9. But now the chord having recovered its rectilinear situation I k L, the following motion will be changed, for the point K, which before advanced with a motion more and more accelerated, though by the force it has acquired it will go on to move the same way as before, till it has advanced near as far forwards, as it was at first drawn backwards; yet the motion of it will henceforth be gradually lessened. The effect of which upon the particles a, b, c, d, e, f, g will be, that by the time the chord has made its utmost advance, and is upon the return, these particles will be put into a contrary rangement; so that f shall be nearer to g, than e to f, and e nearer to f than d to e; and the like of the rest, till you come to the first particles a, b, whose distance will then be nearly or quite what it was at first. All which will appear as follows. The present distance between a and b is such, that the elastic power, by which a repels b, is strong enough to maintain that distance, though a advance with the velocity, with which the string resumes its rectilinear figure; and the motion of the particle a being afterwards slower, the present elasticity between a and b will be more than sufficient to preserve the distance between them. Therefore while it accelerates b it will retard a. The distance b c will still diminish, till b come about as near to c, as it is from a at present; for after the distances a b and b c are become equal, the particle b will continue its velocity superior to that of c by its own power of inactivity, till such time as the increase of elasticity between b and c more than shall be between a and b shall suppress its motion: for as the power of inactivity in b made a greater elasticity necessary on the side of a than on the side of c to push b forward, so what motion b has acquired it will retain by the same power of inactivity, till it be suppressed by a greater elasticity on the side of c, than on the side of a. But as soon as b begins to slacken its pace the distance of b from c will widen as the distance a b has already done. Now as a acts on b, so will b on c, c on d, &c. so that the distances between all the particles b, c, d, e, f, g will be successively contracted into the distance of a from b, and then dilated again. Now because the time, in which the chord describes this present half of its vibration, is about equal to that it took up in describing the former; the particles a, b will be as long in dilating their distance, as before in contracting it, and will return nearly to their original distance. And farther, the particles b, c, which did not begin to approach so soon as a, b, are now about as much longer, before they begin to recede; and likewise the particles c, d, which began to approach after b, c, begin to separate later. Whence it appears that the particles, whose distance began to be lessened, when that of a, b was first enlarged, viz. the particles f, g, should be about their nearest distance, when a and b have recovered their prime interval. Thus will the particles a, b, c, d, e, f, g have changed their situation in the manner asserted. But farther, as the particles f, g or F, G gradually approach each other, they will move by degrees the succeeding particles to as great a length, as the particles A, B did by a like approach. So that, when the chord has made its greatest advance, being arrived into the situation I ϰ L, the particles moved by it will have the rangement noted by the points α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ. Where α, β are at the original distance of the particles in the line A H; ζ, η are the nearest of all, and the distance ν χ is equal to that between α and β.

10. By this time the chord I ϰ L begins to return, and the distance between the particles α and β being enlarged to its original magnitude, α has lost all that force it had acquired by its motion, being now at rest; and therefore will return with the chord, making the distance between α and β greater than the natural; for β will not return so soon, because its motion forward is not yet quite suppressed, the distance β γ not being already enlarged to its prime dimension: but the recess of α, by diminishing the pressure upon β by its elasticity, will occasion the motion of β to be stopt in a little time by the action of γ, and then shall β begin to return: at which time the distance between γ and δ shall by the superior action of δ above β be enlarged to the dimension of the distance β γ, and therefore soon after to that of α β. Thus it appears, that each of these particles goes on to move forward, till its distance from the preceding one be equal to its original distance; the whole chain α, β, γ, δ, ε, ζ, η, having an undulating motion forward, which is stopt gradually by the excess of the expansive power of the preceding parts above that of the hinder. Thus are these parts successively stopt, as before they were moved; so that when the chord has regained its rectilinear situation, the expansion of the parts of the air will have advanced so far, that the interval between ζ η, which at present is most contracted, will then be restored to its natural size: the distances between η and θ, θ and λ, λ and μ, μ and ν, ν and χ, being successively contracted into the present distance of ζ from η, and again enlarged; so that the same effect shall be produced upon the parts beyond ζ η, by the enlargement of the distance between those two particles, as was occasioned upon the particles α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ, by the enlargement of the distance α β to its natural extent. And therefore the motion in the air will be extended half as much farther as at present, and the distance between ν and χ contracted into that, which is at present between ζ and η, all the particles of the air in motion taking the rangement expressed in figure 111. by the points α, β, γ, δ, ε, ζ, η, θ, λ, μ, ν, χ, ϰ, ρ, σ, τ, φ wherein the particles from α to χ have their distances from each other gradually diminished, the distances between the particles ν, χ being contracted the most from the natural distance between those particles, and the distance between α, β as much augmented, and the distance between the middle particles ζ, η becoming equal to the natural. The particles π, ρ, ω
τ, φ which follow χ, have their distances gradually greater and greater, the particles ν, χ, π, ρ, σ, τ, φ being ranged like the particles a, b, c, d, e, f, g, or like the particles ζ, η, θ, λ, μ, ν, χ in the former figure. Here it will be understood, by what has been before explained, that the particles ζ, η being at their natural distance from each other, the particle ζ is at rest, the particles ε, δ, λ, β, ϰ between them and the string being in motion backward, and the rest of the particles η, θ, λ, μ, ν, χ, π, ρ, σ, τ in motion forward: each of the particles between η and χ moving faster than that, which immediately follows it; but of the particles from χ to φ, on the contrary, those behind moving on faster than those, which precede.

11. But now the string having recovered its rectilinear figure, though it shall go on recoiling, till it return near to its first situation I K L, yet there will be a change in its motion; so that whereas it returned from the situation I ϰ L with an accelerated motion, its motion shall from hence be retarded again by the same degrees, as accelerated before. The effect of which change upon the particles of the air will be this. As by the accelerated motion of the chord α contiguous to it moved faster than β, γ, so as to make the interval α β greater than the interval β γ, and from thence β was made likewise to move faster than γ, and the distance between β and γ rendered greater than the distance between γ and δ, and so of the rest; now the motion of α being diminished, β shall overtake it, and the distance between α and β be reduced into that, which is at present between β and γ, the interval between β and γ being inlarged into the present distance between α and β; but when the interval β γ is increased to that, which is at present between α and β γ the distance between γ and δ shall be enlarged to the present distance between γ and β, and the distance between δ and ι inlarged into the present distance between γ and δ; and the same of the rest. But the chord more and more slackening its pace, the distance between α and β shall be more and more diminished; and in consequence of that the distance between β and γ shall be again contracted, first into its present dimension, and afterwards into a narrower space; while the interval γ δ shall dilate into that at present between α and β, and as soon as it is so much enlarged, it shall contract again. Thus by the reciprocal expansion and contraction of the air between α and ζ, by that time the chord is got into the situation I K L, the interval ζ η shall be expanded into the present distance between α and β; and by that time likewise the present distance of α from β will be contracted into their natural interval: for this distance will be about the same time in contracting it self, as has been taken up in its dilatation; seeing the string will be as long in returning from its rectilinear figure, as it has been in recovering it from its situation I ϰ L. This is the change which will be made in the particles between α and ζ. As for those between ζ and χ, because each preceding particle advances faster than that, which immediately follows it, their distances will successively be dilated into that, which is at present between ζ and η. And as soon as any two particles are arrived at their natural distance, the hindermost of them shall be stopt, and immediately after return, the distances between the returning particles being greater than the natural. And this dilatation of these distances shall extend so far, by that time the chord is returned into its first situation I K L, that the particles ι χ shall be removed to their natural distance. But the dilatation of ν χ shall contract the interval τ φ into that at present between ν and χ, and the contraction of the distance between those two particles τ and φ will agitate a part of the air beyond; so that when the chord is returned into the situation I K L, having made an intire vibration, the moved particles of the air will take the rangement expressed by the points, l, m, n, o, p, q, r, s, t, u, w, x, y, z, 1, 2, 3, 4, 5, 6, 7, 8: in which l m, are at the natural distance of the particles, the distance m n greater than l m and n o greater than m n, and so on, till you come to q r, the widest of all: and then the distances gradually diminish not only to the natural distance, as w x, but till they are contracted as much as χ τ was before; which falls out in the points 2, 3, from whence the distances augment again, till you come to the part of the air untouched.

12. This is the motion, into which the air is put, while the chord makes one vibration, and the whole length of air thus agitated in the time of one vibration of the chord our author calls the length of one pulse. When the chord goes on to make another vibration, it will not only continue to agitate the air at present in motion, but spread the pulsation of the air as much farther, and by the same degrees, as before. For when the chord returns into its rectilinear situation I k L, l m shall be brought into its most contracted state, q r now in the state of greatest dilatation shall be reduced to its natural distance, the points w, x now at their natural distance shall be at their greatest distance, the points 2, 3 now most contracted enlarged to their natural distance, and the points 7, 8 reduced to their most contracted state: and the contraction of them will carry the agitation of the air as far beyond them, as that motion was carried from the chord, when it first moved out of the situation I K L into its rectilinear figure. When the chord is got into the situation I ϰ L, l m shall recover its natural dimensions, q r be reduced to its state of greatest contraction, w x brought to its natural dimension, the distance 2 3 enlarged to the utmost, and the points 7, 8 shall have recovered their natural distance; and by thus recovering themselves they shall agitate the air to as great a length beyond them, as it was moved beyond the chord, when it first came into the situation I ϰ L. When the chord is returned back again into its rectilinear situation, l m shall be in its utmost dilatation, q r restored again to its natural distance, w x reduced into its state of greatest contraction, 2 3 shall recover its natural dimension, and 7 8 be in its state of greatest dilatation. By which means the air shall be moved as far beyond the points 7, 8, as it was moved beyond the chord, when it before made its return back to its rectilinear situation; for the particles 7, 8 have been changed from their state of rest and their natural distance into a state of contraction, and then have proceeded to the recovery of their natural distance, and after that to a dilatation of it, in the same manner as the particles contiguous to the chord were agitated before. In the last place, when the chord is returned into the situation I K L, the particles of air from l to δ shall acquire their present rangement, and the motion of the air be extended as much farther. And the like will happen after every compleat vibration of the string.

13. Concerning this motion of sound, our author shews how to compute the velocity thereof, or in what time it will reach to any proposed distance from the sonorous body. For this he requires to know the height of air, having the same density with the parts here at the surface of the earth, which we breath, that would be equivalent in weight to the whole incumbent atmosphere. This is to be found by the barometer, or common weatherglass. In that instrument quicksilver is included in a hollow glass cane firmly closed at the top. The bottom is open, but immerged into quicksilver contained in a vessel open to the air. Care is taken when the lower end of the cane is immerged, that the whole cane be full of quicksilver, and that no air insinuate itself. When the instrument is thus fixed, the quicksilver in the cane being higher than that in the vessel, if the top of the cane were open, the fluid would soon sink out of the glass cane, till it came to a level with that in the vessel. But the top of the cane being closed up, so that the air, which has free liberty to press on the quicksilver in the vessel, cannot bear at all on that, which is within the cane, the quicksilver in the cane will be suspended to such a height, as to balance the pressure of the air on the quicksilver in the vessel. Here it is evident, that the weight of the quicksilver in the glass cane is equivalent to the pressure of so much of the air, as is perpendicularly over the hollow of the cane; for if the cane be opened that the air may enter, there will be no farther use of the quicksilver to sustain the pressure of the air without; for the quicksilver in the cane, as has already been observed, will then subside to a level with that without. Hence therefore if the proportion between the density of quicksilver and of the air we breath be known, we may know what height of such air would form a column equal in weight to the column of quicksilver within the glass cane. When the quicksilver is sustained in the barometer at the height of 30 inches, the height of such a column of air will be about 29725 feet; for in this case the air has about 1/870 of the density of water, and the density of quicksilver exceeds that of water about 13⅔ times, so that the density of quicksilver exceeds that of the air about 11890 times; and so many times 30 inches make 29725 feet. Now Sir Isaac Newton determines, that while a pendulum of the length of this column should make one vibration or swing, the space, which any sound will have moved, shall bear to this length the same proportion, as the circumference of a circle bears to the diameter thereof; that is, about the proportion of 355 to 113[267]. Only our author here considers singly the gradual progress of sound in the air from particle to particle in the manner we have explained, without taking into consideration the magnitude of those particles. And though there requires time for the motion to be propagated from one particle to another, yet it is communicated to the whole of the same particle in an instant: therefore whatever proportion the thickness of these particles bears to their distance from each other, in the same proportion will the motion of sound be swifter. Again the air we breath is not simply composed of the elastic part, by which sound is conveyed, but partly of vapours, which are of a different nature; and in the computation of the motion of sound we ought to find the height of a column of this pure air only, whose weight should be equal to the weight of the quicksilver in the cane of the barometer, and this pure air being a part only of that we breath, the column of this pure air will be higher than 29725 feet. On both these accounts the motion of sound is found to be about 1142 feet in one second of time, or near 13 miles in a minute, whereas by the computation proposed above, it should move but 979 feet in one second.

[14.] We may observe here, that from these demonstrations of our author it follows, that all sounds whether acute or grave move equally swift, and that sound is swiftest, when the quicksilver stands highest in the barometer.

15. Thus much of the appearances, which are caused in these fluids from their gravitation toward the earth. They also gravitate toward the moon; for in the last chapter it has been proved, that the gravitation between the earth and moon is mutual, and that this gravitation of the whole bodies arises from that power acting in all their parts; so that every particle of the moon gravitates toward the earth, and every particle of the earth toward the moon. But this gravitation of these fluids toward the moon produces no sensible effect, except only in the sea, where it causes the tides.

[16.] That the tides depend upon the influence of the moon has been the receiv’d opinion of all antiquity; nor is there indeed the least shadow of reason to suppose otherwise, considering how steadily they accompany the moon’s course. Though how the moon caused them, and by what principle it was enabled to produce so distinguish’d an appearance, was a secret left for this philosophy to unfold: which teaches, that the moon is not here alone concerned, but that the sun likewise has a considerable share in their production; though they have been generally ascribed to the other luminary, because its effect is greatest, and by that means the tides more immediately suit themselves to its motion; the sun discovering its influence more by enlarging or restraining the moon’s power, than by any distinct effects. Our author finds the power of the moon to bear to the power of the sun about the proportion of 4½ to 1. This he deduces from the observations made at the mouth of the river Avon, three miles from Bristol, by Captain Sturmey, and at Plymouth by Mr. Colepresse, of the height to which the water is raised in the conjunction and opposition of the luminaries, compared with the elevation of it, when the moon is in either quarter; the first being caused by the united actions of the sun and moon, and the other by the difference of them, as shall hereafter be shewn.

17. That the sun should have a like effect on the sea, as the moon, is very manifest; since the sun likewise attracts every single particle, of which this earth is composed. And in both luminaries since the power of gravity is reciprocally in the duplicate proportion of the distance, they will not draw all the parts of the waters in the same manner; but must act upon the nearest parts stronger, than upon the remotest, producing by this inequality an irregular motion. We shall now attempt to shew how the actions of the sun and moon on the waters, by being combined together, produce all the appearances observed in the tides.

18. To begin therefore, the reader will remember what has been said above, that if the moon without the sun would have described an orbit concentrical to the earth, the action of the sun would make the orbit oval, and bring the moon nearer to the earth at the new and full, than at the quarters[268]. Now our excellent author observes, that if instead of one moon, we suppose a ring of moons, contiguous and occupying the whole orbit of the moon, his demonstration would still take place, and prove that the parts of this ring in passing from the quarter to the conjunction or opposition would be accelerated, and be retarded again in passing from the conjunction or opposition to the next quarter. And as this effect does not depend on the magnitude of the bodies, whereof the ring is composed, the same would hold, though the magnitude of these moons were so far to be diminished, and their number increased, till they should form a fluid[269]. Now the earth turns round continually upon its own center, causing thereby the alternate change of day and night, while by this revolution each part of the earth is successively brought toward the sun, and carried off again in the space of 24 hours. And as the sea revolves round along with the earth itself in this diurnal motion, it will represent in some sort such a fluid ring.