is as the moving force directly and the quantity of matter moved inversely,

That is, if M denote the moving force, F the accelerating force, Q the quantity of matter moved, and V the velocity; then (art. 245.) FxV, and MxQV; therefore, MxQF, or M

Fx

252. The quantity of matter in all bodies is in the compound ratio of their magnitudes and densities.

For, when their magnitudes are equal, the quantity of matter they contain is evidently as their densities; and when their densities are the same, the quantity of matter they contain is as their magnitudes.

When, therefore, neither their magnitudes nor densities are equal, the quantity of matter they contain is in a ratio compounded of both.

253. Cor. 1. When the bodies are similar, their masses, or quantities of matter, are as the densities and the cubes of their diameters.

For the magnitudes of bodies are as the cubes of their diameters.

254. Cor. 2. The masses are as the magnitudes of the bodies and their specific gravities.

For the densities of bodies are as their specific gravities. 255. Scholium. Let b denote any body, m its magnitude, d its density, s its specific gravity, and a its diameter; we have, from the three last articles, these general proportions, bmdoms a3d;

Ex. Let the proportional quantities of matter of two bodies A and B be required, the diameter and density of A being 4 and 10, and that of B, 6 and 5 respectively.

In the above proportions we find ba'd; therefore, A: B:: 43 × 10: 63 × 5 :: 640 : 1080:: 16: 27; that is, the quantity of matter in A, is to that in B, as 16 to 27.

On the uniform motions of bodies.

PROPOSITION LX.

256. When bodies move uniformly, the spaces described are as the times they have been in motion.

For, since the bodies move uniformly, equal spaces are passed over in equal times; that is, in twice the time twice the space will be described, and in thrice the time thrice the space, and so on: hence when bodies move uniformly, &c.

PROPOSITION LXI.

257. The spaces described in the same time by bodies moving uniformly, are as the velocities with which the bodies

move.

For the greater the velocity the greater will be the space described; increase the velocity and the space will be increased in the same proportion.

PROPOSITION LXII.

258. When bodies move uniformly, the spaces described are in a ratio compounded of their velocities and the times they have been in motion jointly.

For (art. 256.) when the velocities are equal, the spaces described are as the times, and (art. 257.) when the times are equal, the spaces are as the velocities with which they move; when, therefore, neither the times nor the velocities are equal, the spaces described will be in a ratio compounded of both.

Thus let V, v, be the velocities of two bodies moving with different uniform motions, T, t, the times they have been in motion, and let S, s, denote the spaces described in those times; then S:8:: TV: tv, by the proposition.

Ex. Suppose A and B to move uniformly, and let 6: 5 be the ratio of the times they have been in motion, and 2 : 3 the ratio of their velocities; then S:s::6×2:5×3::12:15 :: 4:5; that is, the spaces described are as 4 to 5.

259. Cor. 1. Because s xtv, by division vo ; that is,

t

the velocity is as the space directly and time reciprocally. Er. Let A move through 5 feet in 3", and B through 9 feet in 7"; to find the ratio of their velocities, both moving uniformly.

Here we have V:v::::: 35 : 27.

260. Cor. 2. Again, since s xtv, we have t∞;

time is as the space directly and velocity inversely.

; or the

Er. Let the velocity of A be to that of B as 5:4; to compare the times in which they will describe 9 and 7 feet respectively.

Here Tt::: 1 :: 36: 35.

261. Scholium. If we call the force f, the body or quantity of matter b, its velocity v, the momentum generated in the body m, the space described s, and the time the body has been in motion t; then, the force being supposed impulsive or which is the same, the motion of the body uniform; we have (art. 250.) ƒxmxbv, and (art. 258.) stv; from whence we may easily derive the following proportions.

262. When a body moves uniformly, the space described may be represented by the area of a right-angled parallelogram, one of whose sides represents the time of the body's motion and the other its velocity.

For in computations relating to forces, motions, times, velocities, or the spaces described,, we generally consider only the proportions of those quantities to each other, and not the absolute quantities themselves; in such cases our computations are much, facilitated by substituting mathematical quantities instead of physical, provided they have the same proportions to each other; and then we may transfer our reasoning on the mutual proportions of the exponents to the mutual proportion of the physical quantities which they represent. Thus in uniform motions, the magnitude of the spaces described depends on the velocities with which the bodies move, and on the times they have been in motion, in the very same manner that rectangles depend on their bases and altitudes; and, therefore, in our calculations respecting uniform motions, we may express the spaces described by rectangles whose bases and altitudes have the same proportion to each other with the times and velocities. By thus cailing in geometry to our assistance, we are often enabled more easily to explain the most complicated properties of the doctrine' of motion.

ON THE MOTION OF SOUND.

263. It has been proved from various experiments that sound flies uniformly at the rate of 1142 feet in a second of