This motion is entirely independent of the form of the earth. Now, if we assume that the sun and moon give the equinoctial points a retrograde motion on the ecliptic, we must deduct the influence of the planets. We may then calculate the mean disturbance by subtracting the latter from the former—the difference is settled by both theory and observation to be 50´´.1 annually. This motion of the equinoxes is called the precession of the equinoxes. Its consideration forms a very important element in the estimation of time, as the position of the various fixed stars, though so very distant, are all affected in longitude by this quantity of 50´´.1—being an increase of longitude. Therefore, if we were to calculate the position of any given star in order to get a transit for mean time, or true time, we must take this quantity into consideration. The increase is so great that the earliest astronomers, even with their imperfect modes of observation, detected it. Hipparchus, 128 years before Christ, compared his own observations with those of Timocharis, 153 years before. He found the solution of the problem the same as Diophantus found the solution of the squares and cubes, by analysis. In the time of Hipparchus, the sun was at a point 30° in advance of its present position, for it then entered into the constellation of Aries near the vernal equinox.
At the present time the position of the equinoctial points shows a recession of the whole, 30° 1´ 40´´.2. At this rate of motion the constellations called the Signs of the Zodiac are some distance from the divisions of the ecliptic that bear their names. At the rate of 50´´.1 the whole revolution of the equinoctial points will be accomplished in 25,868 years; but this is again modified because the precession must vary in different centuries for the following reasons: the sun’s motion is direct, the precession retrograde; therefore, the sun arrives at the equator sooner than he does at the same star of observation. Now, the tropical year is 365d. 5h. 48´ 49´´.7; and as the precession is exactly 50´´.1, we must suppose it takes some time for the sun to move through that arc. By direct observation it is found that the time required for such translation is 20´ 19´´.6. By adding this amount to the tropical year we have the sidereal year of 365d. 6h. 9´ 9´´.6 in mean solar days. This amount of precession has been on the increase since the days of its first recorder, Hipparchus, as the augmentation amounts to no less than 0.´´455. By adding that to the known precession we find that the civil year is shorter now by 4´´.21 than in his time; but, as a great division of time, the year can be changed by this cause not more than 43.´´
The action of the moon on the accumulation of matter at the earth’s equator is a source of disturbance that in very accurate observations for time should be eliminated. Thus the moon, with the conjoint action of the sun, depending on relative position, causes the pole of the equator to describe a small ellipse in the heavens with axes of 18´´.5 for the major, and 13´´.674 for the minor; the longer axis being directed to the pole of the ecliptic. This inequality has a period of 19 years,—it being equal to the revolution of the nodes of the lunar orbit. The combination of these disturbances changes, by a small quantity, the position of the polar axis of the earth in regard to the stars, but not in regard to its own surface. With so many disturbing causes, we must add that of Jupiter, whose attraction is diminishing the obliquity of the ecliptic by 0´´.457 according to M. Bessel.
The results of all these forces must affect the position of all the stars and planets as seen from our earth. Their longitudes being reckoned from the equinoxes, the precession of these points would increase the longitude; but as it affects all the stars and planets alike, it would make no real or apparent change in their relative positions. Nutation, however, affects the celestial latitudes and longitudes, as the real motion of the earth’s polar axis changes the relative positions. So great is the change that our present pole star has changed from 12° to 1° 24; in regard to the celestial pole, the gradual approximation will continue until it is with 0° 30´, after which it will leave the pole indefinitely until in 12,934 years α Lyræ will be the pole star.
So far we have given only the causes that affect the meridian, and consequently our standard for time; but that point being established for the yearly and diurnal revolutions, it becomes necessary to find some means to divide the day into minute fractional parts, such as seconds and parts of seconds. This, it has been stated, is effected by means of an isochronous pendulum. On this instrument no comment is required but of the causes that disturb its accuracy much is needed. In 1672, at Cayenne, the astronomer Richter, while taking transits of fixed stars, found his clock lost 2´ 28´´ per day. This was an error that arrested his attention, and he immediately attributed it to some variation in the length of the pendulum—due to other causes than atmospheric changes and expansion. He determined the length of a pendulum beating seconds in that latitude, which was 5° N. in South America. He found that that pendulum was shorter than one beating seconds in Paris, by 0833+ of an inch. Now, if the earth was a sphere, the attraction of gravitation at all places on its surface would be equal, and the oscillations of a pendulum would also be equal, + or - the disturbing effect of centrifugal force—an amount that can be easily determined. The real reason of the variation is found in the configuration of the earth.
The amount of the attraction of gravitation at any point of the earth’s surface is found by the distance traversed by any body during the first second of its fall. The pendulum is a falling body, and may be by the same analysis reasoned on that pertains to the laws of gravitation; the centrifugal force is measured by any deflection from a tangent to the earth’s surface in a second.
It follows that the centrifugal force at the poles, where there is the least motion, would not be equal to the force of gravitation, and at the equator must be exactly equal; but the deflection of a circle from a tangent measures the intensity of the earth’s attraction, and is equal to the versed sine of the arc described during that time, the velocity of the earth’s rotation being known, the value of the arc is deducible. The centrifugal force at the equator is equal to ¹⁄₂₈₉th part of the attraction of gravitation. Again, the uniformity of the earth’s mass becomes an object of consideration. Assuming that the figure of the earth is an ellipsoid of rotation, we will show the relation that form bears to the equal oscillation of a pendulum.
Taking the earth as a homogeneous mass, analysis gives us the certainty that if the intensity of gravitation at the equator be taken as unity, the increase of gravity to the poles eliminating the differences of the centrifugal force must be = to 2.5, the ratio of the centrifugal force to that of gravitation at the equator. Now, taking the 2.5 of .346 = 1/115.2, this then must be the total increase of gravitation. Did we know the exact amount of increase at every point, from the equator to the poles, a perfect map of the form of the earth could be produced from calculation; experiment being from physical causes totally impracticable. The following analysis, quoted from an eminent physicist, gives a very lucid idea of the reasoning:
“If the earth were a homogeneous sphere without rotation, its attraction on bodies on its surface would be everywhere equal. If it be elliptical and of variable density, the force of gravity ought to increase in intensity from the equator to the pole as unity plus a constant quantity multiplied into the square of the sine of the latitude. But for a spheroid in rotation the centrifugal varies by the law of mechanics, as the square of the sine of the latitude from the equator, where it is greatest, to the poles, where it is least. And as it tends to make bodies fly off the surface, it diminishes the force of gravity by a small quantity. Hence, by gravitation, which is the difference of these two forces, the fall of bodies ought to be accelerated from the equator to the poles proportionably to the square of the sine of the latitude, and the weight of the body ought to increase in that ratio.”
Assuming the above reasoning to be correct, it follows, that the rate of descent of falling bodies will be accelerated in the transition from the equator to the poles. Now, it has been before stated that the pendulum is a falling body; therefore, with the same length of pendulum, the oscillations at the pole should be faster than at the equator. Theory, in this case, is verified; for it has been proved by experiments, repeated again and again, that a pendulum oscillating 86,400 times in a mean day at the equator, will give the same number of oscillations at any other point, provided its length is made longer in the exact ratio as the square of the sine of the latitude.