Halley observed a fine comet in 1682, and calculated its...o...b..t 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 pa.s.sed. After immense labour (for the difficulty of the calculation was extreme, and the ma.s.s 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.

[Ill.u.s.tration: FIG. 71.--Well-known model exhibiting the oblate spheroidal form as a consequence of spinning about a central axis. The bra.s.s strip _a_ looks like a transparent globe when whirled, and bulges out equatorially.]

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 circ.u.mference (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

[Ill.u.s.tration: FIG. 72.--Jupiter.]

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 lat.i.tude; 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 ma.s.s 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 ma.s.s 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 h.o.m.ogeneous sphere attracts as if its ma.s.s were concentrated at its centre. For any other figure, such as an oblate spheroid, this is not exactly true. A hollow concentric spherical sh.e.l.l 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 ma.s.s 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 sh.e.l.l, 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 ma.s.s 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 pa.s.s 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 ma.s.s 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...o...b..t (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 superst.i.tion 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 ma.s.s, 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 const.i.tute 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 ma.s.s of intricate detail as well as the main features of the tides.

No. 16. The sun's ma.s.s 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 ma.s.s of the moon.

Knowing the sun's ma.s.s 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 acc.u.mulate, as timely pushes given to a pendulum acc.u.mulate, the amount of disturbance might become excessive, and its calculation would involve a mult.i.tude 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 ma.s.s of the moon. I said that the ma.s.ses 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 ma.s.s.

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 ma.s.s 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 att.i.tude of mind expressed in Pope's famous epigram:--

"Nature and Nature's laws lay hid in Night, G.o.d 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:--