SCHOLIU M.

The laft propofitions refpect the motions of light and founds. For fince light is propagated in right lines, it is certain that it cannot confift in action alone, (by Prop. 41 and 42.) As to founds, fince they arife from tremulous bodies, they can be nothing elfe but pulfes of the air propagated thro' it, (by Prop. 43) And this is confirmed by the tremors, which founds, if they be loud and deep, excite in the bodies near them, as we experience in the found of drums, For quick and fhort tremors are lefs eafily excited. But it is well known, that any founds, falling upon ftrings in unifon with the fonorous bodies, excite tremors in thofe ftrings. This is alfo confirmed from the velocity of founds. For fince the specific gravities of rain-water and quick-filver are to one another as about 1 to 133, and when the mercury in the barometer is at the height of 30 inches of our measure, the specific gravities of the air and of rain-water are to one another as about 1 to 870: therefore the specific gravity of air and quickfilver are to each other as I to 11890. Therefore when the height of the quick-filver is at 30 inches, a height of uniform air, whofe weight would be fufficient to comprefs our air to the denfity we find it to be of, must be equal to 356700 inches or 29725 feet of our meafure. And this is that very height of the medium, which I have called A in the conftruction of the foregoing propofition. A circle whofe radius is 29725 feet is 186768 feet in circumference. And fince a pendulum 39 inches in length compleats one ofcillation, compofed of its going and return, in two feconds of time, as is commonly known; it follows that a pendulum 29725 feet or 356700 inches in length will perform a like ofcillation in 190 feconds. Therefore

in that time a found will go right onwards 186768 feet, and therefore in one fecond 979 feet.

But in this computation we have made no allowance for the craffitude of the folid particles of the air, by which the found is propagated inftantaneously. Becaufe the weight of air is to the weight of water as 1 to 870, and because falts are almoft twice as dense as water; if the particles of air are fuppofed to be of near the fame denfity as thofe of water or falt, and the rarity of the air arifes from the intervals of the particles; the diameter of one particle of air will be to the interval between the centres of the particles, as I to about 9 or 10, and to the interval between the particles themselves as 1 to 8 or 9. Therefore to 979 feet, which, according to the above calculation, a found will advance forward in one fecond of time, we may add 222, or about 109 feet, to compenfate for the craffitude of the particles of the air; and then a found will go forward

about 1088 feet in one fecond of time.

Moreover, the vapors floating in the air, being of another fpring, and a different tone, will hardly, if at all, partake of the motion of the true air in which the founds are propagated. Now if these vapors remain unmoved, that motion will be propagated the fwifter thro' the true air alone, and that in the fubduplicate ratio of the defect of the matter. So if the atmosphere confift of ten parts of true air and one part of vapors, the motion of founds will be swifter in the fubduplicate ratio of 11 to 10, or very nearly in the entire ratio of 21 to 20, than if it were propagated thro' eleven parts of true air and therefore the motion of founds above difcovered must be encreased in that ratio. By this means the found will pafs thro' 1142 feet in one fecond

of time.

These things will be found true in fpring and autumn, when the air is rarefied by the gentle warmth of thofe feafons, and by that means its elastic force be

comes

comes fomewhat more intenfe. But in winter, when the air is condensed by the cold, and its elastic force is fomewhat remitted, the motion of founds will be flower in a fubduplicate ratio of the denfity; and on the other hand, fwifter in the fummer,

Now by experiments it actually appears that founds do really advance in one fecond of time about 1142 feet of English measure, or 1070 feet of French meafure.

The velocity of founds being known, the intervals of the pulses are known alfo. For M. Sauveur, by fome experiments that he made, found that an open pipe about five Paris feet in length, gives a found of the fame tone with a viol-ftring that vibrates a hundred times in one fecond. Therefore there are near 100 pulfes in a space of 1070 Paris feet, which a found runs over in a fecond of time; and therefore one pulfe fills up a fpace of about 10 Paris feet, that is, about twice the length of the pipe. From whence it is probable, that the breadths of the pulfes, in all founds made in open pipes, are equal to twice the length of the pipes.

10

Moreover, from the corollary of prop. 47. appears the reason, why the founds immediately ceafe with the motion of the fonorous body, and why they are heard no longer when we are at a great diftance from the fonorous bodies, than when we are very near them. And befides, from the foregoing principles it plainly ap→ pears how it comes to pafs that founds are fo mightily encreased in speaking-trumpets. For all reciprocal motion uses to be encreafed by the generating cause at each And in tubes hindering the dilatation of the founds, the motion decays more flowly, and recurs more forcibly; and therefore is the more encreased by the new motion impreffed at each return. And these are the principal phænomena of founds.

SECTION IX.

Of the circular motion of fluids.

HYPOTHESIS.

The refiftance, arising from the want of lubri city in the parts of a fluid, is, cæteris paribus, proportional to the velocity with which the parts of the fluid are feparated from each other.

PROPOSITION LI. THEOREM XXXVIII. If a folid cylinder infinitely long, in an uniform and infinite fluid, revolve with an uniform motion about an axis given in pofition, and the fluid be forced round by only this impulfe of the cylinder, and every part of the fluid perfevere uniformly in its motion; I fay, that the periodic times of the parts of the fluid are as their distances from the axis of the cylinder.

Let AFL (Pl.9. Fig. 2.) be a cylinder turning uniformly about the axis S, and let the concentric circles BG M, CHN, DIO, EKP, &c. divide the fluid into innumerable concentric cylindric folid orbs of the

fame

fame thickness. Then, because the fluid is homogeneous, the impreffions which the contiguous orbs make upon each other mutually, will be (by the hypothefis) as their tranflations from each other, and as the contiguous fuperficies upon which the impreffions are made. If the impreffion made upon any orb be greater or lefs on its concave, than on its convex fide, the ftronger impreffion will prevail, and will either accelerate or retard the motion of the orb, according as it agrees with, or is contrary to the motion of the fame. Therefore, that every orb may perfevere uniformly in its motion, the impreffions made on both fides must be equal, and their directions contrary. Therefore fince the impreffions are as the contiguous fuperficies, and as their tranflations from one another; the tranflations will be inversely as the fuperficies, that is, inverfely as the diftances of the fuperficies from the axis. But the differences of the angular motions about the axis, are as those tranflations applied to the distances, or as the tranflations directly and the diftances inverfely; that is, joining these ratio's together, as the fquares of the diftances inverfely. Therefore if there be erected the lines Aa, Bb, Cc, Dd, Ee, &c. perpendicular to the feveral parts of the infinite right line SABCDE Q and reciprocally proportional to the fquares of SA, SB, SC, SD, SE, &c. and thro' the extremities of those perpendiculars there be fuppofed to pafs an hyperbolic curve; the fums of the differences, that is, the whole angular motions, will be as the correspondent fums of the lines Aa, Bb, Cc, Dd, Ee, that is, (if to conftitute a medium uniformly fluid, the number of the orbs be encreafed and their breadth diminished in infinitum) as the hyperbolic area's Aa Q, BbQ, CcQ, DdQ, EeQ, &c. analogous to the fums. And the times, reciprocally proportional to the angular motions, will be alfo reciprocally proportional to thofe areas. Therefore the periodic time of any particle as D, is reciprocally as the area DdQ, that is, (as

appears