astronomers in this case, we can only too readily understand that their
measurements should differ somewhat widely from each other. Let us
picture to ourselves for a moment the central sun, the earth, and the
earth's path, not as they really are, for the mind refuses altogether
to picture the dimensions even of the earth, which is but an atom
compared with the sun, whose own proportions, in turn, mighty though
they are, sink into utter insignificance compared with the enormous
scale of the orbit in which the earth travels around him. Let us reduce
the scale of the entire system to one 500-millionth part of its real
value: even then we have a tolerably large orbit to imagine. We must
picture to ourselves a fiery globe 3 yards in diameter to represent the
sun, and the earth as a one-inch ball circling round that globe at a
distance of about 325 yards, or about 350 paces. The diameter of the
earth's orbit would on this scale, therefore, be somewhat more than a
third of a mile. If we imagine the one-inch ball moving round the fiery
globe once in a year, while turning on its axis once in a day, we find
ourselves under a difficulty arising from the slowness of the resulting
motions. We should have found ourselves under a difficulty arising from
the rapidity of the actual motions if we had considered them instead.
The only resource is to reduce our time-scale, in the same way that we
have reduced our space-scale: but not in the same degree; for if we did
we should have the one-inch ball circling round its orbit, a third of
a mile in diameter, sixteen times in a second, and turning on its axis
five thousand times in a second. Say, instead, that for convenience we
suppose days reduced to seconds. Then we have to picture a one-inch
globe circling once in rather more than six minutes about a globe of
fire 3 yards in diameter, one-sixth of a mile from it, and turning
on its axis once in a second. We must further picture the one-inch
globe as inhabited by some 1,500 millions of creatures far too small
to be seen with the most powerful microscope--in fact, so small that
the tallest would be in height but about the seven-millionth of an
inch--and we must imagine that a few of these creatures undertake the
task of determining from their tiny home swiftly rotating as it rushes
in its orbit around a large globe of fire, 325 yards from them--the
number of yards really intervening between that globe and their home.
If we rightly picture these conditions, which fairly represent those
under which the astronomer has to determine the distance of the sun
from the earth, we shall perceive that the wonder rather is that any
idea of the sun's distance should be obtained at all, than that the
estimates obtained should differ from each other, and that the best of
them should err in measurable degree from the true distance.
Anything like a full explanation of the way in which transits of Venus
across the sun's face are utilised in the solution of the problem of
determining the sun's distance would be out of place in these pages.
But perhaps the following illustration may serve sufficiently, yet
simply, to indicate the qualities of the two leading methods of using
a transit. Imagine a bird flying in a circle round a distant globe
in such a way that, as seen from a certain window (a circular window
suppose), the bird will seem to cross the face of the globe once in
each circuit. Suppose that though the distance of the globe is not
known, the window is known to be exactly half as far again from the
globe as the bird's path is, and that the window is exactly a yard
in diameter. Now in the first place, suppose two observers watch the
bird, one (A) from the extreme right side, and the other (B) from the
extreme left side of the window, the bird flying across from right
to left. A sees the bird begin to cross the face of the globe before
B does,--say they find that A sees this exactly one second before B
does. But A's eye and B's being 3 feet apart, and the bird two-thirds
as far from the globe as the window is, the line traversed by the bird
in this interval is of course only 2 feet in length. The bird then
flies 2 feet in a second (this is rather slow for a bird, but the
principle of the explanation is not affected on that account). Say it
is further observed that he completes a circuit in exactly ten minutes
or six hundred seconds. Thus the entire length of a circuit is 1,200
feet,--whence by the well-known relation between the circumference and
the diameter of a circle, it follows that the diameter of the bird's
path is about 382 feet, and his distance from the centre of the globe
191 feet. So that the distance of the globe from the window, known to
be half as great again, is about 286-1/2 feet.
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.