Of Universal Harmony

...the perpendiculars; and to determine if the earth in motion would throw aside bodies that fall, or those that were already upon it. 137.

XVII. To examine if the earth, turning at a given speed like a wheel, would throw stones off along its tangent, or otherwise. Here one sees the marvelous properties of the angle of contingencyThe angle between a curve and its tangent line; a significant concept in 17th-century geometry regarding how curves relate to straight lines., and the examination of Galileo's arguments. 241.

XVIII. To explain the difference of projections that can be made by the different speeds of the same wheel, and of two or more wheels of various sizes. 146.

XIX. To determine the force of the earth turning in twenty-four hours to throw off stones, and that of other wheels. 148.

XX. If one can demonstrate that the movement of falling bodies is simple and perpendicular; and if the circular motion of the earth would impede the said perpendicular motion, if it is opposed to it. 150.

XXI. Why bodies falling from the top of a ship's mast, or those thrown upward, fall in the same place, whether the ship is moving or remains stationary, and whether one is running or standing still. 153.

XXII. To determine if a cannonball fired horizontally from the top of a tower reaches the ground at the same moment that a ball falls perpendicularly from the top of the said tower. 155.

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I. The ratio of the number of vibrationsoriginal: "retours"; literally "returns," referring to the back-and-forth oscillation of a string. of all kinds of strings is the inverse of their lengths. 157.

II. To explain the different speeds of the parts of each oscillation and return of harmonic strings, and the reason for their diminution. 160.

III. Whether strings and other bodies making oscillations and returns come to rest at the points of their reflection The momentary pause at the furthest point of a vibration before the string moves back in the opposite direction.. 163.

IV. Why a lute string often passes beyond its center, or its line of rest, without stopping there. 165.

V. To determine the duration of each oscillation and return of the said string, and how many it makes before coming to rest. 166. This number V is repeated again in the following Proposition, and the others proceed correctly in their order from then on.

VI. To explain the manner of counting the oscillations and returns of every string of the Lute, Viol, etc., and where the sensitivity original: "subtilité"; referring to the physical limits of human perception. of the eye and the ear ends. 169.

VII. At what moment and in what place of the string's oscillations or returns the sound is produced, and whether it is higher in pitch original: "aigu"; sharp or high-pitched. at the beginning than at the end of the vibrations. 171.

VIII. To explain the other differences and the various forces of each oscillation or return of the strings. 172.

IX. To determine all the ratios of the length of bodies in relation to their sounds. 174.

X. Given several different sounds, to find the cylinders that produce them; and given the cylinders, to find their sounds. Here one sees marvelous observations. 175.

XI. Of what length and thickness the cylinders must be to produce sounds where one can discern the low and high pitch; and why they do not maintain...