In the same manner, if the mass A be doubled, it will be attracted by B with a double force, because B exerts the same degree of attraction on every part of the mass A, and any addition which it may receive will not diminish or otherwise affect the influence of B on its former mass. To express this in general arithmetical symbols let _a_ and _b_ express the space through which A and B respectively would be moved towards each other by their mutual attraction. We would then have A × _a_ = B × _b_. Thus, it is a general law of gravitation, that so long as the distance between two bodies remains the same, each will attract and be attracted by the other, in proportion to its mass; and any increase or decrease of the mass will cause a corresponding increase or decrease in the amount of the attraction. (108.) We shall now explain the law, according to which the attraction is changed, by changing the distance between the bodies. At the distance of one mile the body B attracts A with a certain force. At the distance of two miles, the masses not being changed, the attraction of B upon A will be one-fourth of its amount at the distance of one mile. At the distance of three miles, it will be one-ninth of its original amount; at four miles, it is reduced to a sixteenth, and so on. The following table exhibits the diminution of the attraction corresponding to the successive increase of distance: +-----------+---+----+----+----+----+----+----+----+----+ |Distance | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | &c.| +-----------+---+----+----+----+----+----+----+----+----+ |Attraction | 1 | 1/4| 1/9|1/16|1/25|1/36|1/49|1/64| &c.| +-----------+---+----+----+----+----+----+----+----+----+ In ARITHMETIC, that number which is found by multiplying any proposed number by itself, is called its _square_. Thus 4, that is, 2 multiplied by 2, is the square of 2; 9 that is, 3 times 3, is the square of 3, and so on. On inspecting the above table, it will be apparent, therefore, that the attraction of gravitation decreases in the same proportion as the square of the distance from the attracting body increases, the mass of both bodies in this case being supposed to remain the same; but if the mass of either be increased or diminished, the attraction will be increased or diminished in the same proportion. (109.) Hence the _law of gravitation_ may be thus expressed: “The mutual attraction of two bodies increases in the same proportion as their masses are increased, and as the square of their distance is decreased; and it decreases in proportion as their masses are decreased, and as the square of their distance is increased.” This law may be more clearly expressed by means of general symbols. Let _f_ express the force with which a mass weighing 1 lb. will attract another mass weighing 1 lb., at the distance of 1 foot. The force with which they will mutually attract, when removed to the distance expressed in feet by D, will be _f_/D^2