when the observer at E' recorded the Sun's position. Then if the supposition were correct, the above process would be available. But if E were ten or twelve miles to the east of the supposed longitude, apparent noon would occur a minute or so* earlier than at a place in that longitude. But, in one minute, Venus, as seen in transit, moves over an arc of about two seconds on the Sun's face, so that the observer at E noting her place a minute or so too soon (so far as the comparison with the other observer's record is concerned) would set her two seconds of arc out of place. But our problem is one in which seconds of arc are all-important.

But this is not all. The determination of the exact place of Venus on the Sun's disc at any epoch would be a matter of extreme difficulty. It would be necessary to determine, not merely her distance from the Sun's centre, but her bearing from that point, and a very slight error in either determination would (in so delicate an inquiry) cause a considerable error in the determination of the Sun's distance. There is, indeed, a way of getting over this difficulty which I touch upon in appendix A; but though it gives, in my opinion, the very best method of determining the Sun's distance now available to us, it requires (as will be seen) a preliminary knowledge which was not possessed when the observation of Venus in transit was first proposed as a means for solving the problem we are upon.

Accordingly Halley proposed (in 1716) a plan for

* The exact difference of time would of course depend on the latitude of the station.

evading the observational difficulties. He suggested, that instead of attempting to estimate the position of Venus on the Sun's disc at any moment, the observers at two stations such as E and E', should time the passage of Venus along her chord of transit. Neglecting for a moment the consideration of the Earth's rotation, Venus would seem to the observer at E to describe such a path as Iv m, while to the observer at E' she would seem to describe such a path as l'v'm'. Now if we know the length of time she takes in describing these chords, we know the length of the chords, since the rate of Venus's motion across the Sun's disc (the same of course for both stations on the assumption that the Earth is not rotating) is known from the tables independently of her actual distance. Hence it is a very simple problem in geometry to determine the distance separating the chords 7 m and I'm', and thence as in the former case to determine the distance of Venus, and so that of the Sun.

Nor does the fact that the Earth is rotating prevent us from applying this method; though it causes the problem to be somewhat more complicated. Venus in fact does not describe quite a straight chord across the sun as seen from any station; nor does she move quite uniformly; nor again is her rate of motion across the Sun's face exactly the same as seen from different stations. But all these points are such as the astronomer is quite accustomed to take into account, nor do they in themselves detract one whit from the certainty with which Halley's method can be applied.

But there is one effect of the Earth's rotation which has to be very carefully considered in weighing the value of Halley's method. It is absolutely necessary (since the duration of the transit is to be timed) that at each station the beginning and end of the transit should be visible. Now a transit may last a considerable timeas long indeed as eight hours; and it may not always be easy to find two stations-one far to the north, and the other far to the south, at each of which both the beginning and end of the transit will be favourably seen. For it must be remembered that a large part of the Earth is unfitted for the observer's purposes. We must not place our observers on the open sea, nor in regions where bad weather ordinarily prevails. And this question of the weather is in itself a great difficulty. For transits of Venus can only occur in December or in June, as is obvious from a consideration of fig. 10, where E' and e-the points near which the Earth must be that Venus in conjunction may be near a node-correspond to the Earth's position on about December 8 and June 6. Now, at a northern station in June, or at a southern station in December, fair weather may be commonly expected, but the reverse holds as respects the northern station in a December transit, and the southern station in a June transit. So that the difficulty of finding two stations, one northern the other southern, both well suited for observing both the beginning and end of the transits, and at both of which there is a fair prospect of clear weather at both epochs, is a very serious one.

D

In Appendix A we shall see more about the circumstances here considered, which it will be understood are not such, ordinarily, as to prevent Halley's method from being applied, though they call for the most cautious exercise of judgment in the selection of stations for the purpose. At present it suffices to say that the difficulty led Delisle, in anticipation of the transit of Venus in 1761, to propose another method.

Reverting to fig. 11, it will be clear that as v moves onward in the direction of the arrow* the time must come when the transit is just beginning at some point

FIG. 12.

S

on the Earth's surface, from whence the first view (as it were) is obtained of Venus in transit. Some interval must elapse before the transit has begun for the whole Earth-or at least for all that hemisphere which is turned towards the Sun. And at some point on the Earth's surface-which will clearly be nearly opposite the point just referred to the transit will seem to begin later than at any other station. Now, neglecting matters of detail, and considering Venus as a point for the moment, we may reason in this way on observations of the kind considered:

* We suppose the Earth at rest, or which is the same thing-consider only Venus's motion relatively to a line constantly joining the centres the Sun and Earth.

Suppose Venus at v'(fig. 12) when the transit first begins, so that a line E v s just touches both the Earth and Sun, and that Venus is at v' when the transit has begun for the whole Earth, so that E' V's just touches both the Earth and Sun. Then we know the length of E E', and therefore we know the length of v v' which is less than E E' in the proportion before used, of about five to seven. We also know exactly how long Venus has taken to traverse this arc v v', and therefore, since we know how long she takes to complete the circuit of her orbit, we know what proportion the known length v v bears to the circumference of her orbit. This gives the circumference, and thence the radius of her orbit, whence, as before, we learn the radius of the Earth's orbit.

Here we have supposed the Earth at rest. But as the motions both of the Earth and Venus are known, the relative motion of Venus is known, and so the conditions of the problem are as fully ascertained as in the simpler case actually dealt with.

We thus see that the phenomena presented at the commencement of the transit are sufficient for determining the Sun's distance. So also are the phenomena presented at the end of the transit; since it is obvious that as Venus passes from v to v similar relations will be presented, but in a contrary order.

All that is required, then, for a successful application of Delisle's method is that one observer should have a favourable view of the commencement (or end) of the transit from some place on the Earth where the transit