It is a well-known quirk of pre-modern theories of motion that objects can only have one kind of motion applied at any time. An arrow would therefore first move forward in a straight line, and then stop its forward movement and drop straight down.

Pardon, but do you have a reference for this? Anyone firing an arrow in an arc at 45°, perhaps to hit a distant troop formation or perhaps simply to see how far he can shoot it, could clearly see that his arrow never stops moving away from him, even as it shifts from moving up to moving down.

A reply to /u/link0007

Yes and no. Even as far as the late 1500s you could find books such as this one by Tartaglia (1500-1557), which contain images such as this one. So, it can be said that belief in the Aristotelian theory of trajectory was widespread in that era.

However, in practice, artillerists had developed tools with which they calculate trajectories. Typically these are tables of very crude ranges depending on the cannon, the weight of cannonball, and quality and quantity of gunpowder. But to a large extent, each cannon was unique as there was a large variance in manufacturing of the cannon itself, the cannonball, and the gunpowder. So it was not a matter of "one shot, one hit" rather "adjust each shot consecutively" or "create table of ranges for each class of cannon" or combinations thereof. You can see such examples in Diego Ufano's Tratado dela artilleria here.

Field artillery pieces tend to be fired at fairly flat trajectories as the objective is not to hit the target directly, but rather when you use solid shot you want the ball to bounce and hit as many men/horses/equipment as possible. Bombards fire at higher trajectories so they require better aiming, but their target are fixed positions, walls, encampments, buildings. So there is the opportunity to adjust the shots consecutively.

By the late 1600s and 1700s, you are correct that there started a tendency to experiment and to move towards a more systematic and scientific approach.