108 THE MATHEMATICAL PRINCIPLES [BooK III. diameter of the earth, according to PicarVs mensuration, is 19615800 Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will be higher at the equator than at the poles by 85472 feet, or 17^miles. And its height at the equator will be about 19658600 feet, and at the poles 19573000 feet. If, the density and periodic time of the diurnal revolution remaining the same, the planet was greater or less than the earth, the proportion of the centrifugal force to that of gravity, and therefore also of the diameter be twixt the poles to the diameter at the equator, would likewise remain the game. But if the diurnal motion was accelerated or retarded in any pro portion, the centrifugal force would be augmented or diminished nearly in the same duplicate proportion ; and therefore the difference of the diame ters will be increased or diminished in the same duplicate ratio very nearly. And if the density of the planet was augmented or diminished in any pro portion, the force of gravity tending towards it would also be augmented or diminished in the same proportion : and the difference of the diameters contrariwise would be diminished in proportion as the force of gravity is augmented, and augmented in proportion as the force of gravity is dimin ished. Wherefore, since the earth, in respect of the fixed stars, revolves in 23h. 56', but Jupiter in 9h. 56', and the squares of their periodic times are as 29 to 5, and their densities as 400 to 94 £, the difference of the diameters 29 400 1 of Jupiter will be to its lesser diameter as — X ^^ X ^Tm to 1; or as 1 to 9 f, nearly. Therefore the diameter of Jupiter from east to west is to its diameter from pole to pole nearly as 10 £• to 9