Chapter VI: The Variation of Latitude (1)

If we should desire to classify discoveries in order of merit, we must undoubtedly give a high place to those which are made under direct discouragements. In the last chapter we saw that Schwabe entered upon his work under conditions of this kind, it being the opinion of experienced astronomers who had looked at the facts that there was nothing of interest to be got by watching sun-spots. In the present chapter I propose to deal with a discovery made in the very teeth of the unanimous opinion of the astronomical world by an American amateur, Mr. S. C. Chandler of Cambridge (Massachusetts). It is my purpose to allow him to himself explain the steps of this discovery by giving extracts from the magnificent series of papers which he contributed to the _Astronomical Journal_ on the subject in the years 1891-94, but it may help in the understanding of these extracts if I give a brief summary of the facts. And I will first explain what is meant by the "Variation of Latitude."

[Sidenote: Latitude.]

[Sidenote: Precession.]

We are all familiar with the existence of a certain star in the heavens called the Pole Star, and we know that at any particular place it is seen constantly in the north at a definite height above the horizon, which is the latitude of the place. When watched carefully with a telescope it is found to be not absolutely stationary, but to describe a small circle in the heavens day by day, or rather night by night. These simple facts are bound up with the phenomenon of the earth's rotation in this way: the axis about which it is rotating points to the centre of that little circle, and any change in the position of the axis can therefore be determined by observing these motions of the Pole Star. Such changes may be of two kinds: firstly, we might find that the size of the circle increased or diminished, and this would mean that the earth's axis was pointing farther away from the Pole Star or nearer to it--pointing, that is to say, in a different direction in space. This actually happens (as has been known for some thousands of years) owing to the phenomenon called "precession"; the circle described by our Pole Star is at present getting a little smaller, but it will ultimately increase in size, and after thousands of years become so large that the Pole Star will entirely lose its character as a steady guide to the North.

[Sidenote: Change of latitude.]

[Sidenote: Twenty years ago disbelieved.]

Secondly (and this is what more immediately concerns us), the centre of the circle may alter its position and be no longer at the same height above the horizon of any given place. This would mean that the earth's axis was shifting _in the earth itself_--that the North Pole which our explorers go to seek is not remaining in the same place. That it does not change appreciably in position we know from familiar experience; our climates, for instance, would suffer considerably if there were any large changes. But astronomers are concerned with minute changes which would not have any appreciable effect on climate, and the question has long been before them whether, putting aside large movements, there were any minute variations in position of the North Pole. Twenty years ago the answer to this question would have been given decidedly in the negative; it was considered as certain that the North Pole did not move at all within the limits of our most refined astronomical observations. Accepted theory seemed to indicate that any movements must in any case recur after a period of ten months, and careful discussion of the observations showed that there was no oscillation in such a period. Now we know that the theory itself was wrong, or rather was founded upon a mistaken assumption; and that the facts when properly examined show clearly a distinct movement of the North Pole, not a very large one, for all its movements take place within the area occupied by a moderate-sized room, but still a movement easily measurable by astronomical observations, and Mr. Chandler was the first to point out the law of these movements, and very possibly the first to suspect them.

[Sidenote: Chandler's papers.]

With these few words of explanation I will let Mr. Chandler tell his own story. His first paper appeared in the _Astronomical Journal_ in November 1891, and is courageously headed, "On the Variation of Latitude"--I say courageously, because at that time it was believed that the latitude did _not_ vary, and Mr. Chandler himself was only in possession of a small portion of the facts. They unravelled themselves as he went forward; but he felt that he had firm hold of the end of the thread, and he faced the world confidently in that belief. He begins thus:--

[Sidenote: First signs of change.]

"In the determination of the latitude of Cambridge[5] with the
Almucantar, about six years and a half ago, it was shown that the
observed values, arranged according to nights of observation,
exhibited a decided and curious progression throughout the series,
the earlier values being small, the later ones large, and the range
from November 1884 to April 1885 being about four-tenths of a second.
There was no known or imaginable instrumental or personal cause for
this phenomenon, yet the only alternative seemed to be an inference
that the latitude had actually changed. This seemed at the time too
bold an inference to place upon record, and I therefore left the
results to speak for themselves. The subsequent continuation of the
series of observations to the end of June 1885 gave a maximum about
May 1, while the discussion of the previous observations from May to
November 1884 gave a minimum about September 1, indicating a range of
0".7 within a half-period of about seven months."

Mr. Chandler then gives some figures in support of these statements, presenting them with the clearness which is so well marked a feature of the whole series of papers, and concludes this introductory paper as follows:--

"It thus appears that the apparent change in the latitude of
Cambridge is verified by this discussion of more abundant material.
The presumption that it is real, on this determination alone, would
justify further inquiry.

[Sidenote: Confirmed in Europe.]

"Curiously enough Dr. Kuestner, in his determination of the
aberration from a series of observations coincident in time with
those of the Almucantar, came upon similar anomalies, and his
results, published in 1888, furnish a counterpart to those which I
had pointed out in 1885. The verification afforded by the recent
parallel determinations at Berlin, Prague, Potsdam, and Pulkowa,
which show a most surprising and satisfactory accordance, as to the
character of the change, in range and periodicity, with the
Almucantar results, has led me to make further investigations on the
subject. They seem to establish the nature of the law of those
changes, and I will proceed to present them in due order."

The second paper appeared on November 23, and opens with the following brief statement of his general results at that time:--

[Sidenote: 427 days' period.]

"Before entering upon the details of the investigations spoken of in
the preceding number, it is convenient to say that the general result
of a preliminary discussion is to show a revolution of the earth's
pole in a period of 427 days, from west to east, with a radius of
thirty feet, measured at the earth's surface. Assuming provisionally,
for the purpose of statement, that this is a motion of the north pole
of the principal axis of inertia about that of the axis of rotation,
the direction of the former from the latter lay towards the Greenwich
meridian about the beginning of the year 1890. This, with the period
of 427 days, will serve to fix approximately the relative positions
of these axes at any other time, for any given meridian. It is not
possible at this stage of the investigation to be more precise, as
there are facts which appear to show that the rotation is not a
perfectly uniform one, but is subject to secular change, and perhaps
irregularities within brief spaces of time."

[Sidenote: Contrary to received views.]

It is almost impossible, now that we have become familiar with the ideas conveyed in this paragraph, to understand, or even fully to remember, the impression produced by them at the time; the sensation caused in some quarters, and the ridicule excited in others. They were in flat contradiction to all accepted views; and it was believed that these views were not only theoretically sound, but had been matured by a thorough examination of observational evidence. The only period in which the earth's pole could revolve was believed to be ten mouths; and here was Mr. Chandler proclaiming, apparently without any idea that he was contradicting the laws of dynamics, that it was revolving in fourteen months! The radius of its path had been found to be insensible by careful discussion of observations, and now he proclaimed a sensible radius oL thirty feet. Finally, he had the audacity to announce a _variable_ period, to which there was nothing at all corresponding in the mathematical possibilities. This was the bitterest pill of all. Even after Professor Newcomb had shown us how to swallow the other two, he could not recommend any attempt at the third, as we shall presently see; and Mr. Chandler was fain ultimately to gild it a little before it could be gulped.

[Sidenote: Pulkowa puzzle solved, also Washington.]

But this is anticipating, and it is our intention to follow patiently the evidence adduced in support of the above statements, made with such splendid confidence to a totally disbelieving world. Mr. Chandler first examines the observations of Dr. Kuestner of Berlin, quoted at the end of his last paper, and shows how well they are suited by the existence of a variation in the latitude of 427 days; and that this new fact is added--when the Cambridge (U.S.A.) latitudes were the smallest those of Berlin were the largest, and _vice versa_, as would clearly be the case if the phenomenon was due to a motion of the earth's pole; for if it moved nearer America it must move further from Europe. He then examines a long series of observations made in the years 1864-1873 at Pulkowa, near St. Petersburg, and again finds satisfactory confirmation of his law of variation. Now it had long been known that there was something curious about these observations, but no one could tell what it was. The key offered by Mr. Chandler fitted the lock exactly, and the anomalies which had been a puzzle were removed. This was in itself a great triumph; but there was another to come, which we may let Mr. Chandler describe in his own words:--

"In 1862 Professor Hubbard began a series of observations of [a]
Lyrae at the Washington Observatory with the prime vertical transit
instrument, for the purpose of determining the constants of
aberration and nutation and the parallax of the star. The methods of
observation and reduction were conformed to those used with such
success by W. Struve. After Hubbard's death the series was continued
by Professors Newcomb, Hall, and Harkness until the beginning of
1867. Professor Hall describes these observations as the most
accurate determinations of declination ever made at the Naval
Observatory. The probable error of a declination from a single
transit was +-0".141, and judging from the accidental errors, the
series ought to give trustworthy results. Upon reducing them,
however, it was found that some abnormal source of error existed,
which resulted in anomalous values of the aberration-constant in the
different years, and a negative parallax in all. A careful
verification of the processes of reduction failed to discover the
cause of the trouble, and Professor Hall says that the results must
stand as printed, and that probably some annual disturbance in the
observations or the instrument occurred, which will never be
explained, and which renders all deductions from them uncertain. The
trouble could not be connected with personal equation, the anomalies
remaining when the observations of the four observers who took part
were separately treated. Nor, as Professor Hall points out, will the
theoretical ten-month period in the latitude furnish the explanation.

"It is manifest, however, that if the 427-day period exists, its
effect ought to appear distinctly in declination-measurements of such
high degree of excellence as these presumably were, and, as I hope
satisfactorily to show, actually are. When this variation is taken
into account the observations will unquestionably vindicate the high
expectations entertained with regard to them by the accomplished and
skilful astronomers who designed and carried them out."

[Sidenote: Direction of revolution of Pole.]

[Sidenote: Example of results.]

From this general account I am excluding technical details and figures, and unfortunately a great deal is thereby lost. We lose the sense of conviction which the long rows of accordant figures force upon us, and we lose the opportunities of admiring both the astonishing amount of work done and the beautiful way in which the material is handled by a master. But I am tempted to give one very small illustration of the numerical results from near the end of the paper. After discussing the Washington results, and amply fulfilling the promise made in the preceding extract, Mr. Chandler compares them with the Pulkowa results, and shows that the Earth's Pole must be revolving from west to east, and not from east to west. And then he writes down a simple formula representing this motion, and compares his formula with the observations. He gives the results in seconds of arc, but for the benefit of those not familiar with astronomical measurements we may readily convert these into feet; and in the following tables are shown the distances of the Earth's Pole _in feet_ from its average position,[6] as observed at Washington and at Pulkowa, and the same distances calculated according to the formula which Mr. Chandler was able to write down at this early stage. The signs + and - of course indicate opposite directions of displacement:--

WASHINGTON.

_Deviation of Pole._

+-------------------------------------+
| Date. | Observed.| Formula. |
|-------------------------------------|
| 1864, Dec. 28 | -28 feet | -23 feet |
| 1865, Mar. 19 | - 1 " | -12 " |
| " June 1 | +15 " | +12 " |
| " Aug. 11 | +22 " | +23 " |
| " Oct. 9 | +11 " | +15 " |
| " Dec. 13 | -17 " | - 6 " |
+-------------------------------------+

PULKOWA.

_Deviation of Pole._

+-------------------------------------+
| Date. | Observed.| Formula. |
|-------------------------------------|
| 1865, July 25 | -18 feet | -12 feet |
| " Sept. 9 | + 3 " | + 3 " |
| " Nov. 22 | +26 " | +22 " |
| 1866, Feb. 22 | +18 " | +13 " |
| " June 4 | -11 " | -18 " |
| " July 17 | -16 " | -23 " |
+-------------------------------------+

Of course the figures are not exact in every case, but they are never many feet wrong; and it may well be imagined that it is a difficult thing to deduce, even from the most refined observations, the position of the earth's pole to within a foot. The difficulty is exactly the same as that of measuring the length of an object 300 miles away to within an inch!

Mr. Chandler winds up his second paper thus:--

"We thus find that the comparison of the simultaneous series at
Pulkowa and Washington, 1863-1867, leads to the same conclusion as
that already drawn from the simultaneous series at Berlin and
Cambridge, 1884-1885. The direction of the polar motion may therefore
be looked upon as established with a large degree of probability.

"In the next paper I will present the results derived from PETERS,
STRUVE, BRADLEY, and various other series of observations, after
which the results of all will be brought to bear upon the
determination of the best numerical values of the constants
involved."

[Sidenote: Bradley's observations.]

[Sidenote: Latitude varied in twelve months then.]

The results were not, however, presented in this order. In the next paper, which appeared on December 23, 1891, Mr. Chandler begins, with the work of Bradley, the very series of observations at Kew and Wansted which led to the discoveries of aberration and nutation, and which we considered in the third chapter. He first shows that, notwithstanding the obvious accuracy of the observations, there is some unexplained discordance. The very constant of aberration which Bradley discovered from them differs by half-a-second of arc from our best modern determinations. Attempts have been made to ascribe the discordance to changes in the instrument, but Mr. Chandler shows that such changes, setting aside the fact that Bradley would almost certainly have discovered them, will not fit in with the facts. The facts, when analysed with the skill to which we have become accustomed, are that there is a periodic swing in the results _with a period of about a year_, and not fourteen months, as before, "a result so curious," as he admits, that "if we found no further support, it might lead us to distrust the above reasoning, and throw us back to the possibility that, after all, BRADLEY'S observations may have been vitiated by some kind of annual instrumental error. But it will abundantly appear, when I have had the opportunity to print the deductions from all the other series of observations down to the present time, that the inference of an increase in the period of polar revolution is firmly established by their concurrent testimony." We shall presently return to this curious result, which might well have dismayed a less determined researcher than Mr. Chandler, but which only led him on to renewed exertions.

The results obtained from Bradley's observations may be put in the form of a diagram thus:--

It will be seen that the maxima and minima fall in the spring and autumn, and this fact alone seemed to show that the effect could not be due to temperature, for we should expect the greatest effect in that case in winter and summer. It could not be due to the parallax of the stars for which Bradley began his search, for stars in different quarters of the heavens would then be differently affected, and this was not the case. "There remains," concluded Mr. Chandler after full discussion, "the only natural conclusion of an actual displacement of the zenith, in other words, a change of latitude." And he concludes this paper with the following fine passage:--

"So far, then, as the results of this incomparable series of
observations at Kew and Wansted, considered by themselves alone, can
now be stated, the period of the polar rotation at that epoch appears
to have been probably somewhat over a year, and certainly shorter by
about two months than it is at the present time. The range of the
variation was apparently in the neighbourhood of a second of arc, or
considerably larger than that shown by the best modern observations.

[Sidenote: Bradley's greatness.]

"Before taking leave of these observations for the present I cannot
forbear to speak of the profound impression which a study of them
leaves upon the mind, and the satisfaction which all astronomers must
feel in recognising that, besides its first fruits of the phenomena
of aberration and nutation, we now owe also our first knowledge of
the polar motion to this same immortal work of Bradley. Its
excellence, highly appreciated as it has been, has still been
hitherto obscured by the presence of this unsuspected phenomenon.
When divested of its effects, the wonderful accuracy of this work
must appear in a finer light, and our admiration must be raised to
higher pitch. Going back to it after one hundred and sixty years
seems indeed like advancing into an era of practical astronomy more
refined than that from which we pass. And this leads to a suggestion
worthy of serious practical consideration--whether we can do better
in the future study of the polar rotation, than again to avail
ourselves of Bradley's method, without endangering its elegant
simplicity and effectiveness by attempts at improvement, other than
supplying certain means of instrumental control which would without
doubt commend themselves to his sagacious mind.

[Sidenote: Other puzzles explained.]

"In the next article Bradley's later observations at Greenwich, the
results of which are not so distinct, will be discussed; and also
those of Brinkley at Dublin, 1808-13 and 1818-22. This will bring
again to the surface one of the most interesting episodes in
astronomical history, the spirited and almost acrimonious dispute
between Brinkley and Pond with regard to stellar parallaxes. I hope
to show that the hitherto unsolved enigma of Brinkley's singular
results finds its easy solution in the fact of the polar motion. The
period of his epoch appears to have been about a year, and its range
more than a second. Afterwards will follow various discussions
already more or less advanced towards completion. These include
Bessel's observations at Koenigsberg, 1820-24, with the Reichenbach
circle, and in 1842-44 with the Repsold circle; the latitudes derived
from the polar-point determinations of Struve and Maedler with the
Dorpat circle, 1822-38; Struve's observations for the determination
of the aberration; Peters' observations of _Polaris_, 1841-43, with
the vertical-circle; the results obtained from the reflex zenith-tube
at Greenwich, 1837-75, whose singular anomalies can be referred in
large part to our present phenomenon, complicated with instrumental
error, to which until now they have been exclusively attributed; the
Greenwich transit-circle results, 1851-65, in which case, however, a
similar complication and the large accidental errors of observation
seem to frustrate efforts to get any pertinent results; the Berlin
prime-vertical observations of Weyer and Bruennow, 1845-46, in which I
hope to show that the parallax of [beta] _Draconis_ derived from them
is simply a record of the change of latitude; the conflicting
latitude determinations at Cambridge, England; the Washington
observation of _Polaris_ and other close Polars, 1866-87, with the
transit-circle; also those at Melbourne, 1863-84, a portion of which
have already been drawn upon in the last number of the _Journal_,
and some others. While the list is a considerable one, I shall be
able to compress the statement of results for many of the series into
a short space.

[Sidenote: Provisional nature of results.]

"In connection with this synopsis of the scope of the investigations,
one or two particulars may be of interest, which at the present
writing seem to foreshadow the probable outcome. I beg, however, that
the statement will be regarded merely as a provisional one. First,
while the period is manifestly subject to change, as has already once
or twice been intimated, I have hitherto failed in tracing the
variations to any regular law, expressible in a numerical formula.
Indeed, the general impression produced by a study of these changes
in the length of the period is that the cause which produces them
operates capriciously to a certain degree, although the average
effect for a century has been to diminish the velocity of the
revolution of the pole. How far this impression is due to the
uncertainty of the observations, and to the complication of the
phenomenon with other periodical changes of a purely instrumental
kind, I cannot say. Almost all of the series of any extent which have
been examined, have the peculiarity that they manifest the
periodicity quite uniformly and distinctly for a number of years,
then for a while obscurely. In some cases, however, what at first
appears to be an objective irregularity proves not to be so by
comparison with overlapping series at other observatories.

"Another characteristic which has struck my attention, although
somewhat vaguely, is that the variations in the length of the period
seem to go hand in hand with simultaneous alterations in the
amplitude of the rotation; the shorter periods being apparently
associated with the larger coefficients for the latter. The
verification of these surmises awaits a closer comparative scrutiny,
the opportunity for which will come when the computations are in a
more forward state. If confirmed, these observations will afford a
valuable touchstone, in seeking for the cause of a phenomenon which
now seems to be at variance with the accepted laws of terrestrial
rotation."

[Sidenote: Reception of discovery.]

Let us now for a few moments turn aside from the actual research to see how the announcement was received. It would be ungracious to reprint here any of the early statements of incredulity which found their way into print, especially in Germany. But the first note of welcome came from Simon Newcomb, in the same number of the _Astronomical Journal_ as the paper just dealt with, and the following extract will indicate both the difficulties felt in receiving Mr. Chandler's results and the way in which Newcomb struck at the root of them.

[Sidenote: Newcomb's explanation.]

"Mr. Chandler's remarkable discovery, that the apparent variations in
terrestrial latitudes may be accounted for by supposing a revolution
of the axis of rotation of the earth around that of figure, in a
period of 427 days, is in such disaccord with the received theory of
the earth's rotation that at first I was disposed to doubt its
possibility. But I am now able to point out a _vera causa_ which
affords a complete explanation of this period. Up to the present time
the treatment of this subject has been this: The ratio of the moment
of inertia of the earth around its principal axis to the mean of the
other two principal moments, admits of very accurate determination
from the amount of precession and nutation. This ratio involves what
we might call, in a general way, the solid ellipticity of the earth,
or the ellipticity of a homogeneous spheroid having the same moments
of inertia as the earth.

"When the differential equations of the earth's rotation are
integrated, there appear two arbitrary constants, representing the
position of any assigned epoch of the axis of rotation relative to
that of figure. Theory then shows that the axis of rotation will
revolve round that of figure, in a period of 306 days, and in a
direction from west toward east. The attempts to determine the value
of these constants have seemed to show that both are zero, or that
the axes of rotation and figure are coincident. Several years since,
Sir William Thomson published the result of a brief computation from
the Washington Prime-Vertical observations of [alpha] Lyrae which I
made at his request and which showed a coefficient 0".05. This
coefficient did not exceed the possible error of the result; I
therefore regarded it as unreal.

[Sidenote: The forgotten assumption.]

"The question now arises whether Mr. Chandler's result can be
reconciled with dynamic theory. I answer that it can, because the
theory which assigns 306 days as the time of revolution is based on
the hypothesis that the earth is an absolutely rigid body. But, as a
matter of fact, the fluidity of the ocean plays an important part in
the phenomenon, as does also the elasticity of the earth. The
combined effect of this fluidity and elasticity is that if the axis
of rotation is displaced by a certain amount, the axis of figure
will, by the changed action of the centrifugal force, be moved
toward coincidence with the new axis of rotation. The result is, that
the motion of the latter will be diminished in a corresponding ratio,
and thus the time of revolution will be lengthened. An exact
computation of the effect is not possible without a knowledge of the
earth's modulus of elasticity. But I think the result of
investigation will be that the rigidity derived from Mr. Chandler's
period is as great as that claimed by Sir William Thomson from the
phenomena of the tides."

[Sidenote: But Chandler's work still mistrusted.]

This was very satisfactory. Professor Newcomb put his finger on the assumption which had been made so long ago that it had been forgotten: and the lesson is well worth taking to heart, for it is not the first time that mistaken confidence in a supposed fact has been traced to some forgotten preliminary assumption: and we must be ever ready to cast our eyes backward over all our assumptions, when some new fact seems to challenge our conclusions. It might further be expected that this discovery of the way in which theory had been defective would as a secondary consequence inspire confidence in the other conclusions which Mr. Chandler had arrived at in apparent contradiction to theory; or at least suggest the suspension of judgment. But Professor Newcomb did not feel that this was possible in respect of the _change_ of period, from about twelve months in Bradley's time to fourteen months in ours. We have seen that Mr. Chandler himself regarded this as a "curious result" requiring confirmation: but since the confirmation was forthcoming, he stated it with full confidence, and drew the following remarks from Professor Newcomb in July 22, 1892:--

"The fact of a periodic variation of terrestrial latitudes, and the
general law of that variation, have been established beyond
reasonable doubt by the observations collected by Mr. Chandler. But
two of his minor conclusions, as enumerated in No. 3 of this volume,
do not seem to me well founded. They are--

"1. That the period of the inequality is a variable quantity.

"2. That the amplitude of the inequality has remained constant for
the last half century."

Professor Newcomb proceeds to give his reasons for scepticism, which are too technical in character to reproduce here. But I will quote the following further sentence from his paper:--

"The question now arises how far we are entitled to assume that the
period must be invariable. I reply that, perturbations aside, any
variation of the period is in such direct conflict with the laws of
dynamics that we are entitled to pronounce it impossible. But we know
that there are perturbations, and I do not see how one can doubt
that they have so acted as to increase the amplitude of the variation
since 1840."

[Sidenote: Chandler's reply.]

In other words, while recognising that there may be a way of reconciling one of the "minor" conclusions with theory, Professor Newcomb considers that in this case the other must go. Mr. Chandler's answer will speak for itself. It was delayed a little in order that he might present an immense mass of evidence in support of his conclusions, and was ultimately printed on August 23, 1892.

"The material utilised in the foregoing forty-five series aggregates
more than thirty-three thousand observations. Of these more than
one-third were made in the southern hemisphere, a fact which we owe
principally to Cordoba. It comprises the work of seventeen
observatories (four of them in the southern hemisphere) with
twenty-one different instruments, and by nine distinct methods of
observation. Only three of the series (XXI., XXV., and XXXV.), and
these among the least precise intrinsically, give results
contradictory of the general law developed in No. 267. This degree of
general harmony is indeed surprising when the evanescent character of
the phenomenon under investigation is considered.

"The reader has now before him the means for independent scrutiny of
the material on which the conclusions already drawn, and those which
are to follow, are based. The space taken in the printing may seem
unconscionable, but I hope this will be charged to the extent of the
evidence collected, and not to diffuseness or the presentation of
needless detail; for I have studiously sought to compress the form of
statement without omitting anything essential for searching
criticism. That it was important to do this is manifest, since the
conclusions, if established, overthrow the existing theory of the
earth's rotation, as I have pointed out on p. 21. I am neither
surprised nor disconcerted, therefore, that Professor Newcomb should
hesitate to accept some of these conclusions on the ground (_A. J._,
No. 271) that they are in such conflict with the laws of dynamics
that we are entitled to pronounce them impossible. He has been so
considerate and courteous in his treatment of my work thus far, that
I am sure he will not deem presumptuous the following argument in
rebuttal.

[Sidenote: He "put aside all teachings of theory," and "is not
dismayed."]

"It should be said, first, that in beginning these investigations
last year, I deliberately put aside all teachings of theory, because
it seemed to me high time that the facts should be examined by a
purely inductive process; that the nugatory results of all attempts
to detect the existence of the Eulerian period probably arose from a
defect of the theory itself; and that the entangled condition of the
whole subject required that it should be examined afresh by processes
unfettered by any preconceived notions whatever. The problem which I
therefore proposed to myself was to see whether it would not be
possible to lay the numerous ghosts--in the shape of numerous
discordant residual phenomena pertaining to determinations of
aberration, parallaxes, latitudes, and the like--which had heretofore
flitted elusively about the astronomy of precision during the
century; or to reduce them to tangible form by some simple consistent
hypothesis. It was thought that if this could be done, a study of the
nature of the forces, as thus indicated, by which the earth's
rotation is influenced, might lead to a physical explanation of them.

"Naturally, then, I am not much dismayed by the argument of conflict
with dynamic laws, since all that such a phrase means must refer
merely to the existent state of the theory at any given time. When
the 427-day period was propounded, it was as inconsistent with known
dynamic law as the variation of it now appears to be. Professor
Newcomb's own happy explanation has already set aside the first
difficulty, as it would appear, and advanced the theory by an
important step. Are we so sure yet of a complete knowledge of all the
forces at work as to exclude the chance of a _vera causa_ for the
second?"

[Sidenote: Faraday's words.]

There is a splendid ring of resolution about these words. Let us compare them with a notable utterance of Faraday:--

"The philosopher should be a man willing to listen to every
suggestion, but determined to judge for himself. He should not be
biassed by appearances; have no favourite hypothesis; be of no
school; and in doctrine have no master. He should not be a respecter
of persons, but of things. Truth should be his primary object. If to
these qualities be added industry, he may indeed hope to walk within
the veil of the temple of Nature."

[Sidenote: Chandler's other work at this time.]

[Sidenote: His ultimate satisfactory solution.]

[Sidenote: Interference of two waves.]

Tested by this severe standard, Mr. Chandler fails in no particular, least of all in that of industry. The amount of work he got through about this time was enormous, for besides the main line of investigation, of which we have only had after all a mere glimpse, he had been able to turn aside to discuss a subsidiary question with Professor Comstock; he had examined with great care some puzzling characteristics in the variability of stars; he computed some comet ephemerides; and he was preparing a new catalogue of variable stars--a piece of work involving the collection and arrangement of great masses of miscellaneous material. Yet within a few months after replying as above to Professor Newcomb's criticism, he was able to announce that he had found the key to the new puzzle, and that "theory and observation were again brought into complete accord." We will as before listen to the account of this new step in his own words, but a slight preliminary explanation may help those unaccustomed to the terminology. The polar motion was found to be compounded of _two_ independent motions, both periodic, but having different periods. Now, the general results of such a composition are well known in several different branches of physics, especially in the theory of sound. If two notes of nearly the same pitch be struck at the same time, we hear the resultant sound alternately swell and die away, because the vibrations caused by the two notes are sometimes going in the same direction, and after an interval are going exactly in opposite directions. Diagrammatically we should represent the vibrations by two waves, as below; the upper wave goes through its period seven and a half times between A and D, the lower only six times; and it is easily seen that at A and C the waves are sympathetic, at B and D antipathetic. At A and C the compound vibration would be doubled; at B and D reduced to insensibility. The point is so important that perhaps a numerical illustration of it will not be superfluous. The waves are now represented by rows of figures as below. The first series recurs after every 6, the second after every 7.

First Wave 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1
Second Wave 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3
-------------------------------------------------------------
Combined Effect 2 4 6 8 7 5 3 3 5 7 7 6 4 4 4 6 6 6 5 5 5 5 5 5 5 6 6 6 4 4 4
Great disturbance. Calm.
-----------------------------------------------------------------------------

First Wave 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2 3 4 3 2 1 2
Second Wave 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3 2 1 2 3 4 4 3
-------------------------------------------------------------

Combined Effect 6 7 7 5 3 3 5 7 8 6 4 2 4 6 8 7 5 3 3 5 7 7 6 4 4 4 6 6 6 5 5
Great disturbance.

[Sidenote: Illustration from ocean travel.]

Adding the two rows together, the oscillations at first reinforce one another and we get numbers ranging from 2 to 8 instead of from 1 to 4; but one wave gains on the other, until it is rising when the other is falling, and the numbers add up to a steady series of 5's. It will be seen that there are no less than seven consecutive 5's, and all the variation seems to have disappeared. But presently the waves separate again, and the period of great disturbance recurs; it will be seen that in the "combined effect" the numbers repeat exactly after the 42nd term. Now those unfamiliar with the subject may not be prepared for the addition of one physical wave to another, as though they were numbers, but the analogy is perfect. Travellers by some of the fast twin-screw steamers have had unpleasant occasion to notice this phenomenon, when the engineer does not run the two screws precisely at the same speed; there come times when the ship vibrates violently, separated by periods of comparative stillness. Instances from other walks of life may recur to the memory when once attention is called to the general facts; but enough has been said to explain the point numbered (2) in the subjoined statement. To understand the rest, we must remember that if the two waves are not equal in "amplitude," _i.e._ if the backward and forward motion is not the same in both, they cannot annul one another, but the greater will always predominate. Those interested in following the matter further should have no difficulty in constructing simple examples to illustrate such points. We will proceed to give Mr. Chandler's statements:--

[Sidenote: Chandler's final formulae.]

"We now come upon a new line of investigation. Heretofore, as has
been seen, the method has been to condense the results of each series
of observations into the interval comprised by a single period, then
to determine the mean epoch of minimum and the mean range for each
series, and, finally, by a discussion of these quantities, to
establish the general character of the law of the rotation of the
pole. It is now requisite to analyse the observations in a different
way, and discover whether the deviations from the general provisional
law, in the last column of Table II., are real, and also in what
manner the variation of the period is brought about. The outcome of
this discussion, which is to be presented in the present paper, is
extremely satisfactory. The real nature of the phenomenon is most
distinctly revealed, and may be described as follows:--

"1. The observed variation of the latitude is the resultant curve
arising from two periodic fluctuations superposed upon each other.
The first of these, and in general the more considerable, has a
period of about 427 days, and a semi-amplitude of about 0".12. The
second has an annual period with a range variable between 0".04 and
0".20 during the last half-century. During the middle portion of this
interval, roughly characterised as between 1860 and 1880, the value
represented by the lower limit has prevailed, but before and after
those dates, the higher one. The minimum and maximum of this annual
component of the variation occur at the meridian of Greenwich, about
ten days before the vernal and autumnal equinoxes respectively, and
it becomes zero just before the solstices.

"2. As the resultant of these two motions, the effective variation of
the latitude is subject to a systematic alternation in a cycle of
seven years' duration, resulting from the commensurability of the two
terms. According as they conspire or interfere, the total range
varies between two-thirds of a second as a maximum, to but a few
hundredths of a second, generally speaking, as a minimum.

"3. In consequence of the variability of the coefficient of the
annual term above mentioned, the apparent average period between 1840
and 1855 approximated to 380 or 390 days; widely fluctuated from
1855 to 1865; from 1865 to about 1885 was very nearly 427 days, with
minor fluctuations; afterwards increased to near 440 days, and very
recently fell to somewhat below 400 days. The general course of these
fluctuations is quite faithfully represented by the law of eq. (3),
(No. 267), and accurately, even down to the minor oscillations of
individual periods, by the law of eq. (15), hereafter given, and
verbally interpreted above. This law also gives a similarly accurate
account of the corresponding oscillations in the amplitude. The
closeness of the accordance between observation and the numerical
theory, in both particulars, places the reality of the law beyond
reasonable doubt."

Those who cannot follow the details of the above statement will nevertheless catch the general purport--that the difficulties felt by Professor Newcomb have been surmounted; and this is made clearer by a later extract:--

"A very important conclusion necessarily follows from the agreement
of the values of the 427-day term, deduced from the intervals between
the consecutive values of T in Table XII., namely, that there has
been no discontinuity in the revolution, such as Professor Newcomb
regarded as so probable that he doubted the possibility of drawing
any conclusions from the comparison of observations before and after
1860 (_A. J._, 271, p. 50).

[Sidenote: Theory must go, if it will not fit observation.]

"The present investigation demonstrates that the way out of the
apparently irreconcilable contradiction of theory and observation in
this matter does not lie in the direction of discrediting the
observations, as he is inclined to do. On the contrary, the result is
a beautiful vindication of the trustworthiness of the latter, and, at
the same time, of the theory that demands an invariable rate of
motion; providing a perfectly fitting key to the riddle by showing
that another cause has intervened to produce the variability of the
period. I feel confident that Professor Newcomb will agree with the
reality of the explanation here set forth, and will reconsider his
view that the perturbations in the position of the Pole must be of
the nature of chance accumulations of motion, a view which he then
considered necessary to the maintenance of the constancy in the
period of latitude-variation."

[Sidenote: The final paper.]

The paper from which these words are taken appeared on November 4, 1892. The next paper on the main theme did not appear till a year later, though much work was being done in the meantime on the constant of aberration and other matters arising immediately after the discovery. On November 14, 1893, Mr. Chandler winds up the series of eight papers "On the Variation of Latitude," which he had commenced just two years before. His work was by no means done; rather was it only beginning, for the torch he had lit illuminated many dark corners. But he rightly regarded his discovery as now so firmly established that the series of papers dealing with it as still under consideration might be terminated. In this final paper he first devotes the most careful attention to one point of detail. He had shown earlier in the series that the North Pole must be revolving from West to East, and not from East to West; but this was when the motion was supposed to be simple and not complex, and it was necessary to re-examine the question of direction for each of the components. After establishing conclusively that the original direction holds for each of the components, he almost apologises for the trouble he has taken, thus:--

"It is therefore proved beyond reasonable doubt that the directions
of the rotations is from West to East in both elements; whence the
general form of the equation for the variation of latitude adopted in
_A. J._, 284, p. 154, eq. (19). It may be thought that too much pains
have been here bestowed upon a point which might be trusted to theory
to decide. I cannot think so. One of the most salient results of
these articles has been the proof of the fact that theory has been a
blind guide with regard to the velocity of the Polar rotation,
obscuring truth and misleading investigators for a half a century.
And even if we were certain, which we are not, that the fourteen
months' term is the Eulerian period in a modified form. It would
still be necessary to settle by observation the direction of the
annual motion, with regard to which theory is powerless to inform us.
To save repetition of argument, I must refer to the statement in _A.
J._, 273, pp. 68, 70, of the principles adopted in beginning these
inquiries in 1891."

Finally, he answers one of the few objectors of eminence who still lingered, the great French physicist Cornu:--

[Sidenote: Cornu answered.]

"The ground is now cleared for examination of the only topic
remaining to be covered, to establish, upon the foundation of fact,
every point in the present theory of these remarkable movements of
the earth's axis. This is the question of the possibility that these
movements are not real, but merely misinterpretations of the observed
phenomena; being in whole or in part an illusory effect of
instrumental error due to the influence of temperature. Such a
possibility has been a nightmare in practical astronomy from the
first, frightening us in every series of unexplained residuals,
brought to light continually in nearly all attempts at delicate
instrumental research. A source of danger so subtile could not fail
to be ever present in the mind of every astronomer and physicist who
has given even a superficial attention to the question of the
latitude variations, and there is no doubt that some are even now
thus deterred from accepting these variations as proved facts.
Perhaps the most explicit and forcible statement of the doubts that
may arise on this subject has been given very recently by Mr. Cornu.
The views of so distinguished a physicist, and of others who are
inclined to agree with him, call for careful attention, and cannot be
neglected in the present closing argument upon the theory presented
in these articles. It is unnecessary, for the purpose of disposing of
objections of the sort raised by Cornu, to insist that it is not
sufficient to show that the observed variations, attributed to the
unsteadiness of the Earth's Pole, are near the limit of precision
attainable in linear differential measures, and in the indication of
the direction of gravity by means of the air bubble of the level; or
to show that there are known variations in divided circles and in
levels, dependent on temperature and seasons. Nor need we require of
objectors the difficult, although essential, task--which they have
not distinctly attempted--of showing that these errors are not
eliminated, as they appear to be, by the modes in which astronomers
use their instruments. Neither need we even urge the fact that a
large portion of the data which have been utilised in the present
researches on the latitude were derived by methods which dispense
with levels, or with circles, a part of them indeed with both, and
yet that the results of all are harmonious. On the contrary, let us
admit, although merely for argument's sake, that all the known means
of determining the direction of gravity--including the plumb-line,
the level, and a fluid at rest, whether used for a reflecting surface
or as a support for a floating instrument--are subject to a common
law of periodical error which vitiates the result of astronomical
observation, obtained by whatever methods, and in precisely the same
manner. Now, the observed law of latitude variation includes two
terms, with periods of fourteen and twelve months respectively. Since
the phases of the first term are repeated at intervals of two months
in successive years, and hence in a series of years come into all
possible relations to conditions of temperature dependent on season,
the argument against the reality of this term, on this ground,
absolutely fails, and needs no further notice. As to the second, or
annual term, while the phases, as observed in any given longitude,
are indeed synchronical with the seasons, they are not so as regards
different longitudes. If, therefore, the times of any given phase, as
observed in the same latitude, but in successively increasing
longitudes, occurred at the same date in all of them, there would be
a fatal presumption against the existence of an annual period in the
polar motion. If, on the contrary, they occur at times successively
corresponding to the differences of longitude, the presumption is
equally fatal to the hypothesis that they can possibly be due to
temperature variation as affecting instrumental measurement. But the
facts given in the foregoing section correspond most distinctly to
the latter condition. Therefore, unless additional facts can be
brought to disprove successively these observed results, we may
dismiss for ever the bugbear which has undoubtedly led many to
distrust the reality of the annual component of the
latitude-variation, while they admit the existence of the 427-day
term."

[Sidenote: Consequences of the discovery.]

[Sidenote: Suspected observers acquitted.]

At this point we must leave the fascinating account of the manner in which this great discovery was established, in the teeth of opposition such as might have dismayed and dissuaded a less clear-sighted or courageous man. It is my purpose to lay more stress upon the method of making the discovery than upon its results; but we may afford a brief glance at some of the consequences which have already begun to flow from this step in advance. Some of them have indeed already come before us, especially that large class represented by the explanation of anomalies in series of observations which had been put aside as inexplicable. We have seen how the observations made in Russia, or in Washington, or at Greenwich, in all of which there was some puzzling error, were immediately straightened out when Chandler applied his new rule to them. We in England have special cause to be grateful to Chandler; not only has he demonstrated more clearly than ever the greatness of Bradley, but he has rehabilitated Pond, the Astronomer Royal of the beginning of the nineteenth century; showing that his observations, which had been condemned as in some way erroneous, were really far more accurate than might have been expected; and further he has shown that the beautiful instrument designed by Airy, and called the Reflex Zenith Tube, which seemed to have unaccountably failed in the purpose for which it was designed, was really all the time accumulating observations of this new phenomenon, the Variation of Latitude. Instead of Airy having failed in his design, he had in Chandler's words "builded better than he knew."

[Sidenote: Constant of Aberration improved.]