CHAPTER IX.UNIVERSAL GRAVITATION.

“Nature and Nature’s laws lay hid in night;God said ‘Let Newton be!’ and all was light.”

“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 thePrincipiain 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;100this 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, aftersome 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’sElements of Geometryfor 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 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 areflecting 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 therefracting telescopesof Galilei and Kepler. Telescopes on this principle, differing however in some important particulars from Newton’s, had already been described in 1663 byJames 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 (chapterXII.), 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 (chapterXIII.,§ 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.”101

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 (chapterVII.,§ 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 (chapterVIII.,§ 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.

Fig. 70.—Motion in a circle.

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.

LetArepresent the position at a certain instant of a body which is revolving with uniform speed in a circle of centreO. Then at this instant the body is moving in the direction of the tangentAato the circle. Consequently by Galilei’s First Law (chapterVI., §§ 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 lineAa, and consequently would be found after some time at such a point asa. But actually it is found to be atBon the circle. Hence some influence must have been at work to bring it toBinstead of toa. ButBis nearer to the centre of the circle thanais; hence some influence must be at work tendingconstantly to draw the body towardsO, or counteracting the tendency which it has, in virtue of the First Law of Motion, to get farther and farther away fromO. To express either of these tendencies numerically we want a more complex idea than that of velocity or rate of motion, namelyaccelerationor rate of change of velocity, an idea which Galilei added to science in his discussion of the law of falling bodies (chapterVI.,§§ 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 magnitudedirected 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 (10 × 10)∕4 = 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 thePendulum Clock(chapterVIII.,§ 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 roundthe 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 (chapterVII.,§ 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 areinverselyproportional to the distances, the more distant planet having the less velocity andvice 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 fromthe 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 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.

Thislaw 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 (chapterVI.,§ 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 forterrestrial 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 thedistance 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 fromBin fig. 71, then moving at first horizontally it will describe a curved path and reach the ground at a point such asC, at some distance from the pointA, 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 pointC′beyondC; if the velocity were greater still, it would reach the ground atC″or atC‴; 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 asB 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.”102

Fig. 71.—The moon as a projectile.

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 (chapterVIII.,§ 159); it is known that even at a later stage he was unable 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 assubstantially 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 (chapterVIII.,§ 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 (chapterVII.,§ 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 thedistance 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 byEdmund Halley(chapterX.,§§ 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.103Later 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 sentto the Royal Society, a tract calledPropositiones 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 propositions 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 theDe 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 titlePhilosophiae Naturalis Principia Mathematica.

178. ThePrincipia, 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 bepassed over here, but certain results of special astronomical importance require to be mentioned.

Galilei, as we have seen (chapterVI.,§§ 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 (chapterVI.,§ 133), and that this acceleration is the same for all bodies (chapterVI.,§ 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 ofmass.

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 accustomedto 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. Theweightof a body was thus identified with theforce 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 (chapterVIII.,§ 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.104

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, theaccelerationsproduced are not the same, but since forceis 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,105its 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 thePrincipiahe 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 on the earth’s 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 thesameplanet in different parts of its path, but also indifferentplanets. 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 inwhich 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.106

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 uniformdensity 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 thePrincipia, 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 orperturbationsin its path. Although even in this simplified form the problem of themotion of the planets is one of extreme difficulty (cf. chapterXI.,§ 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 (chapterIII.,§ 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 (chapterVIII.,§ 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 (chapterII.,§§ 40, 48; chapterV.,§ 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;107and 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 themassesof 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 inthe case of the large planets Jupiter and Saturn (cf. chapterVII.,§ 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 thecentre 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 beat restin 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.108

187. The variation of the time of oscillation of a pendulum in different parts of the earth, discovered by Richer in 1672 (chapterVIII.,§ 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 OaAin fig. 72), and then found the condition that these two columns of waterO 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 thepoleBfrom the centre of the earthO, then the actual surface of the earth extends at the equator beyond this circle as far asA, where, according to Newton,aAis about 1∕230 ofO BorO A, and according to modern estimates, based on actual measurement of the earth as well as upon theory (chapterX.,§ 221), it is about 1∕293 ofO 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 the lengthaAis made, for the sake of distinctness, nearly 30 times as great as it should be.

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