METHODS OF FINDING A MERIDIAN LINE.

201. The knowledge of the direction of the meridian is useful for several purposes, but absolutely necessary for adjusting a transit instrument. The first step, and that the most difficult, is to find it nearly: when this is done, it may easily be corrected by help of the transit instrument itself. Either of the two following methods, especially the second, will serve at once for finding it sufficiently near for most purposes, except for the transit instrument.

202. On an horizontal plane describe several concentric circles of a few inches in diameter. In the centre place a wire, a few inches long, at right angles to the horizontal plane. Note in the forenoon the point where the shadow of the top of the wire just reaches any of the circles, and watch in the afternoon the point where the extremity of the shadow again reaches the same circle. The arch intercepted between these two points being bisected by a radius, the radius will be in the direction of the meridian; because the direction of the shadow is in the plane of the vertical circle passing through the sun, and the sun has equal azimuths at equal distances from noon, unless as far as the change of declination interferes.

This meridian may be transferred to any near place, by suspending a plumb line directly over the southern extremity of the line drawn as above, and noting when the shadow falls on that line at this time another plumb line, suspended at the place where the meridian line is required, will, by its shadow, shew the meridian.

The imperfections of this method of finding a meridian line arise from the inexact termination of the shadow, and from the change of the sun's declination in the interval of the two observations. The latter inconvenience is least in June and December near the solstices.

203. The other method is perhaps as simple and exact as can be expected without the assistance of a telescope, and is applicable, even with a transit instrument. Observe when the pole star above the pole, and & Ursa Majoris, called Alioth, are in the same vertical; a plane passing through these stars at that time is nearly in the plane of the meridian.

The pole star and Alioth pass the meridian within about nine minutes of each other, the former being 1° 45′ above the pole, and the latter 33° below it. Alioth passing the meridian below the pole, about nine minutes before the pole star passes above the pole; it follows that the vertical circle passing through the polar star, and approaching the meridian, will be met by the vertical circle passing through Alioth, receding from the meridian, and therefore Alioth and the pole star will be in the same vertical within less than nine minutes of time of the passage of the pole star: and as the pole star changes its azimuth very slowly, the vertical circle passing through these two stars must be nearly in the plane of the meridian.

204. The deviation of this vertical circle from the plane of the meridian may be easily computed for in general the sine of the azimuth is to the sine of the hour angle at the pole, as the sine of the polar distance is to the sine of the zenith distance. Now the mean R. Asc. on Jan. 1, 1810,

therefore the sine of the azimuth of the pole star when Alioth sin. 2° 14'. sin. 1° 45' sin. (co. lat.— 1° 45′)

passes=

(in lat. 53° 23′,) sin. 7' nearly.

This is the azimuth of the pole star, when Alioth is passing the meridian below the pole. When they are in the same vertical,

the common azimuth is somewhat less ; but the difference is so small, that it is scarcely worth notice in this approximation to the meridian, which serves without farther correction for most common purposes. The changes of the right ascension from aberration are not noticed, because the method is only given here for an approximation.

205. The following is a convenient way of practising this method. Suspend two plummets, A and B, (Fig. 29), to each end of a rod CD. Vessels of water should be used for steadying the plummets. A pivot fixed to the middle of the rod should be supported on a socket at E; so that the rod may turn steadily and freely. If Alioth and the pole star be observed in the plane of the plumb lines, that plane will be, in these latitudes, within about 7′ of the meridian. The eye will readily shew when they are nearly in the same vertical, and then the plumb lines, by turning the rod on its socket, may be easily made to pass through them, when exactly on the same vertical." a transit telescope turning The deviation from the

206. Instead of the plumb lines, on an horizontal axis may be used. meridian of the telescope, so adjusted, may be found by observing the transits of a star to the south of the zenith, and of the pole star. The transit of the former will give the sidereal time

a The correction of the azimuth is very easily computed; for the angular motion of the vertical of the pole star is to the angular motion of the vertical of Alioth, as sine polar dist. of pole star sine polar dist. of Alioth sin. 1°45' sin. 34 52

sine zenith dist.

:

sin. zenith dist.

(in lat. 53° 23′) : :

Therefore the azimuth of common vertical is to azi

muth of pole star, when Alioth passeth, as 11 to 12 nearly; and therefore azimuth of common vertical = X 7' 6,4 nearly.

=

b This method can only be used when the polar star passes the meridian above the pole, when it is dark, that is, from the end of August to the end of January. There are no other stars so convenient for this method, although Capella below the pole, and Ursa Minoris above the pole, may serve.

ε

:

nearly, and comparing the time so found with the sidereal time given by the polar star, the difference, which may be considered as entirely the error from the pole star, will give the deviation from the meridian: for the deviation in seconds of a degree is to error in seconds of a degree of sidereal time of transit of pole star, as the sine of the polar distance of the pole star to the sine of the zenith distance. The reason of considering the whole difference, as the error of the pole star, is, that when the deviation from the meridian is small, the error of sidereal time from a star, southward of the zenith is very small, compared to the error from the polar star.

This is a very convenient method of approximating at pleasure to the meridian.

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207. The deviation from the meridian may also be found by comparing the times of continuance of a circumpolar star on the east and west sides of the meridian.a

A quadrant having an azimuth circle is very convenient for ascertaining the meridian, by observing equal altitudes on each side of the meridian, and then bisecting the arch of azimuth. If the sun be used, allowance must be made for the change of declination.

A good clock will serve instead of an azimuth circle, by observing equal altitudes of the sun or a star, half the interval of time corrected (if the sun is observed) will shew when the object was on the meridian, and thence the error of the clock will be ascertained, and so the time of the transit of any star may be computed, and the instrument adjusted at the time of that transit.

a Professor Vince's Practical Astron. p. 82.

CHAPTER XII.

GEOCENTRIC AND HELIOCENTRIC PLACES OF PLANETS-NODES AND INCLINATIONS OF THEIR ORBITS- -MEAN MOTIONS AND PERIODIC TIMES-DISCOVERIES OF KEPLER-ELLIPTICAL MOTIONS OF PLA

NETS.

208. THE fixed stars, as has been noticed, appear in the same place with respect to the ecliptic from whatever part of the solar system they are seen, but not so the planets: their places as seen from the sun and earth are very different, aud as their motions are performed about the sun, it is necessary to deduce from the observatious made at the earth, the observations that would be made by a spectator at the sun. By this we arrive at the true knowledge of their motions, and discover that their orbits are neither circular, nor their motions entirely equable about the sun, although a uniform motion will, in some measure, solve the phænomena of their appearances.

It has before been shewn how the distances and periodic times of the planets are found, on the hypothesis of their orbits being circular, and their motions uniform; it remains to shew how the places of the nodes and inclinations of the orbits may be found nearly, before we proceed to more accurate investigations. For this, it is necessary to find from the geocentric longitude and latitude (computed, from the right ascension and declination observed,) and the distance of the planet from sun known nearly (art. 97 and 101) the heliocentric latitude and longitude.

209. Let S and E (Fig. 30) be the sun and earth, P the