tecting other fragments of the supposed planet, Dr. Olbers examined thrice every year all the little stars in the opposite constellations of the Virgin and the Whale, till his labours were crowned with success on the 29th March, 1807, by the discovery of a new planet in the constellation Virgo, to which he gave the name of Vesta.

As soon as this discovery was made known in England, the planet was observed at Blackheath, on the 26th April, 1807, by S. Groombridge, Esq. an ingenious and active astronomer, who has successfully devoted his leisure and his fortune to the advancement of astronomy. He continued to observe it with his excellent astronomical circle, till the 20th May, when, from its having ceased to become visible on the meridian, he had recourse to equatorial instruments. On the 11th of August, Mr. Groombridge resumed his meridional observations, from which he has computed part of the elements of its orbit; and he had the good fortune to observe the ecliptic opposition of the planet, on the 8th of September, 1808, at 7h. 30', in longitude 11s. 15° 54′ 26′′. His observations were continued till the beginning of November, 1808, and he expected to have found the planet again at its opposition in. February, 1810; but, from a continuance of cloudy weather, and probably from errors in the elements, he did not succeed.

The planet Vesta is of the fifth or sixth magnitude, and may be seen in a clear evening by the naked eye. Its light is more intense, pure, and white than any of the other three; and it is very similar in its appearance to the Georgium Sidus. It is not surrounded with any nebulosity; and even with a power of 636, Dr. Herschel could not perceive its real disc. The orbit of Vesta cuts the orbit of Pallas, but not in the same place where it is cut by that of Ceres. According to the observations of Schroeter, the apparent diameter of Vesta is only 0.488 of a second, one half of what he found to be the apparent diameter of the fourth satellite of Saturn; and yet it is very remarkable, that its light is so intense that Schroeter saw it several times with his naked eye.

M. Burckhardt is of opinion, that Le Monnier had observed this planet as a fixed star, since a small star situated in the same place, and noticed by that astronomer, has since disappeared.

The following are the elements of the orbit of Vesta, computed by Mr. Groombridge from his own observations.

Mean distance

Excentricity in parts of the earth's radius

2.163 0.0953

The following elements are given by Burckhardt, in the Connoissance de Tems for 1809, from the most recent observations on the continent.

Place of ascending node

Place of perihelion

Inclination of orbit

Mean distance

Excentricity

3s. 13° 1' 0"

8s. 9° 42′ 53" 7° 8' 46"

2.373000

0.093221

These elements do not differ greatly from those of Mr. Groombridge, excepting in the place of the perihelion. The writer of this article, however, has just been informed by that able astronomer, that, from a new set of observations, he has found the place of the perihelion of Vesta to be 8s. 130; differing little more than 3° from that of Burckhardt.

The orbits appear to intersect each other in various places; and it is obvious, that the points of intersection must be perpetually shifting, according to the changes in the aphelia of the planets.

THE PROJECTION OF SOLAR ECLIPSES.

Art. 399. AS the determination of the phases of a solar eclipse, by the calculation of parallaxes, is a tedious operation, a geometrical construction is here given; which, though less accurate, will, when carefully performed on a large scale, generally produce a result sufficiently near the truth for the purpose of predicting the circumstances of an eclipse.

400. If a plane be supposed to pass through the earth's centre, perpendicular to the line joining the centres of the sun and earth, it will divide the dark from the enlightened part of the earth's surface very nearly, and is therefore called the plane of illumination. Now, the earth's axis being perpendicular to the plane of the equator, it follows that the inclination of the axis to the plane of illumination is equal to the declination of the sun: hence, when the sun is in the equator, the poles of the earth lie in the circle of illumination. As the sun advances in the ecliptic, the plane of illumination revolves round an axis perpendicular to the plane of the ecliptic, with an angular velocity equal to that of the earth in its orbit. But the earth's axis continues parallel to itself. Hence, to a spectator at the sun, the apparent motion of the pole P (fig. 1) is the same as if the axis Pp of the earth had an annual conical motion PrQs, pnqm about an axis GOF perpendicular to the ecliptic EOC, the angle POG being equal to the sun's greatest declination. As these circles PrQs, pnqm are parallel to the ecliptic, their planes will pass through the sun, and, therefore, to a spectator at the sun, the apparent motion of the poles will be in the straight lines PQ, pq; and as P moves as fast in the circle PrQs as the sun does in the ecliptic, if P be the place of the pole at the equinox, and we take the arc Pv equal to the sun's distance from that equinox, and draw vo perpendicular to PQ, o will be the apparent place of the pole at that time. As the centres of the circles PrQs, pnqm lie in the plane of illumination, it is manifest that one half of each of those circles will be between that plane and the sun, and the other half on the opposite side of the plane; hence, to a spectator at the sun, the poles will appear and disappear by turns, the north pole being visible when the sun is in north declination, and the south pole, when his declination is south.

As this apparent motion of the pole over the enlightened disc of the earth is caused by the motion of the earth in its orbit, or by the revolution of the plane of illumination, while the earth's

axis retains its parallel position, the motion of the pole over the disc will be in a direction contrary to the diurnal motion of the disc; if, therefore, P be the position of the pole at the vernal equinox, and PrQ be its motion over the disc of the earth to the next equinox, the diurnal motion of the disc will be made in the contrary direction.

401. When the sun, and consequently the spectator, who is supposed to be at the sun, is in the equator, the spectator being in the plane of the equator, and, as to sense, in the plane of all the circles parallel to it, they will all appear to be projected upon the plane of illumination into right lines parallel to each other. But when the sun, and consequently the spectator, is out of the equator, the equator, and all the circles parallel to it, are seen obliquely, and as the eye may be considered as at an infinite distance, these circles will appear to be projected into ellipses on the plane of illumination; and the eye having the same relative situation to all these circles, the ellipses must be all similar. When the sun is on the north side of the equator, that part of the ellipse which is the projection of that part of the circle which lies between the north pole and equator on the enlightened hemisphere will be concave to the pole; but, when the sun is on the other side of the equator, that part will be convex. That is, if P (fig. 2) be the north pole on the enlightened hemisphere, the sun being on the north side of the equator, and vxyz, ambn the ellipses into which the equator and any parallel to it are projected; then amb is that part of the ellipse which the place on this parallel describes in the day, and the other part bna is that which is described in the night; and the place is at m at 12 at noon, and at n at midnight. In this case, the other pole p must be considered as being on the other, or dark side of the earth. But if P be supposed on the dark side, and consequently p on the enlightened side, or if the sun be on the south side of the equator, n will be the situation of the place at noon, and m at midnight. For, if Pp (fig. 3) be the axis, LN the plane upon which the circle ab is to be projected, E the sun on that side next to the north pole; then, drawing Eam, Enb, the point a answering to noon, the sun being on the meridian, is projected at m, and the point b answering to midnight, is projected at n; but, when the sun is on the other side of ab, as at e, a is projected to n', and b to m'; therefore n' represents noon, and m' midnight. On account of the great distance of the sun compared with the radius of the earth, the lines Ea, Eb, and ea, eb may be considered as parallel, and therefore the circle ab is orthographically projected upon the plane LN into an ellipse whose minor axis is mn or

m'n'.

402. The next thing to be done is to determine the ellipse into which the circle ab is projected, and its position on the plane of illu

4

mination. Let Pp (fig. 4) represent the axis of the earth, asbt a circle of latitude to any place, LPNp the meridian passing through the sun, and LON the plane upon which the projection is made; then (400) the angle LOP is equal to the sun's declination; draw am, bn, or perpendicular to LO, and (401) mn is the minor axis of the ellipse; let vs be that radius of the circle ab which is parallel to the plane of projection, and it will be projected into a line equal to itself, and consequently it will be the major axis; hence 2vs, or 2va, or 2 cos. lat. is the major axis of the ellipse; but ab : mn (the projection of ab upon LN): : radius: sin. mab or POL, the declination; that is, the axis major axis minor :: radius: sin. declination. And to find the distance Or from the centre of projection to the centre of the ellipse, we have, rad. =1 : cos. Or the dec. : : vO: Orvo x cos. dec. sin. lat. x cos. dec. But, as will be shown (405), the radius of the projection is the horizontal parallax of the moon diminished by the horizontal parallax of the sun; the radius, therefore, thus expressed being multiplied by the quantities whose values are expressed when radius is supposed to be unity, gives the value in terms of that radius; hence, if hor. par. D-hor. par. Oh, then h× cos. lat. = the semiaxis major of the ellipse; hx cos. lat. × sin. dec. = the semiaxis minor; and Or= h x sin. lat. x cos. dec. Hence we have the following construction for the apparent ellipse described by any place on the earth's surface to a spectator at the sun.

403. Let GCFE (fig. 5) be that half of the earth which is illuminated, EC the plane of the ecliptic, GOF perpendicular to it; take GQ=GV equal the sun's greatest declination, join QV, and on it describe the semicircle VKQ, and take Vh the sun's distance from the vernal equinox corresponding to the pole at V, and draw hP at right angles to VQ, and P (400) is the place of the pole, which we will suppose to be on the enlightened disc of the earth. Or make Vc the radius of a line of sines on the sector, and lay down cP equal to the sine of the sun's distance from the nearest solstice, from c toward the right hand, if the sun's longitude is more than 3 signs, and less than 9, but toward the left when his long. is less than 3 signs, or more than 9. Draw Pop, and upon OP take Or=hx sin. lat. x cos. dec; draw bra perpendicular to OP, and take ra=rb= hx cos. lat., and rm=rn=h× cos. lat. x sin. dec. And describe an ellipse ambn, and (402) it will represent the apparent diurnal path of the place to a spectator at the sun, for the given declination of the sun. If OP be made the radius of a line of sines on the sector, and the sine of the latitude laid from O to r, the centre of the ellipse will be determined; and making OC the radius take ra=rb= cos. lat.; and, making ra the radius, lay down rmrn sin. dec. If x and z be the points