Lectures de Potentia Restitutiva, or of Spring

I have already shown here that the power of all springs is proportionate to the degree of flexure original: "flexure"; the state of being bent or curved: namely, one degree of flexure, or one unit of space bent, has one unit of power; two have two; and three have three, and so on. Every point within the space of flexure has a specific power, and consequently, since there are infinite points in that space, there must be infinite degrees of power.

Consequently, all those powers—beginning from zero and ending at the final degree of tension or bending—when added together into one sum or aggregate, will be in duplicate proportion original: "duplicate proportion"; in modern terms, this refers to a squared relationship or a geometric proportion to the space bent or the degree of flexure.

That is to say: the aggregate of the powers of the spring, when stretched from its resting position through all the intermediate points to a certain space (of whatever length you like), is equal or proportional to the square of one (assuming the said space is infinitely divisible into fractions of one). For two spaces, the aggregate is equal or proportional to the square of two, which is four. For three spaces, it is equal or proportional to the square of three, which is nine, and so on.

Consequently, the aggregate of the first space will be one; of the second space will be three; of the third space will be five; and of the fourth will be seven. This continues onward in an arithmetic proportion original: "Arithmetical proportion"; a sequence where the difference between consecutive terms is constant, these being the degrees or increments by which these aggregates exceed one another.

Therefore, when a spring returns from any degree of flexure to which it has been bent by any power, it receives at every point of the space returned an impulse equal to the power of the spring at that specific point of tension. In returning the whole distance, it receives the entire aggregate of all the forces belonging to the greatest degree of that tension from which it returned.

Thus, a spring bent two spaces receives four degrees of impulse in its return: that is, three in the first space returned, and one in the second. If bent three spaces, it receives in its whole return nine