Catapult - Codex Atlanticus Folio 140a

The basic design of the catapult had been used for hundreds of years before Leonardo da Vinci recorded this device in his notes. The construction depicted in his drawing does not show a leaf spring used in the design, but clearly, one would be required to carry the necessary loads without breaking.
I have shown a two-part leaf spring in this interpretation of the design similar to the English longbow used to great effect in earlier battles at Agincourt and Crecy.

The exact materials intended for use are not known but Yew was commonly used for the inner leaf under compression and Hickory for the outer leaf under tension. Leonardo sought to improve the efficiency and range of earlier designs, by using the large wooden leaf spring as an accumulator to power the catapult. Both ends of the bow are connected by a cord to the same side of the central drum which when wound through the worm and wheel arrangement to store the energy for the catapult in the bow arms.

Leonardo's genius was in his use of the worm and wheel to load the leaf spring, this led to a simple design that on the one hand allowed the catapult to be loaded with an easy winding action of the handle which gave a large mechanical advantage, and at the same time stopped the drum unwinding. Release of the catapult was achieved by simply disengaging the worm from the wheel by dropping it out of the way. On release, the drum was freed to spin freely under the power store in the spring arms and swing the main catapult arm with its projectile, to its release point at the top of its arc.

At the end of the catapult arm a dished cup holds the projectile, the dishing holds the projectile in the dish longer until the centrifugal force allowed the projectile to roll out and be released. This would have had the effect of flattening the arc while allowing the throwing arm to accelerate to a faster release speed.