Rough Ways Made Smooth: A series of familiar essays on scientific subjects

by Richard A. Proctor

If we regard the globe as representing the sun; the window of known size as representing our earth of known dimensions; the bird travelling round in a known period and at a distance whose proportion to the window's distance is known, as representing Venus travelling in a known period round the sun and at a distance bearing a known proportion to the earth's; this way of determining the distance of a remote globe illustrates what is called Delisle's method of determining the sun's distance. It requires that the two observers, A and B, should each make exact note of the moment when the bird seemed to begin to cross the disc of the remote globe; and in like manner Delisle's method requires that two observers, widely separated on the earth in a direction nearly parallel to that in which Venus is travelling, should make the most exact note of the moment when Venus begins to cross the sun's face. Also, as all I have said about the bird's beginning to cross the face of the distant globe would apply equally well if said about the end of his seeming passage across that disc, so two observers, widely separated on the earth, can determine the sun's distance by noting the end of her transit instead of the beginning, if they are suitably placed for the purpose. The window of our illustration remains unchanged during the bird's imagined flight, but as the face of the earth turned sunwards (which corresponds to that window) is all the time changing with the earth's rotation, a different pair of stations would have to be selected for observing the end of transit, than would be suitable for observing the beginning.

So much for the method called Delisle's. The other is in principle equally simple. In the imaginary experiment just described we supposed the two observers at the right and left sides of the circular window. Imagine them now to watch the bird from the top and bottom of the window, 3 feet apart. Suppose they note that the two tracks along which, as seen from these two points, the bird seems to cross the face of the distant globe, lie at a distance from each other equal to one-third of the globe's apparent diameter. Now, the bird being twice as far from the globe as from the window, the two tracks on the globe necessarily lie twice as far apart as the two points from which they are seen—or they lie 6 feet apart. The globe's diameter therefore is 18 feet. Knowing thus how large it is, and knowing also how large it looks, the observers know how far from them it lies. So, in the Halleyan method of determining the sun's distance by observing Venus in transit, astronomers are stationed far north and far south on the sunlit half of the earth, corresponding to the window of the imaginary experiment. Venus corresponds to the bird. The observers note along what track she travels across the sun's face. (That they partially determine this by noting how long she is in crossing, in no sense affects the principle of the method.) They thus learn that such and such a portion of the sun's diameter equals the distance separating them,—some six or seven thousand miles perhaps,—whence the sun's diameter is known. And as we know how large he looks, his distance from the earth is determined.

A peculiarity distinguishing this method from the former is that the observers must have a station whence the whole transit can be seen; for practically the place of Venus's track can only be ascertained satisfactorily by timing her passage across the sun's disc, so that the beginning and end must be observed and very carefully timed. This is to some degree a disadvantage; for during a transit lasting several hours the earth turns considerably on her axis, and the face turned sunwards at the beginning is thus very different from the face turned sunwards at the end of transit. It is often exceedingly difficult to find suitable northern and southern stations belonging to both these faces of the earth. On the other hand, the other method has its peculiar disadvantage. To apply it effectively, the observer must know the exact Greenwich time (or any other selected standard time) at his station,—or in other words he must know exactly how far east or west his station is from Greenwich (or some other standard observatory). For all the observations made by this method must be compared together by some absolute time standard. In the Halleyan method the duration of transit only is wanted, and this can be as readily determined by a clock showing local time (or indeed by a clock set going a few minutes before transit began and showing wrong time altogether, so only that it goes at the right rate) as by a clock showing Greenwich, Paris, or Washington time. The clock must not gain or lose in the interval. But a clock which would gain or lose appreciably in four or five hours, would be worthless to the astronomer; and any clock employed for scientific observation might safely be trusted for an interval of that length; whereas a clock which could be trusted to retain true time for several days, is not so readily to be obtained.

We need not consider here the origin of the misapprehension (under which our principal Government astronomer lay for some time), that the Delislean method was alone available during the transit of 1874, the Halleyan method, to use his words, 'failing totally.' The British stations were selected while this misapprehension remained as yet uncorrected. Fortunately the southern stations were suitable for both methods. The northern were not: for this reason, simply, that one set were so situated that night began soon after the beginning of transit, which alone could be observed; while the other set were so situated that night only came to an end a short time before the transit ended, so that the end of transit only could be observed. No doubt when the mistake just mentioned had been clearly recognised,—as it was early in 1873,—measures would have been taken to rectify its effect by occupying some suitable northern stations for observing the whole transit, if Great Britain had been the only nation taking part in the work. Fortunately, however, other nations might be trusted to occupy the northern region, which had so long been overlooked. England simply strengthened the southern observing corps: this could be done without any change by which the Government astronomers would have seemed to admit that 'some one had blundered.' Thus the matter was arranged—America, Russia, and Germany occupying a large number of stations admirably suited for applying the method which had been supposed to 'fail totally.' The British Official astronomers, on whom of course responsibility for adequately observing the transit (or at least for properly applying money granted by the nation for the purpose) alone rested, did in reality all, or nearly all, that was necessary in doubling some of the southern observing parties, and strengthening all of them; for unquestionably other nations occupied suitable northern stations in sufficiently strong force.

It is to be remembered, however, in estimating the probable value of the result which has been deduced from the British observations, that as yet only a portion of these observations has been effectively dealt with. The British observations of the beginning of transit at northern and southern stations give, when combined together, a value of the sun's distance. The British observations of the end of transit at other northern and southern stations give also, when combined together, a value of the sun's distance. And both sets combined give of course a mean value of the sun's distance, more likely on the whole to be correct than either value taken separately. But the British observations of the duration of transit as observed from southern stations do not of themselves give any means of determining the sun's distance. They must be combined with observations of the duration of transit as observed from northern stations; and no British party was stationed where such observations could be obtained. The value, then, of these particular British southern observations can only be educed when comparison is made between them and the northern observations by American, German, and Russian astronomers.

We must not, then, be disheartened if the results of the British operations alone should not seem to be altogether satisfactory. For it may still happen that that portion of the British operations which only has value when combined with the work of other countries may be found to possess extreme value. We had good reason for doubting beforehand whether results of any great value could be obtained by Delisle's method. It was only because Halley's was supposed to fail totally that the Government astronomers ever thought of employing that method, which the experience of former transits had taught us to regard as of very little value.

It may be asked, however, how we are to form an opinion from the result of calculations based on the Delislean operations during the last transit, whether the method in satisfactory or not. If as yet the sun's distance is not exactly determined, a result differing from former results may be better than any of them, many will think; and therefore the method employed to obtain it may be more satisfactory than others. If, they may reason, we place reliance on a certain method to measure for us a certain unknown distance, how can we possibly tell from the distance so determined whether the method is trustworthy or not?

Perhaps the readiest way of removing this difficulty, and also of illustrating generally the principles on which the determination of the most probable mean value of many different estimates depends, is by considering a familiar experience of many, a case in which the point to be determined is the most probable time of day. Suppose that we are walking along a route where there are several clocks, the time shown by our own watch being, for whatever reason, open to question. We find, say, that as compared with our watch time, one clock is two minutes fast, the next three minutes fast, the next one minute slow, and so on, two or three perhaps being as much as six or seven minutes fast, and two or three being as much as three or four minutes slow as compared with the watch. We note, however, that these wider ranges of difference occur only in the case of clocks presumably inferior—cheap clocks in small shops, old clocks in buildings where manifestly the flight of time is not much noted, and so forth. Rejecting these from consideration, we find other clocks ranging from one minute or so before our watch time to four minutes or so after it. Before striking a rough average, however, we consider that some among these clocks are placed where it is on the whole better to be a minute or two before the time than a second late,—as, for instance, at banks, where there may be occasion to send out clerks so as to make sure of reaching certain places (Clearing-House, General Post-office, and so forth) within specified time limits. On the other hand, we note that others of these clocks are placed where it is better to be a minute or two after time than a second before it,—as at railway stations, post-offices, and so on, where it is essential that the public should be allowed time fully up to a specified hour, for some particular service. Taking fair account of such considerations, we might find that most probably the true time lay between half a minute before and two minutes and a half after our watch time. And thus we might infer that in reality the true time was one minute or so later than that shown by our watch. But if we were well acquainted with the characteristics of different clocks along our route, we might infer the time (nay, we might to all intents and purposes know the time) far more accurately than this. We might, for instance, pass six or seven shop-windows where first-class specimens of horological work were shown,—in each window, perhaps, several excellent clocks, with compensated pendulums and other contrivances for securing perfect working. We might find at one of these shops all such clocks showing the same time within two or three seconds; at the next all such clocks also agreeing inter se within two or three seconds, but perhaps their mean differing from the mean at the last shop of the kind by seven or eight seconds; and all six or seven shops, while showing similar agreement as regards the clocks severally displayed at each, agreeing also with each other so closely that ten or twelve seconds would cover the entire range between their several mean times. If this were observed, we should not hesitate to place entire reliance on these special sets of clocks; and we should feel certain that if we took the mean of all their means as the true time (perhaps slightly modifying this mean in order to give due weight to the known superiority of one or other of these clock-shops), we should not be in error by more than five or six seconds, while most probably we should have the time true within two or three seconds.

So far the illustration corresponds well with what had been done during a quarter of a century or so before the last transit of Venus. Several different methods of determining the sun's distance had been applied to correct a value which for many reasons had come to be looked upon with suspicion. This value—95,365,000 miles—was known to be certainly too large. The methods used to test it gave results varying between about 90 million miles and about 96 million miles. But all the methods worthy of any real reliance gave results lying between 91 million miles and 94 million miles. Not to enter more fully into details than would here be suitable, we may pass on at once to say that those most experienced in the matter recognised seven methods of determining the distance, on which chief reliance must be placed. Of these seven methods, six—each applied, of course, by many different observers—were dealt with exhaustively by Professor Newcomb, of the Washington Observatory, a mathematician who has undoubtedly given closer attention to the general problem of determining the sun's distance than any living astronomer. The six methods give six several results ranging from about 92,250,000 miles to about 92,850,000 miles; but when due weight is given to those of the six methods which are undoubtedly the best, the most probable mean value is found to be about 92,350,000 miles. The seventh method, conceived by Leverrier, the astronomer to whom, with our own Adams, the discovery of Neptune was due, and applied by him as he only could have applied it (he alone possessing at once the necessary material and the necessary skill), gives the value, 92,250,000 miles. From this it may fairly be concluded that Newcomb's mean value, which has in fact been accented by all American and Continental astronomers, is certainly within 600,000 miles, and most probably within 300,000 miles of the true mean distance of the sun.

But now, to revert to our illustrative case, let us suppose that after passing the windows of six or seven horologists, from whose clocks we have obtained such satisfactory evidence as to the probable hour, we bethought ourselves of a place where, from what we had heard, a still more exact determination of the hour might be obtained. While still on the way, however, we learn from a friend certain circumstances suggesting the possibility that the clocks at the place in question may not be so correct as we had supposed. Persisting, however, in our purpose, we arrive at the place, and carefully compare the indications of the various clocks there with the time indicated by our watch, corrected (be it supposed) in accordance with the results of our former observations. Suppose now that the hour indicated by the various clocks at this place, instead of agreeing closely with that which we had thus inferred, differs from it by fully half a minute. Is it not clear that instead of being led by this result to correct our former estimate of the probable hour, we should at once infer that the doubts which had been suggested as to the correctness of the various clocks at this place were fully justified? The evidence of the other sets of clocks would certainly not be invalidated by the evidence given by the set last visited, even if the accuracy of these had not been called in question. But if, as supposed, some good reason had been given for doubt on this point,—as for instance, that of late the supervision of the clocks had been interrupted,—we should not hesitate for a moment to reject the evidence given by these clocks, or at least to regard it as only tending to demonstrate what before we had been led to surmise, namely, that these clocks could not be relied upon to show true time. If however, furthermore, we found, not only that the mean of the various times indicated by the clocks at this last-visited place differed thus widely from the time which we had every reason to consider very nearly exact, but that the different clocks here differed as widely from each other, it would be absurd to rely upon their evidence. The circumstance that there was a range of difference of fully half a minute in their indications would of itself suffice to show how untrustworthy they were, at least for the use of any one who wished to obtain the time with great accuracy. Combined with the observed difference between their mean time and that before obtained, this circumstance would prove the inaccuracy of the clocks beyond all possibility of doubt or question.