about the Sun (s) are accurately laid down,-the line ST representing a fixed line from which astronomers measure the motions of the planets around the Sun.* Now, it is seen at once that when near M, Mars is much nearer to the Earth's path than the Sun is; so that when the Earth and Mars are in conjunction in this neighbourhood, it becomes an easier problem to determine the distance of Mars than that of the Sun.

To put the matter simply, the Earth at E looks larger as seen from м than as seen from s; and to say she looks larger is the same as saying that she subtends a greater visual angle; and the visual angle she subtends from Mars precisely measures the displacement which Mars will show as seen from different parts of the Earth.

Supposing Mars thus favourably situated, and that two observers, one at E and the other at E' (fig. 9),

*This line is only introduced to explain the unsymmetrical aspect of the two paths; to show, in fact, that these are intentionally eccentric.

observe the planet whose centre really lies at M, to lie on different points, m and m' of the celestial sphere. Then this arc m m', if measured accurately, would at once give the actual displacement of Mars corresponding to the distance E E' between the observers; for though in the figure m m' is closer to M than to E, yet in reality the celestial sphere on which we thus estimate the place of Mars may be regarded as infinitely far off, so that M E is as a mere point at the centre of this sphere; and therefore the arc m m', as estimated from the Earth, is precisely the same as though it were estimated from M,-or, in other words, this arc measures the angle m M m' and therefore the equal angle E ME. Thus we learn at once

FIG. 9.

S

M

m

the angle E ME', and as we know the base-line E E′, we deduce the distance E M by a very simple process of calculation.

But what is essential to the accuracy of the result is that the arc m m' should be accurately measured. Whatever error we make in this measurement will produce a proportionate error in our final estimate of the distance of Mars.

Now, if there were no stars in the background of the heavens it would be absolutely impossible to measure mm' as accurately as our purpose requires. The problem would be quite as hopeless as the attempt to measure the Sun's distance by a similar process.

But the presence of stars upon the celestial vault, and the certainty which the astronomer possesses that these stars are at a distance incomparably exceeding that of Mars, make the measurement of this arc m m' feasible. The stars serve as index-points. In fig. 9 for example, a star is supposed to be placed at 8; now while it would be hopeless for an astronomer to attempt to determine the direction of either line E M or E'M' in space, without reference to any star, it is quite easy to measure the arcs m s and m's with a very considerable degree of accuracy, and so to determine the difference

m m'.*

And here one point in which the modern possesses an enormous advantage over the ancient astronomer, lies in the fact that spaces on the heavens which are blank, so far as naked eye vision is concerned, are shown by the powers of the telescope to be occupied by multitudes of minute stars, and the minutest star serves quite as well as a large star for such observations as we are here considering. So that the astronomer need be under no anxiety lest Mars, during the period when he is nearest to us, should approach no

* Here I have supposed s, m, and m' to lie all on the same arc, which of course would not ordinarily be the case. It is easily seen, however, that it falls quite within the scope and bearing of ordinary astronomical observation, to measure not only the distances but the bearings of m and m' from s, and so,-two sides and an included angle of the triangle 8 m m' being determined,—to determine the third side m m'. I may notice here in passing that quite a large proportion of the details involved in the various processes applied to the problem considered in this chapter are necessarily left untouched, or are barely mentioned. A volume much larger than the present would be required to exhibit these details in full and in all their bearings.

star near enough to render the required measurements effective.

So soon as the distance of Mars has been calculated the distance of the Sun can be determined by the application of Kepler's third law. There is a preliminary process depending on the circumstance that SE and SM are not the mean distances of the Earth and Mars; but this process is perfectly simple, since observation has shown what is the true figure of each Thus Kepler's third law by showing us the exact relation between the mean distances, shows us the exact relation between s M and SE; and therefore between EM and S E.

orbit.

The plan here described was the first from which astronomers obtained any satisfactory estimate of the Sun's enormous distance. Kepler, after a careful study of Tycho Brahe's observations of Mars, had already confidently stated that the Sun's parallax is not greater than 1' (or in other words that the Sun's distance is not less than 13 millions of miles). But Tycho Brahe's observations were such as we should now call altogether rough. Cassini proposed and carried out a much more exact series of observations. At his suggestion the Paris Academy of Sciences sent Richer to Cayenne, while Cassini himself, Römer, and Picard, observed Mars at different French stations. The parallax of Mars was indeed not measured, for the instrumental means of the observers were insufficient. But Cassini calculated that if the parallax had exceeded 25" the means employed ought to have

exhibited its effects. A parallax of 25" in the case of Mars (situated as when Cassini's observations were made) corresponds to a solar parallax of 10". Cassini expressed his conviction that the solar parallax is not greater than 9" 5-in other words, that the Sun's distance is not less than 85,500,000 miles.

The next application of this method involved the comparison of observations made by Lacaille at the Cape of Good Hope and by several astronomers at different European stations. The parallax deduced was 10', corresponding to a distance of 82,000,000 miles.

I shall presently have occasion to mention other and much more trustworthy results obtained by this method in recent times. For the present, however, I pass on to other methods.

*

The path of Venus lies even nearer to the earth's orbit than that of Mars does. Fig. 10 represents the

* It might seem that as Mars comes into opposition at intervals averaging about 780 days, the method could be applied frequently, and so the results due to it could be rapidly improved upon. As a matter of fact, however, only those observations made when Mars is in opposition near perihelion are of service. From fig. 8, it will be seen that favourable opportunities do not occur at short intervals. The figure shows the successive conjunction-lines of the Earth and Mars between the years 1856 and 1871. It is seen that only the conjunctions of 1860 and 1862 are favourable, and those not so near as they should be to perihelion. (The wide distances separating conjunction-lines in this neighbourhood as compared with the opposite, are due to the relatively rapid motion of Mars near perihelion.) The opposition of 1877 will be exceptionally favourable, as the conjunction-line will fall nearly midway between those of 1860 and 1862. It is necessary for me to remark that fig. 8 is copied from a drawing of my own, illustrating a paper in the Popular Science Review for January 1867. Mr. Lockyer has copied