Notebook on Mechanics and Natural Philosophy

This is proven by the proportionality of the pyramid, which cuts the semidiameter? original: "semidiametro"; the radius of a circle, which Leonardo is relating to the cross-sections of a cone or pyramid. proportionally to their distance from the apex of the said pyramid. Since these circles ① are similar to that proportionality, it follows that the movement of the force setting such circles in motion will be proportional among themselves.

It follows below: if the motion is doubled, the effect is quadrupled; or if it is tripled, it will be 9 times greater. This is because if the motion is 1/2, the effect is 1/4; and if the motion is 1/3, the effect is 1/9; and if the motion is 1/4, the effect is 1/16; and so it continues through its degrees, etc.

And if the force is increased, the motion will be 1/2; and if the force is 1/2, the motion will be 1/4, etc. Because a force of 1/2 cannot put into effect half of the motion, but only 1/4, since the motion resulting from force follows the proportionality of its inverse doubling. Leonardo is observing that doubling a force does not necessarily double the resulting speed in a simple linear fashion, prefiguring modern debates on the relationship between force, mass, and acceleration. The same thing happens with weight; this weight shall be of as much value as its force.

If a man pushes a cart with all his strength through a space of 10 braccia original: "braccia"; a common Italian Renaissance unit of length, literally "arms," measuring approximately 23 inches or 58 centimeters depending on the city., and you give him another companion, it will not be true that they will push it for a space of 20 braccia, nor that it will move with doubled velocity over the said 10 braccia. Leonardo is refuting a common Aristotelian misconception of his time, noting that adding more "movers" does not result in a simple linear increase in distance or speed.