Part 20 (1/2)
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.