Why did the image thus spread out? If it were due to irregularities in the glass a second prism should rather increase them, but a second prism when held in appropriate position was able to neutralise the dispersion and to reproduce the simple round white spot without deviation. Evidently the spreading out of the beam was connected in some definite way with its refraction. Could it be that the light particles after passing through the prism travelled in variously curved lines, as spinning racquet balls do? To examine this he measured the length of the oval patch when the screen was at different distances from the prism, and found that the two things were directly proportional to each other. Doubling the distance of the screen doubled the length of the patch. Hence the rays travelled in straight lines from the prism, and the spreading out was due to something that occurred within its substance. Could it be that white light was compound, was a mixture of several constituents, and that its different constituents were differently bent? No sooner thought than tried. Pierce the screen to let one of the constituents through and interpose a second prism in its path. If the spreading out depended on the prism only it should spread out just as much as before, but if it depended on the complex character of white light, this isolated simple constituent should be able to spread out no more. It did not spread out any more: a prism had no more dispersive power over it; it was deflected by the appropriate amount, but it was not analysed into constituents. It differed from sunlight in being simple. With many ingenious and beautifully simple experiments, which are quoted in full in several books on optics, he clinched the argument and established his discovery. White light was not simple but compound. It could be sorted out by a prism into an infinite number of constituent parts which were differently refracted, and the most striking of which Newton named violet, indigo, blue, green, yellow, orange, and red.



At once the true nature of colour became manifest. Colour resided not in the coloured object as had till now been thought, but in the light which illuminated it. Red glass for instance adds nothing to sunlight. The light does not get dyed red by passing through the glass; all that the red glass does is to stop and absorb a large part of the sunlight; it is opaque to the larger portion, but it is transparent to that particular portion which affects our eyes with the sensation of red. The prism acts like a sieve sorting out the different kinds of light. Coloured media act like filters, stopping certain kinds but allowing the rest to go through. Leonardo's and all the ancient doctrines of colour had been singularly wrong; colour is not in the object but in the light.

Goethe, in his Farbenlehre, endeavoured to controvert Newton, and to reinstate something more like the old views; but his failure was complete.

Refraction analysed out the various constituents of white light and displayed them in the form of a series of overlapping images of the aperture, each of a different colour; this series of images we call a spectrum, and the operation we now call spectrum analysis. The reason of the defect of lenses was now plain: it was not so much a defect of the lens as a defect of light. A lens acts by refraction and brings rays to a focus. If light be simple it acts well, but if ordinary white light fall upon a lens, its different constituents have different foci; every bright object is fringed with colour, and nothing like a clear image can be obtained.



A parallel beam passing through a lens becomes conical; but instead of a single cone it is a sheaf or nest of cones, all having the edge of the lens as base, but each having a different vertex. The violet cone is innermost, near the lens, the red cone outermost, while the others lie between. Beyond the crossing point or focus the order of cones is reversed, as the above figure shows. Only the two marginal rays of the beam are depicted.

If a screen be held anywhere nearer the lens than the place marked 1 there will be a whitish centre to the patch of light and a red and orange fringe or border. Held anywhere beyond the region 2, the border of the patch will be blue and violet. Held about 3 the colour will be less marked than elsewhere, but nowhere can it be got rid of. Each point of an object will be represented in the image not by a point but by a coloured patch: a fact which amply explains the observed blurring and indistinctness.

Newton measured and calculated the distance between the violet and red foci—VR in the diagram—and showed that it was 1/50th the diameter of the lens. To overcome this difficulty (called chromatic aberration) telescope glasses were made small and of very long focus: some of them so long that they had no tube, all of them egregiously cumbrous. Yet it was with such instruments that all the early discoveries were made. With such an instrument, for instance, Huyghens discovered the real shape of Saturn's ring.

The defects of refractors seemed irremediable, being founded in the nature of light itself. So he gave up his "glass works"; and proceeded to think of reflexion from metal specula. A concave mirror forms an image just as a lens does, but since it does so without refraction or transmission through any substance, there is no accompanying dispersion or chromatic aberration.

The first reflecting telescope he made was 1 in. diameter and 6 in. long, and magnified forty times. It acted as well as a three or four feet refractor of that day, and showed Jupiter's moons. So he made a larger one, now in the library of the Royal Society, London, with an inscription:

"The first reflecting telescope, invented by Sir Isaac Newton, and made with his own hands."

This has been the parent of most of the gigantic telescopes of the present day. Fifty years elapsed before it was much improved on, and then, first by Hadley and afterwards by Herschel and others, large and good reflectors were constructed.

The largest telescope ever made, that of Lord Rosse, is a Newtonian reflector, fifty feet long, six feet diameter, with a mirror weighing four tons. The sextant, as used by navigators, was also invented by Newton.

The year after the plague, in 1667, Newton returned to Trinity College, and there continued his experiments on optics. It is specially to be noted that at this time, at the age of twenty-four, Newton had laid the foundations of all his greatest discoveries:—



The Theory of Fluxions; or, the Differential Calculus.

The Law of Gravitation; or, the complete theory of astronomy.

The compound nature of white light; or, the beginning of Spectrum Analysis.



His later life was to be occupied in working these incipient discoveries out. But the most remarkable thing is that no one knew about any one of them. However, he was known as an accomplished young mathematician, and was made a fellow of his college. You remember that he had a friend there in the person of Dr. Isaac Barrow, first Lucasian Professor of Mathematics in the University. It happened, about 1669, that a mathematical discovery of some interest was being much discussed, and Dr. Barrow happened to mention it to Newton, who said yes, he had worked out that and a few other similar things some time ago. He accordingly went and fetched some papers to Dr. Barrow, who forwarded them to other distinguished mathematicians, and it thus appeared that Newton had discovered theorems much more general than this special case that was exciting so much interest. Dr. Barrow, being anxious to devote his time more particularly to theology, resigned his chair the same year in favour of Newton, who was accordingly elected to the Lucasian Professorship, which he held for thirty years. This chair is now the most famous in the University, and it is commonly referred to as the chair of Newton.

Still, however, his method of fluxions was unknown, and still he did not publish it. He lectured first on optics, giving an account of his experiments. His lectures were afterwards published both in Latin and English, and are highly valued to this day.

The fame of his mathematical genius came to the ears of the Royal Society, and a motion was made to get him elected a fellow of that body. The Royal Society, the oldest and most famous of all scientific societies with a continuous existence, took its origin in some private meetings, got up in London by the Hon. Robert Boyle and a few scientific friends, during all the trouble of the Commonwealth.

After the restoration, Charles II. in 1662 incorporated it under Royal Charter; among the original members being Boyle, Hooke, Christopher Wren, and other less famous names. Boyle was a great experimenter, a worthy follower of Dr. Gilbert. Hooke began as his assistant, but being of a most extraordinary ingenuity he rapidly rose so as to exceed his master in importance. Fate has been a little unkind to Hooke in placing him so near to Newton; had he lived in an ordinary age he would undoubtedly have shone as a star of the first magnitude. With great ingenuity, remarkable scientific insight, and consummate experimental skill, he stands in many respects almost on a level with Galileo. But it is difficult to see stars even of the first magnitude when the sun is up, and thus it happens that the name and fame of this brilliant man are almost lost in the blaze of Newton. Of Christopher Wren I need not say much. He is well known as an architect, but he was a most accomplished all-round man, and had a considerable taste and faculty for science.

These then were the luminaries of the Royal Society at the time we are speaking of, and to them Newton's first scientific publication was submitted. He communicated to them an account of his reflecting telescope, and presented them with the instrument.

Their reception of it surprised him; they were greatly delighted with it, and wrote specially thanking him for the communication, and assuring him that all right should be done him in the matter of the invention. The Bishop of Salisbury (Bishop Burnet) proposed him for election as a fellow, and elected he was.

In reply, he expressed his surprise at the value they set on the telescope, and offered, if they cared for it, to send them an account of a discovery which he doubts not will prove much more grateful than the communication of that instrument, "being in my judgment the oddest, if not the most considerable detection that has recently been made into the operations of Nature."

So he tells them about his optical researches and his discovery of the nature of white light, writing them a series of papers which were long afterwards incorporated and published as his Optics. A magnificent work, which of itself suffices to place its author in the first rank of the world's men of science.

The nature of white light, the true doctrine of colour, and the differential calculus! besides a good number of minor results—binomial theorem, reflecting telescope, sextant, and the like; one would think it enough for one man's life-work, but the masterpiece remains still to be mentioned. It is as when one is considering Shakspeare: King Lear, Macbeth, Othello,—surely a sufficient achievement,—but the masterpiece remains.

Comparisons in different departments are but little help perhaps, nevertheless it seems to me that in his own department, and considered simply as a man of science, Newton towers head and shoulders over, not only his contemporaries—that is a small matter—but over every other scientific man who has ever lived, in a way that we can find no parallel for in other departments. Other nations admit his scientific pre-eminence with as much alacrity as we do.

Well, we have arrived at the year 1672 and his election to the Royal Society. During the first year of his membership there was read at one of the meetings a paper giving an account of a very careful determination of the length of a degree (i.e. of the size of the earth), which had been made by Picard near Paris. The length of the degree turned out to be not sixty miles, but nearly seventy miles. How soon Newton heard of this we do not learn—probably not for some years,—Cambridge was not so near London then as it is now, but ultimately it was brought to his notice. Armed with this new datum, his old speculation concerning gravity occurred to him. He had worked out the mechanics of the solar system on a certain hypothesis, but it had remained a hypothesis somewhat out of harmony with apparent fact. What if it should turn out to be true after all!

He took out his old papers and began again the calculation. If gravity were the force keeping the moon in its orbit, it would fall toward the earth sixteen feet every minute. How far did it fall? The newly known size of the earth would modify the figures: with intense excitement he runs through the working, his mind leaps before his hand, and as he perceives the answer to be coming out right, all the infinite meaning and scope of his mighty discovery flashes upon him, and he can no longer see the paper. He throws down the pen; and the secret of the universe is, to one man, known.

But of course it had to be worked out. The meaning might flash upon him, but its full detail required years of elaboration; and deeper and deeper consequences revealed themselves to him as he proceeded.

For two years he devoted himself solely to this one object. During those years he lived but to calculate and think, and the most ludicrous stories are told concerning his entire absorption and inattention to ordinary affairs of life. Thus, for instance, when getting up in a morning he would sit on the side of the bed half-dressed, and remain like that till dinner time. Often he would stay at home for days together, eating what was taken to him, but without apparently noticing what he was doing.

One day an intimate friend, Dr. Stukely, called on him and found on the table a cover laid for his solitary dinner. After waiting a long time, Dr. Stukely removed the cover and ate the chicken underneath it, replacing and covering up the bones again. At length Newton appeared, and after greeting his friend, sat down to dinner, but on lifting the cover he said in surprise, "Dear me, I thought I had not dined, but I see I have."

It was by this continuous application that the Principia was accomplished. Probably nothing of the first magnitude can be accomplished without something of the same absorbed unconsciousness and freedom from interruption. But though desirable and essential for the work, it was a severe tax upon the powers of the man. There is, in fact, no doubt that Newton's brain suffered temporary aberration after this effort for a short time. The attack was slight, and it has been denied; but there are letters extant which are inexplicable otherwise, and moreover after a year or two he writes to his friends apologizing for strange and disjointed epistles, which he believed he had written without understanding clearly what he wrote. The derangement was, however, both slight and temporary: and it is only instructive to us as showing at what cost such a work as the Principia must be produced, even by so mighty a mind as that of Newton.

The first part of the work having been done, any ordinary mortal would have proceeded to publish it; but the fact is that after he had sent to the Royal Society his papers on optics, there had arisen controversies and objections; most of them rather paltry, to which he felt compelled to find answers. Many men would have enjoyed this part of the work, and taken it as evidence of interest and success. But to Newton's shy and retiring disposition these discussions were merely painful. He writes, indeed, his answers with great patience and ability, and ultimately converts the more reasonable of his opponents, but he relieves his mind in the following letter to the secretary of the Royal Society: "I see I have made myself a slave to philosophy, but if I get free of this present business I will resolutely bid adieu to it eternally, except what I do for my private satisfaction or leave to come out after me; for I see a man must either resolve to put out nothing new, or to become a slave to defend it." And again in a letter to Leibnitz: "I have been so persecuted with discussions arising out of my theory of light that I blamed my own imprudence for parting with so substantial a blessing as my quiet to run after a shadow." This shows how much he cared for contemporary fame.

So he locked up the first part of the Principia in his desk, doubtless intending it to be published after his death. But fortunately this was not so to be.

In 1683, among the leading lights of the Royal Society, the same sort of notions about gravity and the solar system began independently to be bruited. The theory of gravitation seemed to be in the air, and Wren, Hooke, and Halley had many a talk about it.

Hooke showed an experiment with a pendulum, which he likened to a planet going round the sun. The analogy is more superficial than real. It does not obey Kepler's laws; still it was a striking experiment. They had guessed at a law of inverse squares, and their difficulty was to prove what curve a body subject to it would describe. They knew it ought to be an ellipse if it was to serve to explain the planetary motion, and Hooke said he could prove that an ellipse it was; but he was nothing of a mathematician, and the others scarcely believed him. Undoubtedly he had shrewd inklings of the truth, though his guesses were based on little else than a most sagacious intuition. He surmised also that gravity was the force concerned, and asserted that the path of an ordinary projectile was an ellipse, like the path of a planet—which is quite right. In fact the beginnings of the discovery were beginning to dawn upon him in the well-known way in which things do dawn upon ordinary men of genius: and had Newton not lived we should doubtless, by the labours of a long chain of distinguished men, beginning with Hooke, Wren, and Halley, have been now in possession of all the truths revealed by the Principia. We should never have had them stated in the same form, nor proved with the same marvellous lucidity and simplicity, but the facts themselves we should by this time have arrived at. Their developments and completions, due to such men as Clairaut, Euler, D'Alembert, Lagrange, Laplace, Airy, Leverrier, Adams, we should of course not have had to the same extent; because the lives and energies of these great men would have been partially consumed in obtaining the main facts themselves.

The youngest of the three questioners at the time we are speaking of was Edmund Halley, an able and remarkable man. He had been at Cambridge, doubtless had heard Newton lecture, and had acquired a great veneration for him.

In January, 1684, we find Wren offering Hooke and Halley a prize, in the shape of a book worth forty shillings, if they would either of them bring him within two months a demonstration that the path of a planet subject to an inverse square law would be an ellipse. Not in two months, nor yet in seven, was there any proof forthcoming. So at last, in August, Halley went over to Cambridge to speak to Newton about the difficult problem and secure his aid. Arriving at his rooms he went straight to the point. He said, "What path will a body describe if it be attracted by a centre with a force varying as the inverse square of the distance." To which Newton at once replied, "An ellipse." "How on earth do you know?" said Halley in amazement. "Why, I have calculated it," and began hunting about for the paper. He actually couldn't find it just then, but sent it him shortly by post, and with it much more—in fact, what appeared to be a complete treatise on motion in general.

With his valuable burden Halley hastened to the Royal Society and told them what he had discovered. The Society at his representation wrote to Mr. Newton asking leave that it might be printed. To this he consented; but the Royal Society wisely appointed Mr. Halley to see after him and jog his memory, in case he forgot about it. However, he set to work to polish it up and finish it, and added to it a great number of later developments and embellishments, especially the part concerning the lunar theory, which gave him a deal of trouble—and no wonder; for in the way he has put it there never was a man yet living who could have done the same thing. Mathematicians regard the achievement now as men might stare at the work of some demigod of a bygone age, wondering what manner of man this was, able to wield such ponderous implements with such apparent ease.

To Halley the world owes a great debt of gratitude—first, for discovering the Principia; second, for seeing it through the press; and third, for defraying the cost of its publication out of his own scanty purse. For though he ultimately suffered no pecuniary loss, rather the contrary, yet there was considerable risk in bringing out a book which not a dozen men living could at the time comprehend. It is no small part of the merit of Halley that he recognized the transcendent value of the yet unfinished work, that he brought it to light, and assisted in its becoming understood to the best of his ability.

Though Halley afterwards became Astronomer-Royal, lived to the ripe old age of eighty-six, and made many striking observations, yet he would be the first to admit that nothing he ever did was at all comparable in importance with his discovery of the Principia; and he always used to regard his part in it with peculiar pride and pleasure.

And how was the Principia received? Considering the abstruse nature of its subject, it was received with great interest and enthusiasm. In less than twenty years the edition was sold out, and copies fetched large sums. We hear of poor students copying out the whole in manuscript in order to possess a copy—not by any means a bad thing to do, however many copies one may possess. The only useful way really to read a book like that is to pore over every sentence: it is no book to be skimmed.

While the Principia was preparing for the press a curious incident of contact between English history and the University occurred. It seems that James II., in his policy of Catholicising the country, ordered both Universities to elect certain priests to degrees without the ordinary oaths. Oxford had given way, and the Dean of Christ Church was a creature of James's choosing. Cambridge rebelled, and sent eight of its members, among them Mr. Newton, to plead their cause before the Court of High Commission. Judge Jeffreys presided over the Court, and threatened and bullied with his usual insolence. The Vice-Chancellor of Cambridge was deprived of office, the other deputies were silenced and ordered away. From the precincts of this court of justice Newton returned to Trinity College to complete the Principia.

By this time Newton was only forty-five years old, but his main work was done. His method of fluxions was still unpublished; his optics was published only imperfectly; a second edition of the Principia, with additions and improvements, had yet to appear; but fame had now come upon him, and with fame worries of all kinds.

By some fatality, principally no doubt because of the interest they excited, every discovery he published was the signal for an outburst of criticism and sometimes of attack. I shall not go into these matters: they are now trivial enough, but it is necessary to mention them, because to Newton they evidently loomed large and terrible, and occasioned him acute torment.



No sooner was the Principia put than Hooke put in his claims for priority. And indeed his claims were not altogether negligible; for vague ideas of the same sort had been floating in his comprehensive mind, and he doubtless felt indistinctly conscious of a great deal more than he could really state or prove.

By indiscreet friends these two great men were set somewhat at loggerheads, and worse might have happened had they not managed to come to close quarters, and correspond privately in a quite friendly manner, instead of acting through the mischievous medium of third parties. In the next edition Newton liberally recognizes the claims of both Hooke and Wren. However, he takes warning betimes of what he has to expect, and writes to Halley that he will only publish the first two books, those containing general theorems on motion. The third book—concerning the system of the world, i.e. the application to the solar system—he says "I now design to suppress. Philosophy is such an impertinently litigious lady that a man had as good be engaged in law-suits as have to do with her. I found it so formerly, and now I am no sooner come near her again but she gives me warning. The two books without the third will not so well bear the title 'Mathematical Principles of Natural Philosophy,' and therefore I had altered it to this, 'On the Free Motion of Two Bodies'; but on second thoughts I retain the former title: 'twill help the sale of the book—which I ought not to diminish now 'tis yours."

However, fortunately, Halley was able to prevail upon him to publish the third book also. It is, indeed, the most interesting and popular of the three, as it contains all the direct applications to astronomy of the truths established in the other two.

Some years later, when his method of fluxions was published, another and a worse controversy arose—this time with Leibnitz, who had also independently invented the differential calculus. It was not so well recognized then how frequently it happens that two men independently and unknowingly work at the very same thing at the same time. The history of science is now full of such instances; but then the friends of each accused the other of plagiarism.

I will not go into the controversy: it is painful and useless. It only served to embitter the later years of two great men, and it continued long after Newton's death—long after both their deaths. It can hardly be called ancient history even now.

But fame brought other and less unpleasant distractions than controversies. We are a curious, practical, and rather stupid people, and our one idea of honouring a man is to vote for him in some way or other; so they sent Newton to Parliament. He went, I believe, as a Whig, but it is not recorded that he spoke. It is, in fact, recorded that he was once expected to speak when on a Royal Commission about some question of chronometers, but that he would not. However, I dare say he made a good average member.

Then a little later it was realized that Newton was poor, that he still had to teach for his livelihood, and that though the Crown had continued his fellowship to him as Lucasian Professor without the necessity of taking orders, yet it was rather disgraceful that he should not be better off. So an appeal was made to the Government on his behalf, and Lord Halifax, who exerted himself strongly in the matter, succeeding to office on the accession of William III., was able to make him ultimately Master of the Mint, with a salary of some L1,200 a year. I believe he made rather a good Master, and turned out excellent coins: certainly he devoted his attention to his work there in a most exemplary manner.

But what a pitiful business it all is! Here is a man sent by Heaven to do certain things which no man else could do, and so long as he is comparatively unknown he does them; but so soon as he is found out, he is clapped into a routine office with a big salary: and there is, comparatively speaking, an end of him. It is not to be supposed that he had lost his power, for he frequently solved problems very quickly which had been given out by great Continental mathematicians as a challenge to the world.

We may ask why Newton allowed himself to be thus bandied about instead of settling himself down to the work in which he was so pre-eminently great. Well, I expect your truly great man never realizes how great he is, and seldom knows where his real strength lies. Certainly Newton did not know it. He several times talks of giving up philosophy altogether; and though he never really does it, and perhaps the feeling is one only born of some temporary overwork, yet he does not sacrifice everything else to it as he surely must had he been conscious of his own greatness. No; self-consciousness was the last thing that affected him. It is for a great man's contemporaries to discover him, to make much of him, and to put him in surroundings where he may flourish luxuriantly in his own heaven-intended way.

However, it is difficult for us to judge of these things. Perhaps if he had been maintained at the national expense to do that for which he was preternaturally fitted, he might have worn himself out prematurely; whereas by giving him routine work the scientific world got the benefit of his matured wisdom and experience. It was no small matter to the young Royal Society to be able to have him as their President for twenty-four years. His portrait has hung over the President's chair ever since, and there I suppose it will continue to hang until the Royal Society becomes extinct.

The events of his later life I shall pass over lightly. He lived a calm, benevolent life, universally respected and beloved. His silver-white hair when he removed his peruke was a venerable spectacle. A lock of it is still preserved, with many other relics, in the library of Trinity College. He died quietly, after a painful illness, at the ripe age of eighty-five. His body lay in state in the Jerusalem Chamber, and he was buried in Westminster Abbey, six peers bearing the pall. These things are to be mentioned to the credit of the time and the country; for after we have seen the calamitous spectacle of the way Tycho and Kepler and Galileo were treated by their ungrateful and unworthy countries, it is pleasant to reflect that England, with all its mistakes, yet recognized her great man when she received him, and honoured him with the best she knew how to give.



Concerning his character, one need only say that it was what one would expect and wish. It was characterized by a modest, calm, dignified simplicity. He lived frugally with his niece and her husband, Mr. Conduit, who succeeded him as Master of the Mint. He never married, nor apparently did he ever think of so doing. The idea, perhaps, did not naturally occur to him, any more than the idea of publishing his work did.

He was always a deeply religious man and a sincere Christian, though somewhat of the Arian or Unitarian persuasion—so, at least, it is asserted by orthodox divines who understand these matters. He studied theology more or less all his life, and towards the end was greatly interested in questions of Biblical criticism and chronology. By some ancient eclipse or other he altered the recognized system of dates a few hundred years; and his book on the prophecies of Daniel and the Revelation of St. John, wherein he identifies the beast with the Church of Rome in quite the orthodox way, is still by some admired.

But in all these matters it is probable that he was a merely ordinary man, with natural acumen and ability doubtless, but nothing in the least superhuman. In science, the impression he makes upon me is only expressible by the words inspired, superhuman.

And yet if one realizes his method of work, and the calm, uninterrupted flow of all his earlier life, perhaps his achievements become more intelligible. When asked how he made his discoveries, he replied: "By always thinking unto them. I keep the subject constantly before me, and wait till the first dawnings open slowly by little and little into a full and clear light." That is the way—quiet, steady, continuous thinking, uninterrupted and unharassed brooding. Much may be done under those conditions. Much ought to be sacrificed to obtain those conditions. All the best thinking work of the world has been thus done.[18] Buffon said: "Genius is patience." So says Newton: "If I have done the public any service this way, it is due to nothing but industry and patient thought." Genius patience? No, it is not quite that, or, rather, it is much more than that; but genius without patience is like fire without fuel—it will soon burn itself out.



NOTES FOR LECTURE IX

The Principia published 1687. Newton died 1727.

THE LAW OF GRAVITATION.—Every particle of matter attracts every other particle of matter with a force proportional to the mass of each and to the inverse square of the distance between them.

SOME OF NEWTON'S DEDUCTIONS.

1. Kepler's second law (equable description of areas) proves that each planet is acted on by a force directed towards the sun as a centre of force.

2. Kepler's first law proves that this central force diminishes in the same proportion as the square of the distance increases.

3. Kepler's third law proves that all the planets are acted on by the same kind of force; of an intensity depending on the mass of the sun.[19]

4. So by knowing the length of year and distance of any planet from the sun, the sun's mass can be calculated, in terms of that of the earth.

5. For the satellites, the force acting depends on the mass of their central body, a planet. Hence the mass of any planet possessing a satellite becomes known.

6. The force constraining the moon in her orbit is the same gravity as gives terrestrial bodies their weight and regulates the motion of projectiles. [Because, while a stone drops 16 feet in a second, the moon, which is 60 times as far from the centre of the earth, drops 16 feet in a minute.]

* * * * *

7. The moon is attracted not only by the earth, but by the sun also; hence its orbit is perturbed, and Newton calculated out the chief of these perturbations, viz.:—

(The equation of the centre, discovered by Hipparchus.)

(a) The evection, discovered by Hipparchus and Ptolemy.

(b) The variation, discovered by Tycho Brahe.

(c) The annual equation, discovered by Tycho Brahe.

(d) The retrogression of the nodes, then being observed at Greenwich by Flamsteed.

(e) The variation of inclination, then being observed at Greenwich by Flamsteed.

(f) The progression of the apses (with an error of one-half).

(g) The inequality of apogee, previously unknown.

(h) The inequality of nodes, previously unknown.

8. Each planet is attracted not only by the sun but by the other planets, hence their orbits are slightly affected by each other. Newton began the theory of planetary perturbations.

9. He recognized the comets as members of the solar system, obedient to the same law of gravity and moving in very elongated ellipses; so their return could be predicted (e.g. Halley's comet).

10. Applying the idea of centrifugal force to the earth considered as a rotating body, he perceived that it could not be a true sphere, and calculated its oblateness, obtaining 28 miles greater equatorial than polar diameter.

11. Conversely, from the observed shape of Jupiter, or any planet, the length of its day could be estimated.

12. The so-calculated shape of the earth, in combination with centrifugal force, causes the weight of bodies to vary with latitude; and Newton calculated the amount of this variation. 194 lbs. at pole balance 195 lbs. at equator.

13. A homogeneous sphere attracts as if its mass were concentrated at its centre. For any other figure, such as an oblate spheroid, this is not exactly true. A hollow concentric spherical shell exerts no force on small bodies inside it.

14. The earth's equatorial protuberance, being acted on by the attraction of the sun and moon, must disturb its axis of rotation in a calculated manner; and thus is produced the precession of the equinoxes. [The attraction of the planets on the same protuberance causes a smaller and rather different kind of precession.]

15. The waters of the ocean are attracted towards the sun and moon on one side, and whirled a little further away than the solid earth on the other side: hence Newton explained all the main phenomena of the tides.

16. The sun's mass being known, he calculated the height of the solar tide.

17. From the observed heights of spring and neap tides he determined the lunar tide, and thence made an estimate of the mass of the moon.

REFERENCE TABLE OF NUMERICAL DATA.

+ -+ -+ + -+ Masses in Solar Height dropped by a Length of Day or System. stone in first second. time of rotation. + -+ -+ + -+ Mercury .065 7.0 feet 24 hours Venus .885 15.8 " 23-1/2 " Earth 1.000 16.1 " 24 " Mars .108 6.2 " 24-1/2 " Jupiter 300.8 45.0 " 10 " Saturn 89.7 18.4 " 10-1/2 " The Sun 316000. 436.0 " 608 " The Moon about .012 3.7 " 702 " + -+ -+ + -+

The mass of the earth, taken above as unity, is 6,000 trillion tons.

Observatories.—Uraniburg flourished from 1576 to 1597; the Observatory of Paris was founded in 1667; Greenwich Observatory in 1675.

Astronomers-Royal.—Flamsteed, Halley, Bradley, Bliss, Maskelyne, Pond, Airy, Christie.



LECTURE IX

NEWTON'S "PRINCIPIA"

The law of gravitation, above enunciated, in conjunction with the laws of motion rehearsed at the end of the preliminary notes of Lecture VII., now supersedes the laws of Kepler and includes them as special cases. The more comprehensive law enables us to criticize Kepler's laws from a higher standpoint, to see how far they are exact and how far they are only approximations. They are, in fact, not precisely accurate, but the reason for every discrepancy now becomes abundantly clear, and can be worked out by the theory of gravitation.

We may treat Kepler's laws either as immediate consequences of the law of gravitation, or as the known facts upon which that law was founded. Historically, the latter is the more natural plan, and it is thus that they are treated in the first three statements of the above notes; but each proposition may be worked inversely, and we might state them thus:—

1. The fact that the force acting on each planet is directed to the sun, necessitates the equable description of areas.

2. The fact that the force varies as the inverse square of the distance, necessitates motion in an ellipse, or some other conic section, with the sun in one focus.

3. The fact that one attracting body acts on all the planets with an inverse square law, causes the cubes of their mean distances to be proportional to the squares of their periodic times.

Not only these but a multitude of other deductions follow rigorously from the simple datum that every particle of matter attracts every other particle with a force directly proportional to the mass of each and to the inverse square of their mutual distance. Those dealt with in the Principia are summarized above, and it will be convenient to run over them in order, with the object of giving some idea of the general meaning of each, without attempting anything too intricate to be readily intelligible.



No. 1. Kepler's second law (equable description of areas) proves that each planet is acted on by a force directed towards the sun as a centre of force.

The equable description of areas about a centre of force has already been fully, though briefly, established. (p. 175.) It is undoubtedly of fundamental importance, and is the earliest instance of the serious discussion of central forces, i.e. of forces directed always to a fixed centre.

We may put it afresh thus:—OA has been the motion of a particle in a unit of time; at A it receives a knock towards C, whereby in the next unit it travels along AD instead of AB. Now the area of the triangle CAD, swept out by the radius vector in unit time, is 1/2bh; h being the perpendicular height of the triangle from the base AC. (Fig. 70.) Now the blow at A, being along the base, has no effect upon h; and consequently the area remains just what it would have been without the blow. A blow directed to any point other than C would at once alter the area of the triangle.

One interesting deduction may at once be drawn. If gravity were a radiant force emitted from the sun with a velocity like that of light, the moving planet would encounter it at a certain apparent angle (aberration), and the force experienced would come from a point a little in advance of the sun. The rate of description of areas would thus tend to increase; whereas in reality it is constant. Hence the force of gravity, if it travel at all, does so with a speed far greater than that of light. It appears to be practically instantaneous. (Cf. "Modern Views of Electricity," Sec. 126, end of chap. xii.) Again, anything like a retarding effect of the medium through which the planets move would constitute a tangential force, entirely un-directed towards the sun. Hence no such frictional or retarding force can appreciably exist. It is, however, conceivable that both these effects might occur and just neutralize each other. The neutralization is unlikely to be exact for all the planets; and the fact is, that no trace of either effect has as yet been discovered. (See also p. 176.)

The planets are, however, subject to forces not directed towards the sun, viz. their attractions for each other; and these perturbing forces do produce a slight discrepancy from Kepler's second law, but a discrepancy which is completely subject to calculation.

No. 2. Kepler's first law proves that this central force diminishes in the same proportion as the square of the distance increases.

To prove the connection between the inverse-square law of distance, and the travelling in a conic section with the centre of force in one focus (the other focus being empty), is not so simple. It obviously involves some geometry, and must therefore be left to properly armed students. But it may be useful to state that the inverse-square law of distance, although the simplest possible law for force emanating from a point or sphere, is not to be regarded as self-evident or as needing no demonstration. The force of a magnetic pole on a magnetized steel scrap, for instance, varies as the inverse cube of the distance; and the curve described by such a particle would be quite different from a conic section—it would be a definite class of spiral (called Cotes's spiral). Again, on an iron filing the force of a single pole might vary more nearly as the inverse fifth power; and so on. Even when the thing concerned is radiant in straight lines, like light, the law of inverse squares is not universally true. Its truth assumes, first, that the source is a point or sphere; next, that there is no reflection or refraction of any kind; and lastly, that the medium is perfectly transparent. The law of inverse squares by no means holds from a prairie fire for instance, or from a lighthouse, or from a street lamp in a fog.

Mutual perturbations, especially the pull of Jupiter, prevent the path of a planet from being really and truly an ellipse, or indeed from being any simple re-entrant curve. Moreover, when a planet possesses a satellite, it is not the centre of the planet which ever attempts to describe the Keplerian ellipse, but it is the common centre of gravity of the two bodies. Thus, in the case of the earth and moon, the point which really does describe a close attempt at an ellipse is a point displaced about 3000 miles from the centre of the earth towards the moon, and is therefore only 1000 miles beneath the surface.

No. 3. Kepler's third law proves that all the planets are acted on by the same kind of force; of an intensity depending on the mass of the sun.

The third law of Kepler, although it requires geometry to state and establish it for elliptic motion (for which it holds just as well as it does for circular motion), is very easy to establish for circular motion, by any one who knows about centrifugal force. If m is the mass of a planet, v its velocity, r the radius of its orbit, and T the time of describing it; 2[pi]r = vT, and the centripetal force needed to hold it in its orbit is

mv^2 4[pi]^2mr ———— or —————- r T^2

Now the force of gravitative attraction between the planet and the sun is

VmS ——-, r^2

where v is a fixed quantity called the gravitation-constant, to be determined if possible by experiment once for all. Now, expressing the fact that the force of gravitation is the force holding the planet in, we write,

4[pi]^2mr VmS —————- = ————-, T^2 r^2

whence, by the simplest algebra,

r^3 VS ——— = ————-. T^2 4[pi]^2

The mass of the planet has been cancelled out; the mass of the sun remains, multiplied by the gravitation-constant, and is seen to be proportional to the cube of the distance divided by the square of the periodic time: a ratio, which is therefore the same for all planets controlled by the sun. Hence, knowing r and T for any single planet, the value of VS is known.

No. 4. So by knowing the length of year and distance of any planet from the sun, the sun's mass can be calculated, in terms of that of the earth.

No. 5. For the satellites, the force acting depends on the mass of their central body, a planet. Hence the mass of any planet possessing a satellite becomes known.

The same argument holds for any other system controlled by a central body—for instance, for the satellites of Jupiter; only instead of S it will be natural to write J, as meaning the mass of Jupiter. Hence, knowing r and T for any one satellite of Jupiter, the value of VJ is known.

Apply the argument also to the case of moon and earth. Knowing the distance and time of revolution of our moon, the value of VE is at once determined; E being the mass of the earth. Hence, S and J, and in fact the mass of any central body possessing a visible satellite, are now known in terms of E, the mass of the earth (or, what is practically the same thing, in terms of V, the gravitation-constant). Observe that so far none of these quantities are known absolutely. Their relative values are known, and are tabulated at the end of the Notes above, but the finding of their absolute values is another matter, which we must defer.

But, it may be asked, if Kepler's third law only gives us the mass of a central body, how is the mass of a satellite to be known? Well, it is not easy; the mass of no satellite is known with much accuracy. Their mutual perturbations give us some data in the case of the satellites of Jupiter; but to our own moon this method is of course inapplicable. Our moon perturbs at first sight nothing, and accordingly its mass is not even yet known with exactness. The mass of comets, again, is quite unknown. All that we can be sure of is that they are smaller than a certain limit, else they would perturb the planets they pass near. Nothing of this sort has ever been detected. They are themselves perturbed plentifully, but they perturb nothing; hence we learn that their mass is small. The mass of a comet may, indeed, be a few million or even billion tons; but that is quite small in astronomy.

But now it may be asked, surely the moon perturbs the earth, swinging it round their common centre of gravity, and really describing its own orbit about this point instead of about the earth's centre? Yes, that is so; and a more precise consideration of Kepler's third law enables us to make a fair approximation to the position of this common centre of gravity, and thus practically to "weigh the moon," i.e. to compare its mass with that of the earth; for their masses will be inversely as their respective distances from the common centre of gravity or balancing point—on the simple steel-yard principle.

Hitherto we have not troubled ourselves about the precise point about which the revolution occurs, but Kepler's third law is not precisely accurate unless it is attended to. The bigger the revolving body the greater is the discrepancy: and we see in the table preceding Lecture III., on page 57, that Jupiter exhibits an error which, though very slight, is greater than that of any of the other planets, when the sun is considered the fixed centre.

Let the common centre of gravity of earth and moon be displaced a distance x from the centre of the earth, then the moon's distance from the real centre of revolution is not r, but r-x; and the equation of centrifugal force to gravitative-attraction is strictly

4[pi]^2 VE ————- (r-x) = ———, T^2 r^2

instead of what is in the text above; and this gives a slightly modified "third law." From this equation, if we have any distinct method of determining VE (and the next section gives such a method), we can calculate x and thus roughly weigh the moon, since

r-x E ——- = ——-, r E+M

but to get anything like a reasonable result the data must be very precise.

No. 6. The force constraining the moon in her orbit is the same gravity as gives terrestrial bodies their weight and regulates the motion of projectiles.

Here we come to the Newtonian verification already several times mentioned; but because of its importance I will repeat it in other words. The hypothesis to be verified is that the force acting on the moon is the same kind of force as acts on bodies we can handle and weigh, and which gives them their weight. Now the weight of a mass m is commonly written mg, where g is the intensity of terrestrial gravity, a thing easily measured; being, indeed, numerically equal to twice the distance a stone drops in the first second of free fall. [See table p. 205.] Hence, expressing that the weight of a body is due to gravity, and remembering that the centre of the earth's attraction is distant from us by one earth's radius (R), we can write

VmE mg = ———, R^2

or

VE = gR^2 = 95,522 cubic miles-per-second per second.

But we already know vE, in terms of the moon's motion, as

4[pi]^2r^3 —————- T^2

approximately, [more accurately, see preceding note, this quantity is V(E + M)]; hence we can easily see if the two determinations of this quantity agree.[20]

All these deductions are fundamental, and may be considered as the foundation of the Principia. It was these that flashed upon Newton during that moment of excitement when he learned the real size of the earth, and discovered his speculations to be true.

The next are elaborations and amplifications of the theory, such as in ordinary times are left for subsequent generations of theorists to discover and work out.

Newton did not work out these remoter consequences of his theory completely by any means: the astronomical and mathematical world has been working them out ever since; but he carried the theory a great way, and here it is that his marvellous power is most conspicuous.

It is his treatment of No. 7, the perturbations of the moon, that perhaps most especially has struck all future mathematicians with amazement. No. 7, No. 14, No. 15, these are the most inspired of the whole.

No. 7. The moon is attracted not only by the earth, but by the sun also; hence its orbit is perturbed, and Newton calculated out the chief of these perturbations.

Now running through the perturbations (p. 203) in order:—The first is in parenthesis, because it is mere excentricity. It is not a true perturbation at all, and more properly belongs to Kepler.

(a) The first true perturbation is what Ptolemy called "the evection," the principal part of which is a periodic change in the ellipticity or excentricity of the moon's orbit, owing to the pull of the sun. It is a complicated matter, and Newton only partially solved it. I shall not attempt to give an account of it.

(b) The next, "the variation," is a much simpler affair. It is caused by the fact that as the moon revolves round the earth it is half the time nearer to the sun than the earth is, and so gets pulled more than the average, while for the other fortnight it is further from the sun than the earth is, and so gets pulled less. For the week during which it is changing from a decreasing half to a new moon it is moving in the direction of the extra pull, and hence becomes new sooner than would have been expected. All next week it is moving against the same extra pull, and so arrives at quadrature (half moon) somewhat late. For the next fortnight it is in the region of too little pull, the earth gets pulled more than it does; the effect of this is to hurry it up for the third week, so that the full moon occurs a little early, and to retard it for the fourth week, so that the decreasing half moon like the increasing half occurs behind time again. Thus each syzygy (as new and full are technically called) is too early; each quadrature is too late; the maximum hurrying and slackening force being felt at the octants, or intermediate 45 deg. points.

(c) The "annual equation" is a fluctuation introduced into the other perturbations by reason of the varying distance of the disturbing body, the sun, at different seasons of the year. Its magnitude plainly depends simply on the excentricity of the earth's orbit.

Both these perturbations, (b) and (c), Newton worked out completely.

(d) and (e) Next come the retrogression of the nodes and the variation of the inclination, which at the time were being observed at Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged for data that he might complete their theory while he had his mind upon it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see list at end of notes above], and then Newton would have no difficulty in gaining such information as the national Observatory could give.

The "inclination" meant is the angle between the plane of the moon's orbit and that of the earth. The plane of the earth's orbit round the sun is called the ecliptic; the plane of the moon's orbit round the earth is inclined to it at a certain angle, which is slowly changing, though in a periodic manner. Imagine a curtain ring bisected by a sheet of paper, and tilted to a certain angle; it may be likened to the moon's orbit, cutting the plane of the ecliptic. The two points at which the plane is cut by the ring are called "nodes"; and these nodes are not stationary, but are slowly regressing, i.e. travelling in a direction opposite to that of the moon itself. Also the angle of tilt is varying slowly, oscillating up and down in the course of centuries.

(f) The two points in the moon's elliptic orbit where it comes nearest to or farthest from the earth, i.e. the points at the extremity of the long axis of the ellipse, are called separately perigee and apogee, or together "the apses." Now the pull of the sun causes the whole orbit to slowly revolve in its own plane, and consequently these apses "progress," so that the true path is not quite a closed curve, but a sort of spiral with elliptic loops.

But here comes in a striking circumstance. Newton states with reference to this perturbation that theory only accounts for 1-1/2 deg. per annum, whereas observation gives 3 deg., or just twice as much.

This is published in the Principia as a fact, without comment. It was for long regarded as a very curious thing, and many great mathematicians afterwards tried to find an error in the working. D'Alembert, Clairaut, and others attacked the problem, but were led to just the same result. It constituted the great outstanding difficulty in the way of accepting the theory of gravitation. It was suggested that perhaps the inverse square law was only a first approximation; that perhaps a more complete expression, such as

A B —— + ——-, r^2 r^4

must be given for it; and so on.

Ultimately, Clairaut took into account a whole series of neglected terms, and it came out correct; thus verifying the theory.

But the strangest part of this tale is to come. For only a few years ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was editing various old papers of Newton's, now in the possession of the Duke of Portland, and he found manuscripts bearing on this very point, and discovered that Newton had reworked out the calculations himself, had found the cause of the error, had taken into account the terms hitherto neglected, and so, fifty years before Clairaut, had completely, though not publicly, solved this long outstanding problem of the progression of the apses.

(g) and (h) Two other inequalities he calculated out and predicted, viz. variation in the motions of the apses and the nodes. Neither of these had then been observed, but they were afterwards detected and verified.

A good many other minor irregularities are now known—some thirty, I believe; and altogether the lunar theory, or problem of the moon's exact motion, is one of the most complicated and difficult in astronomy; the perturbations being so numerous and large, because of the enormous mass of the perturbing body.

The disturbances experienced by the planets are much smaller, because they are controlled by the sun and perturbed by each other. The moon is controlled only by the earth, and perturbed by the sun. Planetary perturbations can be treated as a series of disturbances with some satisfaction: not so those of the moon. And yet it is the only way at present known of dealing with the lunar theory.

To deal with it satisfactorily would demand the solution of such a problem as this:—Given three rigid spherical masses thrown into empty space with any initial motions whatever, and abandoned to gravity: to determine their subsequent motions. With two masses the problem is simple enough, being pretty well summed up in Kepler's laws; but with three masses, strange to say, it is so complicated as to be beyond the reach of even modern mathematics. It is a famous problem, known as that of "the three bodies," but it has not yet been solved. Even when it is solved it will be only a close approximation to the case of earth, moon, and sun, for these bodies are not spherical, and are not rigid. One may imagine how absurdly and hopelessly complicated a complete treatment of the motions of the entire solar system would be.

No. 8. Each planet is attracted not only by the sun but by the other planets, hence their orbits are slightly affected by each other.

The subject of planetary perturbation was only just begun by Newton. Gradually (by Laplace and others) the theory became highly developed; and, as everybody knows, in 1846 Neptune was discovered by means of it.

No. 9. He recognized the comets as members of the solar system, obedient to the same law of gravity and moving in very elongated ellipses; so their return could be predicted.

It was a long time before Newton recognized the comets as real members of the solar system, and subject to gravity like the rest. He at first thought they moved in straight lines. It was only in the second edition of the Principia that the theory of comets was introduced.

Halley observed a fine comet in 1682, and calculated its orbit on Newtonian principles. He also calculated when it ought to have been seen in past times; and he found the year 1607, when one was seen by Kepler; also the year 1531, when one was seen by Appian; again, he reckoned 1456, 1380, 1305. All these appearances were the same comet, in all probability, returning every seventy-five or seventy-six years. The period was easily allowed to be not exact, because of perturbing planets. He then predicted its return for 1758, or perhaps 1759, a date he could not himself hope to see. He lived to a great age, but he died sixteen years before this date.

As the time drew nigh, three-quarters of a century afterwards, astronomers were greatly interested in this first cometary prediction, and kept an eager look-out for "Halley's comet." Clairaut, a most eminent mathematician and student of Newton, proceeded to calculate out more exactly the perturbing influence of Jupiter, near which it had passed. After immense labour (for the difficulty of the calculation was extreme, and the mass of mere figures something portentous), he predicted its return on the 13th of April, 1759, but he considered that he might have made a possible error of a month. It returned on the 13th of March, 1759, and established beyond all doubt the rule of the Newtonian theory over comets.



No. 10. Applying the idea of centrifugal force to the earth considered as a rotating body, he perceived that it could not be a true sphere, and calculated its oblateness, obtaining 28 miles greater equatorial than polar diameter.

Here we return to one of the more simple deductions. A spinning body of any kind tends to swell at its circumference (or equator), and shrink along its axis (or poles). If the body is of yielding material, its shape must alter under the influence of centrifugal force; and if a globe of yielding substance subject to known forces rotates at a definite pace, its shape can be calculated. Thus a plastic sphere the size of the earth, held together by its own gravity, and rotating once a day, can be shown to have its equatorial diameter twenty-eight miles greater than its polar diameter: the two diameters being 8,000 and 8,028 respectively. Now we have no guarantee that the earth is of yielding material: for all Newton could tell it might be extremely rigid. As a matter of fact it is now very nearly rigid. But he argued thus. The water on it is certainly yielding, and although the solid earth might decline to bulge at the equator in deference to the diurnal rotation, that would not prevent the ocean from flowing from the poles to the equator and piling itself up as an equatorial ocean fourteen miles deep, leaving dry land everywhere near either pole. Nothing of this sort is observed: the distribution of land and water is not thus regulated. Hence, whatever the earth may be now, it must once have been plastic enough to accommodate itself perfectly to the centrifugal forces, and to take the shape appropriate to a perfectly plastic body. In all probability it was once molten, and for long afterwards pasty.

Thus, then, the shape of the earth can be calculated from the length of its day and the intensity of its gravity. The calculation is not difficult: it consists in imagining a couple of holes bored to the centre of the earth, one from a pole and one from the equator; filling these both with water, and calculating how much higher the water will stand in one leg of the gigantic V tube so formed than in the other. The answer comes out about fourteen miles.

The shape of the earth can now be observed geodetically, and it accords with calculation, but the observations are extremely delicate; in Newton's time the size was only barely known, the shape was not observed till long after; but on the principles of mechanics, combined with a little common-sense reasoning, it could be calculated with certainty and accuracy.

No. 11. From the observed shape of Jupiter or any planet the length of its day could be estimated.

Jupiter is much more oblate than the earth. Its two diameters are to one another as 17 is to 16; the ellipticity of its disk is manifest to simple inspection. Hence we perceive that its whirling action must be more violent—it must rotate quicker. As a matter of fact its day is ten



hours long—five hours daylight and five hours night. The times of rotation of other bodies in the solar system are recorded in a table above.

No. 12. The so-calculated shape of the earth, in combination with centrifugal force, causes the weight of bodies to vary with latitude; and Newton calculated the amount of this variation. 194 lbs. at pole balance 195 lbs. at equator.

But following from the calculated shape of the earth follow several interesting consequences. First of all, the intensity of gravity will not be the same everywhere; for at the equator a stone is further from the average bulk of the earth (say the centre) than it is at the poles, and owing to this fact a mass of 590 pounds at the pole; would suffice to balance 591 pounds at the equator, if the two could be placed in the pans of a gigantic balance whose beam straddled along an earth's quadrant. This is a true variation of gravity due to the shape of the earth. But besides this there is a still larger apparent variation due to centrifugal force, which affects all bodies at the equator but not those at the poles. From this cause, even if the earth were a true sphere, yet if it were spinning at its actual pace, 288 pounds at the pole could balance 289 pounds at the equator; because at the equator the true weight of the mass would not be fully appreciated, centrifugal force would virtually diminish it by 1/289th of its amount.

In actual fact both causes co-exist, and accordingly the total variation of gravity observed is compounded of the real and the apparent effects; the result is that 194 pounds at a pole weighs as much as 195 pounds at the equator.

No. 13. A homogeneous sphere attracts as if its mass were concentrated at its centre. For any other figure, such as an oblate spheroid, this is not exactly true. A hollow concentric spherical shell exerts no force on small bodies inside it.

A sphere composed of uniform material, or of materials arranged in concentric strata, can be shown to attract external bodies as if its mass were concentrated at its centre. A hollow sphere, similarly composed, does the same, but on internal bodies it exerts no force at all.

Hence, at all distances above the surface of the earth, gravity decreases in inverse proportion as the square of the distance from the centre of the earth increases; but, if you descend a mine, gravity decreases in this case also as you leave the surface, though not at the same rate as when you went up. For as you penetrate the crust you get inside a concentric shell, which is thus powerless to act upon you, and the earth you are now outside is a smaller one. At what rate the force decreases depends on the distribution of density; if the density were uniform all through, the law of variation would be the direct distance, otherwise it would be more complicated. Anyhow, the intensity of gravity is a maximum at the surface of the earth, and decreases as you travel from the surface either up or down.

No. 14. The earth's equatorial protuberance, being acted on by the attraction of the sun and moon, must disturb its axis of rotation in a calculated manner; and thus is produced the precession of the equinoxes.

Here we come to a truly awful piece of reasoning. A sphere attracts as if its mass were concentrated at its centre (No. 12), but a spheroid does not. The earth is a spheroid, and hence it pulls and is pulled by the moon with a slightly uncentric attraction. In other words, the line of pull does not pass through its precise centre. Now when we have a spinning body, say a top, overloaded on one side so that gravity acts on it unsymmetrically, what happens? The axis of rotation begins to rotate cone-wise, at a pace which depends on the rate of spin, and on the shape and mass of the top, as well as on the amount and leverage of the overloading.

Newton calculated out the rapidity of this conical motion of the axis of the earth, produced by the slightly unsymmetrical pull of the moon, and found that it would complete a revolution in 26,000 years—precisely what was wanted to explain the precession of the equinoxes. In fact he had discovered the physical cause of that precession.

Observe that there were three stages in this discovery of precession:—

First, the observation by Hipparchus, that the nodes, or intersections of the earth's orbit (the sun's apparent orbit) with the plane of the equator, were not stationary, but slowly moved.

Second, the description of this motion by Copernicus, by the statement that it was due to a conical motion of the earth's axis of rotation about its centre as a fixed point.

Third, the explanation of this motion by Newton as due to the pull of the moon on the equatorial protuberance of the earth.

The explanation could not have been previously suspected, for the shape of the earth, on which the whole theory depends, was entirely unknown till Newton calculated it.

Another and smaller motion of a somewhat similar kind has been worked out since: it is due to the unsymmetrical attraction of the other planets for this same equatorial protuberance. It shows itself as a periodic change in the obliquity of the ecliptic, or so-called recession of the apses, rather than as a motion of the nodes.[21]

No. 15. The waters of the ocean are attracted towards the sun and moon on one side, and whirled a little farther away than the solid earth on the other side: hence Newton explained all the main phenomena of the tides.

And now comes another tremendous generalization. The tides had long been an utter mystery. Kepler likens the earth to an animal, and the tides to his breathings and inbreathings, and says they follow the moon.

Galileo chaffs him for this, and says that it is mere superstition to connect the moon with the tides.

Descartes said the moon pressed down upon the waters by the centrifugal force of its vortex, and so produced a low tide under it.

Everything was fog and darkness on the subject. The legend goes that an astronomer threw himself into the sea in despair of ever being able to explain the flux and reflux of its waters.

Newton now with consummate skill applied his theory to the effect of the moon upon the ocean, and all the main details of tidal action gradually revealed themselves to him.

He treated the water, rotating with the earth once a day, somewhat as if it were a satellite acted on by perturbing forces. The moon as it revolves round the earth is perturbed by the sun. The ocean as it revolves round the earth (being held on by gravitation just as the moon is) is perturbed by both sun and moon.

The perturbing effect of a body varies directly as its mass, and inversely as the cube of its distance. (The simple law of inverse square does not apply, because a perturbation is a differential effect: the satellite or ocean when nearer to the perturbing body than the rest of the earth, is attracted more, and when further off it is attracted less than is the main body of the earth; and it is these differences alone which constitute the perturbation.) The moon is the more powerful of the two perturbing bodies, hence the main tides are due to the moon; and its chief action is to cause a pair of low waves or oceanic humps, of gigantic area, to travel round the earth once in a lunar day, i.e. in about 24 hours and 50 minutes. The sun makes a similar but still lower pair of low elevations to travel round once in a solar day of 24 hours. And the combination of the two pairs of humps, thus periodically overtaking each other, accounts for the well-known spring and neap tides,—spring tides when their maxima agree, neap tides when the maximum of one coincides with the minimum of the other: each of which events happens regularly once a fortnight.

These are the main effects, but besides these there are the effects of varying distances and obliquity to be taken into account; and so we have a whole series of minor disturbances, very like those discussed in No. 7, under the lunar theory, but more complex still, because there are two perturbing bodies instead of only one.

The subject of the tides is, therefore, very recondite; and though one may give some elementary account of its main features, it will be best to defer this to a separate lecture (Lecture XVII).

I had better, however, here say that Newton did not limit himself to the consideration of the primary oceanic humps: he pursued the subject into geographical detail. He pointed out that, although the rise and fall of the tide at mid-ocean islands would be but small, yet on stretches of coast the wave would fling itself, and by its momentum would propel the waters, to a much greater height—for instance, 20 or 30 feet; especially in some funnel-shaped openings like the Bristol Channel and the Bay of Fundy, where the concentrated impetus of the water is enormous.

He also showed how the tidal waves reached different stations in successive regular order each day; and how some places might be fed with tide by two distinct channels; and that if the time of these channels happened to differ by six hours, a high tide might be arriving by one channel and a low tide by the other, so that the place would only feel the difference, and so have a very small observed rise and fall; instancing a port in China (in the Gulf of Tonquin) where that approximately occurs.

In fact, although his theory was not, as we now know, complete or final, yet it satisfactorily explained a mass of intricate detail as well as the main features of the tides.

No. 16. The sun's mass being known, he calculated the height of the solar tide.

No. 17. From the observed heights of spring and neap tides he determined the lunar tide, and thence made an estimate of the mass of the moon.

Knowing the sun's mass and distance, it was not difficult for Newton to calculate the height of the protuberance caused by it in a pasty ocean covering the whole earth. I say pasty, because, if there was any tendency for impulses to accumulate, as timely pushes given to a pendulum accumulate, the amount of disturbance might become excessive, and its calculation would involve a multitude of data. The Newtonian tide ignored this, thus practically treating the motion as either dead-beat, or else the impulses as very inadequately timed. With this reservation the mid-ocean tide due to the action of the sun alone comes out about one foot, or let us say one foot for simplicity. Now the actual tide observed in mid-Atlantic is at the springs about four feet, at the neaps about two. The spring tide is lunar plus solar; the neap tide is lunar minus solar. Hence it appears that the tide caused by the moon alone must be about three feet, when unaffected by momentum. From this datum Newton made the first attempt to approximately estimate the mass of the moon. I said that the masses of satellites must be estimated, if at all, by the perturbation they are able to cause. The lunar tide is a perturbation in the diurnal motion of the sea, and its amount is therefore a legitimate mode of calculating the moon's mass. The available data were not at all good, however; nor are they even now very perfect; and so the estimate was a good way out. It is now considered that the mass of the moon is about one-eightieth that of the earth.

* * * * *

Such are some of the gems extracted from their setting in the Principia, and presented as clearly as I am able before you.

Do you realize the tremendous stride in knowledge—not a stride, as Whewell says, nor yet a leap, but a flight—which has occurred between the dim gropings of Kepler, the elementary truths of Galileo, the fascinating but wild speculations of Descartes, and this magnificent and comprehensive system of ordered knowledge. To some his genius seemed almost divine. "Does Mr. Newton eat, drink, sleep, like other men?" said the Marquis de l'Hopital, a French mathematician of no mean eminence; "I picture him to myself as a celestial genius, entirely removed from the restrictions of ordinary matter." To many it seemed as if there was nothing more to be discovered, as if the universe were now explored, and only a few fragments of truth remained for the gleaner. This is the attitude of mind expressed in Pope's famous epigram:—

"Nature and Nature's laws lay hid in Night, God said, Let Newton be, and all was light."

This feeling of hopelessness and impotence was very natural after the advent of so overpowering a genius, and it prevailed in England for fully a century. It was very natural, but it was very mischievous; for, as a consequence, nothing of great moment was done by England in science, and no Englishman of the first magnitude appeared, till some who are either living now or who have lived within the present century.

It appeared to his contemporaries as if he had almost exhausted the possibility of discovery; but did it so appear to Newton? Did it seem to him as if he had seen far and deep into the truths of this great and infinite universe? It did not. When quite an old man, full of honour and renown, venerated, almost worshipped, by his contemporaries, these were his words:—

"I know not what the world will think of my labours, but to myself it seems that I have been but as a child playing on the sea-shore; now finding some pebble rather more polished, and now some shell rather more agreeably variegated than another, while the immense ocean of truth extended itself unexplored before me."

And so it must ever seem to the wisest and greatest of men when brought into contact with the great things of God—that which they know is as nothing, and less than nothing, to the infinitude of which they are ignorant.

Newton's words sound like a simple and pleasing echo of the words of that great unknown poet, the writer of the book of Job:—

"Lo, these are parts of His ways, But how little a portion is heard of Him; The thunder of His power, who can understand?"

END OF PART I.



PART II

A COUPLE OF CENTURIES' PROGRESS.



NOTES TO LECTURE X

Science during the century after Newton

The Principia published, 1687

Roemer 1644-1710 James Bradley 1692-1762 Clairaut 1713-1765 Euler 1707-1783 D'Alembert 1717-1783 Lagrange 1736-1813 Laplace 1749-1827 William Herschel 1738-1822

Olaus Roemer was born in Jutland, and studied at Copenhagen. Assisted Picard in 1671 to determine the exact position of Tycho's observatory on Huen. Accompanied Picard to Paris, and in 1675 read before the Academy his paper "On Successive Propagation of Light as revealed by a certain inequality in the motion of Jupiter's First Satellite." In 1681 he returned to Copenhagen as Professor of Mathematics and Astronomy, and died in 1710. He invented the transit instrument, mural circle, equatorial mounting for telescopes, and most of the other principal instruments now in use in observatories. He made as many observations as Tycho Brahe, but the records of all but the work of three days were destroyed by a great fire in 1728.

Bradley, Professor of Astronomy at Oxford, discovered the aberration of light in 1729, while examining stars for parallax, and the nutation of the earth's axis in 1748. Was appointed Astronomer-Royal in 1742.



LECTURE X

ROEMER AND BRADLEY AND THE VELOCITY OF LIGHT

At Newton's death England stood pre-eminent among the nations of Europe in the sphere of science. But the pre-eminence did not last long. Two great discoveries were made very soon after his decease, both by Professor Bradley, of Oxford, and then there came a gap. A moderately great man often leaves behind him a school of disciples able to work according to their master's methods, and with a healthy spirit of rivalry which stimulates and encourages them. Newton left, indeed, a school of disciples, but his methods of work were largely unknown to them, and such as were known were too ponderous to be used by ordinary men. Only one fresh result, and that a small one, has ever been attained by other men working according to the methods of the Principia. The methods were studied and commented on in England to the exclusion of all others for nigh a century, and as a consequence no really important work was done.

On the Continent, however, no such system of slavish imitation prevailed. Those methods of Newton's which had been simultaneously discovered by Leibnitz were more thoroughly grasped, modified, extended, and improved. There arose a great school of French and German mathematicians, and the laurels of scientific discovery passed to France and Germany—more especially, perhaps, at this time to France. England has never wholly recovered them. During the present century this country has been favoured with some giants who, as they become distant enough for their true magnitude to be perceived, may possibly stand out as great as any who have ever lived; but for the mass and bulk of scientific work at the present day we have to look to Germany, with its enlightened Government and extensive intellectual development. England, however, is waking up, and what its Government does not do, private enterprise is beginning to accomplish. The establishment of centres of scientific and literary activity in the great towns of England, though at present they are partially encumbered with the supply of education of an exceedingly rudimentary type, is a movement that in the course of another century or so will be seen to be one of the most important and fruitful steps ever taken by this country. On the Continent such centres have long existed; almost every large town is the seat of a University, and they are now liberally endowed. The University of Bologna (where, you may remember, Copernicus learnt mathematics) has recently celebrated its 800th anniversary.

The scientific history of the century after Newton, summarized in the above table of dates, embraces the labours of the great mathematicians Clairaut, Euler, D'Alembert, and especially of Lagrange and Laplace.

But the main work of all these men was hardly pioneering work. It was rather the surveying, and mapping out, and bringing into cultivation, of lands already discovered. Probably Herschel may be justly regarded as the next true pioneer. We shall not, however, properly appreciate the stages through which astronomy has passed, nor shall we be prepared adequately to welcome the discoveries of modern times unless we pay some attention to the intervening age. Moreover, during this era several facts of great moment gradually came into recognition; and the importance of the discovery we have now to speak of can hardly be over-estimated.

Our whole direct knowledge of the planetary and stellar universe, from the early observations of the ancients down to the magnificent discoveries of a Herschel, depends entirely upon our happening to possess a sense of sight. To no other of our senses do any other worlds than our own in the slightest degree appeal. We touch them or hear them never. Consequently, if the human race had happened to be blind, no other world but the one it groped its way upon could ever have been known or imagined by it. The outside universe would have existed, but man would have been entirely and hopelessly ignorant of it. The bare idea of an outside universe beyond the world would have been inconceivable, and might have been scouted as absurd. We do possess the sense of sight; but is it to be supposed that we possess every sense that can be possessed by finite beings? There is not the least ground for such an assumption. It is easy to imagine a deaf race or a blind race: it is not so easy to imagine a race more highly endowed with senses than our own; and yet the sense of smell in animals may give us some aid in thinking of powers of perception which transcend our own in particular directions. If there were a race with higher or other senses than our own, or if the human race should ever in the process of development acquire such extra sense-organs, a whole universe of existent fact might become for the first time perceived by us, and we should look back upon our past state as upon a blind chrysalid form of existence in which we had been unconscious of all this new wealth of perception.

It cannot be too clearly and strongly insisted on and brought home to every mind, that the mode in which the universe strikes us, our view of the universe, our whole idea of matter, and force, and other worlds, and even of consciousness, depends upon the particular set of sense-organs with which we, as men, happen to be endowed. The senses of force, of motion, of sound, of light, of touch, of heat, of taste, and of smell—these we have, and these are the things we primarily know. All else is inference founded upon these sensations. So the world appears to us. But given other sense-organs, and it might appear quite otherwise. What it is actually and truly like, therefore, is quite and for ever beyond us—so long as we are finite beings.

Without eyes, astronomy would be non-existent. Light it is which conveys all the information we possess, or, as it would seem, ever can possess, concerning the outer and greater universe in which this small world forms a speck. Light is the channel, the messenger of information; our eyes, aided by telescopes, spectroscopes, and many other "scopes" that may yet be invented, are the means by which we read the information that light brings.

Light travels from the stars to our eyes: does it come instantaneously? or does it loiter by the way? for if it lingers it is not bringing us information properly up to date—it is only telling us what the state of affairs was when it started on its long journey.

Now, it is evidently a matter of interest to us whether we see the sun as he is now, or only as he was some three hundred years ago. If the information came by express train it would be three hundred years behind date, and the sun might have gone out in the reign of Queen Anne without our being as yet any the wiser. The question, therefore, "At what rate does our messenger travel?" is evidently one of great interest for astronomers, and many have been the attempts made to solve it. Very likely the ancient Greeks pondered over this question, but the earliest writer known to me who seriously discussed the question is Galileo. He suggests a rough experimental means of attacking it. First of all, it plainly comes quicker than sound. This can be perceived by merely watching distant hammering, or by noticing that the flash of a pistol is seen before its report is heard, or by listening to the noise of a flash of lightning. Sound takes five seconds to travel a mile—it has about the same speed as a rifle bullet; but light is much quicker than that.

The rude experiment suggested by Galileo was to send two men with lanterns and screens to two distant watch-towers or neighbouring mountain tops, and to arrange that each was to watch alternate displays and obscurations of the light made by the other, and to imitate them as promptly as possible. Either man, therefore, on obscuring or showing his own light would see the distant glimmer do the same, and would be able to judge if there was any appreciable interval between his own action and the response of the distant light. The experiment was actually tried by the Florentine Academicians,[22] with the result that, as practice improved, the interval became shorter and shorter, so that there was no reason to suppose that there was any real interval at all. Light, in fact, seemed to travel instantaneously.

Well might they have arrived at this result. Even if they had made far more perfect arrangements—for instance, by arranging a looking-glass at one of the stations in which a distant observer might see the reflection of his own lantern, and watch the obscurations and flashings made by himself, without having to depend on the response of human mechanism—even then no interval whatever could have been detected.

If, by some impossibly perfect optical arrangement, a lighthouse here were made visible to us after reflection in a mirror erected at New York, so that the light would have to travel across the Atlantic and back before it could be seen, even then the appearance of the light on removing a shutter, or the eclipse on interposing it, would seem to happen quite instantaneously. There would certainly be an interval: the interval would be the fiftieth part of a second (the time a stone takes to drop 1/13th of an inch), but that is too short to be securely detected without mechanism. With mechanism the thing might be managed, for a series of shutters might be arranged like the teeth of a large wheel; so that, when the wheel rotates, eclipses follow one another very rapidly; if then an eye looked through the same opening as that by which the light goes on its way to the distant mirror, a tooth might have moved sufficiently to cover up this space by the time the light returned; in which case the whole would appear dark, for the light would be stopped by a tooth, either at starting or at returning, continually. At higher speeds of rotation some light would reappear, and at lower speeds it would also reappear; by noticing, therefore, the precise speed at which there was constant eclipse the velocity of light could be determined.



This experiment has now been made in a highly refined form by Fizeau, and repeated by M. Cornu with prodigious care and accuracy. But with these recent matters we have no concern at present. It may be instructive to say, however, that if the light had to travel two miles altogether, the wheel would have to possess 450 teeth and to spin 100 times a second (at the risk of flying to pieces) in order that the ray starting through any one of the gaps might be stopped on returning by the adjacent tooth.

Well might the velocity of light be called instantaneous by the early observers. An ordinary experiment seemed (and was) hopeless, and light was supposed to travel at an infinite speed. But a phenomenon was noticed in the heavens by a quick-witted and ingenious Danish astronomer, which was not susceptible of any ordinary explanation, and which he perceived could at once be explained if light had a certain rate of travel—great, indeed, but something short of infinite. This phenomenon was connected with the satellites of Jupiter, and the astronomer's name was Roemer. I will speak first of the observation and then of the man.



Jupiter's satellites are visible, precisely as our own moon is, by reason of the shimmer of sunlight which they reflect. But as they revolve round their great planet they plunge into his shadow at one part of their course, and so become eclipsed from sunshine and invisible to us. The moment of disappearance can be sharply observed.