But had the observation of the philosophers gone so far, they had still been at a distance from the truth. They might have perceived a likeness between the track of those bodies in motion and a parabola, and concluded, from analogy, that all projectiles delineated that curve in the air; but they could never have realized their conjectures by mathematical demonstration, without previously knowing the law of _acceleration_ in falling bodies: a discovery reserved for the next century, and for GALILEO[7], one of the greatest ornaments of it. It was he who first investigated the effects of _gravity_ on falling bodies, and upon that foundation demonstrated, that all projectiles would move in a parabola in a non-resisting medium. And as he made little account of the resistance of the air, whose properties were then imperfectly known, he proved that a ball shot horizontally would, in its flight, describe half a parabola; and when the piece had an elevation above the horizon, the ball would describe a whole parabola, supposing it to fall on the plane of the battery. By the same method of reasoning he shewed, that whatever the ranges of the projected body, or the elevations of the piece were, the ball would still trace that curve line, of a greater or lesser amplitude, by the time it descended to the level of the place from whence it came. Thus far went GALILEO, confining his projections to the horizontal plane of the battery; but TORRICELLI his disciple soon after carried the theory farther, by tracing the shot to its fall, whether that place was above or below the plane; and still found, by geometrical deductions, that it flew in a parabola of a larger or a smaller amplitude, according to the angle of elevation of the piece, and the strength of the powder.