this is the only general method of which so much can be said. By some of the others an astronomer can, indeed, estimate the sun’s distance without leaving his observatory—at least, theoretically he can do so. But many years of observation would be required before he would have materials for achieving this result. On the other hand, one good pair of observations of Mars, in the evening and in the morning, from a station near the equator, would give a very fair measure of the sun’s distance. The reason why the station should be near the equator will be manifest, if we consider that at the poles there would be no displacement due to rotation; at the equator the observer would be carried round a circle some twenty-five thousand miles in circumference; and the nearer his place to the equator the larger the circle in which he would be carried, and (cæteris paribus) the greater the evening and morning displacement of the planet.
Both these methods have been successfully applied to the problem of determining the sun’s distance, and both have recently been applied afresh under circumstances affording exceptionally good prospects of success, though as yet the results are not known.
It is, however, when we leave the direct surveying method to which both the observations of Venus in transit and Mars in opposition belong (in all their varieties), that the most remarkable, and, one may say, unexpected methods of determining the sun’s distance present themselves. Were not my subject a wide one, I would willingly descant at length on the marvellous ingenuity with which astronomers have availed themselves of every point of vantage whence they might measure the solar system. But, as matters actually stand, I must be content to sketch these other methods very roughly, only indicating their characteristic features.
One of them is in some sense related to the method by actual survey, only it takes advantage, not of the earth’s dimensions, but of the dimensions of her orbit round the common centre of gravity of herself and the moon. This orbit has a diameter of about six thousand miles; and as the earth travels round it, speeding swiftly onwards all the time in her path round the sun, the effect is the same as though the sun, in his apparent circuit round the earth, were constantly circling once in a lunar month around a small subordinate orbit of precisely the same size and shape as that small orbit in which the earth circuits round the moon’s centre of gravity. He appears then sometimes displaced about 3000 miles on one side, sometimes about 3000 miles on the other side of the place which he would have if our earth were not thus perturbed by the moon. But astronomers can note each day where he is, and thus learn by how much he seems displaced from his mean position. Knowing that his greatest displacement corresponds to so many miles exactly, and noting what it seems to be, they learn, in fact, how large a span of so many miles (about 3000) looks at the sun’s distance. Thus they learn the sun’s distance precisely as a rifleman learns the distance of a line of soldiers when he has ascertained their apparent size—for only at a certain distance can an object of known size have a certain apparent size.
The moon comes in, in another way, to determine the sun’s distance for us. We know how far away she is from the earth, and how much, therefore, she approaches the sun when new, and recedes from him when full. Calling this distance, roughly, a 390th part of the sun’s, her distance from him when new, her mean distance, and her distance from him when full, are as the numbers 389, 390, 391. Now, these numbers do not quite form a continued proportion, though they do so very nearly (for 389 is to 390 as 390 to 391-1/400). If they were in exact proportion, the sun’s disturbing influence on the moon when she is at her nearest would be exactly equal to his disturbing influence on the moon when at her furthest from him—or generally, the moon would be exactly as much disturbed (on the average) in that half of her path which lies nearer to the sun as in that half which lies further from him. As matters are, there is a slight difference. Astronomers can measure this difference; and measuring it, they can ascertain what the actual numbers are for which I have roughly given the numbers 389, 390, and 391; in other words, they can ascertain in what degree the sun’s distance exceeds the moon’s. This is equivalent to determining the sun’s distance, since the moon’s is already known.
Another way of measuring the sun’s distance has been “favoured” by Jupiter and his family of satellites. Few would have thought, when Römer first explained the delay which occurs in the eclipse of these moons while Jupiter is further from us than his mean distance, that that explanation would lead up to a determination of the sun’s distance. But so it happened. Römer showed that the delay is not in the recurrence of the eclipses, but in the arrival of the news of these events. From the observed time required by light to traverse the extra distance when Jupiter is nearly at his furthest from us, the time in which light crosses the distance separating us from the sun is deduced; whence, if that distance has been rightly determined, the velocity of light can be inferred. If this velocity is directly measured in any way, and found not to be what had been deduced from the adopted measure of the sun’s distance, the inference is that the sun’s distance has been incorrectly determined. Or, to put the matter in another way, we know exactly how many minutes and seconds light takes in travelling to us from the sun; if, therefore, we can find out how fast light travels we know how far away the sun is.
But who could hope to measure a velocity approaching 200,000 miles in a second? At a first view the task seems hopeless. Wheatstone, however, showed how it might be accomplished, measuring by his method the yet greater velocity of freely conducted electricity. Foucault and Fizeau measured the velocity of light; and recently Cornu has made more exact measurements. Knowing, then, how many miles light travels in a second, and in how many seconds it comes to us from the sun, we know the sun’s distance.
The first of the methods which I here describe as new methods must next be considered. It is one which Leverrier regards as the method of the future. In fact, so highly does he esteem it, that, on its account, he may almost be said to have refused to sanction in any way the French expeditions for observing the transit of Venus in 1874.
The members of the sun’s family perturb each other’s motions in a degree corresponding with their relative mass, compared with each other and with the sun. Now, it can be shown (the proof would be unsuitable to these pages,10 but I have given it in my treatise on “The Sun”) that no change in our estimate of the sun’s distance affects our estimate of his mean density as compared with the earth’s. His substance has a mean density equal to one-fourth of the earth’s, whether he be 90 millions or 95 millions of miles from us, or indeed whether he were ten millions or a million million miles from us (supposing for a moment our measures did not indicate his real distance more closely). We should still deduce from calculation the same unvarying estimate of his mean density. It follows that the nearer any estimate of his distance places him, and therefore the smaller it makes his estimated volume, the smaller also it makes his estimated mass, and in precisely the same degree. The same is true of the planets also. We determine Jupiter’s mass, for example (at least, this is the simplest way), by noting how he swerves his moons at their respective (estimated) distances. If we diminish our estimate of their distances, we diminish at the same time our estimate of Jupiter’s attractive power, and in such degree, it may be shown (see note), as precisely to correspond with our changed estimate of his size, leaving our estimate of his mean density unaltered. And the same is true for all methods of determining Jupiter’s mass. Suppose, then, that, adopting a certain estimate of the scale of the solar system, we find that the resulting estimate of the masses of the planets and of the sun, as compared with the earth’s mass, from their observed attractive influences on bodies circling around them or passing near them, accords with their estimated perturbing action as compared with the earth’s,—then we should infer that our estimate of the sun’s distance or of the scale of the solar system was correct. But suppose it appeared, on the contrary, that the earth took a larger or a smaller part in perturbing the planetary system than, according to our estimate of her relative mass, she should do,—then we should infer that the masses of the other members of the system had been overrated or underrated; or, in other words, that the scale of the solar system had been overrated or underrated respectively. Thus we should be able to introduce a correction into our estimate of the sun’s distance.
Such is the principle of the method by which Leverrier showed that in the astronomy of the future the scale of the solar system may be very exactly determined. Of course, the problem is a most delicate one. The earth plays, in truth, but a small part in perturbing the planetary system, and her influence can only