of Universal Harmony.

original: "de l'Harmonie Vniuerſelle"

¶ ij

XXIV. That one can represent the squaring of the circle, the doubling of the cube, and all things in the world subject to quantity, by the same means of sounds. 42.
XXV. In what way sound is different from light, and in what way it is similar to it. 44.
XXVI. How the Echo, or the reflection of sounds, is made. 48. Treatise on the Echo. 50.
XXVII. What are the distances and lengths of the vocal line of the Echo: whether one can know the place from which it responds, and of what length the said line must be, to make an Echo of as many syllables as one wishes. 56. See the 22nd Proposition of the third book.
XXVIII. To explain all the figures proper for making artificial Echoes, Conic sections, and their principal properties. 59. This is done in the following Propositions, from the 23rd to the 32nd Proposition of the book on the Voice, and in the fifth Proposition of the book on the utility of harmony; which must be joined to this one.
XXIX. To determine if sounds break, that is to say if they undergo refraction, like light, when they pass through different mediums. 63.
XXX. By how much the sound of the same instrument is deeper original: "graue" in water than in air: and if one can infer from this how much rarer air is than water. 67. See also the first Proposition of the book on utility.
XXXI. Whether a high-pitched original: "aigu" sound is more pleasing and more excellent than a deep one. 71. See also the third Proposition of the 4th book on Composition.
XXXII. To determine if there is any motion in nature, and what is necessary to establish it. 74.
XXXIII. To consider the motions of bodies in general, and the space in which they occur. 76.
XXXIV. To demonstrate whether a string stretched by a peg, or by a weight, is equally stretched in all its parts; and if the force that tensions it communicates its impression sooner and more strongly to the parts that are near it than to those that are further away.

I. To explain the proportion of the speed original: "viteſſe" with which stones and other heavy bodies descend toward the center of the earth; and to show that it is in a squared ratio of the times. 85. On which see the 29th Proposition of the third book, and particularly its second Corollary.
II. If a weight falling from a given space no longer increased the speed acquired at the last point of that space, it would cover a space double the first in an equal time, if it continued its fall at the same speed acquired at the said last point: from which one infers that a falling stone passes through all possible degrees of slowness. 89.
Corollary I. Of the path the weight would take in the last half-second, in falling from the surface of the earth to its center. 91.
Corollary II. To show in what time a stone would fall from the Stars, the Sun, or the Moon, to the surface or the center of the earth. 92.
III. To determine the figure of the motion of heavy bodies that would fall