67. The angle formed by the vertical lines at two places on the same terrestrial meridian, expresses the difference of the latitudes of the places.

Let CZ and CZ', Fig. 1, be the vertical lines at the two places A and A', on the same meridian pAp'. Then Z and Z' being the zeniths of these places, ZQ and Z'Q are the latitudes (55), and consequently ZZ' is the difference of the latitudes. But ZZ' is the measure of the angle ZCZ' formed by the vertical lines.

This is still true if the vertical lines meet at some distance from the centre of the earth, as must be the case if the earth is not a perfect sphere. For, in consequence of the immense distance of the points Z and Z', the angular distance between them is sensibly the same at a little distance from the centre as at the centre itself.

68. Length of a degree of latitude. The length of a degree of latitude or of a degree of the meridian is the distance, expressed in linear units, between two points on the same terrestrial meridian, the difference of whose latitudes is one degree.

The length of a degree of latitude may be obtained by finding the latitudes of two points on the same meridian, that do not differ in latitude more than a few degrees, measuring the distance between them, and then making the proportion; as the difference of the two latitudes is to one degree, so is the measured distance to the length of a degree. For supposing A and A' to be the points, we have the proportion: as ang. ZCZ': 1° : : length of AA': length of a degree; in which the angle ZCZ' is equal to the difference of the latitudes (67). The proportion is rigorously true on the supposition that the earth is a sphere, and consequently AA' the arc of a circle; and for a small deviation in the form of the earth from a sphere, it is not sensibly erroneous, especially for the degree at the middle of the arc. Supposing the earth to be a sphere, the product of the length of a degree by 360, gives its circumference.

The difference of latitude between the two places A and A', may be found without knowing the latitude of either. For if, at the two places, the meridian zenith distances ZM and Z'M of the same star, be observed and corrected for refraction,* we have ZZ' = ZM

Z'M.

* See next chapter.

The distance between the two places is not found by direct measurement. This would be a very tedious operation, and would generally, from irregularities in the earth's surface, be deficient in accuracy. An extent of level ground is selected and a horizontal line BC, Fig. 3, of a few miles in length, called a base line, is measured with the utmost care and precision. Then, supposing A', one of the places, to be visible from B and C, the horizontal angles of the triangle A'BC are carefully measured with a theodolite, or altitude and azimuth instrument. A station D, visible from B and C, being chosen, the angles of the triangle BCD are observed. Another station E, visible from C and D, being taken, the angles of the triangle CDE are observed. Proceeding thus, on both sides of the base line if requisite, the places A and A' become connected by a series of triangles, in which the angles are all known, and also the side BC. From these data, other sides, and then the distance AA', may be computed.

,

69. The length of a degree of latitude increases from the equator to the pole.

This may be inferred from inspection of the following table, which contains the length of a degree of latitude at several different latitudes, selected from measurements which have been made with great care, in various parts of the earth.

A terrestrial meridian is an Ellipse, having the axis of the earth for its less axis and a diameter of the equator for its greater

axis.

The variation in the length of a degree of latitude proves that the meridian is not a circle; and the small amount of that variation shows that its deviation from a circle is not great. As the whole

deviation is not great, a small portion of the meridian in any part may, without sensible error, be regarded as the arc of a circle; the radius of the circle to which the arc appertains, evidently increasing as the length of the degree of latitude increases, that is, from the equator to the pole. Now as the radius of an arc increases, its curvature decreases. The curvature of the meridian must therefore decrease in proceeding from the equator to the pole. This is the case with an ellipse in passing from the extremity of the major axis to that of the minor axis. Hence the form of the meridian corresponds in this respect with that of an ellipse, as epqp', Fig. 4, in which pp', the axis of the earth, is the less axis, and eq, a diameter of the equator, is the greater axis.

Taking into view the actual lengths of a degree at different latitudes, it has been proved, by analytical investigations not adapted to the present work, that the meridians are really ellipses, or very nearly so; in which the less axis, or axis of the earth, is less than the greater, or a diameter of the equator, by about 300 part of the latter.

:

1

Q. Figure and dimensions of the Earth. From measurements which have been made at right angles to the meridian, it appears that the equator and parallels of latitude are circles, or nearly so. It therefore follows from the last article that the form of the earth is that of an oblate spheroid; that is, of a solid, such as would be generated by the revolution of a semi-ellipse pqp', about its minor axis pp'.

From computations made from the most accurate measurements, it has been found that the equatorial diameter of the earth is 7925 miles, and the polar diameter, or axis, is 7899 miles; the difference between them being 26 miles. Consequently the mean diameter is 7912 miles, and the mean circumference 24856 miles.

Hence the mean length of a degree of the meridian is 691⁄2 miles, the mean length of a minute is 11⁄2 miles, and the mean length of a second is 101 feet. It therefore follows, that in changing our position in a north or south direction, by only 101 feet, we make a change of one second in our latitude.

The length of a degree of the equator is 691 miles.

72. Ellipticity or Oblateness of the Earth. It is frequently

found convenient to denote the equatorial radius of the earth by a unit, or 1, and to express other large lengths and distances by means of this unit.

The fraction which expresses the difference between the equatorial and polar radii of the earth, when the equatorial radius is denoted by a unit, is called the ellipticity or oblateness of the earth. It is also sometimes called the compression of the earth. Hence (70), the ellipticity or oblateness is 300.

73. The ellipticity of the earth may be deduced from experiments with a pendulum.

The number of oscillations made in any given time, as for instance in a sidereal day, by the same pendulum retaining the same length, is found to be different at different places on the earth's surface. It is least at the equator, and continually increases towards the poles. A pendulum oscillating sidereal seconds at the equator, and consequently making there 86400 oscillations in a sidereal day, would, on being transported to Philadelphia, make nearly 100 more in the same time. Now the motion of the pendulum depends on the force of gravity; and it is proved, in treatises on mechanics, that the number of oscillations made by the same pendulum in a given time, varies as the square root of that force. Hence it follows that the force of gravity increases from the equator to the poles, and that the law of this increase may be determined by experiments with a pendulum. This increase in the force of gravity, indicating a decrease in the distance from the earth's centre, is connected with the figure of the earth, and formulæ have been obtained which serve to determine the latter from the former. Computations, founded on numerous accurate experiments with a pendulum, made at various places, give for the ellipticity nearly the same value as that obtained from the measurement of degrees of the meridian.*

*Dr. Bowditch, in his excellent Translation of Laplace's Mécanique Céleste, with a Commentary, obtains, from a combination of several of the most accurately measured arcs of the meridian, a result a little less than ; and from a combination of many observations made with the pendulum, a result a little greater than 3. Hence he infers that may be regarded as being very nearly the true value of the ellipticity or oblateness of the earth.

300

1

74. The Eccentricity of the earth is the distance between a focus of any of the elliptical meridians and the centre.

To find the eccentricity. Let f, Fig. 4, be one of the foci, and put efC the eccentricity. Then by conic sections, fp = eC = 1. Hence,

75. The Geocentric Zenith of a place is the point in which a straight line from the earth's centre, passing through the place, meets the celestial sphere.

The Geocentric Latitude of a place, sometimes called the reduced latitude, is the arc of the meridian intercepted between the equator and the geocentric zenith of the place. The difference. between the latitude and the geocentric latitude is called the reduction of latitude.

76. The tangent of the latitude of a place is to the tangent of the geocentric latitude as the square of the equatorial radius of the earth is to the square of the polar radius.

Let Z, Fig. 4, be the zenith of the place A, and z the geocentric zenith. Then ZGQ is the latitude of A, and 2CQ is its geocentric latitude. Let AD be drawn perpendicular to eq. Put = ZGQ= the latitude, and p=2CQ= the geocentric latitude.

Then in the right angled triangle ACD, we have AD = CD_tang ', and in the right angled triangle AGD, we have AD = GD tang 4. Hence CD tang ' = CD: GD tang ; or, CD : GD :: tang: tang '. But by conic sections, CD: GD: 9C2: pC. Consequently tang tang ' : : qC2 : pC2.

Cor. If qC=1, and e pC2 = 1— e2. Hence, tang tang o' = (1 — e2) tang

earth's eccentricity, we have (74),

tang ':: 1:1-e2; or,
ì'

.....(A)

D