1. Galileo had, in Cosimo de Medici, a powerful politician as a patron; he was consequently rewarded for putting his talents to ude in the service of military science. The study of projectile motion was a central focus of the theory of ballistics, and once the use of gunpowder made cannons useful tools of warfare throughout Europe, it was important to control the placement of ammunition fire on the battlefield, a question that ballistics addressed. Before Galileo, it was thought that a ball fired from the mouth of cannon was provided with an impetus (a force) that was consumed as it traveled farther from the point of fire until it was eventually exhausted. Only then did the heaviness of the ball counteract the impressed impulse to allow it to fall to the ground. (See this link to a diagram that illustrates this; it comes from a 1669 publication.) Galileo's work corrected this view.
2. The model for this is the motion of a ball fired from the mouth of a cannon fired horizontally.
3. This is what later became Newton's First Law of Motion: "a body continues in a state of rest, or motion with a constant velocity, unless compelled to change by an unbalanced force."
4. A "semiparabolic line" is one half of a parabolic curve.
5. "Quadruple" means four times as large, "nonuple" is nine times as large.
6. A "duplicate ratio" is one in which the ratio is multiplied by itself. That is, given the ratio A : B, the ratio (A)(A) : (B)(B) is the duplicate ratio. In other words, the duplicate ratio is the ratio of the squares.
7. See the Corollary to Theorem II at the end of the previous reading. Interpreting Galileo's reasoning here in modern terms, the object has been set in motion along the horizontal (AB) at a constant speed and will maintain this speed in this direction as it is not impressed by any other force except gravity which draws it downward at a perpendicular direction. We can then measure in both directions the distance traveled by the object from the point of its fall at B. In the horizontal direction, the object moves at constant speed v, so the distance it travels x is given by x = vt, t being the time starting at B. In the vertical direction, the object is in free fall, so it obeys the laws described in the selection we read previously: the distance y that it falls from the height of B is y = (1/2)gt2, where g is the constant acceleration due to gravity. It follows that the path of the object is decribed by its x- and y-coordinates relative to the position of B. Solving the equation involving x for t and substituting into the equation for y, we get
was known to the Greeks; in particular, it appears as Proposition 3 in
Archimedes' Quadrature of the
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