Lectures de Potentia Restitutiva, or of Spring

the powers—that is, as the root of the impressed powers. Since the impressed powers are represented as the trapezium A four-sided shape with at least two parallel sides; here representing the area under a curve. (or the excess of the triangle or square of the whole space to be passed over the square of the space yet unpassed), we can visualize this geometrically.

If a circle, such as CGGF, is described around the center A with the radius AC (C being the point from which the spring is released), and vertical lines called ordinates are drawn from any point along the path CA (such as from B, B) to the circle (as BG, BG), these lines BG, BG will represent the velocity of the spring returning from C to B, B, and so on. These ordinates are always in the same proportion as the roots of the trapeziums CDEB, CDEB. For if we set AC equal to a and AB equal to b, the line BG will always be equal to $\sqrt{aa - bb}$, because the square of the ordinate is always equal to the rectangle formed by the intercepted parts of the diameter.

Having thus found the velocities—namely BG, BG, and AF—we must now find the corresponding times. On the diameter AC, draw a parabola CHF whose vertex is C and which passes through the point F. The ordinates of this parabola, BH, BH, and AF, are in the same proportion as the roots of the spaces CB, CB, and CA. Then, by making the ratio of GB to HB the same as the ratio of HB to IB, and drawing the curve CIIIF through the points C, I, I, and F, the respective ordinates of this curve shall represent the proportionate time that the spring spends in returning through the spaces CB, CB, and CA.

If the powers or stiffness of the spring are greater than what I previously supposed, and therefore must be expressed by the triangle C d e A, then the velocities will be the ordinates in an ellipse such as C γ γ N, which is larger than the circle. This will also occur if the power remains the same but the mass original: "bulk" moved by the spring is less. In that case, the S-shaped line representing time will meet the line AF at a point such as X, located inside the point F. But if the powers of the spring are weaker than I supposed, then C δ ε A will represent the powers, and C γ γ O will represent the ellipsis of velocity.