43. The scope of the law of gravitation.—In all the domain of physical science there is no other law so famous as the Newtonian law of gravitation; none other that has been so dwelt upon, studied, and elaborated by astronomers and mathematicians, and perhaps none that can be considered so indisputably proved. Over and over again mathematical analysis, based upon this law, has pointed out conclusions which, though hitherto unsuspected, have afterward been found true, as when Newton himself derived as a corollary from this law that the earth ought to be flattened at the poles—a thing not known at that time, and not proved by actual measurement until long afterward. It is, in fact, this capacity for predicting the unknown and for explaining in minutest detail the complicated phenomena of the heavens and the earth that constitutes the real proof of the law of gravitation, and it is therefore worth while to note that at the present time there are a very few points at which the law fails to furnish a satisfactory account of things observed. Chief among these is the case of the planet Mercury, the long diameter of whose orbit is slowly turning around in a way for which the law of gravitation as yet furnishes no explanation. Whether this is because the law itself is inaccurate or incomplete, or whether it only marks a case in which astronomers have not yet properly applied the law and traced out its consequences, we do not know; but whether it be the one or the other, this and other similar cases show that even here, in its most perfect chapter, astronomy still remains an incomplete science.
CHAPTER V
THE EARTH AS A PLANET
44. The size of the earth.—The student is presumed to have learned, in his study of geography, that the earth is a globe about 8,000 miles in diameter and, without dwelling upon the "proofs" which are commonly given for these statements, we proceed to consider the principles upon which the measurement of the earth's size and shape are based.
Fig. 25.—Measuring the size of the earth.
In [Fig. 25] the circle represents a meridian section of the earth; P P' is the axis about which it rotates, and the dotted lines represent a beam of light coming from a star in the plane of the meridian, and so distant that the dotted lines are all practically parallel to each other. The several radii drawn through the points 1, 2, 3, represent the direction of the vertical at these points, and the angles which these radii produced, make with the rays of starlight are each equal to the angular distance of the star from the zenith of the place at the moment the star crosses the meridian. We have already seen, in [Chapter II], how these angles may be measured, and it is apparent from the figure that the difference between any two of these angles—e. g., the angles at 1 and 2—is equal to the angle at the center, O, between the points 1 and 2. By measuring these angular distances of the star from the zenith, the astronomer finds the angles at the center of the earth between the stations 1, 2, 3, etc., at which his observations are made. If the meridian were a perfect circle the change of zenith distance of the star, as one traveled along a meridian from the equator to the pole, would be perfectly uniform—the same number of degrees for each hundred miles traveled—and observations made in many parts of the earth show that this is very nearly true, but that, on the whole, as we approach the pole it is necessary to travel a little greater distance than is required for a given change in the angle at the equator. The earth is, in fact, flattened at the poles to the amount of about 27 miles in the length of its diameter, and by this amount, as well as by smaller variations due to mountains and valleys, the shape of the earth differs from a perfect sphere. These astronomical measurements of the curvature of the earth's surface furnish by far the most satisfactory proof that it is very approximately a sphere, and furnish as its equatorial diameter 7,926 miles.