Range of a stone-thrower and the physics of the crossbow
Why projectile 1 flies farther than 8, and how an arrow's forward flight mirrors the spanning motion
At upper left a horizontal arm carries a row of stones numbered 8 down to 1 together with a counterweight m; Leonardo argues that although the counterweight handles each stone with equal effort, stone 1 is cast farther than 8 in proportion to its greater distance from the machine's pivot. The long note transfers the reasoning to a crossbow, drawn at the right with a weight loading its cord: but for the great resistance of the air, the arrow would fly forward exactly as far as the motion that spanned the bow. He adds that the cord has two motions, spanning and release, and that pushing the bow forward at the instant of release increases the arrow's range while pulling it back does the contrary.
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Range grows with distance from the machine's pivot
Each stone in the numbered row is handled by the counterweight m with equal effort, but is not cast to equal distance. The motion of stone 1, being proportional to the machine's power, is thrown as much farther than 8 as it lies farther from the pivot.
Absent air, the arrow flies forward as far as it was drawn back
If the arrow were fixed to the cord and drawn to the nut by a sufficient counterweight, then, but for the great resistance of the air, it would fly forward exactly as much as it moved backward in spanning the bow. The thicker the air, the shorter its course; the cord itself has two motions, one in spanning and one in release.
Moving the bow forward at release lengthens the shot
If you move the crossbow forward as you release, you add speed to the arrow as though its cord were loaded by a greater weight and the bow more widely drawn, so it travels much farther. Pulling the bow backward at discharge does the contrary, losing much force and range.