A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

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.