111. THE SUN'S DISTANCE FROM THE EARTH.--To the astronomer the sun presents problems of the highest consequence and apparently of very diverse character, but all tending toward the same goal: the framing of a mechanical explanation of the sun considered as a machine; what it is, and how it does its work. In the forefront of these problems stand those numerical determinations of distance, size, mass, density, etc., which we have already encountered in connection with the moon, but which must here be dealt with in a different manner, because the immensely greater distance of the sun makes impossible the resort to any such simple method as the triangle used for determining the moon's distance. It would be like determining the distance of a steeple a mile away by observing its direction first from one eye, then from the other; too short a base for the triangle. In one respect, however, we stand upon a better footing than in the case of the moon, for the mass of the earth has already been found (Chapter IV) as a fractional part of the sun's mass, and we have only to invert the fraction in order to find that the sun's mass is 329,000 times that of the earth and moon combined, or 333,000 times that of the earth alone. If we could rely implicitly upon this number we might make it determine for us the distance of the sun through the law of gravitation as follows: It was suggested in § 38 that Newton proved Kepler's three laws to be imperfect corollaries from the law of gravitation, requiring a little amendment to make them strictly correct, and below we give in the form of an equation Kepler's statement of the Third Law together with Newton's amendment of it. In these equations-- _T_ = Periodic time of any planet; _a_ = One half the major axis of its orbit; _m_ = Its mass; _M_ = The mass of the sun; _k_ = The gravitation constant corresponding to the particular set of units in which _T_, _a_, _m_, and _M_ are expressed. (Kepler) a^{3}/T^{2} = h; (Newton) a^{3}/T^{2} = k (M + m). Kepler's idea was: For every planet which moves around the sun, _a^{3}_ divided by _T^{2}_ always gives the same quotient, _h_; and he did not concern himself with the significance of this quotient further than to note that if the particular _a_ and _T_ which belong to any planet--e. g., the earth--be taken as the units of length and time, then the quotient will be 1. Newton, on the other hand, attached a meaning to the quotient, and showed that it is equal to the product obtained by multiplying the sum of the two masses, planet and sun, by a number which is always the same when we are dealing with the action of gravitation, whether it be between the sun and planet, or between moon and earth, or between the earth and a roast of beef in the butcher's scales, provided only that we use always the same units with which to measure times, distances, and masses.