that more observations were made than were actually necessary for the solution of the problem. One pair of observations, if absolutely exact, would have sufficed to determine the Sun's distance; but as errors must to a greater or less extent affect all observations, it was necessary that many sets should be made in order that the mean result should be taken. Now, in all such cases the amount of reliance which is to be placed on the final or mean result depends on the closeness with which the several results aggregate round the mean. Precisely as we should place little reliance on the deduced mean of a series of ordinary measurements, which differed considerably inter se, so astronomers would not be prepared to accept with confidence a value of the Sun's distance which was the mean of several discordant results.*

Now, the observations made in 1769 were in this respect much more satisfactory than those made in 1761; but, owing to the peculiarity I have mentioned, they did not accord so well by any means as might have been anticipated.

It is not too much to say that in the attempts first made to determine the Sun's distance by means of the transit observations of 1769, no settled principle was adopted. It was not much to be wondered at, therefore, that the estimates of the solar parallax varied

*The total number of observers in 1761 was no less than 63-thus distributed: 13 in North Europe, 8 in England, 15 in France, 6 in Spain, Portugal, and Italy, 16 in Germany, and 3 in other places. In 1769 there were observers at 50 stations in Europe, 6 in Asia, 17 in America, and at one station in Polynesia.

between somewhat wide limits; so that whereas some mathematicians made the solar parallax as great as 9"-2, others obtained a value of only 7" 5. The corresponding values of the Sun's distance are respectively 87,890,780 and 108,984,560 miles,-a range compared with which recent discrepancies seem insignificant.*

During the years 1822 and 1824 Encke re-examined the whole series of observations made on the transits of 1761 and 1769. With wonderful patience, especially when we consider the nature of his materials, he combined together the results of no less than 149 observations made during the former transit, and as many made (at 75 stations) during the second. From the transit of 1761 he deduced for the Sun's parallax the value 8"-49, and from the transit of 1769 he deduced the value 8"-60; † while, by combining the two, he arrived at 8"-5776, the value which was employed during so many years in the Nautical Almanac' and

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* The results of the transit of 1761 are thus summed up in Dr. Bruhn's Life of Encke (Johann Franz Encke, sein Leben und Wirken):— Short obtained a parallax of between 8"-47 and 8"-52; Pingré, 10′′; Rumowski, 8"35; Planmann, 8"-2; Audefredy, 9"-2. From the transit of 1769, adds Bruhn, Wm. Smith deduced a parallax of 7"-5; Hornby, 8"-78; Pingré, 9"-2, 8"-88, and 8"-43; Lalande, 8"-8; Lexell, by Euler's method, between 8"-65 and 8"-86, whence he finally adopted 8"-8. The value of the Sun's distance corresponding to all these results can be at once deduced from Table II., at the end of this work; save only the distance corresponding to 7"-5 (in Bruhn's work it is written 7'-5) which is stated above.

The actual results deduced by Encke were-from the transit of 1761, 8"-490525+0.060712"; from that of 1769, 8"-6030+0′′·0460; from the two combined, 8"-5776±0′′·0370—3·01208p, where dp is the correction for the Sun's semi-diameter (estimated by Encke at 958"-424). Later, Encke deduced the value 8"-5716, by introducing some corrections in his work.

other like works. The corresponding distance, viz., 95,274,000 miles, held its ground during all those years in popular treatises on astronomy.

It is somewhat surprising, considering the evidence which was afforded by the discrepancies between the observations made in 1761 and 1769, that this result should have been regarded with such confidence, since it needed but a brief examination of the basis on which Encke's result was founded to see that no faith whatever could be placed in three at least out of the five numerals in the expression 8" 5776. Delambre regarded 8"-6, very justly, as the most probable value of the solar parallax half a century ago. It was rightly admitted that the observations of Venus in transit afforded the most reliable results; but Delambre, Bessel, and other astronomers of eminence were far from adopting the value 8"-5776 with that implicit confidence which caused the corrections made in recent times to attract so much notice, and so greatly to surprise the general public.

The observation of Venus in transit being admittedly the most trustworthy method of determining the Sun's distance, it might have been supposed that no new results could serve to throw doubt on those deduced by Encke, until the time should come when Venus would again cross the Sun's face-that is, until the year 1874. But the rapid progress of science during the past half-century, though it has not served to alter the relative value of different methods for determining the Sun's distance, has enabled astronomers to

apply some of the less powerful methods in so much more effective a manner than of old, as to obtain more trustworthy results than had followed from the best method of all when less skilfully applied.

Amongst these relatively inferior methods, no less than four are novel; but a very brief description must suffice for each.

It had been noticed by Laplace, towards the close of the last century, that among the perturbations of the Moon there is one which depends on the Sun's distance. Suppose M, M, M, M, to represent the Moon's path

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3

around the Earth, E. Then clearly as the Moon moves from M, through M, to M,, she is disturbed by the Sun's action, which is here greater on the Moon than on the more distant Earth. On the other hand, while the Moon is moving from M, through M, to M,, she is disturbed, because the Sun's action is greater on the Earth than on her. Without entering into an exact investigation of the effects thus produced, it is abundantly evident that the Sun's perturbing effect in the former case will be greater than in the latter, because the radius of the orbit M, M, M, M, clearly bears a greater proportion to M, S than to M, S. The Moon's orbit is

E

indeed so minute compared with the Sun's distance, that the difference is very slight; but still there is a difference. When the Moon is at M, the Sun tends to pull her more strongly away from the Earth than he tends to pull the Earth away from the Moon when the latter is at M,; and a similar preponderance holds for other and corresponding positions of the Moon and Earth. It follows that there is a slight variation in the Moon's motion depending on this cause alone, and readily admitting of being estimated theoretically, while the continued observations made by astronomers on the Moon's motions suffice to show how great the perturbation really is. It is only necessary to compare the theoretical with the observed value to deduce the Sun's distance; only, of course, the accuracy of the result will depend on the number and accuracy of the observations. Laplace, with the best observations available in his time, deduced for the Sun's parallax the value 8′′-6, corresponding closely with the value subsequently deduced by Encke from the transits of Venus. It is remarkable,' wrote Laplace, that an astronomer, without leaving his observatory, by merely comparing his observations with analysis, has thus been enabled to determine the distance of the Earth from the Sun-an element the knowledge of which has been the fruit of long and troublesome voyages in both hemispheres.'

But this method is clearly one which modern astronomers can apply much more effectively, because the observations at their disposal are so much more