Na xei BN be considered as a rectilinear triangle; therefore, by similar triangles, we have BN: Na::ei:ai = Here, as Na is constant, and the ratio of Ba: Ci, the sines of refraction and incidence, is constant also, by the laws of refraction, their difference e i must vary as Ba; and consequently a i will But A T being=BN vary as Ba aNx Ba cause a N is constant, must also vary, as ", which, be fore a i the refraction, will be as AT the tangent of the apparent zenith distance of the celestial body: for the angle of refraction A Na is equal to the apparent zenith distance, in this kind of refraction. This rule will deviate in very small apparent altitudes, where the rays coming to the atmosphere in a direction nearly parallel to the horizon, have a very large portion of it to pass through, which enlarges the refraction so much, that aei cannot without error be considered as a rectilinear angle. On this account, Dr. Bradley, adopting the principle, that the force with which a ray is attracted in passing through the atmosphere is uniform, has deduced a very simple and general rule for the refraction r at any altitude a whatever; viz. as radius r: cotang. a + 3r 57r, the refraction in seconds: this rule is found to agree exceedingly well with observations made at the mean states of the barometer and thermometer. He has also given formulæ, by which the refraction may be found, at any other states of the atmosphere, to as great exactness as need be looked for in such enquiries. 106. The methods given in articles 103. and 104. are both liable to objection; for each of them requires a knowledge of the latitude of the place of observation, which cannot be obtained without an allowance for refraction; an allowance that certainly cannot be made with accuracy until the refraction is determined. An ingenious method, invented by Boscovich, is not liable to this objection, and is therefore now given, as it requires the admission of no other principles than those exhibited in the last artick. He proposes to find the refractions by the variation of the zenith distances of the circumpolar stars, thus: Let A and a be the apparent zenith distances of a star on the meridian below and above the pole, X and the respective refractions, B and b the apparent meridian zenith distances of another star below and above the pole, Z and ≈ the corresponding refractions then the true distances from the pole will be A+ X, a + r, and B + 2, b + ≈ ; and as the distance of the pole from the zenith is equal to half the sum of the greatest and least true zenith distances, we have A+X+a+x=B+Z+b +z; consequently (I) X+rZzB+ b- A -a. Now, supposing (Art. 105.) the refractions to be as the tangents of the zenith distances, we have, tang. A : tang. a :: X: x = X. tang. B X. tang. a tang. A; fora X. tang. b and ≈ = b tang. A -.By substituting these in the equation (I) we obtain X = B+b-A- a x tang. A tang. A+ tang, a-tang. B-tang. 6; whence the other refractions become known. Or, if we adopt Dr. Bradley's theorem, and suppose the refractions to vary as the tangent of the zenith distance diminished by three times the refraction; then, by putting A3 X = M, a 3t=m, B 3 ZN, and b 32n, we have by reduction, X -ax tang. M. tang. Mtang. m — Calig. N lang, a = ,for the true refrac tion at the apparent zenith distance A; and from this we get the other refractions, by the equations = X. tang, m tang. M X. tang. N or more generally by Dr. Bradley's rule. ; In this manner the refraction of all altitudes may be found by means of a few accurate observations, and reduced into a table for general use, as given in table VIII. at the end of this volume. 107. The atmosphere also causes a particular kind of phenomenon, besides the astronomical refractions, which is the morning and evening twilight: it is called twilight, as being between or partaking of two lights, that of the sun and that of the stars; the latin word crepusculum arises from its being doubtful whether it be day or night. The morning twilight begins when the sun is not more than about 18° below our horizon, for then his rays first reach the eastern parts of the air so as to be brought, in part by refraction, and part by reflection, within our horizon; as the sun grows nearer rising, his light diffuses itself farther round, and enlightens a larger portion of air, which thus becomes more and more illuminated, till the sun rises: in like manner, after sun-set the light gradually decreases, till the sun is gotten so low that none of his rays can reach the western parts of the atmosphere so as to be brought by reflection and refraction without our visible horizon. There is some little difference in the duration of twilight, arising from the different density of the atmosphere, and other causes: thus, when the vapours and other particles which reflect the solar rays are carried to a greater height than at other times, the sun may be depressed more than 18°, and yet produce twilight; when these particles are lower in the atmosphere, the sun cannot produce twilight when it is depressed 18°. On these accounts the evening twilight is longer than that in the morning, and the twilight is longer in hot weather than in cold, other circumstances being the same. There are also differences in the time of twilight, owing to the different situations of places upon the earth, or to the change of the sun's apparent place in the heavens: thus, if a place be situated in a parallel sphere, or nearly so, the apparent motion of the sun being either quite or nearly parallel, he will be carried round for some months at a less depression than 18°, during which there will, of course, be no real night. In a right sphere, the twilight is shortest, because the sun rising and setting at right angles to the horizon, he is depressed 18° below the horizon in the shortest period. In any place in an oblique sphere, the nearer it is to one of the poles, the longer the twilight; and, consequently, the nearer the equator, the shorter the twilight. As to the different places of the sun; the twilight is longest in all places in north latitude, when that luminary is in the tropic of cancer; in south latitudes, when the sun is in the tropic of capricorn, The time of the shortest twilight is different in different latitudes in England it is shortest about the eleventh or twelfth of October, and the second or third of March, as will be shewn by calculation in the next chapter (Art. 141.). Equation of Time, 108. It would be far from an easy task to give a precise definition of duration and time; but it is not necessary to attempt it, since from the succession of our own ideas, and from the successive variations of external objects, we easily acquire the ideas of them and of their measures. We conceive true or absolute time to flow uniformly in an unchangeable course, which alone serves to measure with exactness the changes of all other things; and unless we apply to the vulgar measures of time, which are gross and inaccurate, some proper corrections, the conclusions are always found erroneous. Yet however various the flux of time may appear to different intellectual beings, it cannot for a moment be thought to depend upon, or be regulated by, the ideas of any created being whatever it therefore becomes requisite to choose some object as a proper and adequate measure of time. For this purpose we fix upon either some ra. tural object, whose motion is conspicuous and near.; uniform, or some artificial contrivance invented for the purpose of determining this measure of the first kind the sun has been chosen, and of the latter a pendulum clock; but as the motion of the sun is not exactly uniform, and a clock is liable to variation in its motions, as it is affected by the changes in the atmosphere with respect to heat or cold, dryness or moisture, some mode of comparison must be contrived which will enable us to correct the measure of time deduced from the one, by means of that which is gained from the other. This is best done by tracing out the nature and extent of the irregularities in the sun's apparent motions. 109. The astronomical day, at any place, begins when the sun's centre is on the meridian of that place; it is divided into 24 hours, reckoning in a numerical succession from 1 to 24: the first 12 are sometimes distinguished by the mark P.M. signifying post meridiem, or afternoon; and the latter 12 are marked A. M. signifying ante meridiem, or before noon. But astronomers generally reckon through the 24 hours, from noon to noon; and what are by the civil or common way of reckoning called morning hours, are by astronomers reckoned in the succession from 12, or midnight, to 24 hours. Thus 9 o'clock in the morningof February 14th, is, by astronomers, called February the 13th at 21 hours. 110. The sun's daily motion in longitude is the arc of the ecliptic run through in a day; his daily motion in right ascension is the corresponding arc of the equator: the mean daily motion in either circle. is measured by 59′ 8′′ nearly. For 365: 1a :: 360° : 59′ 8′′.2. 111. The interval of time between two successive transits of the sun's centre over the same meridian, |