geographical problems are solved; it is, therefore, of great importance to ascertain the direction of the meridian lime at the place of the observer. The operation in its more scientific and correct shape is one of very considerable nicety; but the following method will determine it, if much accuracy be not required. On the 15th of June, or the 24th of December, plant a stick A B in a position perpendicular to the horizon, (fig. 3.) at an hour or two before the Fig. 3.

sun has arrived at its greatest altitude in the heavens, that is, be at ten or eleven o'clock in the morning; mark accurately the extremity C of the shadow BC cast by the stick; then from the base B of the stick as a centre, and with the length of the shadow B C as a radius, trace a circle G H upon the ground; as the sun gradually arrives at its greatest altitude, the shadow of the stick will become gradually shorter, and will fall within the circumference of the circle which has been traced. The shadow will be at its shortest B E, when the sun is at its greatest altitude, or when it is on the meridian of the place which is the moment of noon; after this the sun will gradually decline, the shadow of the stick will become longer and longer, until at last it again reaches to the circumference of the circle in the point D, at which time in the afternoon the sun is at the same height in the heavens as it was when the shadow of the stick was of the same length B C before noon. Now it so happens, that, on the abovementioned days, the altitude of the sun above the horizon, at one hour or two hours before noon, is equal to its altitude at the same time after noon; and as the sun has in these equal times before and after noon, described equal

spaces in the heavens (supposing those spaces to be measured from the meridian), the middle point of the whole space described by the sun in the sum of those times will be that point in the heavens which the sun occupies when it is noon; at this time the sun is on the meridian. Hence, if the arc C D of the traced circle be divided into two equal parts C F and FD, and the point F of division and the base of the stick be joined, the line B F, joining these two points, and which will be the direction of the shadow of the stick at noon, will be the meridian line. The longest and most accurate meridian line in the world, is that drawn by Cassini (a celebrated astronomer and mathematician) upon the pavement of the church of St. Petronis, at Bologna, in Italy : it is 120 feet in length.

One of the principal objects in mathematical geography is to ascertain the position of any particular spot upon the earth's surface. This term position is strictly a relative one-applied to a body, it has no meaning unless there be some other body or mark, which is fixed, and to which the first body may be referred. If, in the midst of infinite space, there existed but a single body, it could hardly be said to have position, or at any rate the meaning of the term as applied to such a body would pass our comprehension. This may be illustrated by what is said in some books on mechanics, that motion could not

exist if there were but a single body in the universe; by which is meant that in such a condition of things motion could be neither measured nor perceived: it is not intended by such expressions to assert that motion cannot exist independently of other bodies, because the existence of a foreign body cannot really affect the state or condition of motion in any moving body; it only enables us to ascertain the fact of motion, and its measure. In the idea of position, therefore, there is contained a reference to something which is fixed and which is independent of the body, the position of which is required. The distance of any body from this something which is fixed being known, and also the direction given in which that distance is to be measured, its position may be determined.

In order to ascertain the situation of any spot upon the surface of the globe, it is sufficient to fix upon two great

ridian of A passing through p, the pole of the earth, and at right angles to the equator (e q), the plane of the paper is the plane of the meridian, O the centre of the earth, p O its semi-diameter, to which (eq), the equator, is at right angles, HR the rational horizon of A. Then A e, the arc or angular distance of A from the equator measured upon the meridian of A, is the latitude of A. But ep, or the distance of the pole from the equator, is 90°, or a quadrant, and Ar the distance of A from the point r, where the rational horizon meets the surface of the earth, is also 90°, or a quadrant. Hence e p is equal to Ar; if therefore A p, which is common to both ep and Ar, be taken away from each, the remaining quantities Ae and pr will be equal; and as Ae is the latitude of A, it follows that (pr) or the height of the pole above the horizon is equal to the latitude of the place.

Again, Ap is the distance of the pole from A and as A e is the latitude of A, and pe is 90°, Ap is the difference between 90° and the latitude, so that if Ap be known, A e, or the latitude, is found, by subtracting Ap from 90°. Ap is called the complement of the latitude, or the co-latitude.

Again, Ah is 90°, and, therefore, Ae being the latitude, he is the co

latitude,-he, being the height of the equator above the horizon; so that if he be known, the latitude is found by subtracting he from 90°.

It appears, then, that if we can find any one of the above four arcs, viz., Ae, pr, Ap, he, the latitude of A will be known: and the mode of determining these arcs, is by measuring similar arcs of corresponding circles in the heavens. Let ZPRQ HÆ be the circle in the heavens which corresponds with the meridian circle passing through A, and Z, P, R, Q, H, Æ, points in the heavens corresponding with A, p, r, q, h, e. The attention of the geographer is then transferred from the consideration of the several arcs Ae, pr, Ap, he, to the corresponding arcs in the circle in the heavens, ZE, PR, ZP, HÆ: for if any of these be determined in their number of degrees and parts of degrees, the latitude is found directly. Thus it is, that the geographer depends so much upon the science of astronomy for the solution of the most important geographical problems. Persons who are in the slightest degree acquainted with geometry, or with the most simple properties of the circle, will not object to the above-mentioned mode of determining the latitude of places on the earth by means of corresponding arcs in the heavens, that these corresponding arcs are of different magnitudes; for in computing the latitude, we do not so much want the actual admeasurement and linear quantity of the arc of the meridian intercepted between the given place and the equator, as the number of degrees and parts of a degree which it contains, or, in other words, the proportion which this intercepted arc bears to the whole circumference of the meridian circle. And as arcs are the measures of angles, the arcs, Z Æ and A e, are both measures of the same angle at o; and, therefore, although they are unequal in magnitude, yet they mutually bear the same proportion to the circumference of the circles of which they are parts; that is, Z E contains the same number of degrees as Ae: and as the latitude of a place is always expressed in degrees and parts of degrees, the number of degrees contained in the arc in the heavens, ZÆ, which corresponds with the arc of the meridian A e, will be the latitude of A. If, after having ascertained the latitude in this manner,

that is, in degrees and parts of degrees, the actual linear magnitude contained in the latitude, or the geographical distance between A and the equator measured upon the meridian, be required, it may be obtained thus. Let an observer at A travel upon the same meridian in a direction due north or due south, (i. e. from or towards the equator,) until the pole star has, with respect to the observer's horizon, been raised or sunk one degree (which may be known from observation): then as the star is itself stationary, this gain or loss of one degree in its station with respect to the horizon, has been caused by the observer having travelled exactly one degree, measured upon a meridian of the earth, nearer or farther from the north pole. If this space be actually measured, the result, expressed in linear measure, will give the magnitude of a degree of latitude in geographical miles and parts of a mile; the quantity thus found, being multiplied into the number of degrees and parts of a degree, will give the actual linear distance between A and the equator. The process thus conducted is on the supposition that the earth is perfectly spherical. A degree of latitude measured in this manner contains about sixty-nine miles.

How the spaces or arcs Z Æ, PR, ZP, HÆ in the heavens are to be measured by a spectator at the spot A on the surface of the earth, is now to be explained. This is done by means of observations made by the spectator at A, upon some heavenly body, with an instrument adapted for the purpose of measuring circular arcs: by these observations, which are made when the heavenly body is either upon the meridian of the place or not, the angular distance of the body from the zenith or from the horizon is ascertained. Thus if s be the sun (see fig. 5.) on the meridian, its angular distance s Z from the zenith, (called its zenith distance,) or its angular distance s H from the horizon, (called its altitude,) is measured, and ascertained in degrees and parts of degrees.

As, however, A is the place at which these observations are made, the angle Z As is all that can be determined from observation; but this angle is not the measure of the arc Z s, because A is not the centre of the sphere of the heavens ; but the angle ZOs is the proper measure of this arc, since, by the supposition, the

meridian circle and the corresponding one in the heavens have the same centre, O; and it is a well known truth in geometry, that the angle Z A s is greater than the angle Z Os by the angle A So. This conclusion, expressed in common language, may therefore be stated thus: that a spectator at A, looking upon a heavenly body at 8, will see it lower down in the heavens, namely, at s', or farther removed from the zenith point Z, than a spectator situated at the centre of the earth, who would, at the same instant of time, see the same body at s"; the difference of the apparent places which the body, s, will thus occupy in the heavens, as seen from the surface of the earth, and as seen from the centre, is the angle s's s", which is equal to the angle As O. This angle is formed by two lines drawn from the extremities of the earth's radius, or, in geometrical language, is the angle subtended by the earth's radius, at the distance of the body s. This angle is called the parallax of a heavenly body, and increases the zenith distance of s. It is obvious that parallax produces a contrary effect upon Hs, the altitude of s, and that as the zenith distance is increased by the angle A & O, so the horizontal distance or altitude is diminished by the same angle. The general effect, therefore, of parallax is to depress a heavenly body. If, however, the distance of the body upon which an observation is made, be so great, that it would be seen in exactly the same position in the heavens by a spectator at the surface of the earth, and one at the centre, it is evident that the angle s's s" or AsO (the parallax) would be so small as to escape observation, and would, to our senses, vanish. This is the case with the fixed stars; but with respect to the sun and moon and planets, whose distances are not so great, the parallax has an observable effect upon their apparent positions, as they are seen from different parts of the earth's surface, or from the earth's surface and its centre. And this circumstance raises a necessity for correcting the observed distances of these heavenly bodies from the zenith or horizon of a place, in order to arrive at the true distance, as they would be seen from the earth's centre, and that the respec

From a Greek word, and thus applied, signifies simply a change of place.