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, Ist, 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 tonsiderable 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. 203.

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. Lh: radius (=sin. 900 = 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. 20°):: CH (= 3960 miles): Ch. But the natural sine of

57′ = .01658; hence Ch =

3960

01658

238,842 miles, the

distance of the moon from the centre of the earth when her

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. 8. 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?

1

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 verification 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 a point, the distance of which from the planet will be proportionate to the size and distance of the planet from the sun

The magnitude of the sun is such that the shadow cast by each of the primary planets always converges to a point before it reaches any other planet; so that not one of the primary planets can eclipse another. The shadow of any planet which is accompanied by satellites may, on certain occasions, eclipse these satellites; but it is not long enough to eclipse any other body. The shadow of as atellite or moon may also, on certain occasions, fall on the primary and eclipse it.

2. Eclipses of the Moon. An eclipse of the moon being caused by the passage of the moon through the conical shadow of the earth; the magnitude and duration of the eclipse depend upon the length of the moon's path in the shadow.

Let AB and TE be sections of the sun and earth, by a plane perpendicular to the plane of the ecliptic. Let ATV and BEV touchthese sections externally, and BPG and AMN internally. Let these lines be conceived to revolve about the axis CKV; then TVE will form the conical shadow, from every point of which the light of the sun will be excluded, more of it from the parts near TV and EV than from those near PG and MN.

The semi angle of the cone (TVK) sem. diam. sun (CTA) - horizontal parallax of the sun (TCK.) The angle subtended by the se

section = SKV = TSK-KVT = horizontal parallax of the moon + horizontal parallax of the sun - semi-diameter of the sun. The angle of the cone being known, the height of the shadow may be computed. For height of the shadow: radius of earth:: rad.: tang. angle of cone; also the diameter of section of the shadow at the moon is known, for SO: dist. moon :: tang. semi-diam. of section of shadow:: radius.

The height of the shadow varies from 213 to 220 semidiameters of the earth, and nearly varies inversely as the apparent diameter of the sun.

3. When the moon is entirely immersed in the shadow, the eclipse is total; when only part of it is involved, partial; and when it passes through the axis of the shadow, it is said to be central and total. The breadth of the section of the shadow, at the distance of the moon, is about three diameters of the moon; therefore when the moon passes through the axis of the shadow, it may be entirely in the shadow for nearly two hours.

The angle SKV is, when greatest, about 46': therefore, as the moon's latitude is sometimes above 50, it is evident an eclipse of the moon can only take place when it is near its nodes.

The circumstances of an eclipse of the moon can be readily computed. The latitude of the moon at opposition, the time of opposition, the horizontal parallax of the moon, and diameters of the sun and moon are known from the tables. By the tables we can compute the angular velocity of the moon relatively to the sun at rest. Thence we can find the time from the beginning of the eclipse to opposition, and the time from opposition to the end. And, as the time of opposition is known, the times of beginning and ending of the eclipse are known. See Dr. Brinkley's As

tronomy:

4. The greatest distance of the moon, at opposition, from its node, that an eclipse can happen, is about 11 degrees, and is called its ecliptic Umit.