reckoning from the opposite side of the equator, which circumscribes the Northern hemisphere. But Méru is not the North pole; it is true that it is the Nava, Nobeh, or under the ninetieth degree, not from the equator, but from the horizon; or, in other words, it is the zenith and centre of the known world, or old continent, not including the sea; and this centre, according to the Pauránics, in the time of COSMAS INDOPLEUSTES, in the middle of the sixth century, was said to be exactly between China and Greece. We read constantly in the Puranas of countries, mountains, and rivers, some to the North, others to the East, or to the West of Méru; the country of North Curu, beyond Méru, is repeatedly declared to be to the South of the Northern -ocean. All these expressions shew very plainly, that by Méru, the Pauránius did not originally understand the North pole, which they call Siddhapur, which place, the astronomers say, cannot be under the North pole, because it is in the track of the sun; for when the sun is there, it is midnight at Lancá and in India; it must be then under the equator. This is very true; but we are to argue, in the present case, according to the received notions of the Pauránics, who formerly considered the Earth as a flat surface, with an immense convexity in the centre, behind which the sun disappeared gradually, descending so as to graze the surface of the sea at Siddhapura. In the Brahmanda Purána section of the Bhuvana-Cos'a, it is declared, that one-half of the surface (védi) of the earth is on the South of Méru, and the other half on the North. All this is very plain, if we understand it of the old continent; one half of which is South of the elevated plains of little Bokhara, and the other half to the North of it. Then, twelve or fifteen lines lower, the author of the same Purána adds, and

the district of Chittledroog. They both have a reference to those particular stations, and their surveys, with respect to them, may be relatively correct: and if Sera and Chittledroog be laid down right, their respective surveys will fall into their right places on the globe.

It will be unnecessary to state to the Society the imperfect methods that have generally been practised by supposing the earth to be a flat; and yet it has been on this supposition that surveys have been made in general, and corrected by astronomical observation. But although that method of correction may answer for determining the position of places at a great distance, where an error of five or six minutes will be of no very great consequence, yet in laying down the longitudes of places progressively that are not more than twenty miles from one another, it is evident that errors of such a magnitude are not to be overlooked; and an error, even of one mile, would place objects in situations widely different from that which they actually hold on the face of the globe.

If we consider the earth as an exact sphere, we should naturally advert to spherical computation. And having a base actually measured, and reduced to the level, it would be a part of a great circle, while the horizontal angle would be the angle made by two great circles, intersecting each other at the point where the angle was taken. On this hypothesis, the process of extending a survey would be reduced to as great a degree of simplicity as by the method of plane triangles. For then the length of a degree on the meridian could be easily obtained by the celestial arc, and would be equal to a degree in any other direction. The radius of curvature, or the semidiameter of the earth, might also be easily deduced from thence, and being every where the same, the chord of any arc, or the direct distance between two objects subtending that are, could be computed without the trouble of correcting the observed

!

served angles. The difference of longitude of any two points might be as easily had; for, knowing the arc between them (which would always corre spond with a celestial arc,) and the co-latitudes of the two places, the angle at the pole, or difference of longitude, might be found.

But since the earth is not a sphere, but an oblate spheroid, and differing considerably from a sphere, it becomes necessary to determine the length of a degree on the meridian, and a degree at right angles to that meridian, making the point of intersection of the meridian and its perpendicular the middle point of each degree. Now, in determining the measure of those degrees, if the first measurement, or base line, cannot be had in the meridian, two other objects must be chosen therein, and their distance computed trigonometrically, and then compared with the celestial arc. But here the operations, for obtaining this distance, will be attended with some trouble, on account of its being necessary to calculate the chords of the arcs, and the difficulty of determining the angles made by these chords to a sufficient degree of accuracy. For here we are obliged to assume data, and proceed by an approximating method. And, ist, we must either suppose the earth to be a sphere, and by taking the three angles made by the intersections of three great circles of that sphere, find the sides in degrees and minutes: then take double the sines of half the arcs, or the chords, and there will be had the three sides of a plane triangle, defined in parts of the radius. With these three sides determine the three angles, and these are the angles for calculating the direct distances. Hence, by knowing the base in fathoms, the chord subtending that base (or arc) may also be had in fathoms, by computing from the radius of the assumed sphere, which we must suppose to be of some given magnitude. Then having the length of the chord in fathoms, and the angles corrected as the Tartars Kiloman, or the celestial mountains. In Tibet they call them Tangra, or Tangla, according to F. CASSIANO and PURA'N-GIR; the latter accompanied the late LAMA to China, and gave me an accurate journal of his march from TissooLumbo to Siling, or Sining. Tingri, in the language of the Tartars and Moguls, signifies the heavens; and even Tibet is called Tibet-Tingri, or the heavenly country of Tibet. The name of Tien-chan is given by the Chinese to the mountains to the North of Hima: to the Southern part of the circle they give the name of Sioue-chan, or snowy mountains. This range, says DE GUIGNES, runs along the northern limits of India, toward China, encompassing a large space, enclosed, as it were, within a circle of mountains*. The Southern extremity of this circle is close, according to the present Hindu maps, to the last, or Northern range, called Nishad'ha; and this is actually the case with the mountains of Tangrah, near Lassa, which is in the interval between the second and third range. According to F. CASSIANO, the mountains of Tangrah are seen from the summit of Cambálá, several days journey to the Westward of Lassa. The famous PURA'N-GIR left them on the left, in his way from Tissoo-Lumbo to China, at the distance of about twelve coss, and did not fail to worship them. At the distance of seventy-seven coss from the last place, he reckoned Lassa to be about twenty coss to the right; twenty-three coss beyond that, he was near the mountains of Ninjink Tangrá, a portion of that immense circular ridge. In his progress toward the famous temple of Ujuk, or Uzuk, called Souk in the maps, he saw them several times. Close to Ninjink-Tangra he entered the mountains of Lurkinh, called Larkin in the maps.

stration of the above formula has been given by the Astronomer Royal, and may be seen in the Phil. Transactions for the year 1797, p. 450.

HAVING, by this method, got the angles made by the chords to very near the truth, the rest, with respect to distances, is evident. For the chord of the measured arc (or base) may be had, since by computing the lengths of arcs in any direction, on the ellipsoid, the radius of curvature of that are is likewise had, and thence the chord. And that chord forms the side of a plane triangle, from which, and the corrected angles, all the data may be had for proceeding upon each of the sides of the first plane triangle.

Now, to determine any portion of a degree on the earth's surface in the meridian, two points may be taken therein, and the direct distance between them ascertained by the above method. Then, by taking the zenith distance of a known star, when passing the meridian, at each extremity of the distance, the celestial arc becomes known in degrees, minutes, &c. from which the terrestrial arc between the two objects is had in degrees, minutes, &c. also:-and having determined the chord in fathoms, the arc may likewise be determined in fathoms, which being compared with the degrees, minutes, &c. the value of a degree is thereby obtained in fathoms.

THE length of a degree, at right angles to the meridian, is also easily known by spherical computation, having the latitude of the point of intersection, and the latitude of an object any where in a direction perpendicular to the meridian at that point. For then the arc between these two points, and the two celestial arcs or colatitudes, will form a right angled triangle, two sides of which are given to find the third, which is the arc in question. And this will apply either to the sphere or spheroid. That arc being known, in degrees and minutes, and the chord