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77. Earth's atmosphere. From the science of pneumatics we learn that the earth's atmosphere is an elastic medium, the density of which continually decreases as the distance from the general surface of the earth increases. The law of decrease is such that, as the height increases in arithmetical progression, the density decreases nearly in geometrical progression. The actual decrease is such that, at the height of 31⁄2 miles, the density is only about one half as great as at the earth's surface; at the height of 7 miles, about one fourth as great; at the height of 10 miles, about one eighth as great; and thus on. The whole height of the atmosphere is not known. But from the preceding law, it follows that, at the height of 40 or 50 miles, its density must be extremely small, so as to be nearly or quite insensible. The density at any given place varies with the pressure, as indicated by a barometer, and with the temperature, as indicated by a thermometer; but this variation is not great.

If the density of the whole mass of the atmosphere was uniform throughout, and the same that it is at the earth's surface when the barometer stands at 30 inches, and Fahrenheit's thermometer at 50°, which is regarded as being nearly the mean density at the earth, the height would then be 5.13 miles.

78. Refraction. The science of optics teaches us that, when a ray of light passes obliquely from one medium into another of different density, it becomes bent, or refracted; the ray in the second medium, called the refracted ray, taking a different direction from that in the first, which is called the incident ray. Both rays lie in the same plane with a perpendicular to the common surface of the two mediums at the point of passage from one to the other. When the ray passes from a rarer to a denser medium, the refracted ray is bent towards the perpendicular to the common surface, making with it a less angle than that made with it by the incident ray. Thus an incident ray SA, Fig. 5, entering obliquely

a second medium of greater density at A, takes a direction AB, making the angle BAD, which is called the angle of refraction, less than the angle SAE, which is called the angle of incidence. The angle BAC, which expresses the difference between the directions SA and AB, of the incident and refracted rays, is called the refraction.

For the same two mediums, the amount of refraction changes with a change in the angle of incidence. The law of this change is such that the sine of the angle of incidence is to the sine of the angle of refraction in a constant ratio, which is called the index of refraction. Thus if I be the angle of incidence, R the angle of refraction, and m the index of refraction, the value of which for different mediums is determined by experiment, we have sin I: sin Rm: 1; or, m: 1; or, sin I = m sin R. For the passage of a ray of light from a vacuum into air of a mean density, or that which it has when the barometer stands at 30 inches, and the thermometer at 50°, the value of m is 1.000284.

When a ray passes through a medium composed of strata of different densities, bounded by parallel planes, the whole refraction is the same, as if the incident ray had at once entered the last stratum with its first angle of incidence; the direction of the ray in the last stratum being the same in either case. Thus, if a ray SA, Fig. 6, in passing through such a medium, takes the directions. AB, BC, a ray S'A' entering the last stratum at the same angle of incidence with SA, will take a direction A'C', parallel to BC. When the strata are indefinitely thin and their number indefinitely great, or, which amounts to the same, when the density continually varies from A to C, the broken line ABC becomes a curve. The whole refraction is however still the same, provided the density at the surface C remains unchanged: that is, the whole refraction for a given angle of incidence depends entirely on the density at the second surface.

79. Astronomical Refraction. As the density of the earth's atmosphere continually increases from its upper surface to the earth (77), it follows, from the last article, that when a ray of light, from any of the heavenly bodies, enters the atmosphere obliquely, it becomes bent into a curve, concave towards the earth. The

density in the upper parts of the atmosphere being very small, the curve at first deviates very little from a straight line, but the deviation becomes greater as it approaches the earth. Both the straight and curved parts of the ray must necessarily lie in the same vertical plane; for, as the corresponding parts of the atmosphere on each side of a vertical plane may be regarded as of equal density, there is no cause for a deviation to either side. The whole change produced in the direction of the ray in traversing the atmosphere is called the astronomical refraction.

80. Astronomical refraction increases the altitude of a heavenly body, but does not affect the azimuth.

Let SaA, Fig. 7, be a ray which, proceeding from a body S, enters the atmosphere at a, and being bent by refraction, meets the earth's surface at A; and let AS' be a tangent to the curve Aa at A. Then will the ray enter the eye of an observer at A, in the direction S'A, and consequently the body S will appear to be in the more elevated position S'. As the tangent AS' must be in the same vertical plane with the ray AaS, the azimuth of the body is not affected by refraction.

It follows that the altitude of a heavenly body is obtained by subtracting the refraction from the observed altitude, and the zenith distance, by adding the refraction to the observed zenith distance.

81. At the zenith, the refraction is nothing. For, in consequence of the corresponding density of the atmosphere on every side of a vertical line, there is no cause for a ray entering it in that direction to deviate from its rectilineal course.

82. To obtain approximate formula for computing the refraction due to any altitude or zenith distance.

As the upper and under surface of that portion of the atmosphere through which a ray of the heavenly bodies passes in its course to a place on the earth's surface, do not differ much from parallel planes, we may obtain approximate formulæ for the refraction, by assuming the density to be uniform throughout, and the same that it is at the earth's surface (78). Let bd, Fig. 8, be a part of the boundary of the atmosphere on this supposition, Sa a ray from a body S, which being refracted at a, meets the earth's

surface at A, and let C be the centre of the earth, and Z the zenith of the place A. Then, to an observer at A, the body will appear in the direction AS', and the angle SaS' will be the refraction corresponding to the apparent zenith distance ZAS'. Put,







CA radius of the earth, assumed to be a sphere,

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Then, since I = Cac CaA + Aac = R+r, we have (78),

sin (R + r) = m sin R......

(A); or, (App.* 13), sin R cos r + cos R sin r = m sin R; or, dividing by cos R, we have,

tang R cos r + sin r

= m

m tang R.

But since m differs but little from a unit (78), it is evident from equat. (A), that Rr must differ but little from R, and consequently r must be a small angle. Taking therefore the angle instead of its sine (App. 51), and assuming cos r =

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1, we have


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Taking m = 1.000284 (78),and substituting for

206264".8 (App. 51), we have, (m—1). «




its value

Hence, since

p = 3956 (71) and h 5.13 (77), the formulæ (C) and (B) become,

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The degree of accuracy of these formulæ may be tested by finding the latitude of a place from the observed upper and lower

* Appendix to part 1..

meridian altitudes of different circumpolar stars (58), using the formulæ in computing the refractions; which must be subtracted from the observed altitudes to obtain the correct altitudes. If the state of the air is the same or nearly the same as that assumed in finding the formulæ, and if no one of the lower altitudes of the stars employed is less than about 20°, the latitude as obtained from different stars will be sensibly the same. But if the lower altitude of any one of the stars is much under 20°, the latitude found from that star will be decidedly too great. Whence it follows, that, for a low altitude, the refraction computed by the formulæ is too small. It may thus be ascertained that, for altitudes of 20° and upwards, the refractions computed by the formulæ do not err to the amount of a second; but for lower altitudes the error becomes considerable, amounting at the horizon to several minutes.

83. Tables of Refraction. The complete investigation of astronomical refraction is a subject of great difficulty. It has claimed the attention of many eminent mathematicians,* and formulæ have been obtained which give the amount of the refraction with great precision, except for altitudes under 12° or 14°; and for these they give it very nearly. These formulæ take into view the changes in the density of the air at the earth's surface as indicated by the barometer and thermometer. From the formulæ, tables have been computed, from which the refraction corresponding to a given observed altitude is easily obtained. In these tables, the principal columns contain the refractions computed for a density of the air corresponding to some medium heights of the barometer and thermometer. These are called mean refractions. Other columns contain the corrections due to given changes in the states of these instruments.

84. Refraction increases the visible continuance of the heavenly bodies above the horizon.

As refraction increases the altitudes of the heavenly bodies, it must accelerate their rising and retard their setting, and thus render them longer visible. The refraction at the horizon is about

* Laplace, in the Mécanique Céleste; Prof. Bessel, in the Fundamenta Astronomia; Dr. Young, in the Transactions of the Royal Society of London for 1819 and 1824; Ivory, in the same Transactions for 1823; and various others.

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