Lectures de Potentia Restitutiva, or of Spring

degrees of impulse: that is, five in the first space returned, three in the second, and one in the third. So, if bent ten spaces, it receives in its whole return one hundred degrees of impulse: namely, nineteen in the first, seventeen in the second, fifteen in the third, thirteen in the fourth, eleven in the fifth, nine in the sixth, seven in the seventh, five in the eighth, three in the ninth, and one in the tenth.

Now, the comparative velocities of any body moved are in subduplicate proportion original: "subduplicate proportion"; this refers to the square root of the ratio to the aggregates or sums of the powers by which it is moved. Therefore, the velocities of the whole spaces returned are always in the same proportions with those spaces—they being both subduplicate to the powers—and consequently, all the times shall be equal.

Next, for the velocities of the parts of the space returned, they will always be proportionate to the roots of the aggregates of the powers impressed in every one of these spaces. For in the last instance, where the spring is supposed bent ten spaces, the velocity at the end of the first space returned shall be as the root of 19; at the end of the second as the root of 36 (that is, of 19 + 17); at the end of the third as the root of 51 (that is, of 19 + 17 + 15); at the end of the fourth as the root of 64 (that is, of 19 + 17 + 15 + 13); and at the end of the tenth, or the whole, as the root of 100 (that is, as the square root of 19 + 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1, which is equal to 100).

Now, since the velocity is in the same proportion to the root of the space as the root of the space is to the time, it is easy to determine the particular time in which every one of these spaces is passed; for by dividing the spaces by the corresponding velocities, the quotients give the particular times.

To explain this more intelligibly, let A in the fourth figure represent the end of a spring not bent, or at least