Newton and other astronomers had found that the line of the moon’s _apsides_ (that is of her greatest and least distances from the earth) moves round to different parts of the heavens with a velocity twice as great as that which the calculation from the law of gravitation seems at first to give. According to the theory, it appeared that this line ought to move round once in eighteen years; according to observation, it moves round once in nine years. This difference, the only obvious failure of the theory of gravitation, embarrassed mathematicians exceedingly. It is true, it was afterwards discovered that the apparent discrepancy arose from a mistake; the calculation, which is long and laborious, was supposed to have been carried far enough to get close to the truth; but it appeared afterwards that the residue which had been left out as insignificant, produced, by an unexpected turn in the reckoning, an effect as large as that which had been taken for the whole. But this discovery was not made till afterwards; and in the mean time the law of the inverse square appeared to be at fault. Clairault tried to remedy the defect by supposing that the force of the earth’s gravity consisted of a large force varying as the square of the distance, and a very small force varying as the fourth power (the square of the square.) By such a supposition, observation and theory could be reconciled; but on the suggestion of it, Buffon came forward with the assertion that the force _could_ not vary according to any other law than the inverse square. His arguments are rather metaphysical than physical or mathematical. Gravity, he urges, is a quality, an emanation; and all emanations are inversely as the square of the distance, as light, odours. To this Clairault replies by asking, how we know that light and odours have their intensity inversely as the square of the distance from their origin: not, he observes, by measuring the intensity, but by _supposing_ these effects to be material emanations. But who, he asks, supposes gravity to be a material emanation _from_ the attracting body. Buffon again pleads that so many facts prove the law of the inverse square, that a single one, which occurs to interfere with this agreement, must be in some manner capable of being explained away. Clairault replies, that the facts do _not_ prove this law to obtain exactly; that small effects, of the same order as the one under discussion, have been neglected; and that therefore the law is only known to be true, _as far_ as such an approximation goes, and no farther.