an observer at E, the angle zEm, called the zenith distance of the body, may be measured; hence mEC is known, and the difference between zEm and zCm is equal to CmE, the parallax, then if CER, Cm = r, and zEm = z,

hence, if CE and Cm remain the same, the sine of the parallax, CmE, will vary as the sine of the zenith distance zEm; and when zEm 90°, as in fig. 67,

P being the value of the angle CmE in this case; then the parallax is a maximum, for Em is tangent to the earth, and, as the body m is seen in the horizon, it is called the horizontal parallax; hence the sine of the horizontal parallax is equal to the terrestrial radius divided by the distance of the body from the centre of the earth.

327. The length of the mean terrestrial radius is known, the horizontal parallax may be determined by observation, therefore the distance of m from the centre of the earth is known. By this method the dimensions of the solar system have been ascertained with great accuracy. If the distance be very great compared with the diameter of the earth, the parallax will be insensible. If CmE were an angle of the fourth of a second, it would be inappreciable; an arc of 1" 0.000004848 of the radius, the fourth of a second is there1 fore 0.000001212 = ; and thus, if a body be distant from 825082

the earth by 825082 of its semidiameters, or 3265660000 miles, it will be seen in the same position from every point of the earth's surface. The parallax of all the celestial bodies is very small: even that of the moon at its maximum does not much exceed 1o.

328. P being the horizontal parallax, let p be the parallax EmC, fig. 66, at any height. When P is known, p may be found, and the contrary, for if

R

be eliminated, then sin p = sin P sin z, and

when P is constant, sin p varies as sin z.

m

fig.66.

329. The horizontal parallax is determined as follows: let E and E', fig. 66, be two places on the same meridian of the earth's surface; that is, which contemporaneously have the same noon. Suppose the latitudes of these two places to be perfectly known; when a body m is on the meridian, let its zenith distances

zEmz, z'E'm = z',

be measured by two observers in E and E'. Then ECE', the sum of the lati

E

E

C

tudes, is known, and also the angles CEm, CE'm; hence EmE', EmC, and E'mC may be determined; for P is so small, that it may be put for its sine; therefore

sin p P sin z, sin p' P sin z';

and as p and p' are also very small,

p+p' P {sin z + sin z'.}

=

Now, p+p' is equal to the angle EmE', under which the chord of the terrestrial arc EE', which joins the two observers, would be seen from the centre of m, and it is the fourth angle of the quadrilateral CEmE'.

But

and if

then

hence

180°180°+p+ p' += 360°;

p+ p' = z + z' - ;

therefore the two values of p + p' give

which is the horizontal parallax of the body, when the observers are on different sides of Cm; but when they are on the same side,

It requires a small correction, since the earth, being a spheroid, the lines ZE, Z'E' do not pass through C, the centre of the earth.

The parallax of the moon and of Mars were determined in this manner, from observations made by La Caille at the Cape of Good Hope, in the southern hemisphere; and by Wargesten at Stock

holm, which is nearly on the same meridian in the northern hemisphere.

330. The horizontal parallax varies with the distance of the body from the earth; for it is evident that the greater the distance, the less the parallax. It varies also with the parallels of terrestrial latitude, the earth, being a spheroid, the length of the radius decreases from the equator to the poles. It is on this account that, at the mean distance of the moon, the horizontal parallax observed in different latitudes varies; proving the elliptical figure of the earth. The difference between the mean horizontal parallax at the equator and at the poles, from this cause, is 10".3.

331. In order to obtain a value of the moon's horizontal parallax, independent of these inequalities, the horizontal parallax is chosen at the mean distance of the moon from the earth, and on that parallel of terrestrial latitude, the square of whose sine is, because the attraction of the earth upon the corresponding points of its surface is nearly equal to the mass of the earth, divided by the square of the mean distance of the moon from the earth. This is called the constant part of the horizontal parallax. The force which retains

the moon in her orbit may now be determined.

Force of Gravitation at the Moon.

332. If the force of gravity be assumed to decrease as the inverse

m

fig. 67.

square of the distance, it is clear that the force of gravity at E, fig. 67, would be, to the same force at m, the distance of the moon, as the square of Cm to the square of CE; but CE divided by Cm is the sine of the horizontal parallax of the moon, the constant part of which is found by observation to be 57' 4". 17 in the latitude in question; hence the force of gravity, reduced to the distance of the moon, is equal to the force of gravity at E on the earth's surface, multiplied by sin° 57' 4".17, the square of the sine of the constant part of the horizontal parallax.

Since the earth is a spheroid, whose equatorial diameter is greater than its polar diameter, the force of gravity increases from the equa

tor to the poles; but it has the same intensity in all points of the earth's surface in the same latitude.

Now the space through which a heavy body would fall during a second in the latitude the square of whose sine is, has been ascertained by experiments with the pendulum to be 16.0697 feet; but the effect of the centrifugal force makes this quantity less than it would otherwise be, since that force has a tendency to make bodies fly off from the earth. At the equator it is equal to the 288th part of gravity; but as it decreases from the equator to the poles as the square of the sine of the latitude, the force of gravity in that latitude the square whose sine is, is only diminished by two-thirds of or by its 432nd part. But the 432nd part of 16.0697 is 0.0372, and adding it to 16.0697, the whole effect of terrestrial gravity in the latitude in question is 16.1069 feet; and at the distance of the moon it is 16.1069. sin2 57′ 4′′.17 nearly. But in order to have this quantity more exactly it must be multiplied by 33, because it is found by the theory of the moon's motion, that the action of the sun on the moon diminishes its gravity to the earth by a quantity, the constant part of which is equal to the 358th part of that gravity.

Again, it must be multiplied by §, because the moon in her relative motion round the earth, is urged by a force equal to the sum of the masses of the earth and moon divided by the square of Cm, their mutual distance. It appears by the theory of the tides that the mass of the moon is only the of that of the earth which is taken as the unit of measure; hence the sum of the masses of the two bodies is

Then if the terrestrial attraction be really the force that retains the moon in her orbit, she must fall through

333. Let mS, fig. 68, be the small arc which the moon would describe in her orbit in a second, and let C be the centre of the earth. If the attraction of the earth were suddenly to cease, the moon would

fig. 68.

C

E

m

go off in the tangent mT; and at the end of the second she would be in T instead of S; hence the space that the attraction of the earth causes the moon to fall through in a second, is equal to mn the versed sine of the arc Sm.

The arc Sm is found by simple proportion, for the periodic time of the moon is 27 days.32166, or 2360591", and since the lunar orbit without sensible error may be assumed equal to the circumference of a circle whose radius is the mean distance of the moon from the earth; it is

2Cm. π, or if

355
113

355

113'

The arc Sm is so small that it may be taken for its chord, therefore

Again, the radius CE of the earth in the latitude the square of whose sine is, is computed to be 20898700 feet from the mensuration of the degrees of the meridian: and since

mn=

=0.00445983

2(355)a (20898700)

(113) (2360591") sin 57' 4".17

of a foot, which is the measure of the deflecting force at the But the space described by a body in one second from the earth's attraction at the distance of the moon was

moon.