263. To explain from whence the accuracy of this method arises, let us consider Venus and the sun as moving in the equator, and that observations of the total ingress are made at twe places in the terrestrial equator: let AB (Fig. 40) be the equator, S and V discs of the sun and Venus, perpendicular to, and as seen from the equator. To a spectator at A the internal contact (or the total ingress) commences, when to a spectator at B, the edge of Venus is distant from the sun by the angle VBS. The difference then between the times of total ingress, as seen from B and A, is the time of describing VBS by the approach of the sun and Venus to each other, Venus being retrograde and the sun direct. Hence from this difference of times, and the rate at which Venus and the sun approach each other, we find VBS. And the sine of VBS : sine of VSB : : Venus's distance from the sun: Venus's distance from the earth. The relation of Venus and the earth's distance from the sun, as found by the method in art. 97, may be used. Therefore the angle VSB,a the angle subtended by the two places A and B at the sun is known, and consequently the angle the semidiameter of the earth subtends, will be found in a manner similar to that in the note of art. 58.

264. This simplification of the problem may serve for an illustration, and to point out its superior accuracy. But the actual computation of the problem is very complex, principally on account of the inclination of Venus's orbit to the ecliptic, and on account of the situations of the places of observation at a distance from the equator. The accuracy of the method consists in this that the times of internal contact can be observed with great exactness, and thence the angle VBS computed, and therefore ASB.

a For extreme accuracy the distance of the places A and B is to be diminished by the arch of the equator, described in the interval of the ingresses at each place.

At inferior conjunction, the sun and Venus approach each other at the rate of about 240" in an hour, or 4" in a minute. Hence if the time of contact be erroneous at each place 4s of

time, the angle VSB may be erroneous

4 x 8
60

of a second, and therefore the limit of the error of ASB about of a second.a

265. This method then in fact comes to the same as to find the angle at the sun, subtended by two distant places on the earth's surface; but this angle can be determined much more accurately by the times of ingress, than by the micrometer. On account of the difference of the apparent magnitudes of Venus and Mercury, the internal contact of the former can be determined much more accurately than of the latter.

This method requires the difference of longitude of the places to be accurately known, in order to compare the actual times of contact. The longitude of the Cape of Good Hope being well ascertained, observations of the transit of Venus in 1761, made there, were compared with many made in Europe, and the mean result gave the parallax = 8,47 seconds.

266. But it seemed more convenient not to depend on the knowledge of the difference of longitudes of two places. It appeared better to compare the differences of duration at two places, at one of which the duration was lengthened and at the other shortened. If we assume the parallax of the sun, which we know nearly, we can compute the difference of duration at any place from what it would have been, had it been observed

a This comes to the same, as being able to observe a thread of light (the interval between the limbs of Venus and the sun, when the former has justeentered upon the body of the sun) of only of a second in breadth. Thus by the transit of Venus we can probably measure a smaller angle than by any other method.

at the earth's centre. Hence we can compare the difference of duration at two places, at one of which the duration is shortened and at the other lengthened. Thus we shall have a double effect of the parallax, and we can compare the computed result Iwith the difference observed. From the error we can correct the horizontal parallax assumed.

The transit of Venus in 1769 was observed at Wardhus in Lapland, and at the island of Otaheite in the South Sea.

Assuming the sun's parallax 8,83 seconds,

By computation the duration was

This shews the parallax is less than the parallax assumed, and to make the observed and computed difference of durations agree, the parallax must be taken 8",72. This last conclusion points out the accuracy of which the method is susceptible. difference of excess of duration of 17s makes only a difference of of a second in the parallax.

267. The observations of the transit of 1761 were not so well adapted for determining the sun's parallax as those of 1769. From the latter the parallax was ascertained with great exactness. The mean of the results seems to give 8",72 the sun's parallax at the mean distance, which probably is within of a second of the truth. The transit of 1769 occurring in the middle of summer, very many places of high northern latitude were well situate for observing it, but in all those the duration was affected in the same way.

a See Dr. Maskelyne's Method and Computation, page 398 of Professor Vince's Astr. vol. i.

zon.

The duration is most lengthened when the commencement

is near sunset, or when the sun is near the western horizon, and the end near sunrise or when the sun is near the eastern horiThe duration of the transit in June, 1769, was about six hours. That the commencement and end should take place under the circumstances above mentioned, it evidently required that the place of observation should have considerable north latitude. Wardhus near the North Cape is in 70° 22′ N. lat. The commencement was there at 9h 34m in the evening, and end at 15h 27m.

The duration would be most shortened when the commencement was near sunrise, and end near sunset, and the duration being only about six hours, this required that the days should be shorter than the nights, and therefore the place must be or the south side of the equator, and such that the commencement must be after sunrise and end before sunset. Consequently the choice of situations was much circumscribed.

Astronomers were therefore much at a loss for a proper place for observing this transit, when fortunately Otaheite was discovered. The situation of this island was as favourable as could be desired, and the British government, induced by a memorial from the Royal Society, ordered thither a ship with proper persons to make the observation. In consequence of which, the first of the celebrated voyages of Cook took place. The transit commenced at Otaheite about half past nine in the morning, and ended about half past three in the afternoon, and thus happened during the most favourable part of the day.

CHAPTER XV.

THE VELOCITY OF LIGHT, AND ABERRATION OF THE FIXED STARS AND PLANETS-THE EQUATION OF TIME-DIALS.

268. THE velocity of light is the greatest velocity that has yet been ascertained. Astronomy furnishes two methods of measuring it. Without the discoveries in astronomy, the velocity of light would have remained unknown. The eclipses of Jupiter's satellites, and the aberration of the fixed stars, shew us that the velocity of the reflected light of the sun, and the velocity of the direct light of the fixed stars, are equal.

269. The elder Cassini suspected from observations of the eclipses of Jupiter's first satellite, that light was not instantaneous, but progressive. Roemer first fully established this fact, by a great variety of observations of the eclipses of the satellites of Jupiter.

Let the mean motion of a satellite be computed from two eclipses separated by a long interval, Jupiter being at each at its mean distance from the earth. Then an eclipse, when Jupiter is approaching conjunction, and therefore farther from the earth, happens later than is computed by the mean motion so determined. When Jupiter is in opposition, it happens sooner than according to the mean motion so determined.

From a great variety of observations, it appears that the velocity of light is such, that, moving uniformly, it takes sixteen minutes to move over the diameter of the earth's orbit, or eight minutes in moving from the sun to us. This velocity is about