and eighty degrees, the remainders will be the angles OAM and OBM; for it is a known property in geometry, that a line standing upon another line, makes with it two angles, which, taken together, are equal to two right angles.

From the angles OAM and OBM, thus determined, take the angles OAB and OBA, which have been found by caclulation, and there will remain the angles MAB and MBA: so that in the triangle ABM, we shall have these two angles, and the side AB; and consequently the side мв may also be found as before. This is sufficient for our purpose; we have now, in the triangle OMB, the two sides MB and BO, and the included angle OBM, and therefore the side oм, or the distance of the moon from the centre of the earth, may be determined. This might, however, have been done in a shorter way, by first finding the horizontal parallax; but as that method depends upon a theorem in trigonometry, the demonstration of which does not admit of a familiar explanation, I have chosen to follow rather a more prolix manner, for the sake of greater perspicuity.

LETTER XVIII.

THE SAME SUBJECT CONTINUED.

THE distance of the sun from the earth might be determined in nearly the same manner as that of the moon, if his horizontal parallax was not so small as to be scarcely perceptible; for it is well known, that the angle osa, under which the semidiameter of the earth would appear to a spectator in the sun, can never exceed nine seconds, or the four hundredth part of a degree. (Pl. xI. fig. 4.) And as a mistake of one second, in so small an angle, will occasion an error of about seven millions of miles in the distance, it is easy to perceive what an extraordinary degree of skill it must require, to surmount the difficulties attending this delicate subject.

But the mind grows stronger by frequent exertions, and genius and industry conquer difficulties apparently insurmountable. The vast bulk of the earth has been accurately measured; and the stars of heaven, that are visible to the naked eye, have been all numbered; and the immense distance of the sun is now subjected to a rigorous calculation. By means of the transit of Venus over the sun's disc, which happened in the years 1761 and 1769, this problem was resolved with a degree of precision, unlooked for by the Astronomers of ancient times. The person to whom we are indebted for this excellent method, is Dr. Edmund Halley; a man, whose skill and penetra

tion in all mathematical and philosophical enquiries, entitles him to an eminent place in the classes of literature and science. A few extracts from the Dissertation which he presented to the Royal Society upon this subject, will show you the spirit of his method, and enable you to enter into the illustration of it with the greater facility.

"There are many things, he observes, that appear extremely paradoxical, and even quite incredible to the illiterate, which yet, by means of mathematical principles, are easily solved. Scarcely any thing will be thought more hard and difficult than that of determining the distance of the sun from the earth; but this, when we are made acquainted with some exact observations, taken at places fixed upon, and chosen beforehand, for that purpose, may, without much labour, be easily effected. And this is what I am now desirous to lay before this illustrious Society, that I may explain to young Astronomers, who may perhaps live to observe these things, the method by which the immense distance of the sun from the earth may be truly determined, to within, at least, a five hundredth part of what it really is.

"The distance of the sun from the earth is, by various Astronomers, supposed different, according to what was judged most probable, from the best conjectures they could form. Ptolemy, Copernicus and Tycho Brahe, imagined it to be about one thousand two hundred semi-diameters of the earth. Kepler thought it to be nearly three thousand five hundred; which distance is doubled

by Riccioli, whilst Hevelius only increases it by one half. But Venus and Mercury having, by the assistance of the telescope, been seen to pass over the sun's disc, deprived of their borrowed brightness, it is at length found, that the apparent diameters of the planets are much less than they were formerly supposed; and that the semi-diameter of Venus, as seen from the sun, subtends no more than the fourth part of a minute, or fifteen seconds, whilst the semi-diameter of Mercury is seen, at a mean, under an angle of only ten seconds.

"It has been also found, that the semi-diameter of Saturn, seen from the sun, appears under the same angle as that of Mercury; and that the semidiameter of Jupiter, the largest of all the planets, subtends an angle of no more than the third part of a minute. Whence some modern Astronomers, imagining that the semi-diameter of the earth, as seen from the sun, would subtend a mean angle, between the larger one of Jupiter, and the smaller one of Saturn and Mercury, have concluded, that the sun's parallax is about fifteen seconds, or equal to that of Venus, and that his distance from the earth is about fourteen thousand of the earth's semi-diameters.

"But this is an inference, the truth of which may be fairly questioned; for as the moon's diameter is a little more than one-fourth of the diameter of the earth, if the sun's parallax should be supposed fifteen seconds, it would follow that the body of the moon is larger than that of Mercury; that is, that a secondary planet would be greater

than a primary; which should seem to be inconsistent with the uniformity of the mundane system.(0) And, on the contrary, the same regularity and uniformity seems scarcely to admit, that Venus, an inferior planet, that has no satellite, should be greater than the earth, which stands higher in the system, and has such a splendid attendant.

"Let us, therefore, observe a mean, and suppose that the semi-diameter of the earth, as seen from the sun, or, which is the same thing, the sun's horizontal parallax, is twelve seconds and a half; then, according to this supposition, the moon will be less than Mercury, and the earth larger than Venus; and the sun's distance from the earth will be found to be about sixteen thousand five hundred of the earth's semi-diameters. This distance I assent to, at present, as the true one, till it shall become certain what it is, by the experiment I am about to propose.

"Nor am I induced to alter my opinion by the authority of those, however weighty it may be, who are for placing the sun at an immense distance beyond the bounds here assigned, as observations made upon the vibrations of a pendulum, in order to determine those exceeding small angles, are not

(0) Though Dr. Halley was perfectly right in his conjecture of Mercury being greater than the moon; yet latter discoveries have shown, that it is not inconsistent with the mundane system, for a primary to be less than a secondary; as this is the case with the new planets Vesta, Juno, Pallas and Ceres.