ARTS AND aCI. my. VOL. vi.
sun might with modern instruments be measured in the same way as that of the moon or planets above described, but not so well, as a longer time must elapse between the passage of the sun and that of a star nearly in the same parallel. Ptolemy says that Hipparchus computed the moon's parallax from the phenomena of solar eclipses ; that is, lie deduced the value of the moon's parallax from the phenomena of solar eclipses on two suppositions of the sun's parallax : namely, that it was = 0 ; and, again, that it was a definite small quantity. As the circum stances of a solar eclipse vary from the effects of parallax, it is clear that in this way Hipparchus would get something like equations of condition involving the parallaxes of the sun and of the moon, which could be solved as soon as, by the problem of Aristarchus or by any other method, he could determine the relation between these two quantities.
We have thus shown that an approximate knowledge of the distances of the moon, sun, and planets, iu of the magnitude of the earth, requires nothing more than observation and the solution of a triangle one aide of which and the two including angles are known. The magni tudes of these bodies can be immediately calculated from their apparent diameters and true distance ; so that up to this point there is no room for scepticism, if it be granted that the angles of a triangle equal two right angles.
There is a method of ascertaining the parallax by one observer. Let Mars in opposition be the object, and compare it in right ascension with a neighbouring star in the same parallel on the meridian, and also several hours before and again after his transit. The parallax, being wholly in a vertical circle, will not affect the right ascension in the meridian : hence the meridian comparison will give the true difference of right ascension between the planet and star. The other observations (after correcting the place of Mars by his hourly motion, which is known either from the tables or from observations on preceding and succeeding days) present right ascensions of the planet affected by parallax in different ways, and from these effects it is easy to compute the actual value of the horizontal parallax, and consequently the distance of the planet in terms of the earth's radius at the place. This method has recently been proposed by the Astronomer-Royal, as well adapted for determining the parallax of the star from observations made during the oppositions of 1862 and 1864.
Kepler's discoveries *—that the planets move in ellipses round the sun in the focus ; that the area swept by each radius vector in a given time is a constant quantity for the same planet ; and lastly, that the squares of the periodic times are as the cubes of the mean distance— have supplied means for a much more accurate determination of the sun's parallax. Assuming these laws, the forms of the orbits of the
earth and planets, and their relative distances, can be determined from observation : hence, if the parallax of any one planet can be found, the parallaxes of the sun and of all the other planets can be computed. Observations of Mars, for instance, at his opposition, made at the Cape of Good Hope and at Greenwich, will afford a very tolerable value of his parallax, and hence of his distance. Again, as the proportion between the distances of Mars and the earth from the sun at any time is known from the form of their orbits and their periodic times, and the angle between the sun and Mars at the Barth can be observed, the triangle between the sun, earth, and Mars can be completely solved, and hence the distance of the sun and his parallax be computed. These observations can be repeated at every opposition of Mars; and if Mars be compared by the micrometer with stars near the same parallel, there is scarcely a limit to the possible accuracy of the observations.
The observation however by which the parallax of the sun is determined with the greatest certainty, is that of the passage of Venus over the sun's disc, commonly called the Crank of Venus. [TRANSIT OF VENUS.] In the figure let ss be the sun, re the earth, and v and v' two positions of Venus, which is supposed to be moving in the direction v v'. To make the figure simple, we suppose the earth to be at rest, and that v v' represents the excess of the angular motion of Venus above that of the earth. A spectator at a will see the commencement of the transit when Venus is at v, but a spectator at e will only begin to see it when Venus is at v'. The time which must elapse between thew two moments is equal to the time in which Venus peases from v to v', or in which she deecribes the angle ESe round the sue with her rel./fire angular motion. But the angle ESe is twice the parallax ; hence, since the time elapsed betweeu the ingress at the two placer is known from observation and from the difference of longitudes, and the hourly angular motions of Venus and the earth round the sun are known from the tables, and consequently the difference of these motious, the horizontal parallax of the sun can be found. It is evident that as the ingress is accelerated at e and retarded at e, so the egress will be accelerated at a and retarded at e, Venus then moraine the positions v". v*".