pass through it are acted on as if they passed successively through media of increasing density, and are therefore inflected more and more towards the earth as the density augments. In consequence of this it is, that rays from objects, whether celestial or terrestrial, proceed in curves which are concave towards the earth; and thus it happens, since the eye always refers the place of objects to the direction in which the rays reach the eye, that is, to the direction of the tangent to the curve at that point, that the apparent, or observed elevations of objects, are always greater than the true ones. The difference of these elevations, which is, in fact, the effect of refraction, is, for the sake of brevity, called refraction: and it is distinguished into two kinds, horizontal or terrestrial refraction, being that which affects the altitudes of hills, towers, and other objects on the earth's surface; and astronomical refraction, or that which is observed with regard to the altitudes of heavenly bodies. Refraction is found to vary with the state of the atmosphere, in regard to heat or cold, humidity or dryness, &c: so that, determinations obtained for one state of the atmosphere, will not answer correctly for another, without modification. Tables commonly exhibit the refraction at different altitudes, for some assumed mean state. 2. With regard to the horizontal refraction, the following method of determining it has been successfully practised in the English Trigonometrical Survey. Let A, A', be two elevated stations on the surface of the earth, BD the intercepted arc of the earth's surface, c the earth's centre, AH', A'H, the horizontal lines at A, A', produced to meet the opposite vertical lines CH', CH. Let a, a', represent the apparent places of the objects A, A', then is aAA' the refraction observed IL at A, and aa'a the refraction observed at A'; and half the sum of those angles will be the horizontal refraction, if we assume it equal at each station. Now, an instrument being placed at each of the stations A, A', the reciprocal observations are made at the same instant of time, which is determined by means of signals or watches previously regulated for that purpose: that is, the observer at A takes the apparent depression of a', at the same moment that the other observer takes the apparent depression of A. In the quadrilateral ACA'I, the two angles A, A', are right angles, and therefore the angles I and c are together equal to two right angles: but the three angles of the triangle IAA are are together equal to two right angles; and consequently the angles A and A' are together equal to the angle c, which is measured by the arc BD. If therefore the sum of the two depressions HA'a, H'Ad', be taken from the sum of the angles HA'A H'AA', or, which is equivalent, from the angle C, (which, is known, because its measure BD is known); the remainder is the sum of both refractions, or angles an'a, d'an'. Hence this rule, take the sum of the two depressions from the measure of the intercepted terrestrial arc, half the remainder is the refraction. 3. If, by reason of the minuteness of the contained arc BD, one of the objects, instead of being depressed, appears elevated, as suppose A' to a": then the sum of the angles a'aa' and A will be greater than the sun IAA'+IA'A, or than c, by the angle of elevation a"AA; but if from the former sum there be taken the depression HA'A, there will remain the sum of the two refractions. So that in this case the rule becomes as follows: take the depression from the sum of the contained arc and elevation, half the remainder is the refraction. 4. The quantity of this terrestrial refraction is estimated by Dr. Maskelyne at one-tenth of the distance of the object observed, expressed in degrees of a great circle. So, if the distance be 10000 fathoms, its 10th part 1000 fathoms, is the 60th part of a degree of a great circle on the earth, or l′, which therefore is the refraction in the altitude of the object at that distance. But M. Legendre is induced, he says, by several experiments, to allow only 4th part of the distance for the refraction in altitude. So that, on the distance of 10000 fathoms, the 14th part of which is 714 fathoms, he allows only 44" of terrestrial refraction, so many being contained in the 714 fathoms. See his Memoir concerning the Trigonometrical operations, &c. Again, M. Delambre, an ingenious French astronomer, makes the quantity of the terrestrial refraction to be the 11th part of the arch of distance. But the English measurers, especially Col. Mudge, from a multitude of exact observations, determine the quantity of the medium refraction to be the 12th part of the said distance. The quantity of this refraction, however, is found to vary considerably, with the different states of the weather and atmosphere, from the 4th to the th of the contained arc. See Trigonometrical Survey, vol. 1 pa. 160, 355. Scholium Scholium. Having given the mean results of observations on the terrestrial refraction, it may not be amiss, though we cannot enter at large into the investigation, to present here a correct table of mean astronomical refractions. The table which has been most commonly given in books of astronomy is Dr. Bradley's, computed from the rule = 57" x cot (a + 3r), r where a is the altitude, r the refraction, and r = 2′35′′ when a = 20°. But it has been found by numerous observations, that the refractions thus computed are rather too small.Laplace, in his Mecanique Celeste (tome iv pa. 27) deduces a formula which is strictly similar to Bradley's; for it is r=mx tan (z-nr), where ≈ is the zenith distance, and m and h are two constant quantities to be determined from observation. The only advantage of the formula given by the French philosopher, over that given by the English astronomer, is, that Laplace and his colleagues have found more correct coefficients than Bradley had. KR n Now, if R = 57°-2957795, the arc equal to the radius, if we make m = = (where k is a constant coefficient which, as well as n, is an abstract number,) the preceding equation will become =kx tan (z-nr). Here, as the refraction ris R always very small, as well as the correction nr, the trigonometrical tangent of the arc nr may be substituted for we shall have tan nr = k. tan (z— nr). But nr=42-(1z — nr) 2 = - nr) tan ( 마음 Hence, sin (z- 2nr) = This formula is easy to use, when the coefficients n and 1-k 1+k are known: and it has been ascertained, by a mean of many observations, that these are 4 and 99765175 respectively. Thus Laplace's equation becomes sin (z-8r) = 99765175 sin z: and from this the following table has been computed. Besides the refractions, the differences of refraction, for every 10 minutes of altitude, are given; an addition which will render the table more extensively useful in all cases where great accuracy is required, Table PROBLEM XI. To find the Angle made by a Given Line with the 1. The easiest method of finding the angular distance of a given line from the meridian, is to measure the greatest and the least angular distance of the vertical plane in which is the star marked a in Ursa minor (commonly called the pole star), from the said line: for half the sum of these two measures will manifestly be the angle required. 2. Another method is to observe when the sun is on the given line; to measure the altitude of his centre at that time, and correct it for refraction and parallax. Then, in the spherical triangle zps, where z is the zenith of the place of observation, P the elevated pole, and s the centre of the sun, there are supposed given zs the zenith distance, or co-altitude of the sun, PS the co-declination of that lu minary, Pz the co-latitude of the place of observation, and ZPS the hour angle, measured at the rate of 15° to an hour, to find the angle szp between the meridian PZ and the vertical zs, on which the sun is at the given time. And here, as three sides and one angle are known, the required angle is readily found, by saying, as sine zs: sine ZFS :: sine PS : sine pzs; that is, as the cosine of the sun's altitude, is to the sine of the hour angle from noon; so is the cosine of the sun's declination, to the sine of the angle made by the given vertical and the meridian. Note. Many other methods are given in books of Astronomy; but the above are sufficient for our present purpose. The first is independent of the latitude of the place; the second requires it. PROBLEM XII. To find the Latitude of a Place. The latitude of a place may be found by observing the greatest and least altitude of a circumpolar star, and then applying to each the correction for refraction; so shall half the sum of the altitudes, thus corrected, be the altitude of, the pole, or the latitude. For |