of Universal Harmony.

This section continues the index of propositions from the "Book on the Motion of Bodies."

perpendiculars; and to determine if the Earth, in moving, would cast aside the bodies that fall, or that are upon it. 137.
XVII. To examine if the Earth, turning at a given speed like a wheel, would throw stones along its tangent, or otherwise. One sees here the marvelous properties of the angle of contingency The angle between a curve and its tangent line, and the examination of the arguments of Galileo. 241.
XVIII. To explain the difference in 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 stones, and that of other wheels. 148.
XX. Whether one can demonstrate that the motion of falling bodies is simple and perpendicular; and whether the circular motion of the Earth would hinder 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 moves or remains stationary, and whether one runs or stays still. 153.
XXII. To determine if a cannonball fired horizontally from the top of a tower reaches the ground at the same moment as a ball falling perpendicularly from the top of said tower. 155.

I. The ratio of the number of vibrations original: "retours," referring to the back-and-forth oscillations 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 vibration and return of harmonic strings, and the reason for their diminution. 160.
III. Whether strings and other bodies making vibrations and returns rest at the points of their reflection. 163.
IV. Why the Lute string often passes beyond its center, or its line of rest, without stopping there. 165.
V. To determine the duration of each vibration and return of said string, and how many it makes before coming to rest. 166. This number V is repeated again in the proposition that follows, and the others proceed correctly in their order from then on.
VI. To explain the manner of counting the vibrations and returns of every string of the Lute, Viol, etc., and where the subtlety of the eye and ear ends. 169.
VII. At what moment and in what place during the vibrations or returns of the string the sound is made, and if it is higher in pitch original: "aigu" at the beginning than at the end of the tremors. 171.
VIII. To explain the other differences and the different forces of each vibration 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. One sees marvelous observations here. 175.
XI. Of what length and thickness cylinders must be to produce sounds in which one can discern the low and high pitches; and why they do not keep