TUTOR. The periods of the planets, or the times they take to complete their revolutions in their orbits, are exactly known; and the mean distance of the earth from the sun has been also ascertained. Here, then, we have the periods of all, and the mean distance of one, to find the distances of the rest; which may be found by attending to the following proportion: As the square of the period of any one planet, Is to the cube of its mean distance from the sun; So is the square of the period of any other planet, To the cube of its mean distance. The cube root of this quotient will be the distance sought. PUPIL. Here again I find myself at a loss, as I have not learnt to extract the cube root. TUTOR. I will give you [18]Doctor Turner’s rule, which I think will answer your purpose. [Footnote 18: Young Geometrician’s Companion.] “First, having set down the given number, or resolvend, make a dot over the unit figure, and so on over every third figure (towards the left hand in whole numbers, but towards the right hand in decimals); and so many dots as there are, so many figures will be in the root. Next, seek the nearest cube to the first period; place its root in the quotient, and its cube set under the first period. Subtract it therefrom; and to the remainder bring down one figure only of the next period, which will be a dividend. Then, square the figure put in the quotient, and multiply it by 3, for a divisor. Seek how often this divisor may be had in the dividend, and set the figure in the quotient, which will be the second place in the root. Now, cube the figures in the root, and subtract it from the two first periods of the resolvend; and to the remainder bring down the first figure of the next period, for a new dividend. Square the figures in the quotient, and multiply it by 3, for a new divisor; then proceed in all respects as before, till the whole is finished.” The following example will, I trust, make it clear to you. EXAMPLE. It is required to find the cube root of 15625. . . 15625 (25 8 ───── 12) 76 15625 ───── ..... ═════