BOOK this angle becomes insensible for the fixed stars, and very

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Zenith.

Nadir.

Vertical line.

small for the planets.

We may substitute then without error, fig. 3, instead of the preceding figure, assuming for the horizontal plane, with regard to the stars, the plane NEMO passing through the centre of the earth, and parallel to the plane, touching it in A; or, what is the same thing, perpendicular to the radius CA, drawn from this point to the centre of the earth. We may conceive, in the same manner, the plane of the celestial meridian MZN, to be extended indefinitely around the centre C of the earth, through which it ought necessarily to pass, since it passes through the axis P p. This plane determines upon the surface of the earth a circle P A p, which passes through the poles; this circle is the terrestrial meridian of the place A, and at the same time, of all the places situated upon its circumference.

We ought to observe here, that the horizon represented by the circle NEMO, and which passes through the centre of the earth, is called the rational horizon, to distinguish it from the circle, which is a tangent to the surface that bounds the view, and which is called the sensible horizon.

The line drawn from the centre of the globe through the place of an observer, ascertains in the heavens the position of a point Z. This point is perpendicularly over the head of the observer, and is called the zenith; the same line produced through the globe, marks, in the opposite part of the heavens, another point z, which is called the nadir.

The position of the line ZAC, which is called the vertical line, is ascertained by the direction which heavy bodies. take in falling, as that of the horizontal plane is indicated. by the surface which water at rest and of inconsiderable extent, naturally presents. The vertical line, or that which is ascertained by a thread when stretched by a plummit, is perpendicular to a like surface. This is the proper place to point out the precise situation of the antipodes. As gravity tends every where towards the interior of the globe, it acts at a in the direction za opposite to ZA; in both places bodies fall towards the surface of the earth. The people

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placed at a, having their feet opposite to the feet of those BOOK who are at A, are called the antipodes of these last. The zenith of the one is the nadir of the other.

the hori

zon.

It follows from this definition, that the horizon must Change of change its position relatively to the stars, when the observer changes his place upon the surface of the earth. If he removes, for example, from A to a, fig. 4, directly along the same meridian from north to south, the horizontal visual ray NM, will become nm, so that a star E, situated upon the prolongation of the former ray, will appear to be elcvated above the horizon, mn by the angle EC m, which is precisely equal to that formed by the radii CA, C a drawn to the centre of the earth; for the angles ACM and a Cm being right angles, if we take from each the common angle MC a, it is evident that the remaining angles MC m and AC a are equal.

It was upon this principle that Posidonius, having observed that the star known by the name Canopus, appeared in the horizon at Rhodes, while it appeared at Alexandria in Egypt, elevated by the 48th part of the circle, or 7 degrees and a half, concluded that Rhodes was distant from Alexandria, in the direction of the meridian, by the 48th part of that circle. It is true that the Greek philosopher, from being ignorant that Rhodes and Alexandria are not under the same meridian, was wrong in imagining that, by this observation, he had determined the whole circumference of the earth. Still, however, his principle is true; it is the same that is employed at the present time in order to arrive at the most exact determinations: for, before measuring upon the earth the distance between any two points, it is necessary first to find, by means of observations. made upon the same star, what ratio the arc A a of the meridian passing through the two points of observation, bears to the whole circumference.

By this observation is ascertained the relative position, with regard to north and south, of one place a to another A; but in order to determine in a manner more precise, the position of these places, there is required some fixed

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term of comparison. For this purpose we conceive a plane to pass through the centre of the earth at right angles to the axis of rotation, and to determine upon its spherical surface a circumference GEF, fig. 5, every point in which is at the same distance from the Poles P and p. This circle Equator. is called the equator. Now, if an observer be situated upon the equator, the two poles will be found exactly in the horizon; but according as he removes from this circle towards either pole, that pole to which he approaches, will rise above, while the other sinks below the horizon. when he is at a, fig. 4, the pole P appears elevated above the horizon by the angular space PC n, and when he arrives at A, this angle being increased by NC n, becomes PCN.

Height of the pole.

Distances

Thus

The height, or elevation of the pole above the horizon of any place, is equal to the angular distance of that place from the equator, estimated in the direction of the meridian. For the angles ACN and GCP, fig. 5, being right angles, if we take away the common angle ACP, there remains the angle ACG equal to NCP. By inspecting the same figure, we perceive that the height to which the points of the equator rise above the horizon is equal to the complement of the angle ACG.

It is sufficient, therefore, to determine for any place the of places height of the pole above the horizon, in order to find the earth from angular distance of that place from the equator.

upon the

the equa

tor.

In the regions of the globe, where one of the poles appears elevated above the horizon, the stars called circumpolar, that is, those stars which never set, furnish directly the means of determining the height of the pole. As they appear to describe circles about the celestial pole, each must appear equally removed from it in all directions; and as they twice pass the meridian during a diurnal revolution of the earth, namely, once above the pole, and once below it, we have only to measure their angle of elevation in each of these positions, and to take the arithmetical mean between. the results, in order to obtain the elevation of the pole.

By measuring, for example, at Paris, during a long winter's night, the two meridian altitudes of the pole star, we

find that, when it passes above the pole, its altitude is 50° 37'; and that when it passes below, it is 47° 4′; the sum of these being 97° 41'; the mean is about 48° 50', which is within a few seconds of the altitude of the pole at Paris, or of the distance of that city from the equator.

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regard to

dians.

It is not enough to know merely the distance of a place upon the earth from the equator; because this distance is common to all the places which are situated upon a circle traced upon the surface of the globe by a plane parallel to Distance of the plane of the equator, and passing through the place in places with question. In order to distinguish places equally distant from their merithe equator, it is necessary to know their meridians, the meridian being different for each place. The observation of the celestial motions may be here again successfully employed in the manner which we are now to point out. We have seen that the circles of the different meridians, PAP, PLp, PMP, &c. fig. 6, intersect each other in the axis PCp; but since all these meridians turn upon this line, they must also correspond successively to the same star; and the time which elapses between the passage of two meridians, containing between them any angle, will thus be to the time of the entire rotation, as the angle contained by these meridians is to the whole circumference of the circle. Hence, if we could measure the first of these intervals, in order to compare it with the second, we would be able to deduce the angle which the two proposed meridians formed with each other. To obtain this comparison, it is necessary that we should be able to indicate, by a signal visible at the same time at places under the two meridians, the moment at which a star appears upon one of these meridians, this instant must be noted, and a well regulated clock will measure the time which elapses between this passage and that of the same star over the other meridian.

When we have determined by this method the angle which the meridian PLp, passing through the place L, makes with the meridian PAp, passing through a given place A, the place L becomes entirely determined, provided that we already know its distance GL from the equator

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BOOK EGF; for it will necessarily be situated at the intersection of the semicircle PLp, and the parallel LM, drawn at this given distance.

Definitions

and longi

tude.

The shortest distance of a place from the equator is termof latitude ed its latitude. The distance is measured by an arc of the meridian comprehended between the place and the equator. Latitude is north for those places which lie between the north pole and the equator, and it is south for places in the opposite hemisphere.

Apparent

the sun.

The angle contained by two meridians, measured by an arc of the equator, or of a circle parallel to it, is termed the difference of longitude of the places situated under these two meridians. That we may estimate these differences in an absolute manner, it is necessary to assume a first meridian, the choice of which is altogether arbitrary, and has varied at different periods. The absolute longitude of a place is therefore the angle which the meridian of that place forms with the first meridian.

We have just seen that the determination of the difference of longitude of two places upon the earth requires the use of a signal visible at the same time at both places. It is evident, that for places separated by any considerable distance, the only signals sufficiently elevated must be sought among the stars. It is indeed by means of these celestial bodies that the geographer determines the position of places. We must therefore acquire some idea of their motions, particularly of the motions of the sun and moon.

Every attentive observer of the heavens cannot but have motions of remarked that the sun, besides its apparent diurnal motion, which it has in common with all the stars, appears, in the course of a year, to change its place in a twofold manner. First, it appears to rise and to sink alternately towards one or other of the poles, or towards the north and south. Again, if we observe its place among the stars, it appears either that the sun recedes daily towards the east, or that the stars advance in the opposite direction; for the stars which we see at any time set immediately after the sun, are, on the following evening, lost among the rays of the setting sun;