Force as Line and Arc: an Eightfold Proportion
The straight line a d is octuple the arc b c, hence eightfold in power of force
On this recto Leonardo restates the rule linking derivative and primary force to the lengths of their motions, now rendered geometrically. He lets the circumferential arc b c stand for the motion of one force and the straight line a d for the other, concluding that because a d is eight times as long as b c, it is also eight times as great in power of force. The page carries a wheel bearing a suspended weight on a long cord at upper left, the quarter-circle construction of a d and b c at upper right, and a long hinged spring or lath device below.
On this page
Derivative and primary force set as line and arc
The power of the derivative force stands to that of the primary as the motion of the primary stands to that of the derivative. Leonardo assigns the arc b c to one force's motion and the line a d to the other, so that the ratio of the lines gives the ratio of the forces.
The circumferential arc b c against the line a d
The construction pairs a quarter-circle arc, marked b c, with a straight line a d drawn as the vertical and horizontal frame of the quadrant. The proportion of a d to b c is read directly off these lengths.
The 8 : 1 ratio of lengths and forces
The figure is annotated with the numbers 8 and 1. Since a d is octuple to b c in length, Leonardo argues, it is also octuple in power of force, so the forces stand in an eight-to-one ratio.
Weighted wheel and hinged spring apparatus
At upper left a wheel carries a small weight suspended on a long cord, and at the foot of the page a long hinged bar mechanism holds a ball or weight between paired laths. These illustrate the forces the geometric proportion describes.