higher regions of our atmosphere, and even the more enlightened astronomers who, like Tycho, Kepler, and Galilei, recognised them as belonging to the celestial bodies, were unable to give an explanation of their motions and of their apparently quite irregular appearances and disappearances. Newton was led to consider whether a comet's motion could not be explained, like that of a planet, by gravitation towards the sun. If so then, as he had proved near the beginning of the Principia, its path must be either an ellipse or one of two other allied curves, the parabola and hyperbola. If a comet moved in an ellipse which only differed slightly from a circle, then it would never recede to any very great distance from the centre of the solar system, and would therefore be regularly visible, a result which was contrary to observation. If, however, the ellipse was very elongated, as shewn in fig. 73, then the period of revolution might easily be very great, and, during the greater part of it, the comet would be so far from the sun and consequently also from the earth as to be invisible. If so the comet would be seen for a short time and become invisible, only to reappear after a very long time, when it would naturally be regarded as a new comet. If again the path of the comet were a parabola (which may be regarded as an ellipse indefinitely elongated), the comet would not return at all, but would merely be seen once when in that part of its path which is near the sun. But if a comet moved in a parabola, with the sun in a focus, then its positions when not very far from the sun would be almost the same as if it moved in an elongated ellipse (see fig. 73), and consequently it would hardly be possible to distinguish the two cases. Newton accordingly worked out the case of motion in a parabola, which is mathematically the simpler, and found that, in the case of a comet which had attracted much attention in the winter 1680-1, a parabolic path could be found, the calculated places of the comet in which agreed closely with those observed. In the later editions of the Principia the motions of a number of other comets were investigated with a similar

Fig. 73.—An elongated ellipse and a parabola.

result. It was thus established that in many cases a comet's path is either a parabola or an elongated ellipse, and that a similar result was to be expected in other cases. This reduction to rule of the apparently arbitrary motions of comets, and their inclusion with the planets in the same class of bodies moving round the sun under the action of gravitation, may fairly be regarded as one of the most striking of the innumerable discoveries contained in the Principia.

In the same section Newton discussed also at some length the nature of comets and in particular the structure of their tails, arriving at the conclusion, which is in general agreement with modern theories (chapter xiii., § 304), that the tail is formed by a stream of finely divided matter of the nature of smoke, rising up from the body of the comet, and so illuminated by the light of the sun when tolerably near it as to become visible.

191. The Principia was published, as we have seen, in 1687. Only a small edition seems to have been printed, and this was exhausted in three or four years. Newton's earlier discoveries, and the presentation to the Royal Society of the tract De Motu (§ 177), had prepared the scientific world to look for important new results in the Principia, and the book appears to have been read by the leading Continental mathematicians and astronomers, and to have been very warmly received in England. The Cartesian philosophy had, however, too firm a hold to be easily shaken; and Newton's fundamental principle, involving as it did the idea of an action between two bodies separated by an interval of empty space, seemed impossible of acceptance to thinkers who had not yet fully grasped the notion of judging a scientific theory by the extent to which its consequences agree with observed facts. Hence even so able a man as Huygens (chapter viii., §§ 154, 157, 158), regarded the idea of gravitation as "absurd," and expressed his surprise that Newton should have taken the trouble to make such a number of laborious calculations with no foundation but this principle, a remark which shewed Huygens to have had no conception that the agreement of the results of these calculations with actual facts was proof of the soundness of the principle. Personal reasons also contributed to the Continental neglect of Newton's work, as the famous quarrel between Newton and Leibniz as to their respective claims to the invention of what Newton called fluxions and Leibniz the differential method (out of which the differential and integral calculus have developed) grew in intensity and fresh combatants were drawn into it on both sides. Half a century in fact elapsed before Newton's views made any substantial progress on the Continent (cf chapter xi., § 229). In our country the case was different; not only was the Prinicipia read with admiration by the few who were capable of understanding it, but scholars like Bentley, philosophers like Locke, and courtiers like Halifax all made attempts to grasp Newton's general ideas, even though the details of his mathematics were out of their range. It was moreover soon discovered that his scientific ideas could be used with advantage as theological arguments.

192. One unfortunate result of the great success of the Principia was that Newton was changed from a quiet Cambridge professor, with abundant leisure and a slender income, into a public character, with a continually increasing portion of his time devoted to public business of one sort or another.

Just before the publication of the Principia he had been appointed one of the representatives of his University to defend its rights against the encroachments of James II., and two years later he sat as member for the University in the Convention Parliament, though he retired after its dissolution.

Notwithstanding these and many other distractions, he continued to work at the theory of gravitation, paying particular attention to the lunar theory, a difficult subject with his treatment of which he was never quite satisfied.^[11] He was fortunately able to obtain from time to time first-rate observations of the moon (as well as of other bodies) from the Astronomer Royal Flamsteed (chapter x., §§ 197–8), though Newton's continual requests and Flamsteed's occasional refusals led to strained relations at intervals. It is possible that about this time Newton contemplated writing a new treatise, with more detailed treatment of various points discussed in the Principia; and in 1691 there was already some talk of a new edition of the Principia, possibly to be edited by some younger mathematician. In any case nothing serious in this direction was done for some years, perhaps owing to a serious illness, apparently some nervous disorder, which attacked Newton in 1692 and lasted about two years. During this illness, as he himself said, "he had not his usual consistency of mind," and it is by no means certain that he ever recovered his full mental activity and power.

Soon after recovering from this illness he made some

newton.

[To face p. 240.

preparations for a new edition of the Principia, besides going on with the lunar theory, but the work was again interrupted in 1695, when he received the valuable appointment of Warden to the Mint, from which he was promoted to the Mastership four years later. He had, in consequence, to move to London (1696), and much of his time was henceforward occupied by official duties. In 1701 he resigned his professorship at Cambridge, and in the same year was for the second time elected the Parliamentary representative of the University. In 1703 he was chosen President of the Royal Society, an office which he held till his death, and in 1705 he was knighted on the occasion of a royal visit to Cambridge.

During this time he published (1704) his treatise on Optics, the bulk of which was probably written long before, and in 1709 he finally abandoned the idea of editing the Principia himself, and arranged for the work to be done by Roger Cotes (1682–1716), the brilliant young mathematician whose untimely death a few years later called from Newton the famous eulogy, "If Mr. Cotes had lived we might have known something." The alterations to be made were discussed in a long and active correspondence between the editor and author, the most important changes being improvements and additions to the lunar theory, and to the discussions of precession and of comets, though there were also a very large number of minor changes; and the new edition appeared in 1713. A third edition, edited by Pemberton, was published in 1726, but this time Newton, who was over 80, took much less part, and the alterations were of no great importance. This was Newton's last piece of scientific work, and his death occurred in the following year (March 3rd, 1727).

193. It is impossible to give an adequate idea of the immense magnitude of Newton's scientific discoveries except by a free use of the mathematical technicalities in which the bulk of them were expressed. The criticism passed on him by his personal enemy Leibniz that, "Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half," and the remark of his great successor Lagrange (chapter xi., § 237), "Newton was the greatest genius that ever existed, and the most fortunate, for we cannot find more than once a system of the world to establish," shew the immense respect for his work felt by those who were most competent to judge it.

With these magnificent eulogies it is pleasant to compare Newton's own grateful recognition of his predecessors, "If I have seen further than other men, it is because I have stood upon the shoulders of the giants," and his modest estimate of his own performances:—

"I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

194. It is sometimes said, in explanation of the difference between Newton's achievements and those of earlier astronomers, that whereas they discovered how the celestial bodies moved, he shewed why the motions were as they were, or, in other words, that they described motions while he explained them or ascertained their cause. It is, however, doubtful whether this distinction between How and Why, though undoubtedly to some extent convenient, has any real validity. Ptolemy, for example, represented the motion of a planet by a certain combination of epicycles; his scheme was equivalent to a particular method of describing the motion; but if any one had asked him why the planet would be in a particular position at a particular time, he might legitimately have answered that it was so because the planet was connected with this particular system of epicycles, and its place could be deduced from them by a rigorous process of calculation. But if any one had gone further and asked why the planet's epicycles were as they were, Ptolemy could have given no answer. Moreover, as the system of epicycles differed in some important respects from planet to planet, Ptolemy's system left unanswered a number of questions which obviously presented themselves. Then Coppernicus gave a partial answer to some of these questions. To the question why certain of the planetary motions, corresponding to certain epicycles, existed, he would have replied that it was because of certain motions of the earth, from which these (apparent) planetary motions could be deduced as necessary consequences. But the same information could also have been given as a mere descriptive statement that the earth moves in certain ways and the planets move in certain other ways. But again, if Coppernicus had been asked why the earth rotated on its axis, or why the planets revolved round the sun, he could have given no answer; still less could he have said why the planets had certain irregularities in their motions, represented by his epicycles.

Kepler again described the same motions very much more simply and shortly by means of his three laws of planetary motion; but if any one had asked why a planet's motion varied in certain ways, he might have replied that it was because all planets moved in ellipses so as to sweep out equal areas in equal times. Why this was so Kepler was unable to say, though he spent much time in speculating on the subject. This question was, however, answered by Newton, who shewed that the planetary motions were necessary consequences of his law of gravitation and his laws of motion. Moreover from these same laws, which were extremely simple in statement and few in number, followed as necessary consequences the motion of the moon and many other astronomical phenomena, and also certain familiar terrestrial phenomena, such as the behaviour of falling bodies; so that a large number of groups of observed facts, which had hitherto been disconnected from one another, were here brought into connection as necessary consequences of certain fundamental laws. But again Newton's view of the solar system might equally well be put as a mere descriptive statement that the planets, etc., move with accelerations of certain magnitudes towards one another. As, however, the actual position or rate of motion of a planet at any time can only be deduced by an extremely elaborate calculation from Newton's laws, they are not at all obviously equivalent to the observed celestial motions, and we do not therefore at all easily think of them as being merely a description.

Again Newton's laws at once suggest the question why bodies attract one another in this particular way; and this question, which Newton fully recognised as legitimate, he was unable to answer. Or again we might ask why the planets are of certain sizes, at certain distances from the sun, etc., and to these questions again Newton could give no answer.

But whereas the questions left unanswered by Ptolemy, Coppernicus, and Kepler were in whole or in part answered by their successors, that is, their unexplained facts or laws were shewn to be necessary consequences of other simpler and more general laws, it happens that up to the present day no one has been able to answer, in any satisfactory way, these questions which Newton left unanswered. In this particular direction, therefore, Newton's laws mark the boundary of our present knowledge. But if any one were to succeed this year or next in shewing gravitation to be a consequence of some still more general law, this new law would still bring with it a new Why.

If, however, Newton's laws cannot be regarded as an ultimate explanation of the phenomena of the solar system, except in the historic sense that they have not yet been shewn to depend on other more fundamental laws, their success in "explaining," with fair accuracy, such an immense mass of observed results in all parts of the solar system, and their universal character, gave a powerful impetus to the idea of accounting for observed facts in other departments of science, such as chemistry and physics, in some similar way as the consequence of forces acting between bodies, and hence to the conception of the material universe as made up of a certain number of bodies, each acting on one another with definite forces in such a way that all the changes which can be observed to go on are necessary consequences of these forces, and are capable of prediction by any one who has sufficient knowledge of the forces and sufficient mathematical skill to develop their consequences.

Whether this conception of the material universe is adequate or not, it has undoubtedly exercised a very important influence on scientific discovery as well as on philosophical thought, and although it was never formulated by Newton, and parts of it would probably have been repudiated by him, there are indications that some such ideas were in his head, and those who held the conception most firmly undoubtedly derived their ideas directly or indirectly from him.

195. Newton's scientific method did not differ essentially from that followed by Galilei (chapter vi., § 134), which has been variously described as complete induction or as the inverse deductive method, the difference in name corresponding to a difference in the stress laid upon different parts of the same general process. Facts are obtained by observation or experiment; a hypothesis or provisional theory is devised to account for them; from this theory are obtained, if possible by a rigorous process of deductive reasoning, certain consequences capable of being compared with actual facts, and the comparison is then made. In some cases the first process may appear as the more important, but in Newton's work the really convincing part of the proof of his results lay in the verification involved in the two last processes. This has perhaps been somewhat obscured by his famous remark, Hypotheses non fingo (I do not invent hypotheses), dissociated from its context. The words occur in the conclusion of the Principia, after he has been speaking of universal gravitation:—

"I have not yet been able to deduce (deducere) from phenomena the reason of these properties of gravitation, and I do not invent hypotheses. For any thing which cannot be deduced from phenomena should be called a hypothesis."

Newton probably had in his mind such speculations as the Cartesian vortices, which could not be deduced directly from observations, and the consequences of which either could not be worked out and compared with actual facts or were inconsistent with them. Newton in fact rejected hypotheses which were unverifiable, but he constantly made hypotheses, suggested by observed facts, and verified by the agreement of their consequences with fresh observed facts. The extension of gravity to the moon (§ 173) is a good example: he was acquainted with certain facts as to the motion of falling bodies and the motion of the moon; it occurred to him that the earth's attraction might extend as far as the moon, and certain other facts connected with Kepler's Third Law suggested the law of the inverse square. If this were right, the moon's acceleration towards the earth ought to have a certain value, which could be obtained by calculation. The calculation was made and found to agree roughly with the actual motion of the moon.

Moreover it may be fairly urged, in illustration of the great importance of the process of verification, that Newton's fundamental laws were not rigorously established by him, but that the deficiencies in his proofs have been to a great extent filled up by the elaborate process of verification that has gone on since. For the motions of the solar system, as deduced by Newton from gravitation and the laws of motion, only agreed roughly with observation; many outstanding discrepancies were left; and though there was a strong presumption that these were due to the necessary imperfections of Newton's processes of calculation, an immense expenditure of labour and ingenuity on the part of a series of mathematicians has been required to remove these discrepancies one by one, and as a matter of fact there remain even to-day a few small ones which are unexplained (chapter xiii., § 290).

1. According to the unreformed calendar (O.S.) then in use in England, the date was Christmas Day, 1642. To facilitate comparison with events occurring out of England, I have used throughout this and the following chapters the Gregorian Calendar (N.S.;, which was at this time adopted in a large part of the Continent (cf. chapter ii., § 22).
2. From a MS. among the Portsmouth Papers, quoted in the Preface to the Catalogue of the Portsmouth Papers.
3. W. K. Clifford, Aims and Instruments of Scientific Thought,
4. It is interesting to read that Wren offered a prize of 40s. to whichever of the other two should solve this the central problem of the solar system.
5. The familiar parallelogram of forces, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to the Principia, and was, by a curious coincidence, published also in the same year by Varignon and Lami.
6. It is between 13 and 14 billion billion pounds. See chapter x., § 219.
7. As far as I know Newton gives no short statement of the law in a perfectly complete and general form; separate parts of it are given in different passages of the Principia.
8. It is commonly stated that Newton's value of the motion of the moon's apses was only about half the true value. In a scholium of the Principia to prop. 35 of the third book, given in the first edition but afterwards omitted, he estimated the annual motion at 40°, the observed value being about 41°. In one of his unpublished papers, contained in the Portsmouth collection, he arrived at 39° by a process which he evidently regarded as not altogether satisfactory.
9. Throughout the Coppernican controversy up to Newton's time it had been generally assumed, both by Coppernicans and by their opponents, that there was some meaning in speaking of a body simply as being "at rest" or "in motion," without any reference to any other body. But all that we can really observe is the motion of one body relative to one or more others. Astronomical observation tells us, for example, of a certain motion relative to one another of the earth and sun; and this motion was expressed in two quite different ways by Ptolemy and by Coppernicus. From a modern standpoint the question ultimately involved was whether the motions of the various bodies of the solar system relatively to the earth or relatively to the sun were the simpler to express. If it is found convenient to express them—as Coppernicus and Galilei did—in relation to the sun, some simplicity of statement is gained by speaking of the sun as "fixed" and omitting the qualification "relative to the sun" in speaking of any other body. The same motions might have been expressed relatively to any other body chosen at will: e.g. to one of the hands of a watch carried by a man walking up and down on the deck of a ship on a rough sea; in this case it is clear that the motions of the other bodies of the solar system relative to this body would be excessively complicated; and it would therefore be highly inconvenient though still possible to treat this particular body as "fixed."
   
   A new aspect of the problem presents itself, however, when an attempt—like Newton's—is made to explain the motions of bodies of the solar system as the result of forces exerted on one another by those bodies. If, for example, we look at Newton's First Law of Motion (chapter vi., § 130), we see that it has no meaning, unless we know what are the body or bodies relative to which the motion is being expressed; a body at rest relatively to the earth is moving relatively to the sun or to the fixed stars, and the applicability of the First Law to it depends therefore on whether we are dealing with its motion relatively to the earth or not. For most terrestrial motions it is sufficient to regard the Laws of Motion as referring to motion relative to the earth; or, in other words, we may for this purpose treat the earth as "fixed." But if we examine certain terrestrial motions more exactly, we find that the Laws of Motion thus interpreted are not quite true; but that we get a more accurate explanation of the observed phenomena if we regard the Laws of Motion as referring to motion relative to the centre of the sun and to lines drawn from it to the stars; or, in other words, we treat the centre of the sun as a "fixed" point and these lines as "fixed" directions. But again when we are dealing with the solar system generally this interpretation is slightly inaccurate, and we have to treat the centre of gravity of the solar system instead of the sun as "fixed."
   
   From this point of view we may say that Newton's object in the Principia was to shew that it was possible to choose a certain point (the centre of gravity of the solar system) and certain directions (lines joining this point to the fixed stars), as a base of reference, such that all motions being treated as relative to this base, the Laws of Motion and the law of gravitation afford a consistent explanation of the observed motions of the bodies of the solar system.
10. He estimated the annual precession due to the sun to be about 9", and that due to the moon to be about four and a half times as great, so that the total amount due to the two bodies came out about 50", which agrees within a fraction of a second with the amount shewn by observation; but we know now that the moon's share is not much more than twice that of the sun.
11. He once told Halley in despair that the lunar theory "made his head ache and kept him awake so often that he would think of it no more."