HYDRAULIC ARCHITECTURE BOOK I.

PROPERTY OF THE PARALLELOGRAM

13. If one has two Powers original: "Puissances"; in 18th-century mechanics, this term refers to what we now call forces or vectors of effort. expressed by the lines AB and DB, such that the first, AB, is capable of making body B traverse the side BC of a parallelogram in the same time that the second, DB, would make the same body traverse the other side, BE; then these two forces acting together on body B will make it traverse the diagonal BF of the same parallelogram in a time equal to that which each Power, AB or DB individually, would have employed to make body B traverse each side, BC or BE.

When the two forces AB and DB act together on body B along the directions BC and BE, the direction of body B will be composed of these two directions. Now, if one divides the time that each force would take to make body B traverse side BC or BE into a number of equal instants, it is clear that with the two forces acting together on body B, force AB will tend to make it traverse side BC in the same time that force DB tends to make it traverse side BE. If we suppose that in the first instant, force AB would have made body B traverse the space BH, while force DB would have made it traverse the space HI, the body will be found at point I; and the spaces BH and HI, however small one may imagine them, will always be in the same ratio as the forces AB and DB, or as BC and BE (11). Thus, because of the similar triangles original: "triangles semblables"; triangles that have the same shape and proportions but different sizes. BHI and BCF, the body being at I will be at a point on the diagonal BF, and will even have followed it continuously until I. If, in the second instant, force AB would have made body B traverse space IK, in the same time that force DB would have made it traverse space KL, the body will still be found at point L of the Diagonal; but all lines like BH and IK from B to F, taken together, will still be equal to BC. Thus, the time the body took to traverse the Diagonal BF by the two forces acting together will be equal to the time that each force individually would have taken to make body B traverse side BC or BE.

14. Since the forces AB and DB are capable of making body B traverse the spaces BC and BE in equal times, it follows