distance at which a few of the most closely approaching asteroids may
be observed, and though this 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 their 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 for 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 out 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 short
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.
DRIFTING LIGHT-WAVES.
The method of measuring the motion of very swiftly travelling bodies
by noting changes in the light-waves which reach us from them—one
of the most remarkable methods of observation ever yet devised by
man—has recently been placed upon its trial, so to speak; with results
exceedingly satisfactory to the students of science who had accepted
the facts established by it. The method will not be unfamiliar to many
of my readers. The principle involved was first noted by M. Doppler,
but not in a form which promised any useful results. The method
actually applied appears to have occurred simultaneously to several
persons, as well theorists as observers. Thus Secchi claimed in March,
1868, to have applied it though unsuccessfully; Huggins in April, 1868,
described his successful use of the method. I myself, wholly unaware
that either of these observers was endeavouring to measure celestial
motions by its means, described the method, in words which I shall
presently quote, in the number of _Fraser’s Magazine_ for January,
1868, two months before the earliest enunciation of its nature by the
physicists just named.
It will be well briefly to describe the principle of this interesting
method, before considering the attack to which it has been recently
subjected, and its triumphant acquittal from defects charged against
it. This brief description will not only be useful to those readers who
chance not to be acquainted with the method, but may serve to remove
objections which suggest themselves, I notice, to many who have had
the principle of the method imperfectly explained to them.
Light travels from every self-luminous body in waves which sweep
through the ether of space at the rate of 185,000 miles per second.
The whole of that region of space over which astronomers have extended
their survey, and doubtless a region many millions of millions of times
more extended, may be compared to a wave-tossed sea, only that instead
of a wave-tossed surface, there is wave-tossed space. At every point,
through every point, along every line, athwart every line, myriads of
light-waves are at all times rushing with the inconceivable velocity
just mentioned.
It is from such waves that we have learned all we know about the
universe outside our own earth. They bring to our shores news from
other worlds, though the news is not always easy to decipher.
Now, seeing that we are thus immersed in an ocean, athwart which
infinite series of waves are continually rushing, and moreover that
we ourselves, and every one of the bodies whence the waves proceed
either directly or after reflection, are travelling with enormous
velocity through this ocean, the idea naturally presents itself that we
may learn something about these motions (as well as about the bodies
themselves whence they proceed), by studying the aspect of the waves
which flow in upon us in all directions.
Suppose a strong swimmer who knew that, were he at rest, a certain
series of waves would cross him at a particular rate—ten, for
instance, in a minute—were to notice that when he was swimming
directly facing them, eleven passed him in a minute: he would be able
at once to compare his rate of swimming with the rate of the waves’
motion. He would know that while ten waves had passed him on account
of the waves’ motion, he had by his own motion caused yet another wave
to pass him, or in other words, had traversed the distance from one
wave-crest to the next Thus he would know that his rate was one-tenth
that of the waves. Similarly if, travelling the same way as the waves,
he found that only nine passed him in a minute, instead of ten.
Again, it is not difficult to see that if an observer were at rest,
and a body in the water, which by certain motions produced waves,
were approaching or receding from the observer, the waves would come
in faster in the former case, slower in the latter, than if the body
were at rest. Suppose, for instance, that some machinery at the bows
of a ship raised waves which, if the ship were at rest, would travel
along at the rate of ten a minute past the observer’s station. Then
clearly, if the ship approached him, each successive wave would have
a shorter distance to travel, and so would reach him sooner than it
otherwise would have done. Suppose, for instance, the ship travelled
one-tenth as fast as the waves, and consider ten waves proceeding from
her bows—the first would have to travel a certain distance before
reaching the observer; the tenth, starting a minute later, instead of
having to travel the same distance, would have to travel this distance
diminished by the space over which the ship had passed in one minute
(which the wave itself passes over in the tenth of a minute); instead,
then, of reaching the observer one minute after the other, it would
reach him nine-tenths of a minute after the first. Thus it would seem
to him as though the waves were coming in faster than when the ship was
at rest, in the proportion of ten to nine, though in reality they would
be travelling at the same rate as before, only arriving in quicker
succession, because of the continual shortening of the distance they
had to travel, on account of the ship’s approach. If he knew precisely
how fast they _would_ arrive if the ship were at rest, and determined
precisely how fast they _did_ arrive, he would be able to determine at
once the rate of the ship’s approach, at least the proportion between
her rate and the rate of the waves’ motion. Similarly if, owing to the
ship’s recession, the apparent rate of the waves’ motion were reduced,
it is obvious that the actual change in the wave motion would not be
a difference of rate; but, in the case of the approaching ship, the
breadth from crest to crest would be reduced, while in the case of a
receding ship the distance from crest to crest would be increased.
If the above explanation should still seem to require closer attention
than the general reader may be disposed to give, the following,
suggested by a friend of mine—a very skilful mathematician—will be
found still simpler: Suppose a stream to flow quite uniformly, and that
at one place on its banks an observer is stationed, while at another
higher up a person throws corks into the water at regular intervals,
say ten corks per minute; then these will float down and pass the other
observer, wherever he may be, at the rate of ten per minute, _if_
the cork-thrower is at rest. But if he saunters either up-stream or
down-stream, the corks will no longer float past the other at the exact
rate of ten per minute. If the thrower is sauntering down-stream, then,
between throwing any cork and the next, he has walked a certain way
down, and the tenth cork, instead of having to travel the same distance
as the first before reaching the observer, has a shorter distance to
travel, and so reaches that observer sooner. Or in fact, which some may
find easier to see, this cork will be nearer to the first cork than it
would have been if the thrower had remained still. The corks will lie
at equal distances from each other, but these equal distances will be
less than they would have been if the observer had been at rest. If, on
the contrary, the cork-thrower saunters up-stream, the corks will be
somewhat further apart than if he had remained at rest. And supposing
the observer to know beforehand that the corks would be thrown in
at the rate of ten a minute, he would know, if they passed him at a
greater rate than ten a minute (or, in other words, at a less distance
from each other than the stream traversed in the tenth of a minute),
that the cork-thrower was travelling down-stream or approaching him;
whereas, if fewer than ten a minute passed him, he would know that the
cork-thrower was travelling away from him, or up-stream. But also, if
the cork-thrower were at rest, and the observer moved up-stream—that
is, towards him—the corks would pass him at a greater rate than ten a
minute; whereas, if the observer were travelling down-stream, or from
the thrower, they would pass him at a slower rate. If both were moving,
it is easily seen that if their movement brought them nearer together,
the number of corks passing the observer per minute would be increased,
whereas if their movements set them further apart, the number passing
him per minute would be diminished.
These illustrations, derived from the motions of water, suffice in
reality for our purpose. The waves which are emitted by luminous
bodies in space travel onwards like the water-waves or the corks of
the preceding illustrations. If the body which emits them is rapidly
approaching us, the waves are set closer together or narrowed; whereas,
if the body is receding, they are thrown further apart or broadened.
And if we can in any way recognize such narrowing or broadening of the
light-waves, we know just as certainly that the source of light is
approaching us or receding from us (as the case may be) as our observer
in the second illustration would know from the distance between the
corks whether his friend, the cork-thrower, was drawing near to him or
travelling away from him.
But it may be convenient to give another illustration, drawn from
waves, which, like those of light, are not themselves discernible
by our senses—I refer to those aerial waves of compression and
rarefaction which produce what we call sound. These waves are not
only in this respect better suited than water-waves to illustrate
our subject, but also because they travel in all directions through
aerial space, not merely along a surface. The waves which produce
a certain note, that is, which excite in our minds, through the
auditory nerve, the impression corresponding to a certain tone, have
a definite length. So long as the observer, and a source of sound
vibrating in one particular period, remain both in the same place,
the note is unchanged in tone, though it may grow louder or fainter
according as the vibrations increase or diminish in amplitude. But if
the source of sound is approaching the hearer, the waves are thrown
closer together and the sound is rendered more acute (the longer waves
giving the deeper sound); and, on the other hand, if the source of
sound is receding from the hearer, the waves are thrown further apart
and the sound is rendered graver. The _rationale_ of these changes is
precisely the same as that of the changes described in the preceding
illustrations. It might, perhaps, appear that in so saying we were
dismissing the illustration from sound, at least as an independent
one, because we are explaining the illustration by preceding
illustrations. But in reality, while there is absolutely nothing
new to be said respecting the increase and diminution of distances
(as between the waves and corks of the preceding illustration),
the illustration from sound has the immense advantage of admitting
readily of experimental tests. It is necessary only that the rate of
approach or recession should bear an appreciable proportion to the
rate at which sound travels. For waves are shortened or lengthened by
approach or recession by an amount which bears to the entire length of
the wave the same proportion which the rate of approach or recession
bears to the rate of the wave’s advance. Now it is not very difficult
to obtain rates of approach or recession fairly comparable with the
velocity of sound—about 364 yards per second. An express train at
full speed travels, let us say, about 1800 yards per minute, or 30
yards per second. Such a velocity would suffice to reduce all the
sound-waves proceeding from a bell or whistle upon the engine, by
about one-twelfth part, for an observer at rest on a station platform
approached by the engine. On the contrary, after the engine had passed
him, the sound-waves proceeding from the same bell or whistle would be
lengthened by one-twelfth. The difference between the two tones would
be almost exactly three semitones. If the hearer, instead of being on
a platform, were in a train carried past the other at the same rate,
the difference between the tone of the bell in approaching and its
tone in receding would be about three tones. It would not be at all
difficult so to arrange matters, that while two bells were sounding the
same note—_Mi_, let us say—one bell on one engine the other on the
other, a traveller by one should hear his own engine’s bell, the bell
of the approaching engine, and the bell of the same engine receding,
as the three notes—_Do_—_Mi_—_Sol_, whose wave-lengths are as the
numbers 15, 12, and 10. We have here differences very easily to be
recognized even by those who are not musicians. Every one who travels
much by train must have noticed how the tone of a whistle changes as
the engine sounding it travels past. The change is not quite sharp,
but very rapid, because the other engine does not approach with a
certain velocity up to a definite moment and then recede with the same
velocity. It could only do this by rushing through the hearer, which
would render the experiment theoretically more exact but practically
unsatisfactory. As it rushes past instead of through him, there is a
brief time during which the rate of approach is rapidly being reduced
to nothing, followed by a similarly brief time during which the rate of
recession gradually increases from nothing up to the actual rate of the
engines’ velocities added together.[12] The change of tone may be thus
illustrated:—
[Illustration]
A B representing the sound of the approaching whistle, B C representing
the rapid degradation of sound as the engine rushes close past the
hearer, and C D representing the sound of the receding whistle. When
a bell is sounded on the engine, as in America, the effect is better
recognized, as I had repeated occasion to notice during my travels in
that country. Probably this is because the tone of a bell is in any
case much more clearly recognized than the tone of a railway whistle.
The change of tone as a clanging bell is carried swiftly past (by the
combined motions of both trains) is not at all of such a nature as to
require close attention for its detection.
However, the apparent variation of sound produced by rapid approach or
recession has been tested by exact experiments. On a railway uniting
Utrecht and Maarsen “were placed,” the late Professor Nichol wrote,
“at intervals of something upwards of a thousand yards, three groups
of musicians, who remained motionless during the requisite period.
Another musician on the railway sounded at intervals one uniform note;
and its effects on the ears of the stationary musicians have been
fully published. From these, certainly—from the recorded changes
between grave and the more acute, and _vice versâ_,—confirming, even
_numerically_, what the relative velocities might have enabled one to
predict, it appears justifiable to conclude that the general theory is
correct; and that the note of any sound may be greatly modified, if
not wholly changed, by the velocity of the individual hearing it,” or,
he should have added, by the velocity of the source of sound: perhaps
more correct than either, is the statement that the note may be altered
by the approach or recession of the source of sound, whether that be
caused by the motion of the sounding body, or of the hearer himself, or
of both.