But we have to turn these angular measurements into miles. To effect this we have only to work a simple rule of three sum. It will easily be understood that, as the angular diameter of the earth seen from the moon is to the angular diameter of the moon seen from the earth, so is the diameter of the earth in miles to the diameter of the moon in miles. The diameter of the earth we know to be 7912 miles: putting this therefore in its proper place in the proportion sum, and duly working it out by the schoolboy’s rule, we get:—
1°. 54′. 5″ : 31′. 9″ :: 7912 miles. : 2160 miles.
And 2160 miles is therefore the diameter of the lunar globe.
Knowing the diameter, we can easily obtain the other elements of magnitude. According to the well-known relation of the diameter of a sphere to its area, we find the area of the moon to be 14,657,000 square miles: or half that number, 7,328,500 miles, as the area of the hemisphere at any one time presented to our view. And similarly, from the relation of the solidity of a sphere to its diameter, we find the solid contents of the moon to be 5276 millions of cubic miles of matter.
Comparing these data with corresponding dimensions of the earth, we find that the diameter of the moon is 1/3·665; the area 1/13·4245; and the volume 1/49·1865, of the respective elements of the earth. Those who prefer a graphical to a numerical comparison, may judge of the sizes of the two bodies by the accompanying illustration (Fig. 10). To gain an idea of their distance from each other it is necessary to suppose the two discs in the diagram to be 30 inches apart; the real distance of the moon from the earth being about 238,790 miles at its mean position.
Next, we come to what is technically termed the _mass_, but what in common language we may call the _weight_ of the moon. It is important to know this, because the weight of a body taken in connection with its size furnishes us with a knowledge of its density, or the specific gravity of the material of which it is composed. But it is not quite so easy to determine the mass as the dimensions of the moon: to _measure_ it, we have seen is easy enough; to _weigh_ it is a comparatively difficult matter. To solve the problem we have to appeal to Newton’s law of universal gravitation. This law teaches us that every particle of matter in the universe attracts every other particle with a force which is _directly proportional to the mass_, and inversely proportional to the distance of the attracting particles. There are several methods by which this law is applied to the measurement of the mass of the moon. One of the simplest is by the agency of the Tides. We know that the moon, attracting the waters, produces a certain amount of elevation of the aqueous covering of the earth; and we know that the sun produces also a like elevation, but to a much smaller extent, by reason of its much greater distance. Now measuring accurately the heights of the solar and lunar tides, and making allowance for the difference of distance of the sun and moon from the earth, we can compare directly the effect that is due to the sun with the effect that is due to the moon: and since the masses of the two bodies are just in proportion to the effects they produce, it is evident that we have a comparison between the mass of the sun and that of the moon; and knowing what is the sun’s mass we can, by simple proportion, find that of the moon. Another method is as follows:—The moon is retained in her orbital path by the attraction of the earth; if it were not for this attraction she would fly off from her course in a tangential line. She has thus a constant tendency to quit her orbit, which the earth’s attraction as constantly overcomes. It is evident from this that the earth pulls the moon towards itself by a definite amount in every second of time. But while the earth is pulling the moon, the moon is also pulling the earth: they are pulling each other together; and moreover each is exerting a pull which is _proportional to its mass_. Knowing, then, the mass of the earth, which we do with considerable accuracy, we can find what share of the whole pulling force is due to it, the residue being the moon’s share: the proportion which this residue bears to the earth’s share gives us the proportion of the moon’s mass to that of the earth, and hence the mass of the moon.