How Galileo Analyzed Projectile Motion

His insight was that for the projectile (neglecting air resistance), the horizontal motion was like a ball rolling on a smooth horizontal floor, that is, constant velocity; its vertical motion was identical to that of the same ball just falling, which he had established was at constant acceleration. Then he put them together: we've plotted the separate motions as "ghost balls" along the axes.

Galileo's original text (in translation), with the original diagram is here, my lecture is here.