SCHOL. 2. All the irregularities of the moon are greater when the earth is in its perihelion, than when it is in its aphelion, because the effect of the sun's action, whereby they are produced, is inversely as the cube of its distance from the earth. They are also greater when the moon is in conjunction with the sun, than in opposition, for the same reason; for the earth and moon, taken together, are nearer the sun in the former situation of the moon, than they are in the latter.

CHAP. III.

Of the SPHEROIDICAL FORM of the EARTH.

PROP. CLXVII. In the daily revolution of the earth round its axis, the centrifugal force diminishes the weight of bodies more at the equator than in any other place on the surface of the earth, in the duplicate ratio of the semidiameter to the cosine of the latitude of the place.

Let PEPe be the earth, PP the axis, Ee the equator. As the earth revolves upon its axis, Plate 11. every place on its surface, except the two poles, describes a circle, the plane of which is per- Fig. 13. pendicular to the axis, and the radius of which is the distance of that place from the axis. Thus, a body placed at A will in one revolution of the earth describe a circle, the semidiameter of which will be AB, which with the plane in which it lies will be perpendicular to the axis PP. In like manner, CE is the semidiameter of a circle described by the revolution of a place in the equator. But CE is the semidiameter of the earth, and AB is the cosine of latitude of the place A; for AB is the sine of AP, the complement of AE, which is the latitude of the place. And a body at E, revolving in a circle whose radius is CE, performs its revolution in the same time with a body at A, revolving in a circle whose radius is AB. But where the periodical times are equal, the centrifugal forces are as the radii, by Book II. Prop. LXXVII. Whence the body at E has its centrifugal force as much greater than the body at A, as the radius CE is greater than the radius AB; and universally, the centrifugal force at the equator is to the centrifugal force at any other place on the surface of the earth, as the semidiameter of the earth to the cosine of the latitude of the place. And since it is manifest that the gravity must be diminished as much as the centrifugal force is increased, the gravity of a body at the equator is as much less than that of a body at any other place on the earth, as the semidiameter of the earth is greater than the cosine of the latitude of the place.

The

Moreover, if the centrifugal forces at E and A were equal, they would diminish the weights of bodies unequally, on account of the different directions in which these forces act. centrifugal force at A, acting obliquely upon the force of gravitation towards C, can only diminish this force by such a part of its action as is opposite to the direction of gravitation, that is, resolving Ab which may express the centrifugal force at A into Aa, ab, the part of the centrifugal force which will act to diminish the gravity of the body at A, will be to the whole centrifugal force at A, as Aa to Ab. Whereas at E, the whole centrifugal force, acting in direct opposition to the force of gravitation, will operate to diminish the weight of a body at E. Hence the force which acts to diminish the weight of a body, that is, the dimunition at E is

84

E is, to the same at A, as the whole centrifugal force Ab to the part Aa. But Ab is to Aa, as CE to BA for, the triangles, Aab and ABC being similar, Ab is to Aa, as AC or EC to BA. Therefore, from, the different directions in which the centrifugal forces act at E and A, the weight at E is as much more diminished than at A, as EC the semidiameter is greater than AB the cosine of the latitude of the place A.

The centrifugal force then dimishes the force of gravitation in the ratio of EC to AB, both because the centripetal force at E is greater than at A, and because it acts directly at E, but obliquely at A. Therefore the centrifugal force diminishes the weight of a body at E, more than at A, in the duplicate ratio of CE to BA, that is, as much more at E than at A, as the square of CE is greater than the square of BA.

SCHOL. It is found by calculation from this Prop. that gravity at the equator is diminished by the centrifugal force in the ratio of 288 to 289.

COR. 1. If the diurnal motion of the earth round its axis was about 17 times faster than it is, the centrifugal force would, at the equator, be equal to the power of gravity, and all bodies there would entirely lose their weight. But if the earth revolyed still quicker than this,

they would all fly off.

COR. 2. Since a place in the equator describes a circle of 24,930 miles in 24 hours (Prop. III. Cor.) it is evident that the velocity with which that place moves, is at the rate of about 17°3 miles per minute. The velocity in any parallel of latitude decreases in the proportion of the cosine of latitude to radius. Thus, for the latitude of London, say, as Rad. : Cos. 51° 30': velocity of the equator: velocity of London; by logarithms, as 10'00000 : 9'794150 :: 1232046: 1'026196 10'6 miles; that is, the city of London moves about the axis of the earth at the rate of more than 10 miles in a minute of time.

PROP. CLXVIII. The earth is an oblate spheroid, elevated at the equator, and depressed at the poles.

It has been found by observation, that a pendulum, shorter by 2:169 lines, is required to vibrate seconds at the equator than at the poles; but (from Book II. Prop. XLIII. and XLIV.) the lengths of pendulums vibrating in the same time are as the gravities at the places where they vibrate; therefore the gravity at the poles is greater than at the equator. And it has been found by Sir I. Newton, that this difference of gravity is so much greater than wouldarise from the centrifugal force alone, that the ratio of the equatorial diameter of the earth to the polar diameter, must be as 230 to 229, which makes the equatorial diameter exceed the polar by about 34 miles.

COR. I. Hence bodies near the poles are heavier than the same bodies towards the equator; (1.) Because they are nearer the earth's centre, where the whole force of the earth's attraction is accumulated. (2.) Because their centrifugal force is less on account of their diurnal motion being slower. For both these reasons, bodies carried from the poles towards the equator, gradually lose their weight.

COR. 2. The degrees of latitude upon the earth's surface are longer at the poles than at the equator. For an arc of a meridian near the poles is less curved than near the equator,that

that is, it is an arc of a larger circle; whence a degree measured upon that arc must be greater than upon an arc of the same meridian at the equator.

COR. 3. The tendency of a heavy body, on any part of the surface of the earth between the poles and the equator, is not directly towards the centre, but towards some point between the centre and the equator.

SCHOL. The point towards which a body in any given place will tend may be determined. For (by Prop. CLXVII.) as radius EC is to the cosine of latitude of the place AB, so is the centrifugal force at E to the centrifugal force at A in the direction Ab. Produce, therefore, the line BA to b, till Ab has the same ratio to AC, as the quantity last found has to gravity upon the surface of the earth. Complete the parallelogram AbCc; the point sought will be c, and the tendency of the body will be along Ac; thus suppose the latitude of the place 51o 46'. The centrifugal force at the equator is found to be to that of gravity, as I to 289: hence, as radius to the cosine of 51° 46′, so is 1 to 618, which is the centrifugal force at A. Consequently, the centrifugal force at A is to the force of gravity, as '618 to 289: therefore, by the construction, Ab or Cc, is to AC in that ratio. The ratio of AC to Ce being thus found, as AC is to Cc, or as 289 is to 618, so will the sine of the angle of latitude ACc, or 51° 46′, be to 5' nearly, which is the angle required, measuring the deviation of the line of direction of falling bodies at the given latitude from a line drawn to the centre of the earth.

Of the PRECESSION of the EQUINOXES.

DEF. LXIV. A Periodical Year, is the time in which the sun completes its revolution through the ecliptick.

DEF. LXV. A Tropical Year, is the time in which the sun completes its revolution setting out from any solstitial or equinoctial point, and returning to the same.

PROP. CLXIX. The equinoctial points move in antecedentia, or go backwards from east to west, contrary to the order of the signs.

It is found from observation, that the equator and ecliptick'do not always intersect each other in the same points, but that the points of intersection change their place, moving from east to west, whilst the inclination of the planes remains the same. This motion is called the precession of the equinoxes, because it carries the equinoctial points in precedentia signa.

PROP. CLXX. The precession of the equinoxes makes the tropical year shorter than the periodical year.

Plate 11.

Fig. 13.

Plate 11.
Fig. 12.

If, while the sun moves in the order of the signs, the equinoctial point moves in the contrary direction, it is manifest, that the sun must arrive at the solstitial or equinoctial point from which it set out, before it arrives at the same place in the zodiack, or must complete the tropical year sooner than the periodical year.

The tropical year is observed to be 365 days, 5 hours, 49 minutes; the periodical year, 365 days, 6 hours, 4 minutes, 56 seconds.

PROP. CLXXI. The precession of the equinoxes causes the poles of the equator to describe a circle from east to west about the poles of the ecliptick.

In this precession, the plane of the equator revolves from east to west, cutting the ecliptick, which, with its axis, is at rest, in successive points. But while the plane of the equator is revolving, its axis must revolve with it the same way. And, since the plane of the equator is always equally inclined to that of the ecliptick, the axis of the equator must always have the same inclination to the axis of the ecliptick: consequently, the poles of the equator will revolve round the poles of the ecliptick, always preserving the same distance from each other; that is, the poles of the equator will describe a circle about the poles of the ecliptick.

EXP. The precession of the equinoxes, and the revolution of the pole of the equator about that of the ecliptick, may be thus represented on the celestial globe. Let the broad wooden horizon represent the ecliptick; place the axis of the globe perpendicular to the wooden circle; the ecliptick on the globe will then make an angle of 23° 30' with the wooden horizon: consequently, if the wooden horizon represents the ecliptick, the circle which commonly represents the ecliptick will now represent the equator; and the two points in which this circle cuts the wooden horizon will represent the equinoctial points. If the globe, in this position, be turned slowly round from east to west, these points of intersection will move round the same way, while the inclination of the circle which now represents the equator to that which represents the ecliptick remains the same: whence the precession of the equinoxes is properly represented. Again, the axis and poles of the globe now representing those of the ecliptick, the axis and poles. of the ecliptick, marked on the globe, will represent those of the equator; and in turning the globe round from east to west, the points which represent the poles of the equator, will revolve the same way round the poles of the globe which represent those of the ecliptick, and the axis of the supposed equator will always make the same angle with the plane of the supposed ecliptick.

PROP. CLXXII. The precession of the equinoxes is caused by the action of the sun and moon on that excess of matter about the equatorial parts of the earth, by which from a perfect sphere it becomes an oblate spheroid.

Let ADCB be the plane of the ecliptick, S the sun, E the earth, and AFBG a ring encompassing the earth at any distance, as Saturn is encompassed by its ring. Let the half of this ring AGB towards the sun be above the plane of the ecliptick, and the other half below it:

then,

then, a line passing through A and B will be the line of the nodes of this ring. If it be sup posed that this ring moves round its centre E, the same way in which the moon moves round the earth, it is obvious that every point of this ring will be acted upon by the disturbing force of the sun in the same manner as the moon was shewn to be acted upon in Prop. CLVIII. &c. Particularly, the motion of the nodes of this ring, and consequently of the whole ring which moves with these nodes, and its inclination to the plane in which its centre moves, will be affected in the same manner with the orbit of the moon: whence, its nodes when in syzygies will stand still, and its inclination will be greatest; but in all other situations, the nodes will go backwards, and fastest of all when in the quadratures, at which time the inclination of the ring will be the least. This will be the case whatever be the thickness of the ring, or its distance from the centre.

If this ring be supposed to adhere to the earth, it is obvious that it will still have the motions described above, and that in this situation, the earth itself must participate of these motions. Now the earth being an oblate spheroid, having its equatorial diameter longer than that which passes through its poles, this redundancy of matter, by which the form of the earth departs from a perfect sphere may be considered as a portion of the supposed ring, which receives from the action of the sun the motions abovementioned, and communicates them to the earth. Hence the equinoctial points, which are the nodes of the ring, when they are in syzygy, that is, at the equinox, will stand still, and the inclination of the equator to the plane of the ecliptick will be the greatest: in all other situations they will go backwards, and fastest when in quadrature at the solstices; and the inclination of the plane of the equator to that of the ecliptick is then the least.

COR. Hence the axis of the earth, being perpendicular to the plane of the equator, changes therewith its inclination to the plane of the ecliptick twice in every revolution of the earth about the sun. For instance, it increases whilst the earth is moving from the solstitial to the equinoctial, and diminishes as much in its passage from the equinoctial to the solstitial points which phenomenon is called the Nutation of the Poles.

SCHOL. This precession of the equinoxes is found to be 50 seconds of a degree, every year, westward or contrary to the sun's annual motion; so that with respect to the fixed stars, the equinoctial points fall backwards 30 degrees in 2160 years, whence the stars will appear to have gone 30 degrees forward, with respect to the signs of the ecliptick, which are reckoned from the equinoctial point. Thus the stars which were formerly in Aries are now in Taurus, &c. This period is completed in 25,920 years.

CHAP. V.

Of the TIDES.

PROP. CLXXIII. The tides are caused by the attraction of the moon and of the sun.

Let ApLn be the earth, and C its centre; let the dotted circle PN represent a mass of Plate 11. water covering the surface of the earth; let M, m, be the moon; S, s, the sun in different Fig. 14. 85 situations.