Now, drawing CPm, m is the place of P as seen from the centre, and n from the surface; the arc mn is the diurnal parallax of the object when seen from H, in the position P. And, when the planet appears in the horizon at h, the arc rs is the horizontal parallax. 6. The diurnal parallax is equal to the angle subtended at the planet, by the place of the spectator and centre of the Earth; and, therefore, the horizontal parallax is greatest, and is equal to the angle under which the semi-diameter of the Earth would appear at the planet. For, to a spectator at H, (see the last figure) a fixed star in the direction HV is in the zenith, and the distance of the planet from this star is equal to the angle VHP; but at the centre, the distance is equal to the angle VCP, and the difference of these is the angle HPC. Now, CP: CH: : sin. PHV, (or sine CHP): sin. HPC, the parallax; therefore, CH × sin. PHV CP sine of the parallax HPC = As CH is constant, supposing the Earth to be a sphere, the sine of the parallax varies as the sine of the zenith distance directly, and the distance of the body from the centre of the Earth inversely. Hence, a body in the zenith has no parallax, and at h, in the horizon, the parallax is greatest, being then equal to the angle which the semi-diameter of the Earth subtends at the planet. The nearer a body is to the Earth the greater is its parallax; hence, the Moon on this account has the greatest parallax, and the fixed stars, from their vast distance, have no parallax, the semi-diameter of the Earth appearing at that distance no more than a point. 7. The diurnal parallax depresses an object; a planet at rising appears to the eastward of its true place, and at setting to the westward, whence the term diurnal parallax. And for different altitudes of the same body, supposing it to continue at the same distance from the Earth, the sine of the diurnal parallax, or parallax in altitude, is equal to the sine of the horizontal parallax multiplied by the sine of the apparent zenith distance. For the parallax varies as the sine s of the apparent zenith distance; therefore, if p = the horizontal parallax, and radius be unity, we shall have 1:s::p:ps, the sine of the diurnal parallax. To ascertain, therefore, the parallax at all altitudes, we must find it at some altitude. To find the parallax of the Sun, Moon, or any of the planets. Let a body P be observed from two places H and S on the same meridian, (see the fig. page 343,) then the angle HPS is the effect of parallax between the two places. Now, the angle HPS = horizontal parallax × sin. PHV, taking the angle HPC for the sine of HPC, and the parallax or angle SPC = hor. par. X PSr; hence, the horizontal parallax X (sin. < PHV + + sin. PSr) = HPS. Therefore, the horizontal parallax = ∠ HPS, divided by the sum of those two sines. If the distance between the two places be known, in degrees, the angle HPS = VHP + SP - HCS. Supposing the distance between the two places Hand Stobe 74° 46′ 30′′, equal HCS; the zenith distance VHP=320, and SP-440. Then, the angle HPS = 32° 44°-74° 46′30′′ 760-74° 46′ 30′′ = 1°13′30′′ 73.5'. But the horizon Or, the angle which the disc of the Earth subtends at a planet may be obtained; and, hence, the horizontal parallax is also given. Thus, to find the angle which two distant places, in the same terrestrial meridian, subtend at a planet. Let Hand S be two places, P a planet in the celestial meridian of these places. Hv and Sn the directions in which the fixed star, also in the meridian at the same time, is seen at the two places. The star made use of is supposed to be very nearly in the same parallel of declination as the planet, that is, not differing in declination more than a few minutes. Now, because Hv and Sn are parallel, the angle HnS is equal to the angle nHv; therefore, / HPS = HnS + nSP =vHP + PSn = the sum of the apparent distances of the planet and star, (the place to which the planet is vertical being supposed to be between the places of observations.) These distances can be observed with great accuracy by means of a micrometer. We have thus the principal things necessary to enable us to advance by a most important step, viz. to obtain the angle which the disc of the Earth subtends, as seen from the planet. It may easily be demonstrated that this angle, which equals twice the parallax, is = 2 / HPS X Rad.. sin. VHP + sin. PSr See Dr. Brinkley's Elements of Astronomy. Thus to obtain the angle which the earth's disc subtends at the planet, it is necessary to know the angle VHP and PSr, or zenith distances of the planet at the two places. But it is not necessary that these angles should be observed with much precision, since it is easy to see that an error of even a few minutes, in the quantities of these angles, will make no Rad. sensible error in the quantity sin. < VHP + sin. < PSr The above is on the suppositions, 1st, that the star and planet are on the meridian together: 2nd, that the two places are in the same terrestrial. If the star and planet are not in the meridian together, yet their difference of declination being observed, it is the same as if there had been a star on the meridian with the planet. If the two places are not in the same terrestrial meridian, an allowance must be made for the planet's motions in the interval between its passages over the two meridians, and we obtain the difference of declinations that would have been observed at two places under the same meridian. The Cape of Good Hope is nearly in the same meridian with many places in Europe, having observatories for astronomical purposes, and therefore a comparison of the observations made there, with those made in Europe, furnishes us with the means of practising this method. By a comparison of the observations of De La Caille, made at the. Cape of Good Hope, with those made at Greenwich, Paris, Bologna, Stockholm, and Upsal, the angles which the earth's disc subtend at Mars and at the moon, have been obtained with very considerable precision. Comparisons of observations will also furnish the same for the sun and other planets. But knowing the angle which the earth's disc subtends at any one planet, we can readily find it for the sun, or any other planet. The last method that has been described for finding the parallaxes of the bodies in the solar system, yields only to one other method in point of accuracy; viz. to that furnished by the transit of Venus over the Sun's disc. See Dr. Brinkley's Elements of Astronomy, Art. 263. The Moon =2°2′ A planet, therefore, appearing to us as small as the earth • appears to the inhabitants of Saturn and Uranus, would not have been observed except by the assistance of the telescope. 8. The distance of a celestial body from the centre of the earth, is equal to the semi-diameter of the earth, divided by the sine of the horizontal parallax. For, in the triangle AHC (see the fig. p. 343) right angled at H, are given CH and the angles H and h; therefore, as sin. /h: radius (= sin. 90° = 1)::CH: Ch= sem. diam. earth CH sin. h -, the distance of the body from the centre sin. hor. par. of the earth. Hence, as the semi-diameter of the earth has Deen determined to be 3960 miles; when the horizontal parallax of a body is known, its distance from the centre of the earth is easily found. Example. Supposing the horizontal parallax of the moon to be 57, what is its distance from the earth, the semi-diameter of the latter body being 3960 miles? Solution. As sin. / h = 57': radius 1 (= sin. H = sin. 30°): : C H (= 3960 miles): Ch. But the natural sine of 57' = .01658; hence Ch = 3960 .01658 = 238,842 miles, the Jistance of the moon from the centre of the earth when her To 238,842 5.378114 Ex. 2. What is the distance of the moon from the earth, when her horizontal parallax is the greatest, or 61' 32", the semi-diameter of the earth being 3960 miles ? Ex. S. What is the distance of the moon from the earth, when her horizontal parallax is the least, or 53′ 52"? Ex. 4. What is the distance of the sun from the earth, supposing his horizontal parallax to be 8 seconds? CHAPTER XVI. Of Eclipses. 1. The Eclipses of the sun and moon, of all the celestial phenomena, have most and longest engaged the attention of mankind. They are now, in every respect less interesting than formerly: at first they were objects of superstition; next, before the improvements in instruments, they served for perfecting astronomical tables; and last of all, they assisted geography and navigation. Eclipses of the sun still continue to be of importance for geography, and in some measure for the verifica-. tion of astronomical tables. As every planet belonging to the solar system, both primary and secondary, derives its light from the sun, it must cast a shadow to that part of the heavens which is opposite to the sun. This shadow is of course nothing but a privation of light in the space hid from the sun by the opaque body, and will always be proportionate to the relative magnitudes of the sun and planet. If the sun and planet were both of the same size, the form of the shadow cast by the planet would be that of a cylinder, the diameter of which would be the same as that of the sun or planet, and it would never converge to a point. If the planet were larger than the sun, the shadow would continue to spread or diverge; but as the sun is much larger than any of the planets, the shadow cast by any one of these bodies must converge to point, the distance of which from the planet will be propor tionate to the size and distance of the planet from the sur |