Lectures de Potentia Restitutiva, or of Spring

The force of the spring is proportional to the distance it is bent.

Counterbalanced in that posture by a power fixed to it and movable with it, draw the line A B C, and let it represent the path in which the end of the spring is to be moved by additional powers original: "powers"; here meaning forces or weights. Draw at the end of it, C, at right angles the line C D, and let C D represent the power that is sufficient to bend or move the end of the spring from A to C. Then draw the line D A, and from any point of the line A C, such as B, B, draw lines parallel to C D, cutting the line D A at E, E. The lines B E, B E, will represent the respective powers requisite to bend the end of the spring from A to B; these lines B E, B E, and C D will be in the same proportion as the length of the bend of the spring A B, A B, and A C.

And because the spring has a particular power at every point of the line of bending A C, therefore, by imagining infinite lines drawn from every point of A C parallel to C D until they touch the line A D, they will all together fill and compose the triangle A C D. The triangle A C D, therefore, will represent the aggregate The total sum or accumulation of the powers of the spring bent from A to C. The lesser triangles A B E, A B E will represent the aggregate of all the powers of the spring bent from A to B, B.

The spring, being bent to any point of the line A C and let go from there, will exert in its return to A all those powers which are equal to the respective ordinates The vertical lines in the geometric figure representing force at a specific point B E, B E in the triangles, the sum of all which make up the triangles A B E, A B E. And the aggregate of the powers with which it returns from any point, such as from C to any point of the space C A (such as to B, B), is equal to the trapezium A four-sided shape with at least two parallel sides; used here to calculate the area representing work done between two points C D E B, C D E B, or the excesses of the greater triangles above the less.

Having therefore shown an image to represent the flexure The act of bending or the state of being bent and the powers, so as to plainly solve and answer all questions and problems concerning them, in the next place I come to represent the velocities appropriated to the several powers. The velocities, then, being always in a subduplicate proportion A mathematical term meaning the ratio of the square roots of...