It is strange that the problem of determining the sun’s distance, which for many ages was regarded as altogether insoluble, and which even during later years had seemed fairly solvable in but one or two ways, should be found, on closer investigation, to admit of many methods of solution. If astronomers should only be as fortunate hereafter in dealing with the problem of determining the distances of the stars, as they have been with the question of the sun’s distance, we may hope for knowledge respecting the structure of the universe such as even the Herschels despaired of our ever gaining. Yet this problem of determining star-distances does not seem more intractable, now, than the problem of measuring the sun’s distance appeared only two centuries ago. If we rightly view the many methods devised for dealing with the easier task, we must admit that the more difficult—which, by the way, is in reality infinitely the more interesting—cannot be regarded as so utterly hopeless as, with our present methods and appliances, it appears to be. True, we know only the distances of two or three stars, approximately, and have means of forming a vague opinion about the distances of only a dozen others, or thereabouts, while at distances now immeasurable lie six thousand stars visible to the eye, and twenty millions within range of the telescope. Yet, in Galileo’s time, men might have argued similarly against all hope of measuring the proportions of the solar57 system. “We know only,” they might have urged, “the distance of the moon, our immediate neighbour,—beyond her, at distances so great that hers, so far as we can judge, is by comparison almost as nothing, lie the Sun and Mercury, Venus and Mars; further away yet lie Jupiter and Saturn, and possibly other planets, not visible to the naked eye, but within range of that wonderful instrument, the telescope, which our Galileo and others are using so successfully. What hope can there be, when the exact measurement of the moon’s distance has so fully taxed our powers of celestial measurement, that we can ever obtain exact information respecting the distances of the sun and planets? By what method is a problem so stupendous to be attacked?” Yet, within a few years of that time, Kepler had formed already a rough estimate of the distance of the sun; in 1639, young Horrocks pointed to a method which has since been successfully applied. Before the end of the seventeenth century Cassini and Flamsteed had approached the solution of the problem more nearly, while Hailey had definitely formulated the method which bears his name. Long before the end of the eighteenth century it was certainly known that the sun’s distance lies between 85 millions of miles and 98 millions (Kepler, Cassini, and Flamsteed had been unable to indicate any superior limit). And lastly, in our own time, half a score of methods, each subdivisible into several forms, have been applied to the solution of this fundamental problem of observational astronomy.

I propose now to sketch some new and very promising methods, which have been applied already with a degree of success arguing well for the prospects of future applications of the methods under more favourable conditions.

In the first place, let us very briefly consider the methods which had been before employed, in order that the proper position of the new methods may be more clearly recognized.

The plan obviously suggested at the outset for the solution of the problem was simply to deal with it as a problem of surveying. It was in such a manner that the moon’s58 distance had been found, and the only difficulty in applying the method to the sun or to any planet consisted in the delicacy of the observations required. The earth being the only surveying-ground available to astronomers in dealing with this problem (in dealing with the problem of the stars’ distances they have a very much wider field of operations), it was necessary that a base-line should be measured on this globe of ours,—large enough compared with our small selves, but utterly insignificant compared with the dimensions of the solar system. The diameter of the earth being less than 8000 miles, the longest line which the observers could take for base scarcely exceeded 6000 miles; since observations of the same celestial object at opposite ends of a diameter necessarily imply that the object is in the horizon of both the observing stations (for precisely the same reason that two cords stretched from the ends of any diameter of a ball to a distant point touch the ball at those ends). But the sun’s distance being some 92 millions of miles, a base of 6000 miles amounts to less than the 15,000th part of the distance to be measured. Conceive a surveyor endeavouring to determine the distance of a steeple or rock 15,000 feet, or nearly three miles, from him, with a base-line one foot in length, and you can conceive the task of astronomers who should attempt to apply the direct surveying method to determine the sun’s distance,—at least, you have one of their difficulties strikingly illustrated, though a number of others remain which the illustration does not indicate. For, after all, a base one foot in length, though far too short, is a convenient one in many respects: the observer can pass from one end to the other without trouble—he looks at the distant object under almost exactly the same conditions from each end, and so forth. A base 6000 miles long for determining the sun’s distance is too short in precisely the same degree, but it is assuredly not so convenient a base for the observer. A giant 36,000 miles high would find it as convenient as a surveyor six feet high would find a one foot base-line; but astronomers, as a rule, are less than 36,000 miles in height.59 Accordingly the same observer cannot work at both ends of the base-line, and they have to send out expeditions to occupy each station. All the circumstances of temperature, atmosphere, personal observing qualities, etc., are unlike at the two ends of the base-line. The task of measuring the sun’s distance directly is, in fact, at present beyond the power of observational astronomy, wonderfully though its methods have developed in accuracy.

We all know how, by observations of Venus in transit, the difficulty has been so far reduced that trustworthy results have been obtained. Such observations belong to the surveying method, only Venus’s distance is made the object of measurement instead of the sun’s. The sun serves simply as a sort of dial-plate, Venus’s position while in transit across this celestial dial-plate being more easily measured than when she is at large upon the sky. The devices by which Halley and Delisle severally caused time to be the relation observed, instead of position, do not affect the general principle of the transit method. It remains dependent on the determination of position. Precisely as by the change of the position of the hands of a clock on the face we measure time, so by the transit method, as Halley and Delisle respectively suggested its use, we determine Venus’s position on the sun’s face, by observing the difference of the time she takes in crossing, or the difference of the time at which she begins to cross, or passes off, his face.

Besides the advantage of having a dial-face like the sun’s on which thus to determine positions, the transit method deals with Venus when at her nearest, or about 25 million miles from us, instead of the sun at his greater distance of from 90? to 93? millions of miles. Yet we do not get the entire advantage of this relative proximity of Venus. For the dial-face—the sun, that is—changes its position too—in less degree than Venus changes hers, but still so much as largely to reduce her seeming displacement. The sun being further away as 92 to 25, is less displaced as 25 to 92. Venus’s displacement is thus diminished by 25/92nds of its full60 amount, leaving only 67/92nds. Practically, then, the advantage of observing Venus, so far as distance is concerned, is the same as though, instead of being at a distance of only 25 million miles, her distance were greater as 92 to 67, giving as her effective distance when in transit some 34,300,000 miles.

All the methods of observing Venus in transit are affected in this respect. Astronomers were not content during the recent transit to use Halley’s and Delisle’s two time methods (which may be conveniently called the duration method and the epoch method), but endeavoured to determine the position of Venus on the sun’s face directly, both by observation and by photography. The heliometer was the instrument specially used for the former purpose; and as, in one of the new methods to be presently described, this is the most effective of all available instruments, a few words as to its construction will not be out of place.

The heliometer, then, is a telescope whose object-glass (that is, the large glass at the end towards the object observed) is divided into two halves along a diameter. When these two halves are exactly together—that is, in the position they had before the glass was divided—of course they show any object to which they may be directed precisely as they would have done before the glass was cut. But if, without separating the straight edges of the two semicircular glasses, one be made to slide along the other, the images formed by the two no longer coincide.9 Thus, if we are looking at the sun we see two overlapping discs, and by continuing to turn the screw or other mechanism which carries our half-circular glass past the other, the disc-images61 of the sun may be brought entirely clear of each other. Then we have two suns in the same field of view, seemingly in contact, or nearly so. Now, if we have some means of determining how far the movable half-glass has been carried past the other to bring the two discs into apparently exact contact, we have, in point of fact, a measure of the sun’s apparent diameter. We can improve this estimate by carrying back the movable glass till the images coincide again, then further back till they separate the other way and finally are brought into contact on that side. The entire range, from contact on one side to contact on the other side, gives twice the entire angular span of the sun’s diameter; and the half of this is more likely to be the true measure of the diameter, than the range from coincident images to contact either way, simply because instrumental errors are likely to be more evenly distributed over the double motion than over the movement on either side of the central position. The heliometer derived its name—which signifies sun-measurer—from this particular application of the instrument.

It is easily seen how the heliometer was made available in determining the position of Venus at any instant during transit. The observer could note what displacement of the two half-glasses was necessary to bring the black disc of Venus on one image of the sun to the edge of the other image, first touching on the inside and then on the outside. Then, reversing the motion, he could carry her disc to the opposite edge of the other image of the sun, first touching on the inside and then on the outside. Lord Lindsay’s private expedition—one of the most munificent and also one of the most laborious contributions to astronomy ever made—was the only English expedition which employed the heliometer, none of our public observatories possessing such an instrument, and official astronomers being unwilling to ask Government to provide instruments so costly. The Germans, however, and the Russians employed the heliometer very effectively.

Next in order of proximity, for the employment of the62 direct surveying method, is the planet Mars when he comes into opposition (or on the same line as the earth and sun) in the order

Sun____________________________Earth__________Mars,

at a favourable part of his considerably eccentric orbit. His distance then may be as small as 34? millions of miles; and we have in his case to make no reduction for the displacement of the background on which his place is to be determined. That background is the star sphere, his place being measured from that of stars near which his apparent path on the heavens carries him; and the stars are so remote that the displacement due to a distance of six or seven thousand miles between two observers on the earth is to all intents and purposes nothing. The entire span of the earth’s orbit round the sun, though amounting to 184 millions of miles, is a mere point as seen from all save ten or twelve stars; how utterly evanescent, then, the span of the earth’s globe—less than the 23,000th part of her orbital range! Thus the entire displacement of Mars due to the distance separating the terrestrial observers comes into effect. So that, in comparing the observation of Mars in a favourable opposition with that of Venus in transit, we may fairly say that, so far as surveying considerations are concerned, the two planets are equally well suited for the astronomer’s purpose. Venus’s less distance of 25 millions of miles is effectively increased to 34? millions by the displacement of the solar background on which we see her when in transit; while Mars’s distance of about 34? millions of miles remains effectively the same when we measure his displacement from neighbouring fixed stars.

But in many respects Mars is superior to Venus for the purpose of determining the sun’s distance. Venus can only be observed at her nearest when in transit, and transit lasts but a few hours. Mars can be observed night after night for a fortnight or so, during which his distance still remains near enough to the least or opposition distance. Again,63 Venus being observed on the sun, all the disturbing influences due to the sun’s heat are at work in rendering the observation difficult. The air between us and the sun at such a time is disturbed by undulations due in no small degree to the sun’s action. It is true that we have not, in the case of Mars, any means of substituting time measures or time determinations for measures of position, as we have in Venus’s case, both with Halley’s and Delisle’s methods. But to say the truth, the advantage of substituting these time observations has not proved so great as was expected. Venus’s unfortunate deformity of figure when crossing the sun’s edge renders the determination of the exact moments of her entry on the sun’s face and of her departure from it by no means so trustworthy as astronomers could wish. On the whole, Mars would probably have the advantage even without that point in his favour which has now to be indicated.

Two methods of observing Mars for determining the sun’s distance are available, both of which, as they can be employed in applying one of the new methods, may conveniently be described at this point.

An observer far to the north of the earth’s equator sees Mars at midnight, when the planet is in opposition, displaced somewhat to the south of his true position—that is, of the position he would have as supposed to be seen from the centre of the earth. On the other hand, an observer far to the south of the equator sees Mars displaced somewhat to the north of his true position. The difference may be compared to different views of a distant steeple (projected, let us suppose, against a much more remote hill), from the uppermost and lowermost windows of a house corresponding to the northerly and southerly stations on the earth, and from a window on the middle story corresponding to a view of Mars from the earth’s centre. By ascertaining the displacement of the two views of Mars obtained from a station far to the north and another station far to the south, the astronomer can infer the distance of the planet, and thence the dimensions of the solar system. The displacement is64 determinable by noticing Mars’s position with respect to stars which chance to be close to him. For this purpose the heliometer is specially suitable, because, having first a view of Mars and some companion stars as they actually are placed, the observer can, by suitably displacing the movable half-glass, bring the star into apparent contact with the planet, first on one side of its disc, and then on the other side—the mean of the two resulting measures giving, of course, the distance between the star and the centre of the disc.

This method requires that there shall be two observers, one at a northern station, as Greenwich, or Paris, or Washington, the other at a southern station, as Cape Town, Cordoba, or Melbourne. The base-line is practically a north-and-south line; for though the two stations may not lie in the same, or nearly the same, longitude, the displacement determined is in reality that due to their difference of latitude only, a correction being made for their difference of longitude.

The other method depends, not on displacement of two observers north and south, or difference of latitude, but on displacement east and west. Moreover, it does not require that there shall be two observers at stations far apart, but uses the observations made at one and the same stations at different times. The earth, by turning on her axis, carries the observer from the west to the east of an imaginary line joining the earth’s centre and the centre of Mars. When on the west of that line, or in the early evening, he sees Mars displaced towards the east of the planet’s true position. After nine or ten hours the observer is carried as far to the east of that line, and sees Mars displaced towards the west of his true position. Of course Mars has moved in the interval. He is, in fact, in the midst of his retrograde career. But the astronomer knows perfectly well how to take that motion into account. Thus, by observing the two displacements, or the total displacement of Mars from east to west on account of the earth’s rotation, one and the same65 observer can, in the course of a single favourable night, determine the sun’s distance. And in passing, it may be remarked that 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 (c?teris 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 the66 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 slight67 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 R?mer 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. R?mer 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.

68 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 from69 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 be distinguished satisfactorily (at present,70 at any rate) in the case of the nearer members of the solar family. Yet the method is one which, unlike others, will have an accumulative accuracy, the discrepancies which are to test the result growing larger as time proceeds. The method has already been to some extent successful. It was, in fact, by observing that the motions of Mercury are not such as can be satisfactorily explained by the perturbations of the earth and Venus according to the estimate of relative masses deducible from the lately discarded value of the sun’s distance, that Leverrier first set astronomers on the track of the error affecting that value. He was certainly justified in entertaining a strong hope that hereafter this method will be exceedingly effective.

We come next to a method which promises to be more quickly if not more effectively available.

Venus and Mars approach the orbit of our earth more closely than any other planets, Venus being our nearest neighbour on the one side, and Mars on the other. Looking beyond Venus, we find only Mercury (and the mythical Vulcan), and Mercury can give no useful information respecting the sun’s distance. He could scarcely do so even if we could measure his position among the stars when he is at his nearest, as we can that of Mars; but as he can only then be fairly seen when he transits the sun’s face, and as the sun is nearly as much displaced as Mercury by change in the observer’s station, the difference between the two displacements is utterly insufficient for accurate measurement. But, when we look beyond the orbit of Mars, we find certain bodies which are well worth considering in connection with the problem of determining the sun’s distance. I refer to the asteroids, the ring of small planets travelling between the paths of Mars and Jupiter, but nearer (on the whole11) to the path of Mars than to that of Jupiter.

71 The asteroids present several important advantages over even Mars and Venus.

Of course, none of the asteroids approach so near to the earth as Mars at his nearest. His least distance from the sun being about 127 million miles, and the earth’s mean distance about 92 millions, with a range of about a million and a half on either side, owing to the eccentricity of her orbit, it follows that he may be as near as some 35 million miles (rather less in reality) from the earth when the sun, earth, and Mars are nearly in a straight line and in that order. The least distance of any asteroid from the sun amounts to about 167 million miles, so that their least distance from the earth cannot at any time be less than about 73,500,000 miles, even if the earth’s greatest distance from the sun corresponded with the least distance of one of these closely approaching asteroids. This, by the way, is not very far from being the case with the asteroid Ariadne, which comes within about 169 million miles of the sun at her nearest, her place of nearest approach being almost exactly in the same direction from the sun as the earth’s place of greatest recession, reached about the end of June. So that, whenever it so chances that Ariadne comes into opposition in June, or that the sun, earth, and Ariadne are thus placed—

Sun________Earth________Ariadne,

Ariadne will be but about 75,500,000 miles from the earth. Probably no asteroid will ever be discovered which approaches the earth much more nearly than this; and this approach, be it noticed, is not one which can occur in the case of Ariadne except at very long intervals.

But though we may consider 80 millions of miles as a fair average distance at which a few of the most closely approaching asteroids may be observed, and though this72 distance seems very great by comparison with Mars’s occasional opposition distance of 35 million miles, yet there are two points in which asteroids have the advantage over Mars. First, they are many, and several among them can be observed under favourable circumstances; and in the multitude of observations there is safety. In the second place, which is the great and characteristic good quality of this method of determining the sun’s distance, they do not present a disc, like the planet Mars, but a small star-like point. When we consider the qualities of the heliometric method of measuring the apparent distance between celestial objects, the advantage of points of light over discs will be obvious. If we are measuring the apparent distance between Mars and a star, we must, by shifting the movable object-glass, bring the star’s image into apparent contact with the disc-image of Mars, first on one side and then on the other, taking the mean for the distance between the centres. Whereas, when we determine the distance between a star and an asteroid, we have to bring two star-like points (one a star, the other the asteroid) into apparent coincidence. We can do this in two ways, making the result so much the more accurate. For consider what we have in the field of view when the two halves of the object-glass coincide. There is the asteroid and close by there is the star whose distance we seek to determine in order to ascertain the position of the asteroid on the celestial sphere. When the movable half is shifted, the two images of star and asteroid separate; and by an adjustment they can be made to separate along the line connecting them. Suppose, then, we first make the movable image of the asteroid travel away from the fixed image (meaning by movable and fixed images, respectively, those given by the movable and fixed halves of the object-glass), towards the fixed image of the star—the two points, like images, being brought into coincidence, we have the measure of the distance between star and asteroid. Now reverse the movement, carrying back the movable images of the asteroid and star till they coincide again with their73 fixed images. This movement gives us a second measure of the distance, which, however, may be regarded as only a reversed repetition of the preceding. But now, carrying on the reverse motion, the moving images of star and asteroid separate from their respective fixed images, the moving image of the star drawing near to the fixed image of the asteroid and eventually coinciding with it. Here we have a third measure of the distance, which is independent of the two former. Reversing the motion, and carrying the moving images to coincidence with the fixed images, we have a fourth measure, which is simply the third reversed. These four measures will give a far more satisfactory determination of the true apparent distance between the star and the asteroid than can, under any circumstances, be obtained in the case of Mars and a star. Of course, a much more exact determination is required to give satisfactory measures of the asteroid’s real distance from the earth in miles, for a much smaller error would vitiate the estimate of the asteroid’s distance than would vitiate to the same degree the estimate of Mars’s distance: for the apparent displacements of the asteroid as seen either from Northern and Southern stations, or from stations east and west of the meridian, are very much less than in the case of Mars, owing to his great proximity. But, on the whole, there are reasons for believing that the advantage derived from the nearness of Mars is almost entirely counterbalanced by the advantage derived from the neatness of the asteroid’s image. And the number of asteroids, with the consequent power of repeating such measurements many times for each occasion on which Mars has been thus observed, seem to make the asteroids—so long regarded as very unimportant members of the solar system—the bodies from which, after all, we shall gain our best estimate of the sun’s distance; that is, of the scale of the solar system.
* * * * *

Since the above pages were written, the results deduced from the observations made by the British expeditions for74 observing the transit of December 9, 1874, have been announced by the Astronomer Royal. It should be premised that they are not the results deducible from the entire series of British observations, for many of them can only be used effectively in combination with observations made by other nations. For instance, the British observations of the duration of the transit as seen from Southern stations are only useful when compared with observations of the duration of the transit as seen from Northern stations, and no British observations of this kind were taken at Northern stations, or could be taken at any of the British Northern stations except one, where chief reliance was placed on photographic methods. The only British results as yet “worked up” are those which are of themselves sufficient, theoretically, to indicate the sun’s distance, viz., those which indicated the epochs of the commencement of transit as seen from Northern and Southern stations, and those which indicated the epochs of the end of transit as seen from such stations. The Northern and Southern epochs of commencement compared together suffice of themselves to indicate the sun’s distance; so also do the epochs of the end of transit suffice of themselves for that purpose. Such observations belong to the Delislean method, which was the subject of so much controversy for two or three years before the transit took place. Originally it had been supposed that only observations by that method were available, and the British plans were formed upon that assumption. When it was shown that this assumption was altogether erroneous, there was scarcely time to modify the British plans so that of themselves they might provide for the other or Halleyan method. But the Southern stations which were suitable for that method were strengthened; and as other nations, especially America and Russia, occupied large numbers of Northern stations, the Halleyan method was, in point of fact, effectually provided for—a fortunate circumstance, as will presently be seen.

The British operations, then, thus far dealt with, were based on Delisle’s method; and as they were carried out75 with great zeal and completeness, we may consider that the result affords an excellent test of the qualities of this method, and may supply a satisfactory answer to the questions which were under discussion in 1872–74. Sir George Airy, indeed, considers that the zeal and completeness with which the British operations were carried out suffice to set the result obtained from them above all others. But this opinion is based rather on personal than on strictly scientific grounds. It appears to me that the questions to be primarily decided are whether the results are in satisfactory agreement (i) inter se and (ii) with the general tenor of former researches. In other words, while the Astronomer Royal considers that the method and the manner of its application must be considered so satisfactory that the results are to be accepted unquestionably, it appears to me that the results must be carefully questioned (as it were) to see whether the method, and the observations by it, are satisfactory. In the first place, the result obtained from Northern and Southern observations of the commencement ought to agree closely with the result obtained from Northern and Southern observations of the end of transit. Unfortunately, they differ rather widely. The sun’s distance by the former observations comes out about one million miles greater than the distance determined by the latter observations.

This should be decisive, one would suppose. But it is not all. The mean of the entire series of observations by Delisle’s method comes out nearly one million miles greater than the mean deduced by Professor Newcomb from many entire series of observations by six different methods, all of which may fairly be regarded as equal in value to Delisle’s, while three are regarded by most astronomers as unquestionably superior to it. Newcomb considers the probable limits of error in his evaluation from so many combined series of observations to be about 100,000 miles. Sir G. Airy will allow no wider limits of error for the result of the one series his observers have obtained than 200,000 miles. Thus the greatest value admitted by Newcomb falls short76 of the least value admitted by Sir G. Airy, by nearly 700,000 miles.

The obvious significance of this result should be, one would suppose, that Delisle’s method is not quite so effective as Sir G. Airy supposed; and the wide discordance between the several results, of which the result thus deduced is the mean, should prove this, one would imagine, beyond all possibility of question. The Astronomer Royal thinks differently, however. In his opinion, the wide difference between his result and the mean of all the most valued results by other astronomers, indicates the superiority of Delisle’s method, not its inadequacy to the purpose for which it has been employed.

Time will shortly decide which of these views is correct; but, for my own part, I do not hesitate to express my own conviction that the sun’s distance lies very near the limits indicated by Newcomb, and therefore is several hundred thousand miles less than the minimum distance allowed by the recently announced results.