Catapult Range: Widening the Arc to Double the Throw
If a given force sends the dart a third of a mile, how much wider to reach half a mile?
Leonardo poses a ballistics problem about a catapult: if a set force moving through a given opening throws the dart one third of a mile, by how much must the opening (the sweep of the throwing arm) be widened to send the same dart half a mile. The upper-right drawing shows a catapult with a vertical post and a swinging arm releasing a projectile whose flight is traced as an arc rising to a point labelled d. A lower sketch gives an 'analogous figure', a trajectory arching over a mound, illustrating the same throw-and-fall geometry.
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Doubling a catapult's range by opening its arc
Leonardo takes the force a and the opening (arc) b - a at which the engine is set. He asks: if this force drives the dart one third of a mile, how far must he open the 'percussion' of the arm to make the dart travel half a mile. The question links the width of the arm's sweep to the distance thrown.
Trajectory of the thrown dart
The catapult at upper right releases a projectile whose path is drawn as a curved arc climbing to the point labelled d. A second, 'analogous figure' below repeats the flight as a trajectory arching over a mound, so the two diagrams model how far the projectile falls for a given launch.