is a word which expresses that rising and falling of the waters which are observed on all maritime coasts.

There is a certain depth of the waters of the ocean which would obtain if all were at rest: but observation shows that they are continually varying from this level, and that some of these variations are regular and periodical.

1/2. It is observed, that on the shores of the ocean, and in bays, creeks, and harbours, which communicate freely with the ocean, the waters rise up above this mean height twice a day, and as often sink below it, forming what is called a FLOOD and an EBB, a HIGH and a LOW WATER. The whole interval between high and low water is called a TIDE; the water is said to flow and to ebb; and the rising is called the flood-tide, and the falling is called the ebb tide.

2d. It is observed that this rise and fall of the waters is variable in quantity. At Plymouth, for instance, it is sometimes 21 feet between the greatest and least depth of the water in one day, and sometimes only 12 feet.

These different heights of tide are observed to succeed each other in a regular series, diminishing from the greatest to the least, and then increasing from the least to the greatest. The greatest is called a spring tide, and the least is called a neap tide.

3d. This series is completed in about 15 days. More careful observation shows that two series are completed in the exact time of a lunation. For the spring tide in any place is observed to happen precisely at a certain interval of time (generally between two and three days) after new or full moon, and the neap tide at a certain interval after half moon; or, more accurately speaking, it is observed that the spring tide always happens when the moon has got a certain number of degrees eastward of the line of conjunction and opposition, and the neap tide happens when she is a certain number of degrees from her first or last quadrature. Thus the whole series of tides appears to be regulated by the moon.

4th. It is observed that high water happens at new and full moon when the moon has a certain determined position with respect to the meridian of the place of observation, preceding or following the moon's fouthing a certain interval of time; which is constant with respect to that place, but very different in different places.

5th. The time of high water in any place appears to be regulated by the moon; for the interval between the time of high water and the moon's fouthing never changes above three quarters of an hour, whereas the interval between the time of high water and noon changes six hours in the course of a fortnight.

6th. The interval between two succeeding high waters is variable. It is least of all about new and full moon, and greatest when the moon is in her quadratures. As two high waters happen every day, we may call the double of their interval a tide day, as we call the diurnal revolution of the moon a lunar day. The tide day is shortest about new and full moon, being then about $24^h\ 37'$; about the time of the moon's quadratures it is $25^h\ 27'$. These values are taken from a mean of many observations made at Barbadoes by Dr Maffelyne.

7th. The tides in similar circumstances are greatest when the moon is at her smallest distance from the earth, or in her perigee, and, gradually diminishing, are smallest when she is in her apogee.

8th. The same remark is made with respect to the sun's distance, and the greatest tides are observed during the winter months of Europe.

9th. The tides in any part of the ocean increase as the moon, by changing her declination, approaches the zenith of that place.

10th. The tides which happen while the moon is above the horizon are greater than the tides of the same day when the moon is below the horizon.

Such are the regular phenomena of the tides. They are important to all commercial nations, and have therefore been much attended to. It is of the tides, in all probability, that the Bible speaks, when God is said to let bounds to the sea, and to say "this far shall it go, and no farther."

Homer is the earliest profane author who speaks of the tides. Indeed it is not very clear that it is of them that he speaks (in the XIIth book of the Odyssey) when he speaks of Charybdis, which rises and retires thrice in every day. Herodotus and Dionysius Siculus speak more distinctly of the tides in the Red Sea. Pytheas of Marseille is the first who says anything of their cause. According to Strabo he had been in Britain, where he must have observed the tides of the ocean. Plutarch says expressly that Pytheas ascribed them to the moon. It is somewhat wonderful that Aristotle says so little about the tides. The army of Alexander, his pupil, were flanked at their first appearance to them near the Persian Gulph; and we should have thought that Aristotle would be well informed of all that had been observed there. But there are only three passages concerning them in all Aristotle's writings, and they are very trivial. In one place he speaks of great tides observed in the north of Europe; in another, he mentions their having been ascribed by some to the moon; and in a third, he says, that the tide in a great sea exceeds that in a small one.

The Greeks had little opportunity of observing the tides. The conquests and commerce of the Romans gave them more acquaintance with them. Caesar speaks of them in the 4th book of his Gallic War. Strabo, after Posidonius, clasps the phenomena into daily, monthly, and annual. He observes, that the sea rises as the moon gets near the meridian, whether above or below the horizon, and falls again as she rises or falls; also, that the tides increase at the time of new and full moon, and are greatest at the summer solstice. Pliny explains the phenomena at some length; and says, that both the sun and moon are their cause, dragging the waters along with them (B. II. c. 97). Seneca (Nat. Quaest. III. 28.) speaks of the tides with correctness; and Macrobius (Sonn. Scio. I. 6.) gives a very accurate description of their motions.

It is impossible that such phenomena should not excite human curiosity as to their cause. Plutarch (Plaut. Phil. III. 17), Galileo (Synt. Mund. Dial. 4.), Riccioli in his Almagest, ii. p. 374., and Gassendi, ii. p. 27 have collected most of the notions of their predecessors on the subject; but they are of so little importance, that they do not deserve our notice. Kepler speaks more like a philosopher (De Stella Mortis, and Phil. Afron. p. 555.). He says that all bodies attract each other, and that the waters of the ocean would all go to the moon were they not retained by the attraction of the earth; and then goes on to explain their elevation under the moon and on the opposite side, because the earth is less attracted by the moon than the nearer waters, but more than the waters which are more remote.

The honour of a complete explanation of the tides was reserved for Sir Isaac Newton. He laid hold of this class of phenomena as the most incontrovertible proof of universal gravitation, and has given a most beautiful and symmetrical view of the whole subject; contenting himself, however, with merely exhibiting the chief consequences of the general principle, and applying it to the phenomena with singular address. But the wide steps taken by this great philosopher in his investigation leave ordinary readers frequently at fault: many of his affirmations require the greatest mathematical knowledge to satisfy us of their truth. The academy of Paris therefore proposed to illustrate this among other parts of the principles of natural philosophy, and published the theory of the tides as a prize problem. This produced three excellent dissertations, by M. Laurin, Dan. Bernoulli, and Euler. Aided by these, and chiefly by the second, we shall here give a physical theory, and accommodate it to the purposes of navigation by giving the rules of calculation. We have demonstrated in our dissertations on the physical principles of the celestial motions, that it is an unexpected fact, that every particle of matter in the solar system is actually deflected toward every other particle; and that that the deflection of a particle of matter toward any distant sphere is proportional to the quantity of matter in that sphere directly, and to the square of the distance of the particle from the centre of that sphere inversely; and having found that the heaviness of a piece of terrestrial matter is nothing but the supposed opponent to the force which we exert in carrying this piece of matter, we conceive it as possessing a property, that is, distinguishing quality, manifested by its being gross or heavy. This is heaviness, gravity; and the manifestation of this quality, or the event in which it is seen, whether it be directly falling, or deflecting in a parabolic curve, or stretching a coiled spring, or breaking a rope, or simply pressing on its support, is gravitation; and the body is said to gravitate. When all obstacles are removed from the body, as when we cut the string by which a stone is hung, it moves directly downwards, tendit ad terram. Si d'findatur funis, tenderet lapis ad terram. Dum vero funis integer perficit, lapis terram negeti nisi confestur. By some metaphysical process, which it is needless at present to trace, this nisus ad motum has been called a tendency in our language. Indeed the word has now come to signify the energy of any active quality in those cases where its simplest and most immediate manifestation is prevented by some obstacle. The stone is now laid to tend toward the earth, though it does not actually approach it, being withheld by the string. The stretching the string in a direction perpendicular to the horizon is conceived as a full manifestation of this tendency. This tendency, this energy of its heaviness, is therefore named by the word which distinguishes the quality; and it is called gravitation, and it is said to gravitate.

But Sir Isaac Newton discovered that this deflection of a heavy body differs in no respect from that general deflection observed in all the bodies of the solar system. For 16 feet, which is the deflection of a stone in one second, has the very same proportion to 1/36th of an inch, which is the simultaneous deflection of the moon, that the square of the moon's distance from the centre of the earth has to the square of the stone's distance from it, namely, that of 3600 to 1.

Thus we are enabled to compare all the effects of the mutual tendencies of the heavenly bodies with the tendency of gravity, whose effects and measures are familiar to us.

If the earth were a sphere covered to a great depth with water, the water would form a concentric spherical shell; for the gravitation of every particle of its surface would then be directed to the centre, and would be equal. The curvature of its surface therefore would be everywhere the same, that is, it would be the uniform curvature of a sphere.

It has been demonstrated in former articles, after Sir Isaac Newton, that the gravitation of a particle C (fig. 1.) to the centre O, is to that of a particle E at the surface as CO to EO. In like manner the gravitation of o is to that of p as oO to pO. If therefore EO and OP are two communicating canals, of equal lengths, the water in both would be in equilibrium, because each column would exert the same total pressure at O. But if the gravitation of each particle in pO be diminished by a certain proportion, such as 1/36th of its whole weight, it is plain that the total pressure of the column pO will be 1/36th part less than that of the column EO. Therefore they will no longer be in equilibrium. The weight of the column EO will prevail; and if a hollow tower Pp be built at the mouth of the pit pO, the water will sink in EO and rise in pO, till both are again in equilibrium, exerting equal total pressures at O. Or we may prevent the sinking at E by pouring in more water into the tower Pf. The same thing must happen in the canal fC perpendicular to EO, if the gravitation of every particle be diminished by a force acting in the direction CF, and proportional to the distance of the particle from C, and such, that when cC is equal to oO, the force acting on c is equal to the force acting on o. In order that the former equilibrium may be restored after this diminution of the gravitation of the column fC, it is plain that more water must be poured into the oblique tower Pf. All this is evident when we consider the matter hydrostatically. The gravitation of the particle c may be represented by oO; but the diminution of the pressure occasioned by this at O is represented by Cc.

Hence we can collect this much, that the whole diminution of pressure at C is to the whole diminution of pressure at O as the sum of all the lines cC to the sum of all the lines oO, that is, as fC to pO. But the weight of the small quantity of water added in each tower is diminished in the same proportion; therefore the quantity added at Ff must be to the quantity added at Pp as fC to pO. Therefore we must have Ff : Pp = fC : pO, and the points E, F, P, p must be in the circumference of an ellipse, of which PO and EO are the transverse and conjugate semiaxes.

What we have here supposed concerning the diminution of gravity in these canals is a thing which really obtains in nature. It was demonstrated, when treating of the precession of the equinoxes, that if the sun or moon lie in the direction OP, at a very great distance, there results from the unequal gravitation of the different particles of the earth a diminution of the gravity of each particle; which diminution is in a direction parallel to OP, and proportional to the distance of the particle from a plane passing through the centre of the earth at right angles to the line OP.

Thus it happens that the waters of the ocean have their equilibrium disturbed by the unequal gravitation of their different particles to the sun or to the moon; and this equilibrium cannot be restored till the waters come in from all hands, and rise up around the line joining the centres of the earth and of the luminary. The spherical ocean must acquire the form of a prolate spheroid generated by the revolution of an ellipse round its transverse axis. The waters will be highest in that place which has the luminary in its zenith, and in the antipodes to that place; and they will be most depressed in all those places which have the luminary in their horizon. P and P' will be the poles, and EOQ will be the equator of this prolate spheroid.

Mr Ferguson, in his Astronomy, affirms another cause of this arrangement, viz. the difference of the centrifugal forces of the different particles of water, while the earth is turning round the common centre of gravity of the earth and moon. This, however, is a mistake. It would be just if the earth and moon were attached to the ends of a rod, and the earth kept always the same face toward the moon.

It is evident that the accumulation at P and P', and the depression at the equator, must augment and diminish in the same proportion with the disturbing force. It is also evident that its absolute quantity may be discovered by our knowledge of the proportion of the disturbing force to the force of gravity.—Now this proportion is known; for the proportion of the gravitation of the earth's centre to the sun or moon, to the force of gravity at the earth's surface, is known; and the proportion of the gravitation of the earth's centre to the luminary, to the difference of the gravitations of the centre and of the surface, is also known, being very nearly the proportion of the distance of the luminary to twice the radius of the earth.

Although this reasoning, by which we have ascertained the elliptical form of the watery spheroid, be sufficiently convincing, it is very imperfect, being accommodated to one condition only of equilibrium, viz. the equilibrium of the canals There are several other conditions equally necessary to which this lax reasoning will not apply, such as the direction of the whole remaining gravitation in any point F. This must be perpendicular to the surface, &c. &c. Nor will this mode of investigation ascertain the eccentricity of the spheroid without a most intricate process. We must therefore take the subject more generally, and show the proportion and directions of gravity in every point of the spheroid. We need not, however, again demonstrate that the gravitation of a particle placed anywhere without a perfect spherical shell, or a sphere consisting of concentric spherical shells, either of uniform density, or of densities varying according to some function of the radius, is the same as if the whole matter of the shell or sphere were collected in the centre. This has been demonstrated in the article Astronomy. We need only remind the reader of some consequences of this theorem which are of continual use in the present investigation.

1st. The gravitation to a sphere is proportional to its quantity of matter directly, and to the square of the distance of its centre from the gravitating particle inversely.

2nd. If the spheres be homogeneous and of the same density, the gravitations of particles placed on their surfaces, or at distances which are proportional to their diameters, are as the radii; for the quantities of matter are as the cubes of the radii, and the attractions are inversely as the squares of the radii; and therefore the whole gravitations are as \( \frac{r^3}{r^2} \), or as \( r \).

3rd. A particle placed within a sphere has no tendency to the matter of the shell which lies without it, because its tendency to any part is balanced by an opposite tendency to the opposite part. Therefore,

4th. A particle placed anywhere within a homogeneous sphere gravitates to its centre with a force proportional to its distance from it.

It is a much more difficult problem to determine the gravitation of particles to a spheroid. To do this in general terms, and for every situation of the particle, would require a chain of propositions which our limits will by no means admit; we must content ourselves with as much as is necessary for merely ascertaining the ratio of the axes. This will be obtained by knowing the ratio of the gravitation at the pole to that at the equator. Therefore

Let NMSQ (fig. 2) be a section through the axis of an oblate homogeneous spheroid, which differs very little from a sphere. NS is the axis, MQ is the equatorial diameter, O is the centre, and NMSQ is the section of the inscribed sphere. Let P be a particle situated at any distance without the sphere in its axis produced; it is required to determine the gravitation of this particle to the whole matter of the spheroid?

Draw two lines PAC, PBD, very near to each other, cutting off two small arches AB, CD; draw GAa, HBb, ICc, KDd, perpendicular to the axis; also draw OE and AL perpendicular to PAC, and OF perpendicular to PD, cutting PC in f. Join OA.

Let OA, the radius of the inscribed sphere, be \( r \), and OP the distance of the gravitating particle be \( d \), and MM, the elevation of the equator of the spheroid, or the ellipticity, be \( e \). Also make AE = \( x \), and OE = \( y \), \( y = \sqrt{r^2 - x^2} \).

Then AE = \( \frac{x}{y} \) and \( Ff = \frac{y}{x} = \frac{\sqrt{r^2 - x^2}}{x} \).

Suppose the whole figure to turn round the axis OP. The little space ABba will generate a ring of the redundant matter; so will CDdc. This ring may be considered as consisting of a number of thin rings generated by the revolution of AAa. The ring generated by AAa is equal to a parallelogram whose base is the circumference described by AA and whose height is AAa. Therefore let \( c \) be the circumference of a circle whose radius is \( r \). The ring will be AAa × c × AG. But because MM : AAa = MO : AG = \( r : AG \), and AAa = MM × \( \frac{AG}{r} \), \( c = \frac{AG}{r} \). Therefore the surface of this ring is \( c \times \frac{AG}{r} \times AG^2 \).

We have supposed the spheroid to be very nearly spherical, that is, \( e \) exceedingly small in comparison of \( r \). This being the case, all the particles in AAa, and consequently all the particles in the ring generated by the revolution of AAa, will attract the remote particle P with the same force that A does very nearly. We may say the same thing of the whole matter of the ring generated by the revolution of ABba. This attraction is exerted in the direction PA by each individual particle. But every action of a particle A is accompanied by the action of a particle A' in the direction PA'. These two compose an attraction in the direction PA. The whole attraction in the directions similar to PA is \( c \times \frac{AG^2}{r} \times \frac{PA^2}{GH} \times GH \), for GH measures the number of parallel plates of which the solid ring is composed.

This being decomposed in the direction PC is \( c \times \frac{AG^2}{r} \times \frac{PG}{PA^2} \times GH \). But \( \frac{AG^2}{PA^2} = \frac{OE^2}{PO^2} \), and \( \frac{PG}{PA} = \frac{PE}{PO} \). Therefore the attraction of the ring, estimated in the direction PO, is \( c \times \frac{OE^2}{r} \times \frac{PE}{PO^2} \times GH \).

Farther, by the nature of the circle, we have HG : AB = AG : AO; also AB : BL = AO : OE. But PA : AG = PO : OE, and OE = \( \frac{AG \times PO}{PA} \). Therefore

\[ AB : BL = AO : \frac{AG \times PO}{PA} = AO \times PA : PO \times AG \]

Also BL : LA = EO : EA,

And LA : Ff = PA : PE; ultimately PA : PE. Therefore, by equality, HG : Ff = AG : AO. PA : EO. PA : AO. PO. AG. EA. PE.

Or HG : Ff = EO. PA^2 : PO. EA. PE.

And HG = Ff × \( \frac{EO}{PO} \times \frac{PA^2}{PE} \times \frac{EA}{EO} \).

Now substitute this value of HG in the formula expressing the attraction of the ring. This changes it to \( c \times \frac{OE^2}{r} \times \frac{PE}{PO^2} \times \frac{OE}{PA^2} \times \frac{PA^2}{PE} \times \frac{EO}{EA} \times Ff \); or \( c \times \frac{OE^2}{r} \times \frac{PC^2}{PO^2} \times \frac{Ff}{EA} \times Ff \).

In like manner, the attraction of the ring generated by the revolution of CDdc is \( c \times \frac{OE^2}{r} \times \frac{PC^2}{PO^2} \times \frac{Ff}{EA} \times Ff \).

Therefore the attraction of both is \( c \times \frac{OE^2}{r} \times \frac{PC^2}{PO^2} \times \frac{Ff}{EA} \times Ff \times \frac{PA^2 + PC^2}{r} \times \frac{Ff}{d^2} \times \frac{Ff}{x} \times \frac{PA^2 + PC^2}{r} \).

But \( PA^2 + PC^2 = 2PE^2 + 2EA^2 = 2PE^2 + 2x^2 \). Therefore the attraction is \( 2c \times \frac{OE^2}{r} \times \frac{PC^2}{PO^2} \times \frac{Ff}{EA} \times Ff \times \frac{PA^2 + PC^2}{r} \times \frac{Ff}{d^2} \times \frac{Ff}{x} \times \frac{PA^2 + PC^2}{r} \).

But \( PA^2 + PC^2 = 2PE^2 + 2EA^2 = 2PE^2 + 2x^2 \). Therefore \( Ff = \frac{y}{x} \times \frac{x}{y} = \frac{y^2}{x^2} \). Therefore the attraction of the two rings is

\[2 \frac{e}{r^4} \times r^2 - x^2 \times PE^2 + x^2 \times \frac{x}{x}. \]

But \(PE^2 = PO^2 - OE^2 = d^2 - (r^2 - x^2) = d^2 - r^2 + x^2.\)

Therefore the attraction of the two rings is

\[2 \frac{e}{r^4} \times r^2 - x^2 \times d^2 - r^2 + 2x^2 = 2 \frac{e}{r^4} \times\]

\[r^2 - r^2 + 2x^2 - d^2 + x^2 + r^2 - 2x^2 = 2 \frac{e}{r^4} \times\]

\[x^2 + 3x^2 - r^2 - d^2 + x^2 - 2x^2 = 2 \frac{e}{r^4} \times\]

The attraction of the whole shell of redundant matter will be had by taking the fluent of this formula, which is

\[2 \frac{e}{r^4} \times \left( r^2 d^2 + \frac{3}{3} r^3 - r^4 - \frac{d^3}{3} x^3 - \frac{2}{5} x^5 \right),\]

and then make \(x = r.\) This gives \(2 \frac{e}{r^4} \times \left( d^2 r^3 + r^5 - r^5 - \frac{d^2 r^3}{3} - \frac{r^5}{5} \right),\) which is \(= 2 \frac{e}{r^4} \times \left( \frac{1}{3} d^2 r^3 - \frac{2}{5} r^5 \right),\)

\[= \frac{4}{3} \frac{e r^3}{d^2} - \frac{4}{5} \frac{e r^5}{d^4}.\]

To this add the attraction of the inscribed sphere, which is \(\frac{c}{d^2},\) and we have the attraction of the whole spheroid

\[= \frac{7}{3} \frac{e r^3}{d^2} + \frac{4}{5} \frac{e r^5}{d^4} - \frac{4}{5} \frac{e r^5}{d^4}.\]

Cor. 1. If the particle P is situated precisely in N, the pole of the spheroid, the attraction of the spheroid, is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

If the spheroid is not oblate, but oblong, and if the greater semiaxis be \(r,\) and the depression at the equator be \(e,\) the analysis is the same, taking \(e\) negatively. Therefore the attraction for a particle in the pole, or the gravitation of a particle in the pole, is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

But if the polar semiaxis be \(r + e,\) and the equatorial radius be \(r,\) so that this oblong spheroid has the same axis with the former oblate one, the gravitation of a particle in the pole is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Cor. 2. If a number of parallel planes are drawn perpendicular to the equator of an oblong spheroid, whose longer semiaxis is \(r + e,\) and equatorial radius \(r,\) they will divide the spheroid into a number of similar ellipses; and since the ellipse through the axis has \(r + e\) and \(r\) for its two semiaxes, and the radius of a circle of equal area with this ellipse is a mean proportional between \(r\) and \(r + e,\) and therefore very nearly \(r + \frac{1}{2} e,\) when \(e\) is very small in comparison of \(r,\) a particle on the equator of the oblong spheroid will be as much attracted by these circles of equal areas, with their corresponding ellipses, as by the ellipses. Now the attraction at the pole of an oblate spheroid is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Therefore putting \(\frac{1}{2} e\) in place of \(e,\) the attraction on the equator of the oblong spheroid will be equal to \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Thus we have ascertained the gravitations of a particle situated in the poles, and of one situated in the equator, of a homogeneous oblong spheroid. This will enable us to solve the following problem:

If the particles of a homogeneous oblong fluid spheroid attract each other with a force inversely as the squares of their distances, and if they are attracted by a very distant body by the same law, and if the ratio of the equatorial gravity to this external force be given; to find what must be the proportion of the semiaxis, so that all may be in equilibrium, and the spheroid preserve its form?

Let \(r\) be the equatorial radius, and \(r + e\) be the polar semiaxis. Then the gravitation at the pole is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

And the gravitation at the equator is \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Now by the gravitation towards the distant body placed in the direction of the polar axis, the polar gravitation is diminished, and the equatorial gravitation is increased; and the increase of the equatorial gravitation is to the diminution of the polar gravitation as NO to \(2mO.\) Therefore if the whole attraction of the oblong spheroid for a particle on its equator be to the force which the distant body exerts there, as G to P, and if the spheroid is very nearly spherical, the absolute weight at the equator will be \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

And the absolute weight at the pole will be \(\frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Their difference is \(\frac{7}{3} \frac{e c e}{d^4} + \frac{2}{5} \frac{e r^3}{d^2}.\)

Now if we suppose this spheroid to be composed of similar concentric shells, all the forces will decrease in the same ratio. Therefore the weight of a particle in a column reaching from the equator to the centre will be to the weight of a similarly situated particle of a column reaching from the pole to the centre, as the weight of a particle at the equator to the weight of a particle at the pole. But the whole weights of the two columns must be equal, that they may balance each other at the centre. Their lengths must therefore be reciprocally as the weights of similarly situated particles; that is, the polar semiaxis must be to the equatorial radius, as the weight of a particle at the equator to the weight of a particle at the pole. Therefore we must have \(\frac{7}{3} \frac{e c e}{d^4} + \frac{2}{5} \frac{e r^3}{d^2} : \frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4} = \frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4} : \frac{7}{3} \frac{e r^3}{d^2} + \frac{8}{5} \frac{e c e}{d^4}.\)

Hence we derive \(2 \frac{P}{G} = \frac{8}{5} e,\) or \(4G : 15P = r : e.\)

This determines the form of the fluid spheroid when the ratio of \(G\) to \(P\) is given.

It is well known that the gravitation of the moon to the earth is to the disturbing force of the sun as \(178,725\) to \(1\) very nearly. The lunar gravitation is increased as it approaches the earth in the reciprocal duplicate ratio of the distances. The disturbing force of the sun diminishes in the simple ratio of the distances; therefore the weight of a body on the surface of the earth is to the disturbing force of the sun on the same body, in a ratio compounded of the ratio of \(178,725\) to \(1,\) the ratio of \(3600\) to \(1,\) and the ratio of \(60\) to \(1;\) that is, in the ratio of \(386,400\) to \(1.\) If the mean radius of the earth be \(209,430,000\) feet, the difference of the axis, or the elevation of the pole of the watery spheroid produced by the gravitation to the sun, will be \(\frac{1}{4} \times \frac{386,400}{209,430,000}\) feet, or very nearly \(24\frac{1}{2}\) inches. This is the tide produced by the sun on a homogeneous fluid sphere.

It is plain, that if the earth consists of a solid nucleus of the same density with the water, the form of the solar tide will be the same. But if the density of the nucleus be different, the form of the tide will be different, and will depend both on the density and on the figure of the nucleus.

If the nucleus be of the same form as the surrounding fluid, the whole will still maintain its form with the same proportion of the axis. If the nucleus be spherical, its action on the surrounding fluid will be the same as if all the matter of the nucleus by which it exceeds an equal bulk of the fluid were collected at the centre. In this case, the ocean cannot maintain the same form: for the action of this central body being proportional to the square of the distance inversely, will augment the gravity of the equatorial fluid more than it augments that of the circumpolar fluid; and the ocean, which was in equilibrium (by supposition), must now become more protuberant at the poles. It may, however, ever be again balanced in an elliptical form, when it has acquired a just proportion of the axes. The process for determining this is tedious, but precisely similar to the preceding.

If the density of the nucleus exceed that of the fluid about \( \frac{1}{5} \), we shall have \( r : e = G : 3 P \), which is nearly the form which has been determined for the earth, by the mensuration of degrees of the meridian, and by the vibration of pendulums. The curious reader will do well to consult the excellent dissertations by Clairaut and Boscovich on the Figure of the Earth, where this curious problem is treated in the most complete manner. Mr Bernoulli, in his dissertation on the Tides, has committed a great mistake in this particular. On the other hand, if the nucleus be less dense than the waters, or if there be a great central hollow, the elevation produced by the sun will exceed 24½ inches.

It is needless to examine this any farther. We have collected enough for explaining the chief affections of the tides.

- It is known that the earth is not a sphere, but swelled out at the equator by the diurnal rotation. But the change of form is so very small in proportion to the whole bulk, that it cannot sensibly affect the change of form afterwards induced by the sun on the waters of the ocean. For the disturbing force of the sun would produce a certain protuberance on a fluid sphere; and this protuberance depends on the ratio of the disturbing force to the force of gravity at the surface of this sphere. If the gravity be changed in any proportion, the protuberance will change in the same proportion. Therefore if the body be a spheroid, the protuberance produced at any point by the sun will increase or diminish in the same proportion that the gravity at this point has been changed by the change of form. Now the change of gravity, even at the pole of the terrestrial spheroid, is extremely small in comparison with the whole gravity. Therefore the change produced on the spheroid will not sensibly differ from that produced on the sphere; and the elevations of the waters above the surface, which they would have assumed independent of the sun's action, will be the same on the spheroid as on the sphere. For the same reason, the moon will change the surface already changed by the sun, in the same manner as she would have changed the surface of the undisturbed ocean. Therefore the change produced by both these luminaries in any place will be the same when acting together as when acting separately; and it will be equal to the sum, or the difference of their separate changes, according as these would have been in the same or in opposite directions.

Let us now consider the most interesting circumstances of the form of an elliptical tide, which differs very little from a sphere.

Let \( T \) (fig. 2.) be a point in the surface of the inscribed sphere, and let \( Z \) express the angular distance \( TOQ \) from the longer axis of the surrounding spheroid \( S \) \( m \) \( N \) \( q \). Let \( TR, TW \) be perpendicular to the equatorial diameter and to the axis, so that they are the cosine and the sine of \( TOQ \) to the radius \( TO \) or \( QO \). Let \( S'qN \) be a section of the circumscribed sphere. Draw \( OT \) cutting the spheroid in \( Z \) and the circumscribed sphere in \( V \). Also let \( s \) \( o \) \( n \) be a section of a sphere which has the same capacity with the spheroid, and let it cut the radius in \( r \). Then,

1. The elevation \( TZ \) of the point \( Z \) of the spheroid above the inscribed sphere is \( = Qg \times \text{col}^2Z \), and the depression \( TZ \) below the circumscribed sphere is \( = Qg \times \text{fin}^2Z \).

Produce \( RT \) till it meet the surface of the spheroid in \( V \). The minute triangle \( VIZ \) may be considered as a rectilineal, right-angled at \( Z \), and therefore similar to \( OTR \).

Therefore \( OT : TR = TV : TZ \). But in the ellipse \( OO' \), or \( O'T : TR = Qg : TV \). Therefore \( OT^2 : TR^2 = Qg : TZ \), and \( TZ = \frac{Qg \cdot TR^2}{OT^2} = \frac{Qg \times \text{col}^2Z}{1} = Qg \times \text{col}^2Z \).

And in the very same manner it may be shown, that \( TZ = Qg \times \text{fin}^2Z \).

2. The elevation of the point \( T \) above another point \( T' \), whose angular distance \( TO'T' \) from the point \( T \) is \( 90^\circ \), is \( = Qg \times \text{col}^2Z - \text{fin}^2Z \). Call the angle \( QOT' Z' \). Then \( T'Z' = Qg \times \text{col}^2Z' \), and \( TZ - T'Z' = Qg \times \text{col}^2Z - \text{col}^2Z' \). But the arch \( QT' \) is the complement of \( QF \), and therefore \( \text{col}^2Z' = \text{fin}^2Z \). Therefore \( TZ - T'Z' = Qg \times \text{col}^2Z - \text{fin}^2Z \).

3. \( Qo = \frac{1}{4} Qg \). For the inscribed sphere is to the spheroid as \( OO \) to \( Oq \). But the inscribed sphere is to the sphere \( s \) \( o \) \( n \) as \( OO^3 \) to \( Oo^3 \). Therefore because the sphere \( s \) \( o \) \( n \) is equal to the spheroid \( S \) \( q \) \( N \), we have \( OO : Oq = OO : Oo^3 \), and \( Oo \) is the first of two mean proportions between \( OO \) and \( Oq \). But \( Qg \) is very small in comparison with \( OO \). Therefore \( Qo \) is very nearly \( \frac{1}{4} Qg \).

Since \( s \) \( o \) \( n \) is the sphere of equal capacity, it is the form of the undisturbed ocean. The best way therefore of conceiving the changes of form produced by the sun or moon, or by both together, is to consider the elevations or depressions which they produce above or below this surface. Therefore,

4. The elevation \( TZ \) of the point \( Z \) above the equi-capacious sphere is evidently \( = Qg \times \text{col}^2Z - \frac{1}{4} Qg \). Also the depression \( TZ \) of the point \( Z' \) is \( = Qg \times \text{fin}^2Z' - \frac{1}{4} Qg \).

N.B. Either of these formulæ will answer for either the elevation above, or the depression below, the natural ocean; For if \( \text{col}^2Z \) is less than \( \frac{1}{4} \), the elevation given by the formula will be negative; that is, the point is below the natural surface. In like manner, when \( \text{fin}^2Z' \) is less than \( \frac{1}{4} \), the depression is negative, and the point is above the surface. But if \( \text{col}^2Z \) be \( \frac{1}{4} \), or \( \text{fin}^2Z' \) be \( \frac{1}{4} \), the point is in the natural surface. This marks the place where the spheroid and the equal sphere intersect each other, viz. in \( P \), the arch \( P' \) being \( 54^\circ 44' \) very nearly, and \( PS = 35^\circ 16' \).

Let \( S \) represent the whole elevation of the pole of the solar tide above its equator, or the difference between high and low water produced by the sun; and let \( M \) represent the whole elevation produced by the moon. Let \( x \) and \( y \) represent the zenith distances of the sun and moon with respect to any point whatever on the ocean. Then \( x \) and \( y \) will be the arches intercepted between that point and the summits of the solar and lunar tides. Then the elevation produced by both luminaries in that plane is \( S \cdot \text{col}^2x + M \cdot \text{col}^2y - \frac{1}{4} S + \frac{1}{4} M \); or, more concisely, \( S \cdot \text{col}^2x + M \cdot \text{col}^2y - \frac{1}{4} S + \frac{1}{4} M \), and the depression is \( S \cdot \text{fin}^2x + M \cdot \text{fin}^2y - \frac{1}{4} S + \frac{1}{4} M \).

Let the sun and moon be in the same point of the heavens. The solar and lunar tides will have the same axis; the cosines of \( x \) and \( y \) will each be \( 1 \), and the elevation at the compound pole will be \( S + M - \frac{1}{4} S + \frac{1}{4} M = \frac{3}{4} S + \frac{3}{4} M \). The depression at any point \( 90^\circ \) from this pole will be \( \frac{1}{4} S + \frac{1}{4} M \), and the whole tide is \( S + M \).

Let the moon be in quadrature, as in \( a \) (fig. 3). The appearance at \( s \) will be known, by considering that in this place the cosine of \( x \) is \( 1 \), and the cosine of \( y \) is \( 0 \). Therefore the elevation at \( s = S - \frac{1}{4} S + \frac{1}{4} M = \frac{3}{4} S - \frac{1}{4} M \). The depression at \( a = S - \frac{1}{4} S + \frac{1}{4} M = \frac{3}{4} S - \frac{1}{4} M \). The difference or whole tide \( = S - M \). In like manner, the whole elevation at \(a\) above the inferiord sphere is \(M - S\).

Hence we see that the whole tide, when the moon is in quadrature, is the difference of \(S\) and \(M\). We also see, that if \(M\) exceeds \(S\), the water will be higher at \(a\) than at \(v\). Now it is a matter of observation, that in the quadratures it is high water under the moon, and low water under the sun. It is also a matter of observation, that in the free ocean, the ebb tide, or the water at \(s\), immediately under the sun, is below the natural surface of the ocean. Hence we must conclude, that \(\frac{1}{2}S\) is less than \(\frac{1}{2}M\), or that \(M\) is more than double of \(S\). This agrees with the phenomena of nutation and precession, which seem to make \(S = \frac{1}{2}\) of \(M\).

In all other positions of the sun and moon, the place of high water will be different. It is high water where the sum of the elevations produced by both luminaries above the natural ocean is greatest; and the place of low water is where the depression below the natural ocean is greatest. Therefore, in order that it may be high water, we must have

\[S \cdot \text{cof}^2 x + M \cdot \text{cof}^2 y - \frac{1}{3}S + M\] a maximum; or, neglecting the constant quantity \(\frac{S + M}{3}\), we must have

\[S \cdot \text{cof}^2 x + M \cdot \text{cof}^2 y\] a maximum.

In like manner, to have low water in a place where the zenith distances of the sun and moon are \(v\) and \(w\), we must have \(S \cdot \text{fin}^2 v + M \cdot \text{fin}^2 w\) a maximum.

**Lemma 1.** If we consider the fines and cosines of angles as numerical fractions of the radius \(r\), then we have \(\text{cof}^2 Z = \frac{1}{4} + \frac{1}{4} \cdot \text{cof}^2 Z\), and \(\text{fin}^2 Z = \frac{1}{4} - \frac{1}{4} \cdot \text{cof}^2 Z\).

Let \(a\) and \(b\) (fig. 3.) be a quadrant of a circle of which \(O\) is the centre, and \(Ox\) is the radius. On \(Ox\) describe the semicircle \(OMS\), cutting \(Om\) in \(M\). Draw \(MS\) and produce it till it cut the quadrant in \(n\). Also draw \(MC\) to the centre of the semicircle, and \(MD\) and \(n\) perpendicular to \(Ox\).

It is plain that \(MS\) is perpendicular to \(OM\); and if \(Ox\) be radius, \(MS\) is the fine of the angle \(OM\), which we may call \(Z\); \(OM\) is its cosine; and because \(Ox : OM = OM : OD\), and \(Ox : OD = OM : OM^2\), and \(OD\) may represent \(\text{cof}^2 Z\). Now \(OD = OC + CD\). If \(Ox = r\), then \(OC = \frac{1}{2}r\), \(CD = CM \cdot \text{cof}^2 MCD\), \(= CM \cdot \text{cof}^2 MOD\), \(= \frac{1}{2} \cdot \text{cof}^2 Z\). Therefore \(\text{cof}^2 Z = \frac{1}{4} + \frac{1}{4} \cdot \text{cof}^2 Z\).

In like manner, because \(Ox : MS = M : D\), \(D\) is \(\text{fin}^2 Z\). This is evidently \(\frac{1}{4} - \frac{1}{4} \cdot \text{cof}^2 Z\).

**Lemma 2.** \(\text{cof}^2 Z = \text{fin}^2 Z = \text{cof}^2 Z\). For, because \(MS\) is perpendicular to \(OM\), the arch \(sn\) is double of the arch \(sm\), and because \(MD\) is parallel to \(nd\), \(sd\) is \(2zD\), and \(dD = \text{fin}^2 Z\). Therefore \(Ox = \text{cof}^2 Z = \text{fin}^2 Z\).

But \(OD\) is the cosine of \(ns\), \(= \text{cof}^2 Z\), and \(Z = \text{fin}^2 Z = \text{cof}^2 Z\).

By the first Lemma we see, that in order that there may be high water at any place, when the zenith distances of the sun and moon are \(x\) and \(y\), we must have \(S \cdot \text{cof}^2 x + M \cdot \text{cof}^2 y = 0\), or \(S \cdot \text{fin}^2 x + M \cdot \text{fin}^2 y = 0\), which gives us \(\text{fin}^2 x + \text{fin}^2 y = M : S\).

In like manner, the place of low water requires \(\text{fin}^2 v + \text{fin}^2 w = M : S\).

From this last circumstance we learn, that the place of low water is \(o\), removed \(90^\circ\) from the place of high water; whereas we might have expected, that the spheroid would have been most protuberant on that side on which the moon is: For the fines of \(v\) and of \(w\) have the same proportion with the fines of \(x\) and of \(y\). Now we know that the fine of the double of any arch is the same with the fine of the double of its complement. Therefore if low water be really distant \(90^\circ\) from high water, we shall have \(\text{fin}^2 x : \text{fin}^2 y = \text{fin}^2 v : \text{fin}^2 w\). But if it is at any other place, the fines cannot have this proportion.

Now let \(s\) be the point of the earth's surface which has the sun in the zenith, and \(m\) the point which has the moon in the zenith. Let \(b\) be any other point. Draw \(Ob\) cutting the semicircle \(OMs\) in \(H\). Make \(CM\) to \(CS\) as the disturbing force of the moon to that of the sun; and draw \(Sw\) parallel, and \(SaMr\) perpendicular to \(HH'\). Join \(MH\) and \(MH'\). The angle \(HCs\) is double of the angle \(HOs\), and \(CHs\) is double of \(MHs\), or of its equal \(MOH\). Because \(HM\) is a semicircle, \(HM\) is perpendicular to \(MO\). Therefore \(HH'\) be considered as radius, \(HM\) is the sine, and \(HM\) is the cosine of \(MHs\). And \(Cr = MC \cdot \text{cof}^2 y = M \cdot \text{cof}^2 y\). And \(Ct = SC \cdot \text{cof}^2 x\). Therefore \(tr\) or \(Sw\) is \(= S \cdot \text{cof}^2 x + M \cdot \text{cof}^2 y\). Therefore \(tr\) or \(Sw\) will express the whole difference of elevation between \(b\) and the points that are \(90^\circ\) degrees from it on either side (by Lemma 2.), and if \(b\) be the place of high water, it will express the whole tide, because the high and low waters were shown to be \(90^\circ\) afunder. But when \(b\) is the place of high water, \(Sw\) is a maximum. Because the place of the moon, and therefore the point \(M\), is given, \(Sw\) will be a maximum when it coincides with \(SM\), and \(CH\) is parallel to \(SM\).

This suggested to us the following new, and not inelegant, solution of the problem for determining the place of high water.

Let \(Q\) and \(g\) (fig. 4. and 5.) be a section of the terraqueous globe, by a plane passing through the sun and moon, and let \(O\) be its centre. Let \(s\) be the point which is immediately under the sun, and \(m\) the place immediately under the moon. Bifect \(Os\) in \(C\), and describe round \(C\) the circle \(CLO\), cutting \(Om\) in \(M\). Take \(Cs\) to represent the disturbing force of the moon; and make \(Cs\) to \(CS\) as the force of the moon to that of the sun (supposing this ratio to be known). Join \(MS\), and draw \(CH\) parallel to it. Draw \(OHb\), and \(OL\) perpendicular to it. And lastly, draw \(CI\) perpendicular to \(SM\). Then we say that \(m\) and its opposite \(m'\) are the places of high water, \(l\) and \(l'\) are the places of low water, \(MS\) is the height of the tide, and \(MI, SI\) are the portions of this tide produced by the moon and sun.

For it is plain, that in this case the line \(Sw\) of the last proposition coincides with \(MS\), and is a maximum. We may also observe, that \(MC : CS = \text{fin}. MSC : \text{fin}. SMC\), \(= \text{fin}. HCS : \text{fin}. MCH\), \(= \text{fin}^2 b Os : \text{fin}^2 b Om\), \(= \text{fin}^2 x : \text{fin}^2 y\), or \(M : S = \text{fin}^2 x : \text{fin}^2 y\), agreeably to what was required for the maximum.

It is also evident, that \(MI = MC \cdot \text{cof}. CMI\), \(= M \cdot \text{cof}^2 y\), and \(SI = SC \cdot \text{cof}. ISC\), \(= S \cdot \text{cof}^2 x\); and therefore \(MS\) is the difference of elevation between \(b\) and the points \(l\) and \(l'\), which are \(90^\circ\) from it, and is therefore the place of low water; that is, \(MS\) is the whole tide.

The elevation of every other point may be determined in the same way, and thus may the form of the spheroid be completely determined.

If we suppose the figure to represent a section through the earth's equator (which is the case when the sun and moon are in the equator), and farther suppose the two luminaries to be in conjunction, the ocean is an oblong spheroid, whose axis is in the line of the syzygies, and whose equator coincides with the fix hour circle. But if the moon be in any other point of the equator, the figure of the ocean will be very complicated. It will not be any figure of revolution; because neither its equator (or most depressed part), part), nor its meridians, are circles. The most depressed part of its equator will be in that section through the axis which is perpendicular to the plane in which the luminaries are situated. And this greatest depression, and its shortest equatorial diameter, will be constant, while its other dimensions vary with the moon's place. We need not inquire more minutely into its form; and it is sufficient to know, that all the sections perpendicular to the plane passing thro' the sun and moon are ellipses.

This construction will afford us a very simple, and, we hope, a very perspicuous explanation of the chief phenomena of the tides. The well-informed reader will be pleased with observing its coincidence with the algebraic solution of the problem given by Daniel Bernoulli, in his excellent dissertation on the Tides, which shared with M. Laurin and Euler the prize given by the Academy of Sciences at Paris, and with the ease and perspicuity with which the phenomena are deducible from it, being in some sort exhibited to the eye.

In our application, we shall begin with the simplest cases, and gradually introduce the complicating circumstances which accommodate the theory to the true state of things.

We begin, therefore, by supposing the earth covered, to a proper depth, with water, forming an ocean concentric with its solid nucleus.

In the next place, we suppose that this ocean adopts in an instant the form which is consistent with the equilibrium of gravity and the disturbing forces.

Thirdly, We suppose the sun stationary, and the moon to move eastward from him above $12\frac{1}{2}$ every day.

Fourthly, We suppose that the solid nucleus turns round its proper axis to the eastward, making a rotation in 24 solar hours. Thus any place of observation will successively experience all the different depths of water.

Thus we shall obtain a certain Succession of phenomena, precisely similar to the succession observed in nature, with this sole difference, that they do not correspond to the contemporaneous situations of the sun and moon. When we shall have accounted for this difference, we shall presume to think that we have given a just theory of the tides.

We begin with the simplest case, supposing the sun and moon to be always in the equator. Let the series begin with the sun and moon in conjunction in the line $Ox$. In this case the points $s$, $m$, and $b$ coincide, and we have high water at 12 o'clock noon and midnight.

While the moon moves from $s$ to $Q$, $Om$ cuts the upper semicircle in $M$; and therefore $CH$, which is always parallel to $MS$, lies between $MC$ and $Cs$. Therefore $b$ is between $m$ and $s$, and we have high water after 12 o'clock, but before the moon's loathing. The same thing happens while the moon moves from $o$ to $q$, during her third quarter.

But while the moon moves from her first quadrature in $Q$ to opposition in $o$ (as in fig. 5.), the line $mo$ drawn from the moon's place, cuts the lower semicircle in $M$ and $CH$, parallel to $SM$, again lies between $M$ and $s$, and therefore $b$ lies between $m$ and $o$. The place of high water is to the eastward of the moon, and we have high water after the moon's fouthing. The same thing happens while the moon is moving from her last quadrature in $g$ to the next syzygy. In short, the point $H$ is always between $M$ and $s$, and the place of high water is always between the moon and the nearest syzygy. The place of high water overtakes the moon in each quadrature, and is overtaken by the moon in each syzygy. Therefore during the first and third quarters, the place of high water gradually falls behind the moon for some time, and then gains upon her again, so as to overtake her in the next quadrature. But during the second and fourth quarters, the place of high water advances before the moon to a certain distance, and then the moon gains upon it, and overtakes it in the next syzygy.

If therefore we suppose the moon to advance uniformly along the equator, the place of high water moves unequally, slowest in the times of new and full moon, and swiftest in the time of the quadratures. There must be some intermediate situations where the place of high water neither gains nor loses upon the moon, but moves with the same velocity.

The rate of motion of the point $b$ may be determined as follows: Draw $Ci$, $Sn$, making very small and equal angles with $HC$ and $MS$. Draw $nC$, and about $S$, with the distance $Sn$, describe the arch $nv$, which may be considered as a straight line perpendicular to $Sn$, or to $MS$.

Then, because $SM$ and $Sn$ are parallel to $CH$ and $Ci$, the points $n$ and $i$ are contemporaneous situations of $M$ and $H$, and the arches $nM$, $iH$, are in the ratio of the angular motions of $m$ and $b$. Also, because $nv$ and $nM$ are perpendicular to $Sn$ and $nC$, the angle $nvM$ is equal to the angle $SnC$, or $SMC$. Also, because the angles $nvM$ and $MIC$ are right angles, and the angles $nvM$, $CMi$, are also equal, the triangles $nvM$, $CMi$, are similar. Therefore

$$nM : nv = MC : MI.$$ And

$$nv : iH = nS : iC,$$ or $$MS : MC;$$ therefore

$$nM : iH = MS : MI.$$ Therefore the angular motion of the moon is to the angular motion of the place of high water as $MS$ to $MI$.

Therefore, when $MS$ is perpendicular to $SC$, and the point $I$ coincides with $S$, the motion of high water is equal to that of the moon. But when $MS$ is perpendicular $SC$, $HC$ is also perpendicular to $Cs$, and the angle $bOs$ is $4c'$, and the high water is in the octant. While the moon passes from $s$ to $m'$, or the high water from $s$ to $b$, the point $I$ falls between $M$ and $S$, and the motion of high water is slower than that of the moon. The contrary obtains while the moon moves from $m'$ to $Q$, or the high water from the octant to the quadrature.

It is evident, that the motion of $b$ in the third quarter of the lunation, that is, in passing from $o$ to $q$, is similar to its motion from $s$ to $Q$. Also, that its motion from $Q$ to $o$ must retard by the same degrees as it accelerated in passing from $s$ to $Q$, and that its motion in the last quarter from $q$ to $s$ is similar to its motion from $Q$ to $o$.

At new and full moon the point $I$ coincides with $C$, and the point $M$ coincides with $s$. Therefore the motion of the high water at full and change is to the motion of the moon as $sC$ to $sS$. But when the moon is in quadrature, $I$ coincides with $C$, and $M$ with $o$. Therefore the motion of the moon is to that of high water as $OS$ to $OC$ or $sC$. Therefore the motion of high water at full and change is to its motion in the quadratures as $OS$ to $sC$, or as the difference of the disturbing forces to their sum. The motion of the tide is therefore slowest in the syzygies and swiftest in the quadratures; yet even in the syzygies it passes the sun along with the moon, but more slowly.

Let the interval between the morning tide of one day and that of the next day be called a tide day. This is always greater than a solar day, or 24 hours, because the place of high water is moving faster to the eastward than the sun. It is less than a lunar day, or 24h. 50', while the high water passes from the second to the third octant, or from the fourth to the first. It is equal to a lunar day when high water is in the octants, and it exceeds a lunar day while high water passes from the first to the second octant, or from the third to the fourth.

The difference between a solar day and a tide day is called, called the priming or the retardation of the tides. This is evidently equal to the time of the earth's describing in its rotation an angle equal to the motion of the high water in a day from the sun. The smallest of these retardations is to the greatest as the difference of the disturbing forces to their sum. Of all the phenomena of the tides, this seems liable to the fewest and most inconceivable derangements from local and accidental circumstances. It therefore affords the best means for determining the proportion of the disturbing forces. By a comparison of a great number of observations made by Dr Mafkeleyne at St Helena and at Barbadoes (places situated in the open sea), it appears that the shortest tide-day is 24 h. 37', and the longest is 25 h. 27'. This gives \( M - S : M + S = 37 : 87 \), and \( S : M = 2 : 4.96 \); which differs only 1 part in 124 from the proportion of 2 to 5, which Daniel Bernoulli collected from a variety of different observations. We shall therefore adopt the proportion of 2 to 5 as abundantly exact. It also agrees exactly with the phenomena of the nutation of the earth's axis and the precession of the equinoxes; and the astronomers affect to have deduced this proportion from these phenomena. But an intelligent reader of their writings will perceive more sense than justice in this assertion. The nutation and precession do not afford phenomena of which we can assign the share to each luminary with sufficient precision for determining the proportion of their disturbing forces; and it is by means of many arbitrary combinations, and without necessity, that D'Alembert has made out this ratio. We cannot help being of opinion, that D'Alembert has accommodated his distribution of the phenomena to this ratio of 2 to 5, which Daniel Bernoulli (the best philosopher and the most candid man of that illustrious family of mathematicians) had, with so much sagacity and justness of inference, deduced from the phenomena of the tides. D'Alembert could not but see the value of this inference; but he wanted to show his own address in deducing it proprio motu forsooth from the nutation and precession. His procedure in this resembles that of his no less vain countryman De la Place, who affects to be highly pleased with finding that Mr Bode's discovery that Meyer had seen the Georium sidus in 1756, perfectly agreed with the theory of its motions which he (De la Place) had deduced from his own doctrines. Any well-informed mathematician will see, that De la Place's data afforded no such precision; and the book on the Elliptical Motions of the Planets, to which he alludes, contains no grounds for his inference. This observation we owe to the author of a paper on that subject in the Transactions of the Royal Society of Edinburgh.

We hope that our readers will excuse this occasional observation, by which we wish to do justice to the merit of a modest man, and one of the greatest philosophers of his time. Our only claim in the present dissertation is the making his excellent performance on the tides accessible to an English reader not much versed in mathematical researches; and we are sorry that our limits do not admit anything more than a sketch of it. But to proceed.

Assuming 2 : 5 as the ratio of SC to CM, we have the angle CMS = 23° 34' nearly, and \( m' o b' = 110° 47' \); and this is the greatest difference between the moon's place and the place of high water. And when this obtains, the moon's elongation \( m' o s \) is 56° 47' from the nearest syzygy. Hence it follows, that while the moon moves uniformly from 56° 47' west elongation to 56° 47' east, or from 123° 13' east to 123° 13' west, the tide day is shorter than the lunar day; and while the moves from 56° 47' east to 123° 13', or from 123° 13' west to 56° 47', the tide-day is longer than the lunar day.

Vol. XVIII. Part II.

We now see the reason why

The swelling tides obey the moon.

The time of high water, when the sun and moon are in the equator, is never more than 47 minutes different from that of the moon's fouthing (+ or — a certain fixed quantity, to be determined once for all by observation.)

It is now an easy matter to determine the hour of high water corresponding to any position of the sun and moon in the equator. Suppose that on the noon of a certain day the moon's distance from the sun is \( m \). The construction of this problem gives us \( h \), and the length of the tide day. Call this \( T \). Then say \( 360° : s m = T : t \), and \( t \) is the hour of high water.

Or, if we choose to refer the time of high water to the moon's fouthing, we must find the value of \( m b \) at the time of the moon's fouthing, and the difference \( d \) between the tide day and a mean lunar day \( L \), and say \( 360° : m b = d : s \), the time of high water before the moon's fouthing in the first and third quarters, but after it in the second and fourth.

The following table by Daniel Bernoulli exhibits these times for every 10th degree of the moon's elongation from the sun. The first or leading column is the moon's elongation from the sun or from the point of opposition. The second column is the minutes of time between the moon's fouthing and the place of high water. The marks — and + distinguish whether the high water is before or after the moon's fouthing. The third column is the hour and minute of high water. But we must remark, that the first column exhibits the elongation, not on the noon of any day, but at the very time of high water. The two remaining columns express the heights of the tides and their daily variations.

| \( m.s. \) | \( m.b. \) | \( s.b. \) | \( M.S. \) | \( M.o. \) | |----------|-----------|-----------|-----------|-----------| | 0 | 0 | 0 | 1000 | 13 | | 10 | 11\(\frac{1}{2}\) | 0.28\(\frac{1}{2}\) | 987 | 38 | | 20 | 22 | 0.58 | 949 | 62 | | 30 | 31\(\frac{1}{2}\) | 1.28\(\frac{1}{2}\) | 887 | 81 | | 40 | 40 | 2.2 | 806 | 91 | | 50 | 45 | 2.35 | 715 | 105 | | 60 | 46\(\frac{1}{2}\) | 3.13\(\frac{1}{2}\) | 610 | 92 | | 70 | 40\(\frac{1}{2}\) | 3.59\(\frac{1}{2}\) | 518 | 65 | | 80 | 25 | 4.55 | 453 | 24 | | 90 | 0 | 6.2 | 429 | 24 | | 100 | 25 | 7.5 | 453 | 65 | | 110 | 40\(\frac{1}{2}\) | 8.0\(\frac{1}{2}\) | 518 | 92 | | 120 | 40\(\frac{1}{2}\) | 8.40\(\frac{1}{2}\) | 610 | 105 | | 130 | 45 | 9.25 | 715 | 91 | | 140 | 40 | 10.2 | 806 | 81 | | 150 | 31\(\frac{1}{2}\) | 10.31\(\frac{1}{2}\) | 887 | 62 | | 160 | 22 | 11.2 | 949 | 38 | | 170 | 11\(\frac{1}{2}\) | 11.31\(\frac{1}{2}\) | 987 | 13 | | 180 | 0 | 12.2 | 1000 | |

The height of high water above the low water constitutes what is usually called the tide. This is the interesting circumstance in practice. Many circumstances render it almost impossible to say what is the elevation of high water above the natural surface of the ocean. In many places the surface at low water is above the natural surface of the ocean. This is the case in rivers at a great distance from their their mouths. This may appear absurd, and is certainly very paradoxical; but it is a fact established on the most unexceptionable authority. One instance fell under our own observation. The low water mark at spring tide in the harbour of Alloa was found by accurate levelling to be three feet higher than the top of the stone pier at Leith, which is several feet above the high water mark of this harbour. A little attention to the motion of running waters will explain this completely. Whatever checks the motion of water in a canal must raise its surface. Water in a canal runs only in consequence of the declivity of this surface: (See River). Therefore a flood tide coming to the mouth of a river checks the current of its waters, and they accumulate at the mouth. This checks the current farther up, and therefore the waters accumulate there also; and this checking of the stream, and consequent rising of the waters, is gradually communicated up the river to a great distance. The water rises everywhere, though its surface still has a slope. In the mean time, the flood tide at the mouth passes by, and an ebb succeeds. This must accelerate even the ordinary course of the river. It will more remarkably accelerate the river now raised above its ordinary level, because the declivity at the mouth will be so much greater. Therefore the waters near the mouth, by accelerating, will sink in their channel, and increase the declivity of the canal beyond them. This will accelerate the waters beyond them; and thus a stream more rapid than ordinary will be produced along the whole river, and the waters will sink below their ordinary level. Thus there will be an ebb below the ordinary surface as well as a flood above it, however sloping that surface may be.

Hence it follows, that we cannot tell what is the natural surface of the ocean by any observations made in a river, even though near its mouth. Yet even in rivers we have regular tides, subjected to all the varieties deduced from this theory.

We have seen that the tide is always proportional to MS. It is greatest therefore when the moon is in conjunction or opposition, being then S₁, the sum of the separate tides produced by the sun and moon. It gradually decreases as the moon approaches quadrature; and when she is at Q or q, it is S₀, or the difference of the separate tides. Supposing S₁ divided into 1000 equal parts, the length of MS is expressed in these parts in the fourth column of the foregoing table, and their differences are expressed in the fifth column.

We may here observe, that the variations of the tides in equal small times are proportional to the sine of twice the distance of the place of high water from the moon. For since Mₙ is a constant quantity, on the supposition of the moon's uniform motion, Mᵥ is proportional to the variation of MS. Now Mₙ : Mᵥ = MC : CI = 1 : sin. 2 y, and Mₙ and MC are constant quantities.

Thus we have seen with what ease the geometrical construction of this problem not only explains all the interesting circumstances of the tides, but also points them out, almost without employing the judgment, and exhibits to the eye the gradual progress of each phenomenon. In these respects it has great advantages over the very elegant algebraic analysis of Mr Bernoulli. In that process we advance almost without ideas, and obtain our solutions as detached facts, without perceiving their regular series. This is the usual pre-eminence of geometrical analysis; and we regret that Mr Bernoulli, who was eminent in this branch, did not rather employ it. We doubt not but that he would have shown still more clearly the connection and gradual progress of every particular. His aim, however, being to instruct those who were to calculate tables of the different affections of the tides, he adhered to the algebraic method. Unfortunately it did not present him with the easiest formulae for practice. But the geometrical construction which we have given suggests several formulae which are exceedingly simple, and afford a very ready mode of calculation.

The fundamental problems are to determine the angle O b or m O b, having m O s given; and to determine MS.

Let the given angle m O s be called a; and, to avoid the ambiguity of algebraic signs, let it always be reckoned from the nearest syzygy, so that we may always have a equal to the sum of x and y. Also make d² = S₁ × fin.² 2 a, which represents the S₁ of fig. 4. or fin.² 2 y, and make p = S × fin. 2 a / M + S × cof. 2 a', which is the expression of S₁ of that figure, or of tan. 2 y. Then we shall have,

1. Sin. y = √(1 - √(1 - d²) / 2). For we shall have cof. 2 y = √(1 - d²). But fin.² y = I / 2 - I / 2 cof. 2 y = I / 2 - √(1 - d²),

and sin. y = √(1 - √(1 - d²) / 2).

2. Tan. y = p / 1 + √(1 + p²). For because p is = tan. 2 y, √(1 + p²) is the secant of 2 y, and 1 + √(1 + p²) : x = p : tan. y.

These processes for obtaining y directly are abundantly simple. But it will be much more expeditious and easy to content ourselves with obtaining 2 y by means of the value of its tangent, viz. S₁ × fin. 2 a / M₁ + S₁ × cof. 2 a. Or, we may find x by means of the similar value of its tangent M₁ d / S₁ d of fig. 4.

There is still an easier method of finding both 2 x and 2 y, as follows.

Make M₁ + S₁ : M₁ - S₁ = tan. a : tan. b. Then b is the difference of x and y, as a is their sum. For this analogy evidently gives the tangent of half the difference of the angles CSM and CMS of fig. 4, or of 2 x and 2 y. Therefore to a, which is half the sum of 2 x + 2 y, add b, and we have 2 x = a + b, or x = a + b / 2, and y = a - b / 2.

By either of these methods a table may be readily computed of the value of x or y for every value of a.

But we must recollect that the values of S₁ and M₁ are by no means constant, but vary in the inverse triplicate ratio of the earth's distance from the sun and moon; and the ratio of 2 to 5 obtains only when these luminaries are at their mean distances from the earth. The forces corresponding to the perigean medium and apogean distances are as follow.

| Apogean | Sun. | Moon. | |---------|------|-------| | Medium | 1,901| 4,258 | | Perigean| 2 | 5 |

Hence we see that the ratio of S₁ to M₁ may vary from 1,901 : 5,025 to 2,105 : 4,258, that is, nearly from 1 : 3 to 1 : 2, or from 2 : 6 to 2 : 4. The solar force does not vary much, and may be retained as constant without any great error. But the change of the moon's force has great effects on the tides both as to their time and their quantity. I. In respect of their Time.

1. The tide day following a spring tide is 24 h. 27' when the moon is in perigee, but 24 h. 33' when she is in apogee.

2. The tide day following neap tide is 25 h. 15', and 25 h. 40' in these two situations of the moon.

3. The greatest interval of time between high water and the moon's southing is 39' and 61'; the angle y being 9° 45' in the first case, and 15° 15' in the second.

II. In respect of their Heights.

1. If the moon is in perigee when new or full, the spring tide will be 8 feet instead of 7, which corresponds to her mean distance. The very next spring tide happens when she is near her apogee, and will be 6 feet instead of 7. The neap tides happen when she is at her mean distance, and will therefore be 3 feet.

But if the moon be at her mean distance when new or full, the two succeeding spring tides will be regular or 7 feet, and one of the neap tides will be 4 feet and the other only 2 feet.

Mr Bernoulli has given us the following table of the time of high water for these three chief situations of the moon, namely, her perigee, mean distance, and apogee. It may be had by interpolation for all intermediate positions with as great accuracy as can be hoped for in phenomena which are subject to such a complication of disturbances. The first column contains the moon's elongation from the sun. The columns P, M, A, contain the minutes of time which elapse between the moon's southing and high water, according as she is in perigee, at her mean distance, or in apogee. The sign — indicates the priority, and + the posteriority, of high water to the moon's southing.

| D | P | M | A | |---|---|---|---| | 0 | 0 | 0 | 0 | | 10 | 9½ | 11½ | 14 | | 20 | 18 | 22 | 27½ | | 30 | 26 | 31½ | 39½ | | 40 | 33 | 40 | 50 | | 50 | 37½ | 45 | 56 | | 60 | 38½ | 46½ | 58 | | 70 | 33½ | 40½ | 50½ | | 80 | 22 | 25 | 31 | | 90 | 0 | 0 | 0 | | 100 | 21 | 25 | 31 | | 110 | 33½ | 40½ | 50½ | | 120 | 38½ | 46½ | 58 | | 130 | 37½ | 45 | 56 | | 140 | 33 | 40 | 50 | | 150 | 26 | 31½ | 39½ | | 160 | 18 | 22 | 27½ | | 170 | 9½ | 11½ | 14 | | 180 | 0 | 0 | 0 |

The reader will undoubtedly be making some comparison in his own mind of the deductions from this theory with the actual state of things. He will find some considerable resemblances; but he will also find such great differences as will make him very doubtful of its justness. In very few places does the high water happen within 4ths of an hour of the moon's southing, as the theory leads him to expect; and in no place whatever does the spring tide fall on the day of new and full moon, nor the neap tide on the day of her quadrature. These always happen two or three days later. By comparing the difference of high water and the moon's southing in different places, he will hardly find any connecting principle. This shows evidently that the cause of this irregularity is local, and that the justness of the theory is not affected by it. By considering the phenomena in a navigable river, he will learn the real cause of the deviation. A flood tide arrives at the mouth of a river. The true theoretical tide differs in no respect from a wave. Suppose a spring tide actually formed on a fluid sphere, and the sun and moon then annihilated. The elevation must sink, pressing the under waters aside, and causing them to rise where they were depressed. The motion will not stop when the surface comes to a level; for the waters arrived at that position with a motion continually accelerated. They will therefore pass this position as a pendulum passes the perpendicular, and will rise as far on the other side, forming a high water where it was low water, and a low water where it was high water; and this would go on for ever, oscillating in a time which mathematicians can determine, if it were not for the viscosity, or something like friction, of the waters. If the sphere is not fluid to the centre, the motion of this wave will be different. The elevated waters cannot sink without diffusing themselves sidewise, and occasioning a great horizontal motion, in order to fill up the hollow at the place of low water. This motion will be greatest about half way between the places of high and low water. The shallower we suppose the ocean the greater must this horizontal motion be. The resistance of the bottom (though perfectly smooth and even) will greatly retard it all the way to the surface. Still, however, it will move till all be level, and will even move a little farther, and produce a small flood and ebb where the ebb and flood had been. Then a contrary motion will obtain; and after a few oscillations, which can be calculated, it will be intangible. If the bottom of the ocean (which we still suppose to cover the whole earth) be uneven, with long extended valleys running in various directions, and with elevations reaching near the surface, it is evident that this must occasion great irregularities in the motion of the undermost waters, both in respect of velocity and direction, and even occasion small inequalities on the surface, as we see in a river with a rugged bottom and rapid current. The deviations of the under currents will drag with them the contiguous incumbent waters, and thus occasion greater superficial irregularities.

Now a flood arriving at the mouth of a river, must act precisely as this great wave does. It must be propagated up the river (or along it, even though perfectly level) in a certain time, and we shall have high water at all the different places in succession. This is distinctly seen in all rivers. It is high water at the mouth of the Thames at three o'clock, and later as we go up the river, till at London bridge we have not high water till three o'clock in the morning, at which time it is again high water at the Nore. But, in the mean time, there has been low water at the Nore, and high water about half way to London; and while the high water is proceeding to London, it is ebbling at this intermediate place, and is low water there when it is high water at London and at the Nore. Did the tide extend as far beyond London as London is from the Nore, we should have three high waters with two low waters interposed. The most remarkable instance of this kind is the Maranon or Amazon river in South America. It appears by the observations of Condamin and others, that between Para, at the mouth of the river, and the conflux of the Ma- Tide and Maragon, there are seven coexistent high waters, with six low waters between them. Nothing can more evidently show that the tides in these places are nothing but the propagation of a wave. The velocity of its superficial motion, and the distance to which it will sensibly go, must depend on many circumstances. A deep channel and gentle acclivity will allow it to proceed much farther up the river, and the distance between the successive summits will be greater than when the channel is shallow and steep. If we apply the ingenious theory of Chevalier Buat, delivered in the article River, we may tell both the velocity of the motion and the interval of the successive high waters. It may be imitated in artificial canals, and experiments of this kind would be very instructive. We have said enough at present for our purpose of explaining the irregularity of the times of high water in different places, with respect to the moon's fouthing. For we now see clearly, that something of the same kind must happen in all great arms of the sea which are of an oblong shape, and communicate by one end with the open ocean. The general tide in this ocean must proceed along this channel, and the high water will happen on its shores in succession. This also is distinctly seen. The tide in the Atlantic ocean produces high water at new and full moon at a later and later hour along the south coast of Great Britain in proportion as we proceed from Scilly islands to Dover. In the same manner it is later and later as we come along the east coast from Orkney to Dover. Yet even in this progress there are considerable irregularities, owing to the sinuosities of the shores, deep indented bays, prominent capes, and extensive ridges and valleys in the channel.

A similar progress is observed along the coasts of Spain and France, the tide advancing gradually from the south, turning round Cape Finisterre, ranging along the north coast of Spain, and along the west and north coasts of France.

The attentive consideration of these facts will not only satisfy us with respect to this difficulty, but will enable us to trace a principle of connection amidst all the irregularities that we observe.

We now add, that if we note the difference between the time of high water of spring tide, as given by theory, for any place, and the observed time of high water, we shall find this interval to be very nearly constant throughout the whole series of tides during a lunation. Suppose this interval to be forty hours. We shall find every other phenomenon succeed after the same interval. And if we suppose the moon to be in the place where she was four hours before, the observation will agree pretty well with the theory, as to the succession of tides, the length of tide day, the retardations of the tides, and their gradual diminution from spring to near tide. We say pretty well; for there still remain several small irregularities, different in different places, and not following any observable law. These are therefore local, and owing to local causes. Some of these we shall afterwards point out. There is also a general deviation of the theory from the real series of tides. The neap tides, and those adjoining, happen a little earlier than the corrected theory points out. Thus at Brest (where more numerous and accurate observations have been made than at any other place in Europe), when the moon changes precisely at noon, it is high water at 3h. 28'. When the moon enters her second quarter at noon, it is high water at 8h. 45', instead of 9h. 45', which theory assigns.

Something similar, and within a very few minutes equal, to this is observed in every place on the sea-coast. This is therefore something general, and indicates a real defect in the theory.

But this arises from the same cause with the other general deviation, viz. that the greatest and least tides do not happen on the days of full and half moon, but a certain time after. We shall attempt to explain this.

We set out with the supposition, that the water acquired in an instant the elevation competent to its equilibrium. But this is not true. No motion is instantaneous, however great the force; and every motion and change of motion produced by a sensible or finite force increases from nothing to a sensible quantity by infinitely small degrees. Time elapses before the body can acquire any sensible velocity; and in order to acquire the same sensible velocity by the action of different forces acting similarly, a time must elapse inversely proportional to the force. An infinitely small force requires a finite time for communicating even an infinitely small velocity; and a finite force, in an infinitely small time, communicates only an infinitely small velocity; and if there be any kind of motion which changes by insensible degrees, it requires a finite force to prevent this change. Thus a bucket of water, hanging by a cord lapped round a light and easily moveable cylinder, will run down with a motion uniformly accelerated; but this motion will be prevented by hanging an equal bucket on the other side, so as to act with a finite force. This force prevents only infinitely small accelerations.

Now let ALKF (fig. 6.) be the solid nucleus of the earth, surrounded by the spherical ocean LBG. Let this be raised to a spheroid BHCG by the action of the moon at M, or in the direction of the axis CM. If all be at rest, this spheroid may have the form precisely competent to its equilibrium. But let the nucleus, with its spheroidal ocean, have a motion round C in the direction AFKL from west to east. When the line of water BA is carried into the situation q infinitely near to BA, it is no longer in equilibrium; for q is too elevated, and the part now come to B is too much depressed. There is a force tending to depress the waters at q, and to raise those now at B; but this force is infinitely small. It cannot therefore restore the shape competent to equilibrium till a sensible time has elapsed; therefore the disturbing force of the moon cannot keep the summit of the ocean in the line MC. The force must be of a certain determinate magnitude before it can in an instant undo the instantaneous effect of the rotation of the waters and keep the summit of the ocean in the same place. But this effect is possible; for the depression at q necessary for this purpose is nearly as the distance from B, being a depression, not from a straight line, but from a circle described with the radius CB. It is therefore an infinitesimal of the first order, and may be restored in an instant, or the continuation of the depression prevented by a certain finite force. Therefore there is some distance, such as BR, where the disturbing force of the moon may have the necessary intensity. Therefore the spherical ocean, instead of being kept continually accumulated at B and D, as the waters turn round, will be kept accumulated at y and y', but at a height somewhat smaller. It is much in this way that we keep melted pitch or other clay my matter from running off from a brush, by continually turning it round, and it hangs protuberant, not from the lowest point, but from a point beyond it, in the direction of its motion. The facts are very similar. The following experiment will illustrate this completely, and is quite a parallel fact. Conceive GDH, the lower half of the ellipse, to be a supple heavy rope or chain hanging from a roller with a handle. The weight of the rope makes it hang in an oblong curve, just as the force of the moon raises the waters of the ocean. Turn the roller very slowly, and the rope, unwinding at one side and winding up on the other side of the roller, will continue to form the same curve; but turn the roller very briskly in the direction FKL, and the rope will now hang like the curve u'y', considerably advanced. advanced from the perpendicular, so far, to wit, that the force of gravity may be able in an instant to undo the infinitely small elevation produced by the turning.

We are very anxious to give this circumstance clearly conceived, and its truth firmly established; because we have observed it to puzzle many persons not accustomed to such difficulties; we therefore hope that our readers, who have got over the difficulty, will indulge us while we give yet another view of this matter, which leads to the same conclusion.

It is certain that the interval between high and low water is not sufficient for producing all the accumulation necessary for equilibrium in an ocean so very shallow. The horizontal motion necessary for gathering together so much water along a shallow sea would be prodigious. Therefore it never attains its full height; and when the waters, already raised to a certain degree, have passed the situation immediately under the moon, they are still under the action of accumulating forces, although these forces are now diminished. They will continue rising, till they have so far past the moon that their situation subjects them to depressing forces. If they have acquired this situation with an accelerated motion, they will rise still farther by their inherent motion, till the depressing forces have destroyed all their acceleration, and then they will begin to fall again. It is in this way that the nutation of the earth's axis produces the greatest inclination, not when the inclining forces are greatest, but three months after. It is thus that the warmest time of the day is a considerable while after noon, and that the warmest season is considerably after midsummer. The warmth increases till the momentary waste of heat exceeds the momentary supply. We conclude by saying, that it may be demonstrated, that, in a sphere fluid to the centre, the time of high water cannot be less, and may be more, than three lunar hours after the moon's southing. As the depth of the ocean diminishes, this interval also diminishes.

It is perhaps impossible to assign the distance by at which the summit of the ocean may be kept while the earth turns round its axis. We can only see, that it must be less when the accumulating force is greater, and therefore less in spring tides than in neap tides; but the difference may be insensible. All this depends on circumstances which we are little acquainted with: many of these circumstances are local; and the situation of the summit of the ocean, with respect to the moon, may be different in different places.

Nor have we been able to determine theoretically what will be the height of the summit. It will certainly be less than the height necessary for perfect equilibrium. Daniel Bernoulli says, that, after very attentive consideration, he is convinced that the height at new or full moon will be to the theoretical height as the cosine of the angle BC to radius or that the height at y will be $Bb \times \frac{C}{C_0}$.

The result of all this reasoning is, that we must always suppose the summit of the tide is at a certain distance eastward from the place assigned by the theory. Mr Bernoulli concludes, from a very copious comparison of observations at different places, that the place of high water is about 20 degrees to the eastward of the place assigned by the theory. Therefore the table formerly given will correspond with observation, if the leading column of the moon's elongation from the sun be altered accordingly. We have inserted it again in this place, with this alteration, and added three columns for the times of high water. Thus changed it will be of great use.

We have now an explanation of the acceleration of the neap tides, which should happen 6 hours later than the spring tides. They are in fact tides corresponding to positions of the moon, which are 20° more, and not the real spring and neap tides. These do not happen till two days after; and if the really greatest and least tides be observed, the least will be found 6 hours later than the first.

| Moon's Southing | Time of High Water | |-----------------|-------------------| | Perigee | M. Dist. | Apogee | | Perigee | M. Dist. | Apogee |

This table is general; and exhibits the time of high water, and their difference from those of the moon's southing, in the open sea, free from all local obstructions. If therefore the time of high water in any place on the earth's equator (for we have hitherto considered no other) be different from this table (supposed correct), we must attribute the difference to the distinguishing circumstances of the situation. Thus every place on the equator should have high water on the day that the moon, situated at her mean distance, changes precisely at noon, at 22 minutes past noon; because the moon passes the meridian along with the sun by supposition. Therefore, to make use of this table, we must take the difference between the first number of the column, intitled time of high water, from the time of high water at full and change peculiar to any place, and add this to all the other numbers of that column. This adapts the table to the given place. Thus, to know the time of high water at Leith when the moon is 50° east of the sun, at her mean distance from the earth, take 22' from 4h. 30', there remains 4.08. Add this to 2h. 45', and we have 6h. 56' for the hour of high water. (The hour of high water at new and full moon for Edinburgh is marked 4h. 30' in Macleay's tables, but we do not pretend to give it as the exact determination. This would require a series of accurate observations.)

It is by no means an easy matter to ascertain the time of high water with precision. It changes so very slowly, that we may easily mistake the exact minute. The best method is to have a pipe with a small hole near its bottom, and a float with a long graduated rod. The water gets in by the small hole and raises the float, and the smoothness of the hole prevents the sudden and irregular starts which waves would occasion. Instead of observing the moment of high water, observe the height of the road about half an hour before, and wait after high water till the rod comes again to that height; take the middle between them. The water rises TID

We have now given in sufficient detail the phenomena of the tides along the equator, when the sun and moon are both in the equator, shewing both their times and their magnitude. When we recollect that all the sections of an oblong spheroid by a plane passing through an equatorial diameter are ellipses, and that the compound tide is a combination of two such spheroids, we perceive that every section of it through the centre, and perpendicular to the plane in which the sun and moon are situated, is also an ellipse, whose shorter axis is the equatorial diameter of a spring tide. This is the greatest depression in all situations of the luminaries; and the points of greatest depression are the lower poles of every compound tide. When the luminaries are in the equator, these lower poles coincide with the poles of the earth. The equator, therefore, of every compound tide is also an ellipse; the whole circumference of which is lower than any other section of this tide, and gives the place of low water in every part of the earth. In like manner, the section through the four poles, upper and lower, gives the place of high water. These two sections are terrestrial meridians or hour circles, when the luminaries are in the equator.

Hence it follows, that all that we have already said as to the times of high and low water may be applied to every place on the surface of the earth, when the sun and moon are in the equator. But the heights of tide will diminish as we recede from the equator. The heights must be reduced in the proportion of radius to the cosine of the latitude of the place. But in every other situation of the sun and moon all the circumstances vary exceedingly. It is very true, that the determination of the elevation of the waters in any place whatever is equally easy. The difficulty is, to exhibit for that place a connected view of the whole tide, with the hours of flood and ebb, and the difference between high and low water. This is not indeed difficult; but the process by the ordinary rules of spherical trigonometry is tedious. When the sun and moon are not near conjunction or opposition, the shape of the ocean resembles a tunip, which is flat and not round in its broadest part. Before we can determine with precision the different phenomena in connection, we must ascertain the position or attitude of this tunip; marking on the surface of the earth both its elliptical equators. One of these is the plane passing through the sun and moon, and the other is perpendicular to it, and marks the place of low water. And we must mark in like manner its first meridian, which passes through all the four poles, and marks on the surface of the earth the place of high water. The position of the greatest section of this compound spheroid is frequently much inclined to the earth's equator; nay, sometimes is at right angles to it, when the moon has the same right ascension with the sun, but a different declination. In these cases the ebb tide on the equator is the greatest possible; for the lower poles of the compound spheroid are in the equator. Such situations occasion a very complicated calculus. We must therefore content ourselves with a good approximation.

And first, with respect to the times of high water. It will be sufficient to conceive the sun and moon as always in one plane, viz. the ecliptic. The orbits of the sun and moon are never more inclined than 5 degrees. This will make very little difference; for when the luminaries are so situated that the great circle through them is much inclined to the equator, they are then very near to each other, and the form of the spheroid is little different from what it would be if they were really in conjunction or opposition. It will therefore be sufficient to consider the moon in three different situations.

1. In the equator. The point of highest water is never farther... ther from the moon than $15^\circ$, when she is in apogee and the sun in perigee. Therefore if a meridian be drawn thro' the point of highest water to the equator, the arch $mb$ of fig. 4 will be represented on the equator by another arch about $\frac{2}{3}$ of this by reason of the inclination of the equator and ecliptic. Therefore, to have the time of high water, multiply the numbers of the columns which express the difference of high water and the moon's southing by $\frac{2}{3}$, and the products give the real difference.

2. Let the moon be in her greatest declination. The arch of right ascension corresponding to $mb$ will be had by multiplying $mb$, or the time corresponding to it in the table, by $\frac{2}{3}$.

3. When the moon is in a middle situation between these two extremes, the numbers of the table will give the right ascension corresponding to $mb$ without any correction, the distance from the equator compensating for the obliquity of the ecliptic arch $mb$.

The time of low water is not so easily found; and we must either go through the whole trigonometrical process, or content ourselves with a less perfect approximation. The trigonometrical process is not indeed difficult: We must find the position of the plane through the sun and moon. A great circle through the moon perpendicular to this is the line of high water; and another perpendicular circle cutting this at right angles is the circle of low water.

But it will be abundantly exact to consider the tide as accompanying the moon only.

Let NOS (fig. 7.) be a section of the terraqueous globe, of which N and S are the north and south poles and POQ the equator. Let the moon be in the direction OM, having the declination BQ. Let D be any place on the earth's surface. Draw the parallel LDC of latitude. Let BFB be the ocean, formed into a spheroid, of which BB is the axis and F the equator.

As the place D is carried along the parallel CDL by the rotation of the earth, it will pass in succession through different depths of the watery spheroid. It will have high water when at C and L, and low water when it crosses the circle fOF. Draw the meridian NDG, and the great circle BdB. The arch GQ, when converted into lunar hours (each about 62 minutes), gives the duration of the flood $ac$ and of the subsequent ebb $cd$, which happen while the moon is above the horizon; and the arch EG will give the durations of the flood and of the ebb which happen when the moon is below the horizon. It is evident that these two floods and two ebbs have unequal durations. When D is at C it has high water; and the height of the tide is CC'. For the spheroid is supposed to touch the sphere on the equator fOF, so that of CC' is the difference between high and low water. At L the height of the tide is LL'; and if we describe the circle LNq, C'q is the difference of these high waters, or of these tides.

Hence it appears, that the two tides of one lunar day may be considerably different, and it is proper to distinguish them by different names. We shall call that a superior tide which happens when the moon is above the horizon during high water. The other may be called the inferior tide. The duration of the superior tide is measured by $2GQ$, and that of the inferior tide by $2EG$, and $4GO$ measures the difference between the whole duration of a superior and an inferior tide.

From this construction we may learn in general, 1. When the moon has no declination, the durations and also the heights of the superior and inferior tides are equal in all parts of the world. For in this case the tide equator f coincides with the meridian NOS, and the poles B b of the watery spheroid are on the earth's equator.

2. When the moon has declination, the duration and also the height of a superior tide at any place is greater than that of the inferior; or is less than it, according as the moon's declination and the latitude of the place are of the same or opposite names.

This is an important circumstance. It frequently happens that the inferior tide is found the greatest when it should be the least; which is particularly the case at the Nore. This shows, without further reasoning, that the tide at the Nore is only a branch of the regular tide. The regular tide comes in between Scotland and the continent; and after travelling along the coast reaches the Thames, while the regular tide is just coming in again between Scotland and the continent.

3. If the moon's declination is equal to the colatitude of the place, or exceeds it, there will be only one tide in a lunar day. It will be a superior or an inferior tide, according as the declination of the moon and the latitude of the place are of the same or opposite kinds. For the equator of the tide cuts the meridian in f and F. Therefore a place which moves in the parallel ef has high water when at $e_9$, and 12 lunar hours afterwards, has low water when at $f$. And any place k which is still nearer to the pole N has high water when at $k$, and 12 lunar hours afterwards has low water at m. Therefore, as the moon's declination extends to $30^\circ$, all places farther north or south than the latitude $60^\circ$ will sometimes have only one tide in a lunar day.

4. The sine of the arch GO, which measures $\frac{1}{2}$th of the difference between the duration of a superior and inferior tide, is = tan. lat. $\times$ tan. decl. For in the spherical triangle dOG

$$\text{Rad} \cdot \cotan. dOG = \tan. dG : \sin. GO,$$

and

$$\text{Sin. GO} = \tan. dOQ \times \tan. dG = \tan. decl. \times \tan. lat.$$

Hence we see, that the difference of the durations of the superior and inferior tides of the same day increase both with the moon's declination and with the latitude of the place.

The different situations of the moon and of the place of observation affect the heights of the tides no less remarkably. When the point D comes under the meridian NBQ in which the moon is situated, there is a superior high water, and the height of the tide above the low water of that day is CC'. When D is at L, the height of the inferior tide is LL'. The elevation above the inscribed sphere is M $\times$ cof. $y$, y being the zenith distance of the moon at the place of observation. Therefore at high water, which by the theory is in the place directly under the moon, the height of the tide is as the square of the cosine of the moon's zenith or nadir distance.

Hence we derive a construction which solves all questions relating to the height of the tides with great facility, free from all the intricacy and ambiguities of the algebraic analysis employed by Bernoulli.

With the radius CQ = M (the elevation produced by the moon above the inscribed sphere) describe the circle pQPE (fig. 8.) to represent a meridian, of which P and p are the poles, and EQ the equator. Bisect CP in O; and round O describe the circle PBOD. Let M be the place over which the moon is vertical, and Z be the place of observation. MQ is the moon's declination, and ZQ is the latitude of the place. Draw MCm, ZCN, cutting the small circle in A and B. Draw AGI perpendicular to CP, and draw CI, which will cut off an arch E = QM. MZ and N are the moon's zenith and nadir distances. Draw the diameter BD, and the perpendiculars IK, GH, and AF. Also draw OA, PA, AB, ID.

Then DF is the superior tide, DK is the inferior tide, and DH is the arithmetical mean tide. For the angles PCA, BDA, standing on BA, are equal.

Also the angles DB, CN, are equal, being supplements of the angle ICB. Therefore, if BD be made radius, DA and DI are the lines of the zenith and nadir distances of the moon.

But \( BD : DA = DA : DF \). Therefore \( DF = M \cdot \text{cof.}^2 y \), the height \( Z \) of the superior tide. Also \( DK = M \cdot \text{cof.}^2 y \), the height \( N \) of the inferior tide.

Also, because IA is bisected in G, KF is bisected in H, and \( DH = \frac{DK + DF}{2} \), the medium tide.

Let us trace the relation of the consequences of the various positions of Z and M, as we formerly considered the results of the various situations of the sun and moon.

First, then, let Z retain its place, and let M gradually approach it from the equator. When M is in the equator, A and I coincide with C, and the three points F, K, and H, coincide in i.

As M approaches to Z, A and I approach to B and D; DF increases, and DK diminishes. The superior or inferior tide is greatest when the moon is in M or in N; and DF is then \( = M \). As the moon passes to the northward of the place, the superior and inferior tides both diminish till I comes to D; at which time MQ is equal to ZP, and there is no inferior tide. This however cannot happen if \( ZP \) is greater than \( 30^\circ \), because the moon never goes farther from the equator. M still going north, we have again a perpendicular from I on BD, but below I, indicating that the inferior tide, now measured by DK, belongs to the hemispheroid next the moon. Also, as M advances from the equator northward, DH diminishes continually. First, while H lies between O and B, because G approaches O; and afterwards, when G is above O and H lies between O and D. It is otherwise, however, if \( ZQ \) is greater than \( 45^\circ \); for then DB is inclined to EQ the other way, and DH increases as the point G rises.

In the next place, let M retain its position, and Z proceed along the meridian.

Let us begin at the equator, or suppose Q the place of observation. BD then coincides with CP, and the three lines DF, DK, and DH, all coincide with PG, denoting the two equal tides Qg and Ec and their medium, equal to each other. As Z goes northward from Q, BOD detaches itself from COP; the line DF increases, while DK and DH diminish. When Z has come to M, F and B coincide with A, and DK and DH are still more diminished. When Z passes M, all the three lines DF, DK, and DH, continue to diminish. When Z comes to latitude \( 45^\circ \), DB is parallel to IA and EQ, and the point H coincides with O. This situation of Z has the peculiar property that DH (now DO) is the same, whatever be the declination of the moon. For IA being always parallel to DB, OK and OF will be equal, and DO will be half of DK and DF however they may vary. When Z gets so far north that ZP is \( = MQ \), the diameter bd falls on I; so that Jk vanishes, and we have only df. And when Z goes still farther north, dk appears on the other side of I. When Z arrives at the pole, BD again coincides with PC, D with C, and DF, DK, and DH, coincide with CG.

These variations of the points F, K, and H, indicate the following phenomena.

1. The greatest tides happen when the moon is in the zenith or nadir of the place of observation; for then the point B coincides with A, and DF becomes DB; that is, \( = M \), indicating the full tide BB'.

2. When the moon is in the equator, the superior and inferior tides have equal heights, \( = M \cdot \text{cof.}^2 \text{lat.} \). For then A and I coincide with C, and the points F and K coincide in i, and \( Di = DB \cdot \text{cof.}^2 BDC = M \cdot \text{cof.}^2 \text{lat.} \).

3. If the place of observation is in the equator, the inferior and superior tides are again equal, whatever is the moon's declination: For then B coincides with C, and the points F, K, and H, coincide with G; and \( PG = PC \cdot \text{cof.}^2 APG = M \cdot \text{cof.}^2 \text{decl. moon.} \)

4. The superior tides are greater or less than the inferior tides according as the latitude and declination are of the same or of opposite names. For by making \( QZ = ZQ \), and drawing \( ZC \), cutting the small circle in \( P \), we see that the figure is reversed. The difference between the superior and inferior tides is \( KF \), or \( IA \cdot \text{cof.}^2 \) of the angle formed by IA and DB; that is, of the angle BD \( ^\circ \), which is the complement of twice \( ZQ \); because \( BOC = 2ZCQ \). Now \( TA = 2GA = 2OA \cdot \text{fin.} 2MQ = PC \cdot \text{fin.} 2MQ = M \cdot \text{fin.} 2 \text{decl.} \). Therefore the difference of the superior and inferior tides is \( M \cdot \text{fin.} 2 \text{declin. fin.} 2 \text{lat.} \).

5. If the colatitude be equal to the declination, or less than it, there will be no inferior tide, or no superior tide, according as the latitude of the place and declination of the moon are of the same or opposite names.

For when \( PQ = MQ \), D coincides with I, and IK vanishes. When \( PQ \) is less than \( MQ \), the point D is between C and I, and the point Z never passes through the equator of the watery spheroid; and the low water of its only tide is really the summit of the inferior tide.

6. At the pole there is no daily tide; but there are two monthly tides \( = M \cdot \text{fin.}^2 \text{declin.} \) and it is low water when the moon is in the equator.

7. The medium tide, represented by DH, is \( = M \cdot \text{cof.}^2 \text{lat.} \times \frac{1 + \text{cof.}^2 \text{lat.}}{2} \cdot \text{cof.}^2 \text{declin.} \). For \( DH = DO + OH \).

Now \( OH \) is equal to \( OG \cdot \text{cof.} GOH = OG \cdot \text{cof.}^2 ZQ \). And \( OG = OA \cdot \text{cof.} GOA = OA \cdot \text{cof.}^2 MQ \). Therefore \( OH = OA \cdot \text{cof.}^2 ZQ \cdot \text{cof.}^2 MQ = OA + OA \cdot \text{cof.}^2 ZQ \cdot \text{cof.}^2 MQ = M \cdot \text{cof.}^2 ZQ \cdot \text{cof.}^2 MQ \). Let this for the future be called \( m \).

N.B. The moon's declination never exceeds \( 30^\circ \). Therefore \( \text{cof.}^2 MQ \) is always a positive quantity, and never less than \( \frac{1}{2} \), which is the cofine of \( 60^\circ \). While the latitude is less than \( 45^\circ \), \( \text{cof.}^2 \text{lat.} \) is also a positive quantity. When it is precisely \( 45^\circ \), the cofine of its double is \( \frac{1}{2} \); and when it is greater than \( 45^\circ \), the cofine of its double is negative. Hence we see,

1. That the medium tides are equally affected by the northern and southern declinations of the moon.

2. If the latitude of the place is \( 45^\circ \), the medium tide is always \( \frac{1}{2} M \). This is the reason why the tides along the coasts of France and Spain are so little affected by the declination of the moon.

3. If the latitude is less than \( 45^\circ \), the mean tides increase as the moon's declination diminishes. The contrary happens if \( ZQ \) is greater than \( 45^\circ \). For DH increases or diminishes while the point G separates from C according as the angle COD is greater or less than COB; that is, according as PCZ is greater or less than ZCO.

4. When Z is in the equator, H coincides with G, and the effect of the moon's declination on the height of the tides is the most sensible. The mean tide is then \( = M \cdot \text{cof.}^2 MQ \).

All that we have now said may be said of the solar tide, putting S in place of M.

Also the same things hold true of spring tides, putting \( M + S \) in place of M. But in order to ascertain the effects of declination and latitude on other tides, we must make a much more complicated construction, even tho' we suppose both luminaries in the ecliptic. For in this case the two depressed poles of the watery spheroid are not in the poles of the earth; and therefore the sections of the ocean, made by meridiana, are by no means ellipses.

In a neap tide, the moon is vertical at B (fig. 7 or 8.), and the sun at some point of \( F \), 92° from B. If O be this point, the construction for the heights of the tides may be made by adding to both the superior and inferior tides for any point D, the quantity \( M + S - D'F \) or \( DK \times \sin^2 dO = M + S - \text{tide} \times \frac{\sin^2 2Q}{\text{col}^2 MQ} \), as is evident.

But if the sun be vertical at \( d \), \( d \) will be the highest part of the circle \( OF \), and no correction is necessary. But in this case the circle of high water will be inclined to the meridian in an angle equal to \( dBO \) (fig. 7.), and neither the times nor elevations of high water will be properly ascertained, and the error in time may be considerable in high latitudes.

The inaccuracies are not so great in intermediate tides, and respect chiefly the time of high water and the height of low water.

The exact computation is very tedious and peculiar, so that it is hardly possible to give any account of a regular progress of phenomena; and all we can do is, to ascertain the precise heights of detached points. For which reasons, we must content ourselves with the construction already given. It is the exact geometrical expression of Bernoulli's analysis, and its consequences now related contain all that he has investigated. We may accommodate it very nearly to the real state of things, by supposing PC equal, not to CO of fig. 4., but to MS, exhibiting the whole compound tide. And the point B, instead of representing the moon's place, must represent the place of high water.

Thus have we obtained a general, though not very accurate, view of the phenomena which must take place in different latitudes and in different declinations of the sun and moon, provided that the physical theory which determines the form and position of the watery spheroid be just. We have only to compute, by a very simple process of spherical trigonometry, the place of the pole of this spheroid. The second construction, in fig. 8., shows us all the circumstances of the time and height of high water at any point. It will be recollected, that in computing this place of the pole, the anticipation of 20 degrees, arising from the inertia of the waters, must be attended to.

Were we to institute a comparison of this theory with observation, without farther consideration, we should still find it unfavourable, partly in respect of the heights of the tides, and more remarkably in respect of the time of low water. We must again consider the effects of the inertia of the waters, and recollect, that a regular theoretical tide differs very little in its progress from the motion of a wave. Even along the free ocean, its motion much resembles that of any other wave. All waves are propagated by an oscillatory motion of the waters, precisely similar to that of a pendulum. It is well known, that if a pendulum receive a small impulse in the time of every descent, its vibrations may be increased to infinity. Did the successive actions of the sun or moon just keep time with the natural propagation of the tides, or the natural oscillations of the waters, the tides would also augment to infinity: But there is an infinite odds against this exact adjustment. It is much more probable that the action of to-day interrupts or checks the oscillation produced by yesterday's action, and that the motion which we perceive in this day's tide is what remains, and is compounded with the action of to-day. This being the case, we should expect that the nature of any tide will depend much on the nature of the preceding tide. Therefore we should expect that the superior and inferior tides of the same day will be more nearly equal than the theory determines. The whole course of observation confirms this. In latitude 45°, the superior and inferior tides of one day may differ in the proportion of 2:1 to 1, and the tides corresponding to the greatest and least declinations of the moon may differ nearly as much. But the difference of the superior and inferior tides, as they occur in the list of observations at Rochefort, is not the third part of this, and the changes made by the moon's declination is not above one-half. Therefore we shall come much nearer the true measure of a spring tide, by taking the arithmetical mean, than by taking either the superior or inferior.

We should expect less deviation from the theory in the gradual diminution of the tides from spring tide to neap tide, and in the gradual changes of the medium tide by the declination of the moon; because the successive changes are very small; and when they change in kind, that is, diminish after having for some time augmented, the change is by insensible degrees. This is most accurately confirmed by observation. The vast collection made by Cassini of the Observations at Bretz being examined by Bernoulli, and the medium of the two tides in one day being taken for the tide of that day, he found such an agreement between the progression of these medium tides and the progression of the lines MS of fig. 4., that the one seemed to be calculated by the other. He found no less agreement in the changes of the medium tides by the moon's declination.

In like manner, the changes produced by the different distances of the moon from the earth, were found abundantly conformable to the theory, although not so exact as the other. This difference or inferiority is easily accounted for: When the moon changes in her mean distance, one of the neap tides is uncommonly small, and therefore the successive diminutions are very great, and one tide sensibly affects another. The same circumstance operates when the changes in apogee, by reason of a very large spring tide. And the changes corresponding both to the sun's distance from the earth and his declination agreed almost exactly.

All these things considered together, we have abundant reason to conclude, that not only the theory itself is just in principle (a thing which no intelligent naturalist can doubt), but also that the data which are assumed in the application are properly chosen; that is, that the proportion of 2 to 5 is very nearly the true proportion of the mean solar and lunar forces. If we now compute the medium tide for any place in succession, from spring tide to neap tide, and still more, if we compute the series of times of their occurrence, we shall find as great an agreement as can be desired. Not but that there are many irregularities; but these are evidently so anomalous, that we can ascribe them to nothing but circumstances which are purely local.

This general rule of computation must be formed in the following manner:

The spring tide, according to theory, being called A, and the neap tide B, recollect that the spring tide, according to the regular theory, is measured by \( M + S \). Recollect also, that when the lunar tide only is considered, the superior spring tide is \( M \times \sin^2 ZM \) (fig. 8.). But when we consider the action of two adjoining tides on each other, we find it safer to take the medium of the superior and inferior tides for the measure; and this is \( M \times \frac{1 + \text{cof}^2 ZQ \times \text{cof}^2 MQ}{2} \). Let this be called \( m \). This Tide, being totally the effect of M as modified by latitude and declination, may be taken as its proper measure, by which we are to calculate the other tides of the monthly series from spring tide to neap tide.

In like manner, we must compute a value for S, as modified by declination and latitude; call this r. Then say,

\[ M + S : A = m + s : A \times \frac{m + s}{M + S}. \]

This fourth proportional will give the spring tide as modified for the given declination of the luminaries, and the latitude of the place.

Now recollect, that the medium tide, when the luminaries are in the equator, is \( A \times \text{cof.}^2 \text{lat.} \). Therefore let F be the spring tide observed at any place when the luminaries are in the equator; and let this be the medium of a great many observations made in these circumstances. This gives \( A \cdot \text{cof.}^2 \text{lat.} \) (as modified by the peculiar circumstances of the place) = F. Therefore the fourth proportional now given changes to \( F \times \frac{m + s}{M + S} \cdot \text{cof.}^2 \text{lat.} \).

And a similar substitute for B is \( G \times \frac{m - s}{M - S} \cdot \text{cof.}^2 \text{lat.} \).

Lastly, To accommodate our formulae to every distance of the earth from the sun and moon, let D and d be the mean distances of the sun and moon, and d and d' their distances at the given time; and then the two substitutes become

\[ \Delta d^3 \frac{M + S}{d^3} \times F \times \frac{m + s}{(M + S) \cdot \text{cof.}^2 \text{lat.}}. \]

The half sum of these two quantities will be the MC, and their half difference will be the SC, of fig. 4, with which we may now operate, in order to find the tide for any other day of the menstrual series, by means of the elongation \( a \) of the moon from the sun; that is, we must say \( MC + CS : \)

\[ MC - CS = \tan a : \tan b; \quad \text{then } x = \frac{a + b}{2}, \quad \text{and } y = \frac{a - b}{2}. \]

And MS, the height of the tide, is \( MC \times \text{cof.}^2 y + CS \times \text{cof.}^2 x \).

Such is the general theory of the tides, deduced from the principle of universal gravitation, and adjusted to that proportion of the solar and lunar forces which is most consistent with other celestial phenomena. The comparison of the greatest and least daily retardations of the tides was with great judgment preferred to the proportion of spring and neap tides, selected by Sir Isaac Newton for this purpose. This proportion must depend on many local circumstances. When a wave or tide comes to the mouths of two rivers, and finds a tide up each, and another tide of half the magnitude comes a fortnight after; the proportion of tides sent up to any given places of these rivers may be extremely different. Nay, the proportion of tides sent up to two distant places of the same river can hardly be the same; nor are they the same in any river that we know. It can be demonstrated, in the strictest manner, that the farther we go up the river, where the declivity is greater, the neap tide will be smaller in proportion to the spring tide. But it does not appear that the time of succession of the different tides will be much affected by local circumstances. The tide of the second day of the moon being very little less than that of the first, will be nearly as much retarded, and the intervals between their arrivals cannot be very different from the real intervals of the undisturbed tides; accordingly, the succession of the highest to the highest but one is found to be the same in all places, when not disturbed by different winds. In like manner, the succession of the lowest and the lowest but one is found equally invariable; and the highest and the lowest tides observed in any place must be accounted the spring and neap tides of that place, whether they happen on the day of full and half moon or not. Nay, we can see here the explanation of a general deviation of the theory which we formerly noticed. A low tide, being less able to overcome obstructions, will be sooner stopped, and the neap tides should happen a little earlier than by the undisturbed theory.

With all these corrections, the theory now delivered will be found to correspond, with observation, with all the exactness that we can reasonably expect. We had an opportunity of comparing it with the phenomena in a place where they are very singular, viz. in the harbour of Biftefelt in Iceland. The equator of the watery sphere frequently passes through the neighbourhood of this place, in a variety of positions with respect to its parallel of diurnal revolution, and the differences of superior and inferior tides are most remarkable and various. We found a wonderful conformity to the most diversified circumstances of the theory.

There is a period of 18 years, reflecting the tides in Iceland, taken notice of by the ancient Saxons; but it is not distinctly described. Now this is the period of the moon's nodes, and of the greatest and least inclination of her orbit to the equator. It is therefore the period of the positions of the equator of the tides which ranges round this island, and very sensibly affects them.

Hitherto we have supposed the tides to be formed on an ocean completely covering the earth. Let us see how those may be determined which happen in a small and confined sea, such as the Caspian or the Black Sea. The determination in this case is very simple. As no supply of water is supposed to come into the basin, it is susceptible of a tide only by sinking at one end and rising at the other. This may be illustrated by fig. 6, where C_x, C_y, are two perpendicular planes bounding a small portion of the natural ocean. The water will sink at z and rise at x, and form a surface of r parallel to the equilibrated surface yz. It is evident that there will be high water, or the greatest possible rise at r, when the basin comes to that position where the tangent is most or all inclined to the diameter. This will be when the angle CB is 45° nearly, and therefore three lunar hours after the moon's fourth; at the same time, it will be low water at the other end. It is plain that the rise and fall must be exceedingly small, and that there will be no change in the middle. The tides of this kind in the Caspian Sea, in latitude 45°, whose extent in longitude does not exceed eight degrees, are not above seven inches; a quantity so small, that a slight breeze of wind is sufficient to check it, and even to produce a rise of the waters in the opposite direction. We have not met with any accounts of a tide being observed in this sea.

It should be much greater, though still very small, in the Mediterranean Sea. Accordingly, tides are observed there, but still more remarkably in the Adriatic, for a reason which will be given by and by. We do not know that tides have been observed in the great lakes of North America. These tides, though small, should be very regular.

Should there be another great basin in the neighbourhood of z, lying east or west of it, we should observe a curious phenomenon. It would be low water on one side of the shore z when it is high water on the other side of this partition. If the tides in the Euxine and Caspian Seas, or in the American lakes which are near each other, could be observed, this phenomenon should appear, and would be one of the prettiest examples of universal gravitation that can be Tide. Something like it is to be seen at Gibraltar. It is high water on the east side of the rock about 10 o'clock at full and change, and it is high water on the west side, not a mile distant, at 12. This difference is perhaps the chief cause of the singular current which is observed in the Straits mouth. There are three currents observed at the same time, which change their directions every 12 hours. The small tide of the Mediterranean proceeds along the Barbary shore, which is very uniform all the way from Egypt, with tolerable regularity. But along the northern side, where it is greatly obstructed by Italy, the islands, and the east coast of Spain, it sets very irregularly; and the perceptible high water on the Spanish coast differs four hours from that of the southern coast. Thus it happens, that one tide ranges round Europa point, and another along the shore near Ceuta, and there is a third current in the middle different from both. Its general direction is from the Atlantic Ocean into the Mediterranean Sea, but it sometimes comes out when the ebb tide in the Atlantic is considerable.

Suppose the moon over the middle of the Mediterranean. The surface of the sea will be level, and it will be half tide at both ends, and therefore within the Straits of Gibraltar. But without the Straits it is within half an hour of high water. Therefore there will be a current setting in from the Atlantic. About three and an half hours after, it is high water within and half ebb without. The current now sets out from the Mediterranean. Three hours later, it is low water without the Straits and half ebb within; therefore the current has been setting out all this while. Three hours later, it is half flood without the Straits and low water within, and the current is again setting in, &c.

Were the earth fluid to the centre, the only sensible motion of the waters would be up and down, like the waves on the open ocean, which are not brushed along by strong gales. But the shallowness of the channel makes a horizontal motion necessary, that water may be supplied to form the accumulation of the tide. When this is formed on a flat shelving coast, the water must flow in and out, on the flats and sands, while it rises and falls. These horizontal motions must be greatly modified by the channel or bed along which they move. When the channel contracts along the line of flowing water, the wave, as it moves up the channel, and is checked by the narrowing shores, must be reflected back, and keep a-top of the waters still flowing in underneath. Thus it may rise higher in these narrow seas than in the open ocean. This may serve to explain a little the great tides which happen on some coasts, such as the coast of Normandy. At St Malo the flood frequently rises 50 feet. But we cannot give any thing like a full or satisfactory account of these singularities. In the Bay of Fundy, and particularly at Annapolis Royal, the water sometimes rises above 100 feet. This seems quite inexplicable by any force of the sun and moon, which cannot raise the waters of the free ocean more than eight feet. These great floods are unquestionably owing to the proper timing of certain oscillations or currents adjoining, by which they unite, and form one of great force. Such violent motions of water are frequently seen on a small scale in the motions of brooks and rivers; but we are too little acquainted with hydraulics to explain them with any precision.

We have seen that there is an oscillation of waters formed under the sun and moon; and that in consequence of the rotation of the earth, the inertia and the want of perfect fluidity of the waters, and obstructions in the channel, this accumulation never reaches the place where it would finally settle if the earth did not turn round its axis. The consequence of this must be a general current of the waters from east to west. This may be seen in another way. The moon in her orbit round the earth has her gravity to the earth diminished by the sun's disturbing force, and therefore moves in an orbit less incurvated than she would describe independent of the sun's action. She therefore employs a longer time. If the moon were so near the earth as almost to touch it, the same thing would happen. Therefore suppose the moon turning round the earth, almost in contact with the equator, with her natural undisturbed periodic time, and that the earth is revolving round its axis in the same time, the moon would remain continually above the same spot of the earth's surface (suppose the city of Quito), and a spectator in another planet would see the moon always covering the same spot. Now let the fun act. This will not affect the rotation of the earth, because the action on one part is exactly balanced by the action on another. But it will affect the moon. It will move more slowly round the earth's centre, and at a greater distance. It will be left behind by the city of Quito, which it formerly covered. And as the earth moves round from west to east, the moon, moving more slowly, will have a motion to the west with respect to Quito. In like manner, every particle of water has its gravity diminished, and its diurnal motion retarded; and hence arises a general motion or current from east to west. This is very distinctly perceived in the Atlantic and Pacific Oceans. It comes round the Cape of Good Hope, ranges along the coast of Africa, and then sets directly over to America, where it meets a similar stream which comes in by the north of Europe. Meeting the shores of America, it is deflected both to the south along the coast of Brazil, and to the north along the North American shores, where it forms what is called the Gulf Stream, because it comes from the Gulf of Mexico. This motion is indeed very slow, this being sufficient for the accumulation of seven or eight feet on the deep ocean; but it is not altogether insensible.

We may expect differences in the appearances on the western shores of Europe and Africa, and on the western shore of America, from the appearances on the eastern coasts of America and of Asia, for the general current obstructs the waters from the western shores, and sends them to the eastern shores. Also when we compare the wide opening of the northern extremity of the Atlantic Ocean with the narrow opening between Kamtchatka and America, we should expect differences between the appearances on the west coasts of Europe and of America. The observations made during the circumnavigations of Captain Cook and others show a remarkable difference. All along the west coast of North America the interior tide is very trifling, and frequently is not perceived.

In the very same manner, the disturbing forces of the sun and moon form a tide in the fluid air which surrounds this globe, consisting of an elevation and depression, which move gradually from east to west. Neither does this tide ever attain that position with respect to the disturbing planets which it would do were the earth at rest on its axis. Hence arises a motion of the whole air from east to west; and this is the principal cause of the trade-winds. They are a little accelerated by being heated, and therefore expanding. They expand more to the westward than in the opposite direction, because the air expands on that side into air, which is now cooling and contracting. These winds very evidently follow the sun's motion, tending more to the south or north as he goes south or north. Were this motion considerably affected by the expansion of heated air, we should find the air rather coming northward and southward from the torrid zone. zone, in consequence of its expansion in that climate. We repeat it, it is almost solely produced by the aerial tide, and is necessary for the very formation of this tide. We cannot perceive the accumulation. It cannot affect the barometer, as many think, because, though the air becomes deeper, it becomes deeper only because it is made lighter by the gravitation to the sun. Instead of pressing more on the cistern of the barometer, we imagine that it presses less; because, like the ocean, it never attains the height to which it tends. It remains always too low for equilibrium, and therefore it should press with less force on the cistern of a barometer.

There is an appearance precisely similar to this in the planet Jupiter. He is surrounded by an atmosphere which is arranged in zones or belts, probably owing to climate differences of the different latitudes, by which each seems to have a different kind of sky. Something like this will appear to a spectator in the moon looking at this earth. The general weather and appearance of the sky is considerably different in the torrid and temperate zones. Jupiter's belts are not of a constant shape and colour; but there often appear large spots or tracts of cloud, which retain their shape during several revolutions of Jupiter round his axis. To judge of his rotation by one of these, we should say that he turns round in 9.55. There is also a brighter spot which is frequently seen, occupying one certain situation on the body of Jupiter. This is surely adherent to his body, and is either a bright coloured country, or perhaps a tract of clouds hovering over some volcano. This spot turns round in 9.51. And thus there is a general current in his atmosphere from east to west.

Both the motion of the air and of the water tend to diminish the rotation of the earth round its axis: for they move slower than the earth, because they are retarded by the luminaries. They must communicate this retardation to the earth, and must take from it a quantity of motion precisely equal to what they want, in order to make up the equilibrated tide. In all probability this retardation is compensated by other causes; for no retardation can be observed. This would have altered the length of the year since the time of Hipparchus, giving it a smaller number of days. We see causes of compensation. The continual washing down of soil from the elevated parts of the earth must produce this effect, by communicating to the valley on which it is brought to rest the excess of diurnal velocity which it had on the mountain top.

While we were employed on this article, a book was put into our hands called Studies of Nature, by Mr Saint Pierre. This author scorns the Newtonian theory of the tides, as erroneous in principle, and as quite insufficient for explaining the phenomena; and he attributes all phenomena of the tides to the liquefaction of the ices and snows of the circumpolar regions, and the greater length of the polar than of the equatorial axis of the earth. He is a man of whom we wish to speak with respect, for his constant attention to final causes, and the proof thence resulting of the wisdom and goodness of God. For this he is entitled to the greatest praise, that it required no small degree of fortitude to resist the influence of national example, and to retain his piety in the midst of a people who have drunk the very dregs of the atheism of ancient Greece. This is a species of merit rarely to be met with in a Frenchman of the present day; but as a philosopher, M. de St Pierre can lay claim to no other merit except that of having collected many important facts. The argument which he employs to prove that the earth is a prolate spheroid, is a direct demonstration of the truth of the contrary opinion; and the melting of the ice and snows at the poles cannot produce the smallest motion in the waters. Were there even ten times more ice and snow floating on the northern sea than there is, and were it all to melt in one minute, there would be no flux from it; for it would only fill up the space which it formerly occupied in the water. Of this any person will be convinced, who shall put a handful of snow squeezed hard into a jar of water, and note the exact height of the water. Let the snow melt, and he will find the water of the same height as before.