right ones, therefore the sum of the other two angles at O and C are equal to two right angles, and consequently the angles, O A B, O BA, are together equal to the angle, C, or arc, SS; therefore if the sum of the two depressions or angles, H B a + GA b, be taken from the sum of the angles, HBA + GAB, or the angle, C, the remainder is the sum of both refractions or angles, a BA + bAB; therefore half the difference between the sum of the two depressions and the contained arc, SS (or angle, C), is the refraction.

If one of the objects (B), instead of being depressed, is elevated, suppose to the point, R, then the sum of the angles, d AB + dBA, will be greater than the sum, OAB + OBA (or angle, C), by the angle of elevation, R A G; but if from the sum, dA B + dBA, we take the depression, H Ba, there will remain d AB + a BA, the sum of the two refractions: therefore, if the depression be subtracted from the sum of the contained arc and elevation, half the remainder is the refraction in the case.

It is almost unnecessary to remark that the distance between the places of observation, A and B, should be known sufficiently near to give the contained arc, SS, true to a very few seconds of a degree; the refraction, however, is generally too minute to be of consequence in the operations with a common theodolite, which are usually confined to moderate distances.

Mr. Dalby's illustration of the method of ascertaining the allowance to be made for curvature may be useful to the student; I shall, therefore, transcribe it.

Let A be the place of the instrument, O an object on a distant hill, C the centre of the earth, CSCS, the earth's radius, SS, the contained arc of the earth's circumference, and AH the horizontal line meeting, CS produced; the angle, CA H, being a right one. Make CBC A, then

P

A

8

S

B

the points A and B are the same height above the earth's surface, and BO will be what the object, O, is higher than the station, A. Now, suppose the observed depression of the object, O, to be 2' the angle, H A O, and its distance 17230 yards A B, or the arc, SS; then, taking 69 1-6th miles = 1 degree, we have 69 1-6th x 1760: 60′:: 17230: 8'5 nearly the arc, SS, or angle, C. And because the triangle, ABC, is isosceles, the angle, B A C + angle, C, is equal to a right one; but the angle, B A C + BA H, is also a right one; therefore, BAH= angle, C=4'25; and, allowing 75 of a minute for the effect of refraction, we have 2+75 = 2'75 the depression, HAO, corrected for refraction: therefore, BAH-HAO BAO, or 4'25 275 1'5 angle, — = BAO; whence (supposing A BO to be a right angle), it will be radius: tangent 1'5:: 17230 (A B): 7·52 yards= BO, or what the object, O, is higher than the station, A.

Hence it appears that, when two objects (A and B) are on the same level, or at equal distances from the centre of the earth, the true angular depression of one below the horizon of the other will always be equal to half the number of degrees in the arc contained between them, supposing the earth to be a sphere.

The correction, 75 of a minute, is between 1-11th and 1-12th of the contained arc; in some publications, however, we find 1-10th has been adopted, and others 1-14th; but neither can be depended upon as very correct.

This example is sufficient to point out the method of computation when the object, O, is below the point, B, or above the horizontal line, A H.

LEVELLING BY THE MOUNTAIN BAROMETER.

The barometer is an instrument for measuring the pressure of the atmosphere, and elasticity of the air, at any time. It is commonly made of a glass tube, nearly three feet long, close at one end, and filled with mercury. When the tube is full, by stopping the open end with the finger, then inverting the tube, and immersing that end into a bason of quicksilver, on removing the finger from the orifice, the fluid in the tube will descend into the bason, till what remains in the tube be of the same weight with a column of the atmosphere, which is commonly between 28 and 31 inches of quicksilver; leaving an entire vacuum in the upper end of the tube. For, as the upper end of the tube is quite void of air, there is no pressure downwards but from the column of quicksilver, and, therefore, that will be an exact balance to the counter pressure of the whole column of atmosphere, acting on the orifice of the tube by the quicksilver in the bason.

The upper part of the tube has a graduated scale attached to it, furnished with a vernier, reading commonly to onehundredth of an inch; but in the best barometers it should read to the five-hundredth part of an inch. Again, this scale is variously graduated: in ordinary barometers only, between the heights 27 and 31 inches, while such as are intended for measuring the greatest altitudes-as of the Himalaya Mountains for instance-should be graduated as low as 16 inches. By means of the scale and vernier the height of the mercury in the tube can be accurately measured, by observing the division that is tangential to the surface of the fluid.

Mountain barometers have always marked upon them such readings as these:

Of these, capacity indicates the ratio between the diameter of the tube and the diameter of the cistern; which is usually obtained by trial. A certain quantity of mercury is first poured into the tube, which it fills to the height, say of 14.4 inches: this same quantity is then transferred to the cistern, and found to rise to 2 inches. The capacity is, therefore, as 14.4 to 2, or 72 to 1.

The neutral point denotes the height at which the mercury stood in the tube above the zero mark of the cistern when the barometer was made: the difference between the observed reading of the barometer and the "neutral point" is to be diminished in the proportion of the "capacity," and the remainder applied to the observed reading, to be added when it is above the standard (N P), and subtracted when below. This is termed correction for capacity, and must be made whenever very great accuracy of measurement is required.

Temperature 55° is the generally assumed mean temperature of the air, as a basis for calculation.

Torricelli, who invented the barometer in the year 1643, soon discovered that upon ascending a hill the quicksilver fell in the tube-obviously because the column of air supporting it was shortened thereby; and this observation was, not long after, applied to the measurement of heights. But on account of the great difference of temperature, in low and elevated situations, corrections are necessary to render the results from barometrical observations satisfactory. Before Monsieur De Luc (upwards of a century after Torricelli's discovery) began his experiments with the barometer, a mean of the two temperatures shown by the thermometer attached to the barometer, and the heights.

of the mercury in the barometer, at the bottom and top of a hill, were thought sufficient to determine its height. M. De Luc, however, found that an additional or detached thermometer was also necessary (see Récherches sur les Modifications de l'Atmosphère); and this was subsequently confirmed by the experiments of General Roy and Sir G. Shuckburgh.

There are various experiments by which the real altitude which corresponds to any given height of the mercury may be deduced: the readiest is that which results from the known specific gravity of air, with respect to the whole pressure of the atmosphere on the surface of the earth. The investigation is thus given at vol. ii. p 261, "Hutton's Mathematics," in accordance with the law, that, when the elevation increases in an arithmetical ratio, the weight or density of the atmosphere, and, consequently, the height of the mercurial column, diminish in a geometrical ratio:

"Because the terms of an arithmetical series are proportional to the logarithms of the terms of a geometrical series, different altitudes above the earth's surface are as the logarithms of the densities, or the weights of air, at those altitudes.

66

"So that, if D denote the density at the altitude A, and d denote the density at the altitude a; then A being as the log. of D, and a as the log. of d, the difference of alt. A-a will be as the log. D-log. d, or

And if A = 0, or D the density at the surface of the earth; then any altitude above the surface, a, is as the log. of

D

d

“ Assume h, so that a = h × log., where h will be of one constant value for all altitudes; and, to determine that value, let a case be taken in which we know the altitude, a,