Table of Propositions

IV. Bodies descending to the center of the earth cannot describe a semicircle: where one sees the line they would describe, if one assumes the daily motion of the earth. 96.

V. To explain the utilities and the practices that can be deduced from the preceding Propositions, as much for MechanicsThe study of machines and the laws of motion, as for several other things; and how one can measure all sorts of heights by the fall of weights, and find the fall in a given time, or the time required, when the fall is given. 99.

VI. To determine if the stars have fallen from the same place by a straight motion, which changed into the circular motion they now have, as GalileoGalileo Galilei (1564–1642), the Italian astronomer who championed the heliocentric model imagines, and to give the manner of computing their falls, their distances, and their circular motions. 105.

VII. To explain the movements of weights on planes inclined to the horizon, and the proportion of their speed: and to examine if falling bodies pass through all possible degrees of slowness. 108.

VIII. To demonstrate if a body can descend by an inclined plane to the center of the earth; and the manner of describing a line so inclined, that the weight always presses upon it equally at every point. 113.

IX. To explain another easier geometrical way to describe a plane of equal inclination: and to examine the figure of the motion of a globe rolling on a horizontal plane, and if rolling is faster than sliding. 119.

X. The plane being inclined on the horizon by a given angle, to determine the force which can support the given weight on the said plane. 121. But the entire Treatise on Mechanics added at the end of the following third book, determines much more exactly and amply everything belonging to this subject, and to several mechanical difficulties.

XI. To determine if the speed of falling bodies increases according to the ratio of the line cut in mean and extreme ratioNow commonly known as the Golden Ratio; where one sees several properties of this section, and the manner of cutting this line to infinity. 125. Add here the 18th Proposition of the fourth book of Instruments. 125. On which see the Notice placed at the end of the fifth book of Composition.

XII. To examine if falling bodies always increase their speed, or if they decrease it; and if there is some point of equality at which they begin to descend at an equal speed. 128.

XIII. To explain several experiments on the fall of bodies toward the center of the earth by the circular line. 131.

XIV. To explain how much faster a ball, which descends or rises through a quarter-circle, goes, and how much heavier it is in one place than in another, and of what length it must be to make each of its turns, or returns, in a given time. 133.

XV. To give the manner of making clocks and watches within the time of one minute of an hour, which divide the day, the hour, and the minutes into as many equal parts as one wishes, and the utility of these clocks. 135.

XVI. To explain how circular motions hinder, or help