north, the other south of the moon, at some distance apart; Mr. Ferguson says, "at such a distance from each other, that the arch of the celestial meridian included between their two zeniths, may be at least 80 or 90 degrees."

Let ABD be a meridian of the earth; (Plate VII, Fig. 12.) C, its centre; A, a place in north latitude; B, a place in south latitude.Z, the zenith at A, z, the zenith at B, M the moon. Let an observer at each of these stations with a good instrument, take the exact zenith distance of the moon's centre, when she passes the meridian.From the sum of these two zenith distances subtract the sum of the two latitudes, the remainder is the sum of the two parallaxes. In triangle A B C the sides A C and B C are known, being each equal to a semi-diameter of the earth; the angle A C B is the sum of the two latitudes; the angle CA Bis equal to the angle C BA, (Euclid, B. I, Prop V); therefore subtract the angle A C B from 180°, and half the remainder is the angle B 4 C, or the angle ABC. All the angles therefore, and two sides being given, the side A B may be easily found.

The angle CA M is the supplement of ZA M, and the angle C B M is the supplement of 2 B M.

Subtract the angle C A B from the angle C AM, the remainder is the angle B A M; and subtract the angle C B A from the angle C BM, the remainder is the angle A B M. The sum of these angles taken from 180° leaves the angle AM B. In the triangle A M B all the angles and the side AB may be considered as given to find the side A M, or BM Suppose the side A M found. Then in the triangle A CM, the sides A Cand AM and the angle between them, CAM, being known, the side CM, the distance of the moon from the earth, may be easily found by oblique trigonometry.

SECTION III.

PARALLAX OF THE SUN.

Aristarchus, an astronomer of Samos, who flourished about the middle of the third century before Christ, proposed to find the sun's parallax, by observing the instant the moon is dichotomized, or wheu exactly one half of her disk appears illuminated. This is a little before her first, and a little after her last quarter. The moon, as seen from the sun, is then at her greatest elongation, and the angle at the moon is a right angle. The angle, which the distance of the moon from the sun subtends at the earth, is taken by observation. If then the distance of the moon from the earth be known, the parallax is easily ascertained, and the distance of the sun from the earth may be found by a common problem in rectangular trigonometry. But it is impossible to be very accurate in determining the time, when the moon is dichotomized; and a small error in ascertaining this time will make so great a difference in the sun's parallax, that dependence cannot be placed on this method.

Hipparchus proposed, by observing the exact time the moon is in passing the earth's shadow in a lunar eclipse, to obtain a triangle for finding the sun's parallax. But this method, like the former, is subject to great and unavoidable errors. Indeed all attempts to ascertain the parallax of the sun, prior to the seventeenth century, can scarcely be called approximations to the truth. The method then suggested, will be the subject of the following article."

THE TRANSIT OF VENUS.

No improvement in modern astronomy can be compared to that of determining the magnitude and distance of the planets by the transit of Venus. The manner in which this may be effected was first suggested by Dr. Halley. When, in the

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earlier part of his life, this great astronomer was at the Island of St. Helena, for the purpose of viewing the stars around the south pole, he had an accurate observation of Mercury passing over the disk of the sun. He immediately formed an idea, that such transits might be used for find

ing the sun's parallax.

But Mercury is too near the sun to be conveniently used for the intended purpose. It is necessary, therefore, to have recourse to Venus.

The transit of Venus happens but seldom. Horrox, a young English astronomer and his friend, Mr. Crabtree, as far as we know, were the first who had a view of the singular and pleasing phenomenon, Venus passing over the sun's disk. This was on the 24th of November, O. S. 1639. But their observations were imperfect, the sun going down, in England, during the transit.

The next transit was on the 6th of June, 1761. Doctor Halley, in a paper communicated to the royal society, in the year 1691, gave particular directions for observing this, and the following transit, in 1769, though he knew they must, happen some time after his death.

The exact periodical times and relative distances of the planets; the heliocentrick or angular motion of the earth and Venus in their orbits, and, of course, the excess of Venus's angular motion over that of the earth; the latitude of Venus; the direction and extent of her path over the sun's disk; and the duration of the transit, as viewed from the centre of the earth, were deduced from observation on her motion, or were calculated before the transit of 1761; and may be considered as data in any transit.

If the distance of the earth from the sun be assumed at 100,000, the distances of the other principal planets would

be,

When the true distance of the earth from the sun is known, the true distance of the other planets may be easily found; for the great la of Kepler applies, and as the square of the periodical time of the earth is to the cube of its distance; so is the square of the periodical time of another primary planet to the cube of its distance. More con cisely; as the relative distance of the earth from the sun is to the true distance, so is the relative distance of any other primary planet from the sun to its true distance.

The apparent motion of a planet in a transit is always retrograde.

The transit of Venus may be taken by one observer. The method given in Enfield's philosophy, seems one of the best for this purpose. The principle is; (Plate VIII. Fig 3.) let & be the sun, VA BLM N, a part of Venus's orbit, C D, a part of the earth's orbit; let the observer's station on the earth at C be such, that the sun may be on the meridian about the middle of the transit, as calculated for the earth's centre. This would be at a, had the earth no diurnal rotation. But, on account of such rotation, let it be at b, where Venus at Vis seen at the commencement of the tran sit at I just entering on the sun's disk. Tosuch an observer, unaffected by diurnal rotation, the egress of Venus would appear at K, when she has passed to B. But as the observer is carried, during the transit, by such rotation eastward; suppose to c. Venus would seem to leave the sun when she arrives at A, making the apparent transit shorter than that by calculation in the proportion of VA to V B. Hence the difference between the observed and computed transit, is the time, in which Venus, by the excess of her heliocentrick motion, would pass from A to B in her orbit, measuring the angle A K B or c Kb.

The difference of time must be reduced to Venus's heliocentrick motion from the earth. This may be done by proportion; for as one hour is to Venus's heliocentrick horary motion from the earth; so is the difference between the observed

and computed duration of the transit, to her motion from the earth during that time, viz. the arch, A B.

By knowing the latitude of the place of observation, and the duration of the transit, the length of the line b c may be easily obtained, and may be compared with the semi-diame ter of the earth. By this comparison, with proper allowances for the direction of the observer's motion and that of Venus, the angle, which a semi-diameter of the earth subtends at the sun, equal to the sun's horizontal parallax, may be obtained.

But, lest the computed duration of the transit should not be perfectly correct, an observer ought to be stationed at or near the meridian opposite the former, and so near the enlightened pole, that the beginning of the transit may be observed before sunset, and the end after sunrise. Such an observer, as seen from the sun or Venus, would appear to move, during the transit, in a direction contrary to the former observer. Let D be the place of the earth in its orbit, (Pl. VIII, fig. 3,) and d the place of the observer at the commencement of the transit, when Venus, at L, appears first to touch the sun at I. If the observer were stationary at d, the transit would end, when Venus arrives at M. But during the transit the observer must be carried, by the diurnal motion of the earth, some distance, as to e; Venus must, therefore, pass in her orbit to N, before her apparent egress from the sun's disk, making the observed transit longer than that by computation. From the difference between the duration of the observed and computed transit, the parallax is obtained as before. If there be an error in the computed duration of the transit, the result of the two operations will be unequal. The error, that increases the one, must diminish the other, so that the mean between the two results may be taken for the exact transit. The mean between many results has been taken, and from them the parallax ascertained to a degree of accuracy, equal, it may seem, to the most sanguine expectation of the scientific Halley. 1