Chapter XIV: Introduction (10)

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.

19. BUT as the water of the sea does not move round with so much swiftness, as would carry it about the center of the earth in the circle it now describes, without being supported by the body of the earth; it will be necessary to consider the water under three distinct cases. The first case shall suppose the water to move with the degree of swiftness, required to carry a body round the center of the earth disingaged from it in a circle at the distance of the earth’s semidiameter, like another moon. The second case is, that the waters make but one turn about the axis of the earth in the space of a month, keeping pace with the moon; so that all parts of the water should preserve continually the same situation in respect of the moon. The third case shall be the real one of the waters moving with a velocity between these two, neither so swift as the first case requires, nor so slow as the second.

20. IN the first case the waters, like the body which they equalled in velocity, by the action of the moon would be brought nearer the center under and opposite to the moon, than in the parts in the middle between these eastward or westward. That such a body would so alter its distance by the moon’s action upon it, is clear from what has been mentioned of the like changes in the moon’s motion caused by the sun[270]. And computation shews, that the difference between the greatest and least distance of such a body would not be much above 4½ feet. But in the second case, where all the parts of the water preserve the same situation continually in respect of the moon, the weight of those parts under and opposite to the moon will be diminished by the moon’s action, and the parts in the middle between these will have their weight increased: this being effected just in the same manner, as the sun diminishes the attraction of the moon towards the earth in the conjunction and opposition, but increases that attraction in the quarters. For as the first of these consequences from the sun’s action on the moon is occasioned by the moon’s being attracted by the sun in the conjunction more than the earth, and in the opposition less than it, and therefore in the common motion of the earth and moon, the moon is made to advance toward the sun in one case too fast, and in the other is left as it were behind; so the earth will not have its middle parts drawn towards the moon so strongly as the nearer parts, and yet more forcibly than the remotest: and therefore since the earth and moon move each month round their common center of gravity[271], while the earth moves round this center, the same effect will be produced, on the parts of the water nearest to that center or to the moon, as the moon feels from the sun when in conjunction, and the water on the contrary side of the earth will be affected by the moon, as the moon is by the sun, when in opposition[272]; that is, in both cases the weight of the water, or its propensity towards the center of the earth, will be diminished. The parts in the middle between these will have their weight increased, by being pressed towards the center of the earth through the obliquity of the moon’s action upon them to its action upon the earth’s center, just as the sun increases the gravitation of the moon in the quarters from the same cause[273]. But now it is manifest, that where the weight of the same quantity of water is least, there it will be accumulated; while the parts, which have the greatest weight, will subside. Therefore in this case there would be no tide or alternate rising and falling of the water, but the water would form it self into an oblong figure, whose axis prolonged would pass through the moon. By Sir ~ISAAC NEWTON~’s computation the excess of this axis above the diameters perpendicular to it, that is, the height of the waters under and opposite to the moon above their height in the middle between these places eastward or westward caused by the moon, is about 8⅔ feet.

21. THUS the difference of height in this latter supposition is little short of twice that difference in the preceding. But the case of the sea is a middle between these two: for a body, which should revolve round the center of the earth at the distance of a semidiameter without pressing on the earth’s surface, must perform its period in less than an hour and half, whereas the earth turns round but once in a day; and in the case of the waters keeping pace with the moon it should turn round but once in a month: so that the real motion of the water is between the motions required in these two cases. Again, if the waters moved round as swiftly as the first case required, their weight would be wholly taken off by their motion; for this case supposes the body to move so, as to be kept revolving in a circle round the earth by the power of gravity without pressing on the earth at all, so that its motion just supports its weight. But if the power of gravity had been only 1/289 part of what it is, the body could have moved thus without pressing on the earth, and have been as long in moving round, as the earth it self is. Consequently the motion of the earth takes off from the weight of the water in the middle between the poles, where its motion is swiftest, 1/289 part of its weight and no more. Since therefore in the first case the weight of the waters must be intirely taken off by their motion, and by the real motion of the earth they lose only 1/289 part thereof, the motion of the water will so little diminish their weight, that their figure will much nearer resemble the case of their keeping pace with the moon than the other. Upon the whole, if the waters moved with the velocity necessary to carry a body round the center of the earth at the distance of the earth’s semidiameter without bearing on its surface, the water would be lowest under the moon, and rise gradually as it moved on with the earth eastward, till it came half way toward the place opposite to the moon; from thence it would subside again, till it came to the opposition, where it would become as low as at first; afterwards it would rise again, till it came half way to the place under the moon; and from hence it would subside, till it came a second time under the moon. But in case the water kept pace with the moon, it would be highest where in the other case it is lowest, and lowest where in the other it is highest; therefore the diurnal motion of the earth being between the motions of these two cases, it will cause the highest place of the water to fall between the places of the greatest height in these two cases. The water as it passes from under the moon shall for some time rise, but descend again before it arrives half way to the opposite place, and shall come to its least height before it becomes opposite to the moon; then it shall rise again, continuing so to do till it has passed the place opposite to the moon, but subside before it comes to the middle between the places opposite to and under the moon; and lastly it shall come to its lowest, before it comes a second time under the moon. If A (in fig. 112, 113, 114.) represent the moon, B the center of the earth, the oval C D E F in fig. 112. will represent the situation of the water in the first case; but if the water kept pace with the moon, the line C D E F in fig. 113. would represent the situation of the water; but the line C D E F in fig. 114. will represent the same in the real motion of the water, as it accompanies the earth in its diurnal rotation: in all these figures C and E being the places where the water is lowest, and D and F the places where it is highest. Pursuant to this determination it is found, that on the shores, which lie exposed to the open sea, the high water usually falls out about three hours after the moon has passed the meridian of each place.

22. LET this suffice in general for explaining the manner, in which the moon acts upon the seas. It is farther to be noted, that these effects are greatest, when the moon is over the earth’s equator[274], that is, when it shines perpendicularly upon the parts of the earth in the middle between the poles. For if the moon were placed over either of the poles, it could have no effect upon the water to make it ascend and descend. So that when the moon declines from the equator toward either pole, it’s action must be something diminished, and that the more, the farther it declines. The tides likewise will be greatest, when the moon is nearest to the earth, it’s action being then the strongest.

23. THUS much of the action of the moon. That the sun should produce the very same effects, though in a less degree, is too obvious to require a particular explanation: but as was remarked before, this action of the sun being weaker than that of the moon, will cause the tides to follow more nearly the moon’s course, and principally shew it self by heightening or diminishing the effects of the other luminary. Which is the occasion, that the highest tides are found about the conjunction and opposition of the luminaries, being then produced by their united action, and the weakest tides about the quarters of the moon; because the moon in this case raising the water where the sun depresses it, and depressing it where the sun raises it, the stronger action of the moon is in part retunded and weakened by that of the sun. Our author computes that the sun will add near two feet to the height of the water in the first case, and in the other take from it as much. However the tides in both comply with the same hour of the moon. But at other times, between the conjunction or opposition and quarters, the time deviates from that forementioned, towards the hour in which the sun would make high water, though still it keeps much nearer to the moon’s hour than to the sun’s.

24. AGAIN the tides have some farther varieties from the situation of the places where they happen northward or southward. Let _p_ P (in fig. 115.) represent the axis, on which the earth daily revolves, let _h_ _p_ H P represent the figure of the water, and let _n_ B N D be a globe inscribed within this figure. Suppose the moon to be advanced from the equator toward the north pole, so that _h_ H the axis of the figure of the water _p_ A H P E _h_ shall decline towards the north pole N; take any place G nearer to the north pole than to the south, and from the center of the earth C draw C G F; then will G F denote the altitude to which the water is raised by the tide, when the moon is above the horizon: in the space of twelve hours, the earth having turned half round its axis, the place G will be removed to _g_; but the axis _h_ H will have kept its place preserving its situation in respect of the moon, at least will have moved no more than the moon has done in that time, which it is not necessary here to take into consideration. Now in this case the height of the water will be equal to _g_ _f_, which is not so great as G F. But whereas G F is the altitude at high water, when the moon is above the horizon, _g_ _f_ will be the altitude of the same, when the moon is under the horizon. The contrary happens toward the south pole, for K L is less than _k_ _l_. Hence is proved, that when the moon declines from the equator, in those places, which are on the same side of the equator as the moon, the tides are greater, when the moon is above the horizon, than when under it; and the contrary happens on the other side of the equator.

25. NOW from these principles may be explained all the known appearances in the tides; only by the assistance of this additional remark, that the fluctuating motion, which the water has in flowing and ebbing, is of a durable nature, and would continue for some time, though the action of the luminaries should cease; for this prevents the difference between the tide when the moon is above the horizon, and the tide when the moon is below it from being so great, as the rule laid down requires. This likewise makes the greatest tides not exactly upon the new and full moon, but to be a tide or two after; as at Bristol and Plymouth they are found the third after.

26. THIS doctrine farther shews us, why not only the spring tides fall out about the new and full moon, and the neap tides about the quarters; but likewise how it comes to pass, that the greatest spring tides happen about the equinoxes; because the luminaries are then one of them over the equator, and the other not far from it. It appears too, why the neap tides, which accompany these, are the least of all, for the sun still continuing over the equator continues to have the greatest power of lessening the moon’s action, and the moon in the quarters being far removed toward one of the poles, has its power thereby weakned.

27. MOREOVER the action of the moon being stronger, when near the earth, than when more remote; if the moon, when new suppose, be at its nearest distance from the earth, it shall when at the full be farthest off; whence it is, that two of the very largest spring tides do never immediately succeed each other.

28. BECAUSE the sun in its passage from the winter solstice to the summer recedes from the earth, and passing from the summer solstice to the winter approaches it, and is therefore nearer the earth before the vernal equinox than after, but nearer after the autumnal equinox than before; the greatest tides oftner precede the vernal equinox than follow it, and in the autumnal equinox on the contrary they oftner follow it than come before it.

29. THE altitude, to which the water is raised in the open ocean, corresponds very well to the forementioned calculations; for as it was shewn, that the water in spring tides should rise to the height of 10 or 11 feet, and the neap tides to 6 or 7; accordingly in the Pacific, Atlantic and Ethiopic oceans in the parts without the tropics, the water is observed to rise about 6, 9, 12 or 15 feet. In the Pacific ocean this elevation is said to be greater than in the other, as it ought to be by reason of the wide extent of that sea. For the same reason in the Ethiopic ocean between the tropics the ascent of the water is less than without, by reason of the narrowness of the sea between the coasts of Africa and the southern parts of America. And islands in such narrow seas, if far from shore, have less tides than the coasts. But now in those ports where the water flows in with great violence upon fords and shoals, the force it acquires by that means will carry it to a much greater height, so as to make it ascend and descend to 30, 40 or even 50 feet and more; instances of which we have at Plymouth, and in the Severn near Chepstow; at St. Michael’s and Auranches in Normandy; at Cambay and Pegu in the East Indies.

30. AGAIN the tides take a considerable time in passing through long straits, and shallow places. Thus the tide, which is made on the west coast of Ireland and on the coast of Spain at the third hour after the moon’s coming to the meridian, in the ports eastward toward the British channel falls out later, and as the flood passes up that channel still later and later, so that the tide takes up full twelve hours in coming up to London bridge.

31. IN the last place tides may come to the same port from different seas, and as they may interfere with each other, they will produce particular effects. Suppose the tide from one sea come to a port at the third hour after the moon’s passing the meridian of the place, but from another sea to take up six hours more in its passage. Here one tide will make high water, when by the other it should be lowest; so that when the moon is over the equator, and the two tides are equal, there will be no rising and falling of the water at all; for as much as the water is carried off by one tide, it will be supplied by the other. But when the moon declines from the equator, the same way as the port is situated, we have shewn that of the two tides of the ocean, which are made each day, that tide, which is made when the moon is above the horizon, is greater than the other. Therefore in this case, as four tides come to this port each day the two greatest will come on the third, and on the ninth hour after the moon’s passing the meridian, and the two least at the fifteenth and at the twenty first hour. Thus from the third to the ninth hour more water will be in this port by the two greatest tides than from the ninth to the fifteenth, or from the twenty first to the following third hour, where the water is brought by one great and one small tide; but yet there will be more water brought by these tides, than what will be found between the two least tides, that is, between the fifteenth and twenty first hour. Therefore in the middle between the third and ninth hour, or about the moon’s setting, the water will be at its greatest height; in the middle between the ninth and fifteenth, as also between the twenty first and following third hour it will have its mean height; and be lowest in the middle between the fifteenth and twenty first hour, that is, at the moon’s rising. Thus here the water will have but one flood and one ebb each day. When the moon is on the other side of the equator, the flood will be turned into ebb, and the ebb into flood; the high water falling out at the rising of the moon, and the low water at the setting. Now this is the case of the port of Batsham in the kingdom of Tunquin in the East Indies; to which port there are two inlets, one between the continent and the islands which are called the Manillas, and the other between the continent and Borneo.

32. THE next thing to be considered is the effect, which these fluids of the planets have upon the solid part of the bodies to which they belong. And in the first place I shall shew, that it was necessary upon account of these fluid parts to form the bodies of the planets into a figure something different from that of a perfect globe. Because the diurnal rotation, which our earth performs about its axis, and the like motion we see in some of the other planets, (which is an ample conviction that they all do the like) will diminish the force, with which bodies are attracted upon all the parts of their surfaces, except at the very poles, upon which they turn. Thus a stone or other weighty substance resting upon the surface of the earth, by the force which it receives from the motion communicated to it by the earth, if its weight prevented not, would continue that motion in a straight line from the point where it received it, and according to the direction, in which it was given, that is, in a line which touches the surface at that point; insomuch that it would move off from the earth in the same manner, as a weight fasten’d to a string and whirled about endeavours continually to recede from the center of motion, and would forthwith remove it self to a greater distance from it, if loosed from the string which retains it. And farther, as the centrifugal force, with which such a weight presses from the center of its motion, is greater, by how much greater the velocity is, with which it moves; so such a body, as I have been supposing to lie on the earth, would recede from it with the greater force, the greater the velocity is, with which the part of the earth’s surface it rests upon is moved, that is, the farther distant it is from the poles. But now the power of gravity is great enough to prevent bodies in any part of the earth from being carried off from it by this means; however it is plain that bodies having an effort contrary to that of gravity, though much weaker than it, their weight, that is, the degree of force, with which they are pressed to the earth, will be diminished thereby, and be the more diminished, the greater this contrary effort is; or in other words, the same body will weigh heavier at either of the poles, than upon any other part of the earth; and if any body be removed from the pole towards the equator, it will lose of its weight more and more, and be lightest of all at the equator, that is, in the middle between the poles.

33. THIS now is easily applied to the waters of the seas, and shews that the water under the poles will press more forcibly to the earth, than at or near the equator: and consequently that which presses least, must give place, till by ascending it makes room for receiving a greater quantity, which by its additional weight may place the whole upon a ballance. To illustrate this more particularly I shall make use of fig. 116 In which let A C B D be a circle, by whose revolution about the diameter A B a globe should be formed, representing a globe of solid earth. Suppose this globe covered on all sides with water to the same height, suppose that of E A or B F, at which distance the circle E G F H surrounds the circle A C B D; then it is evident, if the globe of earth be at rest, the water which surrounds it will rest in that situation. But if the globe be turned incessantly about its axis A B, and the water have likewise the same motion, it is also evident, from what has been explained, that the water between the circles E H F G and A D B C will remain no longer in the present situation, the parts of it between H and D, and between G and C being by this rotation become lighter, than the parts between E and A and between B and F; so that the water over the poles A and B must of necessity subside, and the water be accumulated over D and C, till the greater quantity in these latter places supply the defect of its weight. This would be the case, were the globe all covered with water. And the same figure of the surface would also be preserved, if some part of the water adjoining to the globe in any part of it were turned into solid earth, as is too evident to need any proof; because the parts of the water remaining at rest, it is the same thing, whether they continue in the state of being easily separable, which denominates them fluid, or were to be consolidated together, so as to make a hard body: and this, though the water should in some places be thus consolidated, even to the surface of it. Which shews that the form of the solid part of the earth makes no alteration in the figure the water will take: and by consequence in order to the preventing some parts of the earth from being entirely overflowed, and other parts quite deserted, the solid parts of the earth must have given them much the same figure, as if the whole earth were covered on all sides with water.

34. FARTHER, I say, this figure of the earth is the same, as it would receive, were it entirely a globe of water, provided that water were of the same density as the substance of the globe. For suppose the globe A C B D to be liquified, and that the globe E H F G, now entirely water, by its rotation about its axis should receive such a figure as we have been describing, and then the globe A C B D should be consolidated again, the figure of the water would plainly not be altered, by such a consolidation.

35. BUT from this last observation our author is enabled to determine the proportion between the axis of the earth drawn from pole to pole, and the diameter of the equator, upon the supposition that all the parts of the earth are of equal density; which he does by computing in the first place the proportion of the centrifugal force of the parts under the equator to the power of gravity; and then by considering the earth as a spheroid, made by the revolution of an ellipsis about its lesser axis, that is, supposing the line M I L K to be an exact ellipsis, from which it can differ but little, by reason that the difference between the lesser axis M L and the greater I K is but very small. From this supposition, and what was proved before, that all the particles which compose the earth have the attracting power explained in the preceding chapter, he finds at what distance the parts under the equator ought to be removed from the center, that the force, with which they shall be attracted to the center, diminished by their centrifugal force, shall be sufficient to keep those parts in a ballance with those which lie under the poles. And upon the supposition of all the parts of the earth having the same degree of density, the earth’s surface at the equator must be above 17 miles more distant from the center, than at the poles[275].