Conical Refraction
Chapter XIII of Life of Sir William Rowan Hamilton
(Vol. 1, Hodges Figgis, Dublin, 1882).
By Robert Perceval Graves.
In one of Hamilton's letters to Lord Adare he speaks of having copied
out his Third Supplement to his Theory of Systems of Rays as many as
ten times, in the endeavour to perfect it; but this letter does not
record that, while thus giving final shape to his work, he had arrived
by means of his general method at an optical result of a most
remarkable nature. This, however, was the fact. He had made the
theoretical discovery of Conical Refraction. And when he presented
the concluding part of his Third Supplement to the Royal Irish Academy
on the 22nd of October, 1832, it contained a statement of the
discovery, which he then orally announced. Of the position in optical
science of this discovery, the unscientific reader will gather a
correct notion from the following passage which I reproduce from the
memoir published in the Dublin University Magazine of January,
1842.
`The law of the reflexion of light at ordinary mirrors appears to have
been known to Euclid; that of ordinary refraction at a surface of
water, glass, or other uncrystallized medium, was discovered at a much
later age by Snellius; Huyghens discovered, and Malus confirmed, the
law of extraordinary refraction produced by uniaxal crystals, such as
Iceland spar; and finally the law of the extraordinary double
refraction at the faces of biaxal crystals, such as topaz or
arragonite, was found in our own time by Fresnel. But even in these
cases of extraordinary or crystalline refraction, no more than
two refracted rays had ever been observed or even suspected to
exist, if we except a theory of Cauchy, that there might possibly be a
third ray, though probably imperceptible to our senses.
Professor Hamilton, however, in investigating by his general method
the consequences of the law of Fresnel, was led to conclude that there
ought to be in certain cases, which he assigned, not merely two, nor
three, nor any finite number, but an infinite number, or a
cone of refracted rays within a biaxal crystal,
corresponding to and resulting from a single incident ray; and
that in certain other cases, a single ray within such a crystal should
give rise to an infinite number of emergent rays, arranged in a
certain other cone. He was led, therefore, to anticipate from theory
two new laws of light, to which he gave the names of Internal and
External Conical Refraction.'
So sure was Hamilton's grasp of his mathematical results, and of the
necessary correspondence with them of physical phenomena (the truth of
the undulatory theory being supposed), that on the day succeeding the
above-mentioned meeting of the Royal Irish Academy, he requested his
friend Mr. Lloyd, afterwards Provost of Trinity College, Dublin, and
then Professor of Natural Philosophy, to institute experiments for the
purpose of verifying his theoretical anticipations. The task was
promptly undertaken, and besides the letters between Hamilton and
Lloyd which record its progress, others from Hamilton to Airy and
Herschel, with their replies, are in existence, which are of great
interest. Being full of mathematical formulae, they are more suited
for a collection of the scientific correspondence of the subject of
this memoir, which I hope may some day see the light, than for the
present work. Here it must suffice to give an outline of their
contents, indicating the history of the discovery and its
verification, and one or two letters of general statement.
The earliest letter of the series which remains is Hamilton's reply to
Lloyd's inquiry respecting the angle of the cone, for arragonite, in
the case of external conical refraction. It commences thus:-
`November 3, 1832, Saturday morning. - Mrs. Hemans
and some of the young Graveses came here yesterday evening, just as I
had finished my calculation respecting the arragonite, and I had only
time to write as answer, ``3°.'' [1]
I showed the cabalistic
note to Mrs. Hemans, and she admitted that we professors had attained
the perfection of letter-writing.'
He then enters upon a consideration of some of Lloyd's observations in
comparison with his own results of theoretical calculation, and
prepares Lloyd for finding that the cone would not be exactly
circular. On the following day he suggests to Lloyd an easy
experimental verification by means of a slit in a card.
On the 25th of October, Hamilton had written to Airy, offering to
propose him as an Honorary member of the Royal Irish Academy, and
stated in general terms that he had arrived at some new results from
Fresnel's theory. On the 4th November Airy replies -
`I am much obliged by your note of October 25. I should highly value
the honour of being a member of your Academy, and I should esteem it
much more because it originated with you.... I shall be glad in
time to hear of the new results of Fresnel's theory which you allude
to.'
On the 6th November, Lloyd reports some unsuccessful experiments, and
his intention to try another way of his own devising, and also that
suggested by Hamilton, but concludes by saying, `I almost despair of
doing anything with so thin a plate [of arragonite].'
On the 10th of November Hamilton writes thus to Lloyd:-
`Just after the evening when I gave to the Royal Irish Academy an
account of my last optical results, I wrote to Professor Airy, and
among other things I mentioned that I had arrived at a new consequence
from Fresnel's theory, without stating what that consequence was. I
now enclose a letter received from him yesterday, in which he
expresses a wish to be informed of it; and if you should, as you
seemed to think likely, be prevented by want of apparatus or of
leisure from making soon any decisive experiment on the point, I
believe it will be well to mention the theoretical result to Airy.'
To this letter Professor Lloyd sent the following reply. It is
impossible not to be struck by the pure unselfish zeal for science
which it displays.
`I fear it would be wholly impossible to obtain experimentally any
decisive result connected with your theoretical conclusion, without
better means than I have at present at my disposal. The angle of
divergence produced by diffraction in the minutest apertures, when
they are so close as they must be in my specimen, is far greater than
the angle we seek. The specimens I showed you the other day are fine,
but I find they belong to a form of crystallization which the
mineralogists term macled, that is, in fact, they are composed
of several distinct crystals crossing each other. They would be
therefore wholly unfit for the purpose. I am quite sure your
conclusion can be readily tested by anyone having access to fair
specimens; but as that is not the case here, you had better refer the
matter to Airy, or some one else, as soon as possible.''
But happily the honour of bringing these experiments to a successful
termination was not to pass from Professor Lloyd. Within a few days
he had procured a better specimen of the required crystal, and he has
the pleasure of thus writing to Hamilton:
`Trinity College, December 14. - Dear Hamilton, I
write this line to say that I have found the cone. At least I
have almost no doubt on the subject; but must still verify it by
different methods of observation.
`I have no time to say more at present than that I observed it in a
fine specimen of arragonite which I received from Dollond in London
since I saw you last.'
On the 18th of December Hamilton communicated this verification of his
theoretical anticipation to both Airy and Herschel. I give a
transcript of his letter to the last; it is an interesting though not
a full statement of the discovery and the verification.
From W. R. Hamilton to
Sir J. F. W. Herschel.
`Dublin Observatory, December 18, 1832.
`You are aware that the fundamental principle of my optical methods
does not essentially require the adoption of either of the two great
theories of light in preference to the other. However I naturally
feel an interest in applying my general methods to Fresnel's theory of
biaxal crystals; and when in October I was finishing my Third
Supplement for the Royal Irish Academy, I deduced, from such
application, some results respecting the focal lengths and aberrations
of lenses formed of such crystals. In the course of these
calculations I was led to transform in various ways Fresnel's law of
velocity, or, in other words, to study his curved wave: and I found,
what he seems to have not suspected, that the wave has 1st, four
cusps (at the ends of the optic axes) at each of which the tangent
planes are (not, as he thought, two, but) infinite in number;
and 2nd, four circles of plane contact, along each of which the
wave is touched, in the whole extent of the circle, by a plane
(parallel to one of the circular sections of the surface of
elasticity); somewhat as a plum can be laid down on a table so as to
touch and rest on the table in a whole circle of contact, and has, in
the interior of the circular space, a sort of conical cusp. Hence I
was led to expect that under certain circumstances, easily deduced
and assigned by me from these geometrical properties, a single
incident and unpolarized ray would undergo not double but
conical refraction. I announced this expectation to the Royal
Irish Academy at their monthly meeting in October, when I was giving
an account of the results of my Third Supplement; and I applied to
Professor Lloyd, son of our Provost here, to submit the matter to
experiment. For some time he could do nothing decisive, not having
any biaxal crystal of sufficient size and purity; but having lately
obtained from Dollond a fine piece of arragonite, and having treated
it according to my theoretical indications, he has perceived a curious
and beautiful set of new phenomena, which, so far as they have yet
been examined, appear to agree with the theory, and at any rate are
worthy of study. I thought this intelligence would interest you, and
I am,' &c.
On the same day Lloyd writes to Hamilton as follows:-
`Tuesday, 3 o'clock; College. - I am happy to tell you that
since I saw you this morning I succeeded in projecting the cone on a
screen of roughened glass, and observing a section of it so large as
two inches in diameter; you will easily conceive that the phenomenon
is most striking. The appearance is exactly the same as that we saw
when looking through the aperture. Its deviation from an exact
circle, however, is of course more distinctly seen. I traced the
boundary of this section on the screen, and then measured the distance
as accurately as I could. Three such measurements gave me for the
angle of the cone 6° 24´, 6° 22´, 5° 56´, which
you see are tolerably near. The mean (6° 14´) corresponds
pretty well with the measurements of the extreme circle, taken
yesterday. The difference between it and the theoretical result is
probably the effect of diffraction, and I must now try and correct for
this perturbation. This mode of exhibiting the phenomenon is decisive
as well as beautiful, and I am sure you will be glad to see it when
you next come in to town.'
On the 23rd of December, Professor Airy writes:-
`I have duly received your letter concerning double refraction, and
that informing me of my election as Honorary Member of the Royal Irish
Academy (of which I had not received an official notice). I beg you
to say to the authorities of that Body that I am very much gratified
with the honour which they have done me, and I hope it may prove the
cause of greater personal acquaintance with many of its members than I
at present possess.
`I am very much interested with your discovery of the circular
contact of the tangent plane with Fresnel's double wave surface. I
was well aware (a long time ago) that the point of the surfaces, which
in the principal section is the intersection of the circle and the
ellipse, is in the surfaces the meeting of two dimples
(external and internal), and that these dimples near their point of
meeting become ultimately two opposite cones; the outer one diverging
in a sort of trumpet mouth. But I had no idea that the mouth of the
trumpet could be touched by one plane. Now as to the consequences of
this I am extremely puzzled.... Arragonite is a bad substance, I
should imagine; I should think topaz likely to make a wider
cone; [2] perhaps your
formulae will show you at once. Let me beg you to communicate as
soon as possible (if Professor Lloyd does not object) the phenomena
which he has observed. I have to thank him for a copy of his
excellent optical treatise.'
I regret that I have not been able to find the letter from Professor
Lloyd and its enclosure (presumably a note from Mr. Mac Cullagh,
F.T.C.D.), to which the following important letter is a reply.
From W. R. Hamilton to
Professor Lloyd.
`Observatory, January 1, 1833.
`I have just received your letter and the enclosed note and write in
some haste. Mr. Mac Cullagh's last conclusion that the conical
refraction at emergence required the internal ray along the optic axis
to be unpolarized, or to be formed by the superposition of rays
polarized in infinitely various planes, is exactly the same with the
conclusion which I had formed in October, and I distinctly remember
mentioning it to you in our interview on the 23rd of that month; and
it was for that reason I wished to have the luminous point in contact
with the crystal. But I have not yet tried to determine the exact law
connecting the internal plane of polarization of an internal polarized
ray with the position of the corresponding emergent ray of the cone,
though the determination will not be difficult, and the result
probably very nearly the same as in that other connected
question [3]
which we talked of the other day, and which we had both resolved by
different methods. What has hindered me from setting about this
little problem has been my being much engaged and interested in
Cauchy's theory of light. [4] As to the finite
magnitude of the emergent cone, for
a single internal common ray, I certainly expect a finite magnitude,
that is, a finite angle (though the cone of rays is not of revolution
nor even of the second degree), but not a conical shell of finite
thickness, such as one may consider as approximately resulting from a
finite but thin internal cylinder of rays. As this last is nearly the
case of the experiment, there must no doubt be a thickness in the cone
of the order of the aperture besides the angular divergence; and this
may, as you say, account for part of what you observed, but I scarcely
think it will account for the whole. It is much for theory to have
predicted the facts of conical refraction, but I suspect that
the exact laws of it depend on things as yet unknown. You see
my pleasure at perceiving so great a confirmation of theory does not
make me sanguine enough to believe as yet the coincidence absolute and
rigorous. As to rays inclined a little to the optic axis all round,
it was in fact from considering them and passing to the limit that I
first deduced my expectation of conical refraction. When you are
drawing up your Paper I shall be glad if your plan leads you (when you
are speaking of my having requested you to try experiments) to mention
distinctly the following facts, which constitute all my merit, such as
it is, on the subject.
-
`I announced to you on the 23rd of October last, having on the
preceding evening announced to a general meeting of the Royal Irish
Academy, that I had discovered two new geometrical properties
of Fresnel's wave; one property being the existence of four
conoidal cusps at the intersections of circle and ellipse in
the plane of the greatest and least axes; and the other property being
the existence of four finite circles of plane contact, each of
the four planes of these circles being parallel to one of the two
circular sections of the surface of elasticity.
-
`I announced to you on the same day, and had done so to the Academy on
the evening before, my expectation of a new kind of refraction,
namely conical refraction, which ought to happen in two
distinct cases; one at emergence, when a single ray of light from a
point within a biaxal crystal proceeded along an optic axis
(from centre to cusp of Fresnel's wave) and then emerged; the
other at entering when a single ray of common light from a
point without falls on a biaxal crystal and enters so
that the plane wave within, or the tangent plane to the curved wave
within, is parallel to either of the two circular sections of the
surface of elasticity.
-
`I requested you to try experiments to confirm or refute the
theoretical expectations which I had deduced from Fresnel's
principles.
`You intended I know to mention the third, but you might not have
thought of distinctly putting the two others on record, which yet may
save some controversy with others hereafter. I expect on Thursday
evening to leave the neighbourhood of Dublin for a few days, but to
return early next week.'
That Hamilton was ready to make known the work done by Mac Cullagh in
the same field with himself is proved by the following passage in a
letter of his to Professor Airy, written, as a shorthand draft of it
shows, a few days later, viz., January 4, 1833.
`I hear from Lloyd that Mac Cullagh (another of our young Fellows, a
Paper by whom I once showed you) had deduced the same results by his
geometrical methods, having however previously heard of my theory of
conical refraction.' [5]
This letter to Airy communicated at length the results of Professor
Lloyd with respect to external conical refraction, together with some
views of Hamilton's own as to the `vibrations', `interference,' and
`polarization,' involved in the experiment.
In Professor Airy's answer, after referring to polarization, he
expresses strongly his conviction that if the phenomenon of external
conical refraction be true in fact, it has no connexion with the
theory of Hamilton. He then ably sketches what he considers possible
results, but shows that he has misconceived Hamilton's statement.
To this letter Hamilton sent a reply on the 21st of January, and, not
hearing in return from Airy, another on the 1st of February; in these
letters he modestly, and it may be in accordance with the fact,
supposes that some ambiguity in his own expressions may have caused
his correspondent's failure correctly to appreciate the results
arrived at by himself and Professor Lloyd, [6]
and after re-stating and explaining them he quietly adds, `I
believe that if you consider the thing you will come to the same
conclusion with me.' On the very day on which he had despatched the
second of these letters Hamilton received from Professor Airy a
letter dated January 28, which handsomely acknowledged that he
had been convinced by Hamilton's explanation; the following are
its terms:-
From Professor Airy to
W. R. Hamilton.
`Allow me to thank you for your last note, which is all comprehensible
and all true; and if I had not been very dull, I might perhaps have
guessed at some of it before. You had not mentioned to me anything
about the cusp-ray, and therefore there were parts of the previous
letter which were altogether mysterious to me, and were likely to
remain so, except I could divine or you explain.'
It will be seen [7] that, not long after, Professor
Airy followed up this private amende by a public testimony, still
fuller, though couched in fewer terms, to the character of Hamilton's
discovery as a scientific feat.
The following letter of this date to Herschel is so clear a statement
of almost everything connected with this discovery that I feel I ought
not to suppress it, though aware that its production involves some
repetition.
From W. R. Hamilton to
Sir John Herschel.
`Observatory, January 29, 1833.
`My dear Sir,
`Professor Lloyd read to the Royal Irish Academy, last night, a paper
``On the Phenomena presented by Light in its Passage along the Axes of
Biaxal Crystals,'' in which he gave an account of some recent
additional experiments, confirming my theoretical conclusions
respecting Conical Refraction. Those conclusions were chiefly the
following:- 1. A single plane wave within a biaxal crystal,
parallel to a circular section of the surface of elasticity,
corresponds in general to an infinite number of internal
ray-directions; in such a manner that a single incident ray in air
will give an internal cone of rays (of the 2nd degree), and will
emerge (from a plane face) as an external cylinder of rays, if the
external incident wave have that direction which corresponds to the
foregoing internal wave. In this kind of internal conical
refraction one refracted ray of the cone is determined by the
ordinary law of the sines, using the mean index 1/b; and the
greatest angular deviation in the cone, from this ray, is in the plane
of the optic axes, and is
for ray E in arragonite, if we use Rudberg's elements. Professor
LLoyd has lately observed an emergent cylinder corresponding to this
theory, from his measures upon which the angle of the cone appeared to
be 1° 52´. He used a fine piece of arragonite, procured from
Dollond, thickness = 0.49 inch; the incident ray was of solar light,
and it passed through two small holes, the first in a screen at some
distance from the crystal, the second in a thin metallic plate,
adjoining the first surface of the crystal; the emergent cylinder of
rays was received on silver paper, and produced on the paper a small
white annulus of which the size was the same at different distances of
the paper from the arragonite. The emergent light was polarized
according to a law which agrees with Fresnel's principles. Great care
was necessary in the adjustment of the holes; when the adjustment was
slightly disturbed, two opposite quadrants of the circle appeared more
faint than the two others, and the two pairs were of complementary
colours.
2. `I conclude also, from Fresnel's principles, that a single
interior cusp-ray (often called an optic axis, but not
normal to a circular section of the surface of elasticity, and on the
contrary normal to a circular section of Fresnel's ellipsoid - one of
those two rays of which each has but a single value for the velocity
of light along it -) ought, on emerging into air, to undergo,
not bifurcation as Fresnel thought, but (external)
conical refraction. If the internal
incidence be perpendicular, the
equation in rectangular co-ordinates of the emergent cone may be put
under the form
for ray E, with Rudberg's elements for arragonite; this cone
therefore is of the 4th degree (whereas the internal cone was of the
2nd), but it does not differ much from a circular cone. In Professor
Lloyd's experiments the normal to the refracting face was Fresnel's
axis a, bisecting the acute angle between the two cusp-rays, and the
internal incidence was about 10°; which made the theoretical
angle of the emergent cone somewhat more than 3° instead of
2° 57´. He has sent to the Annals of Philosophy
[8] a sketch of his experimental results
which appear to agree sufficiently with the theory, as to the position
and magnitude and polarization of the emergent cone, in this
external conical refraction. More lately he has taken new
measures which appear to agree still better; and he has made those
experimental verifications, which I have attempted in this letter to
describe, of the other (the internal) kind of conical
refraction. The appearances in direct vision, or when the light is
received on a screen, are interesting enough, and vary prettily with
the shape and size of the aperture, in the phenomena of external
conical refraction. Figures will be given in the fuller memoir in the
Transactions of our Irish Academy.
`The experimental establishment of these new consequences from
Fresnel's principles, must, I think, be considered as interesting. My
Third Supplement, in which, besides endeavouring in other ways to
perfect my optical methods, I treat of the connexion of my
mathematical view with the undulatory theory of light, is in the
press, but gets on very slowly. Whenever it is printed, which can
scarcely be in less than two or three months, I shall present you with
a copy. Meanwhile believe me,' &c.
In the February and March numbers of the London and Edinburgh
Philosophical Magazine, pp. 112 and 207, were contained two Papers
giving Professor Lloyd's earliest published account of his
experiments, the first of them describing external, the second,
internal conical refraction. They prove that Hamilton was fortunate
in his coadjutor. The conduct of the experiments called for much
ingenuity in devising physical arrangements and the utmost nicety of
observation; and these Papers furnish full evidence of the exercise of
both by Professor Lloyd. They show also that he was more than a mere
verifier; he took note of a phenomenon that had not been predicted,
and ascertained the law to which it conformed. When investigating the
case of external conical refraction, he discovered, by observation
with a tourmaline plate, that all the rays of the cone were polarized
in different planes, and detected the remarkable law that `the angle
between the planes of polarization of any two rays of the cone is half
the angle contained by the planes passing through the rays themselves
and its axis': this law he also proved to be a necessary consequence
of Fresnel's theory. Upon the phenomenon being communicated to
Hamilton, he likewise, by means of his own methods, deduced the same
law from the theory, and subsequently predicted the corresponding
phenomenon in the case of internal conical refraction together with
its analogous law. In this latter case, the prediction of the
phenomenon and its law received its experimental verification at the
hands of Professor Lloyd: in the former case, it has been seen, he had
observed the unpredicted phenomenon, and had preceded Hamilton in
deducing its law from theory. It has become necessary thus distinctly
to put on record the amount of credit due to Professor Lloyd in this
particular, because it has been overlooked by Professor Tait in the
lucid account of the discovery which is contained in his article on
Hamilton in the North British Review of September, 1866. The
omission arose very naturally from the circumstance that these laws
for polarization in both kinds of conical refraction are given in
Hamilton's Third Supplement, which was communicated to the Academy
previously to Lloyd's researches, but which remained unpublished (as
appears from the Introduction) for many subsequent months. The
correspondence in my hands proves that the part of the paper
concerning polarization must have been inserted at a date subsequent
to the 2nd January, 1833. But that Hamilton was willing to leave with
Lloyd the credit of the priority which has been here assigned to him
is proved by the fact, which I have received on the best authority,
that he requested and obtained permission to circulate the private
copies of his friend's paper (in which the above-mentioned facts are
recorded) along with those of his own memoir.
In the XVIIth volume of the Transactions of the Royal Irish
Academy, Part I., which was published in the summer of 1833, may be
found both Hamilton's Third Supplement, containing his theoretical
discovery of Conical Refraction, and Professor Lloyd's perfected
account of the experimental manifestations of both kinds of it,
accompanied by plates of diagrams representing the phenomena. To
these Papers the scientific reader is referred for full information on
the subject. They link the names of Hamilton and Lloyd in an enduring
bond.
I may fitly conclude this statement by again borrowing a passage from
the memoir of Hamilton, published in the Dublin University
Magazine for January, 1842:-
`This result excited at the time a very considerable sensation among
scientific men in England and on the Continent; it was thought a happy
boldness to have thus seized and brought forth into view, by dint of
reasoning, a new class of phenomena, to which nothing similar had been
before observed, and which even seemed, in the words used by an
eminent English philosopher, to be ``in the teeth of all analogy.''
At the Cambridge meeting of the British Association, in 1833, the
attention of the mathematical and physical section was largely given
to the subject, and Herschel, Airy and others,
spoke warmly in praise
of the discovery. In the introductory discourse with which the
proceedings of that meeting were opened, Professor Whewell made it a
topic, and expressed himself in the following words:- ``In the way of
such prophecies, few things have been more remarkable than the
prediction, that under particular circumstances a ray of light must be
refracted into a conical pencil, deduced from the theory by Professor
Hamilton, and afterwards verified experimentally by Professor
Lloyd.'' [9] Previously, in the same year,
Professor Airy had publicly
recorded his impression upon the subject as follows:- ``Perhaps the
most remarkable prediction that has ever been made is that lately made
by Professor Hamilton.'' [10] More lately, Professor
Plücker, of Bonn, in an article on the general form of luminous
waves, published in the nineteenth volume of Crelle's Journal,
has used these words:- ``Aucune experience de physique a fait tant
d'impression sur mon esprit, que celle de la refraction conique. Un
rayon de lumière unique entrant dans un crystal et en sortant
sous l'aspect d'un cone [11] lumineux: c'etait une chose inouie et sans aucune analogie.
Mr. Hamilton l'annonça, en partant de la forme de l'onde, qui
avait été deduite par des longs calculs d'une theorie
abstraite. J'avoue que j'aurois désesperé de voir
confirmé par l'experience un résultat si extraordinaire,
prédit par la seule théorie que las genie de Fresnel avait
nouvellement créée. Mais Mr. Lloyd ayant
démontré
que les experiences etaient en parfaite concordance avec les
predictions de Mr. Hamilton, tout préjugé contre une
théorie si merveilleusement soutenue, a dû disparaitre.''
And it seems to be in part to this subject that reference is made in a
passage of the article, attributed to Sir John Herschel, on the
Inductive Sciences, in the number for last June [1841] (p. 233) of the
Quarterly Review, where mention is made of ``a sound induction
enabling us to predict, bearing not only stress, but torture: of
theory actually remanding back experiment to read her lesson anew;
informing her of facts so strange, as to appear to her impossible, and
showing her all the singularities she would observe in critical cases
she never dreamed of trying.'' '
In the Bridgewater Treatise of Mr. Babbage the author not only bears
his testimony to the merits of Hamilton and Lloyd, but manifests his
appreciation of the remarkable character of the discovery by weaving
it as a typical example into the argument of his book. It has more
recently been characterized as in its own sphere to be classed with
that prediction of the existence of the planet Neptune which has
immortalized the names of Adams and Le Verrier. Yet it will be seen
by his letter to Coleridge of February 3, 1833, that Hamilton himself
looked upon this and all similar predictions as `a subordinate and
secondary result,' when compared with the object he had in
view, - `to introduce harmony and unity into the contemplations and
reasoning of optics, regarded as a branch of pure science.'
Appendix:
Note on Conical Refraction: Hamilton and Mac Cullagh
As I find that the relative positions of Hamilton and Professor Mac
Cullagh in regard to the discovery of Conical Refraction are still,
from time to time, matter of discussion, I feel it necessary to add as
a note the following statement:-
To the August number of the Philosophical Magazine for 1833
(p. 114) was communicated a Paper by Mr. Mac Cullagh entitled, `Note
on the subject of Conical Refraction,' which commences with the
following paragraphs:-
`When Professor Hamilton announced his discovery of Conical
Refraction, he does not seem to have been aware that it is an obvious
and immediate consequence of the theorems published by me, three years
ago, in the Transactions of the Royal Irish Academy, vol. xvi.,
part ii., p. 65, &c. The indeterminate cases of my own theorems
which, optically interpreted, mean conical refraction, of course
occurred to me at the time, but they had nothing to do with the
subject of that Paper; and the full examination of them, along with
the experiments they might suggest, was reserved for a subsequent
essay, which I expressed my intention of writing. Business of a
different nature, however, prevented me from following up the inquiry.
`I shall suppose the reader to have studied the passage in pp. 75,
76, of the volume referred to. he will see that when the section of
either of the two ellipsoids employed there is a circle, the
semiaxes - answering to OR, Or and to OQ, Oq, in the general
statement - are infinite in number, giving of course an
infinite number of corresponding rays. And this is conical
refraction.'
The note then gives geometrical deductions from his previously
published geometrical theorems which correspond with the two cases of
conical refraction.
Hamilton was hurt by the terms in which the first of these paragraphs
was couched; he meditated a reply to it, and informed Professor Lloyd
of his intention. From the latter, early in the month, he received
the following reply:-
From Professor Lloyd to
W. R. H.
`Killiney, August 9, 1833.
`Shortly after I left you on Thursday last I met Mac Cullagh, and
thought it better to avail myself of the liberty you allowed me, and
mentioned that you were about to answer his note. I did not enter
further into the subject, but in the few words which followed he
mentioned that he had explicitly stated to you, at the time of
his first publication, his intention of writing a supplemental essay
on Fresnel's Theory, and that he had made a similar communication to
my father.
`I took no further notice of this at the time, but on my return to the
country I though it would save much embarrassment and recrimination
to make you aware of this fact, which probably has escaped your
recollection. I therefore wrote a short note to Mac Cullagh,
yesterday morning, to inquire whether it was to this he referred in
the passage in his last not, on which you have dwelt so much in your
reply, and to ask permission, if it were so, to state the fact to you.
I received last night his distinct affirmation to both these
points, and along with it some further details which lead me to hope
that the matter may be adjusted in a less hostile manner. In this
hope I now write to urge you to take no further step in this matter
until I see you. I shall be in town on Monday morning, when you will
probably come in to attend the Academy, if not for this business,
which I cannot but regard as of much importance both to you and Mac
Cullagh. I trust I shall then be able to adjust the matter to the
satisfaction of both parties; but if not, it will not be too late for
you to persevere in your present intention of a reply.'
To this note Hamilton briefly replies on the same day:-
From W. R. Hamilton to
Professor Lloyd.
`Observatory, August 10, 1833.
`It is very friendly in you to take so much trouble about the matter,
and what you state in your last note is very important. It has quite
escaped my recollection that Mac Cullagh mentioned to me any intention
of writing a supplemental essay on Fresnel; but of course I do not
doubt his word. I still think I ought to state distinctly that I was
(until very lately) under the impression that he had not in any degree
anticipated me, and that he lately mentioned to me that he had
suppressed his own expectations. But certainly I am anxious not to
appear nor to be hostile to him; and I fully intend to be at the
Academy on Monday next, in the hope of meeting you and him, if you
think it well to do so.'
Later in the month, on the 22nd, Lord Adare writes to Hamilton as
follows:-
`Dear Professor, I hear Mac Cullagh has published in the Phil.
Mag. a Paper in which he says he had arrived at Conical Refraction
some time ago. Of course this will not pass without some remarks from
you.'
In answer, Hamilton gives his friend the following interesting account
of what had been passing:-
From W. R. H. to
Viscount Adare.
`Observatory, August 29, 1833.
`When I saw Mac Cullagh's remarks in the Phil. Mag. for this
month, I was certainly a little offended, for they seemed to insinuate
that I might have got the hint from his Memoir; and I amused myself
writing an answer in a somewhat satirical vein. But I took the
precaution of showing it to Professor Lloyd, who, on receiving it,
immediately came here in great alarm lest Mac Cullagh and I should get
into an unpleasant controversy. I asked Lloyd, but this of course is
entre nous, whether he really thought from his long
acquaintance that Mac Cullagh was an honest man; and he assured me that
he had the highest opinion of his honour. He said, too, that
Mac Cullagh had lately brought some things to his recollection which
agreed with Mac Cullagh's recent statements of his having thought
something odd would arise in connexion with the circular sections of
the two ellipsoids in the theory, though he did not communicate his
thoughts to others, nor develop them himself. In particular, Lloyd
remembers that Mac Cullagh complained to him some years ago, that on
his asking a Dublin optician for crystals, he was shown the crystal of
a watch. But Mac Cullagh did not then tell Lloyd what he wanted the
crystals for, nor (so far as I can learn) had he any distinct
expectation himself. However, Lloyd's assurances of his confidence in
Mac Cullagh's honour changed a good deal my state of feeling; though I
still thought of writing to the Magazine, and indeed Lloyd himself
said that some unguarded expressions in Mac Cullagh's remarks required
some notice to be taken of them. But before the time expired within
which I should have written, if at all, Lloyd brought me a message
from Mac Cullagh that he was very sorry for having unintentionally
offended me; that the obnoxious sentences were written in great haste,
to save the post and the month, and were sent later than the body of
his little Paper (though they are printed at the beginning), under the
influence of a friend who urged him to make some claim, which he had
not at first intended to do, but merely to deduce geometrically the
two cases of conical refraction from his own theorems and methods; and
finally that he was willing to publish in the next number of the
Magazine an explanation, a copy of which was shown me, containing a
statement that he had not only not communicated his thoughts to
others, but had not perfectly developed them himself; until by hearing
of my results he was led to resume the inquiry, and to deduce the
demonstrations which he gave in the last number. You will easily
suppose that I was quite pacified by this, and thought it needless to
indulge the world with the spectacle of a battle between us, which
would no doubt have furnished rare entertainment.
`When all was over, I thanked Lloyd for the trouble he had taken, and
hinted that having reconciled us it would be well not to mention to
Mac Cullagh the doubts which I felt for a while with respect to his
truth and honour. He laughed at this, and said, that would
indeed be drawing the line upon the crystal, in allusion to one of
the blunders which he was pleased to attribute at Cambridge to me and
metaphysics.'
The September number of the Philosophical Magazine accordingly
contains an `Additional Note on Conical Refraction, by J. MacCullagh,
F.T.C.D.,' which I transcribe:-
`The introductory part of my note which appeared in your last number
was written in haste, and I have reason to think it may not be rightly
understood. You will therefore allow me to add a few observations
that seem to be wanting.
`The principal thing pointed out in the Paper that I published some
time ago in the Transactions of the Royal Irish Academy is a
very simple relation between the tangent planes of Fresnel's wave
surface and the sections of two reciprocal ellipsoids. Now this
relation depends upon the axes of the sections, and therefore
naturally suggested to me the peculiar cases of circular section in
which every diameter is an axis. Thus a new inquiry was opened to my
mind. And accordingly, without caring just then to obtain final
results, which seemed to me to be an easy matter at any time, I
expressed in conversation my intention of returning to the subject of
Fresnel's Theory in a supplementary Paper. The design was
interrupted, and I was prevented from attending to it again, until I
was told that Professor Hamilton had discovered cusps and circles of
contact on the wave surface. This reminded me of the cases of
circular section, and the details given in my last note were
immediately deduced.'
Among Hamilton's papers I find the following note from Mac
Cullagh:-
From J. Mac Cullagh to
W. R. Hamilton.
`Tuam, September 5, 1833.
`My dear Hamilton, I have not seen the last number of the
Phil. Mag. though I ordered it to be sent to me, and I am
uneasy to know whether the second Note has been published or not.
I made it clearer and more precise by the alteration of a word or
two in what regards myself; what relates to you was retained
verbatim, and I hope you will find it completely to your
satisfaction. If you should think it necessary to say anything
yourself, perhaps you would defer doing so until we meet, which
may take place in three or four weeks. In the meantime I am anxious
to hear from you, as I suppose the Phil. Mag. has gone
astray.'
On the back of the above letter is the short-hand draft by Hamilton of
his reply:-
`My dear Mac Cullagh, I have just seen your ``Additional Note'' in the
Phil. Mag., and have no intention of troubling the editors
with any remarks of my own on the subject. They [will] know the rest
from some other Papers from you which have not yet been
printed.' [12]
Then follows a generalisation by Hamilton `of your curious theorem
about a refracting hyperboloid:' [13] and he
concludes, `On
going to the Academy the last day that I saw you, I found they had
broken up for the summer, so that I was too late to propose the
insertion of any note to my Third Supplement, and the appearance of
your own communications in the Magazine seem to make it unnecessary.
Believe me, &c.'
In his Introduction, however, to his Third Supplement, printed in
part i. of the xvii.th vol. of the Transactions of the Royal
Irish Academy, and dated June 1833, Hamilton had thus put on record
the researches of Mac Cullagh in this part of Fresnel's Theory:-
`I am informed that James Mac Cullagh, Esq., F.T.C.D., who published
in the last preceding volume of these Transactions a series of
elegant Geometrical Illustrations of Fresnel's Theory has, since he
heard of the experiments of Professor Lloyd, employed his own
geometrical methods to confirm my results respecting the existence of
those conoidal cusps and circles on Fresnel's wave from which I had
been led to the expectation of conical refraction. And on my lately
mentioning to him that I had connected these cusps and circles on
Fresnel's wave with circles and cusps of the same kind on a certain
other surface discovered by M. Cauchy, by a general theory of
reciprocal surfaces, which I stated last year at a general meeting of
the Royal Irish Academy, Mr. Mac Cullagh said that he had arrived
independently at similar results, and put into my hands a Paper on the
subject; which I have not yet been able to examine, but which will I
hope be soon presented to the Academy and published in their
Transactions.'
To this I may add the acknowledgment, which immediately follows, of
the approximation made by Professor Airy to the result arrived at by
Hamilton:-
`I ought also to mention that on my writing in last November to
Professor Airy, and communicating to him my results respecting the
cusps and circles on Fresnel's wave, and my expectation of conical
refraction, which had not then been verified, Professor Airy replied
that he had long been aware of the existence of the conoidal cusps,
which indeed it is surprising that Fresnel did not perceive.
Professor Airy, however, had not perceived the existence of the
circles of contact, nor had he drawn from either cusps or circles any
theory of conical refraction.'
The statements with which the Introduction concludes ought perhaps
here to be given to the reader; in continuation with the paragraph
last quoted, it proceeds:-
`This latter theory was deduced by my general methods from the
hypothesis of transversal vibrations in a luminous ether, which
hypothesis seems to have been first proposed by Dr. Young, but to
have been independently framed and far more perfectly developed by
Fresnel; and from Fresnel's other principle of the existence of three
rectangular axes of elasticity within a biaxal crystallized medium.
The verification therefore of this theory of conical refraction by the
experiments of Professor Lloyd must be considered as affording a new
and important probability in favour of Fresnel's views; that is, a new
encouragement to reason from those views in combining and predicting
appearances.
`The length to which the present Supplement has already extended
obliges me to reserve for a future communication many other results
deduced by me by my general methods from the principle of the
characteristic function; and especially a general theory of the focal
lengths and aberrations of optical instruments of revolution.'
In the Third Report of the Proceedings of the British
Association for the Advancement of Science, giving the proceedings of
the Meeting at Cambridge in June, 1833, but corrected up to the time of
printing in 1834, is to be found, at p. 360, a report of Professor
Hamilton's oral statement of `Results of a view of a Characteristic
Function in Optics.' This embraces some results relating to optical
instruments of revolution, as well as Conical Refraction; and it
concludes, at p. 369, with a reference to the independent researches
of Mac Cullagh and Cauchy. It is followed by a similar report of
Professor Lloyd's oral statement of his verifying experiments.
At the close of an article, [14]
dated September, 1833, contributed by
Hamilton to the November Number of the Dublin University Review
for 1833, p. 823, Mr. Mac Cullagh's claim in this matter is also put
on record.
These statements of Hamilton with regard to Mac Cullagh's work are all
in perfect consistency with one another.
Finally, in the xvii.th volume of the Transactions of the Royal
Irish Academy, part ii., p. 248, Mr. Mac Cullagh put a satisfactory
close to his action in the matter, by appending to his Paper entitled
Geometrical Propositions applied to the Wave Theory of Light
(Read, June 24, 1832), a note dated April 2, 1834, which is
here transcribed:-
`The curves of contact on biaxal surfaces and the conical
intersections and nodes were lately discovered by Professor Hamilton,
who deduced from these properties a theory of conical refraction which
has been verified by the experiments of Professor Lloyd. See
Transactions Royal Irish Academy, vol. xvii., part i., and the
present Paper, Art. 55--58.
`The indeterminate cases of circular section - at least the case of
the nodes - had occurred to me long ago; but having neglected to
examine the matter attentively, I did not perceive the properties
involved in it.'
I have now brought forward or referred to all the facts and
contemporary records respecting the question of priority and mutual
independence which have come within my cognizance. The reader will
see that proceeding by different paths (Hamilton by that of his own
Algebraical method, Mac Cullagh by that of Geometry), Hamilton
independently completed his theoretical discovery and foresaw the
corresponding physical facts: Mac Cullagh, when working independently,
advanced far in the right direction, but stopped short of deducing all
the connected mathematical properties, and failed to anticipate the
physical phenomena to which his theorems might have conducted him.
It may be added with truth that by nothing was Hamilton more
distinguished, from the beginning to the end of his scientific career,
than by his scrupulous anxiety to award to all labourers in the same
fields with himself the shares to which they had a just title in the
priority and independence of discovery.
Notes
- This finally proved to be, quam proxime,
the angle ascertained by experiment: see letter to
Herschel, infra, p. 634.
[The page number here refers to Life of Sir William Rowan
Hamilton by Robert Perceval Graves, vol I, Dublin University
Press, 1882.]
- This I am informed is a mistake.
- In the case of external conical refraction.
- The correspondence on this subject shows that
Hamilton solved it the day after he wrote this letter, viz., January 2.
- See Note in the Appendix.
- In a letter to another correspondent, dated
January 22, 1832, Hamilton writes, `Airy has just answered the letter
that I wrote to him from Bayly Farm; but whether the fault was mine
or his, he has quite mistaken what I meant.'
- See p. 636.
[The page number here refers to Life of Sir William Rowan
Hamilton by Robert Perceval Graves, vol I, Dublin University
Press, 1882.]
- The Philosophical Magazine.
- Report of third Meeting of the British Association,
1833.
- London and Edinburgh Philosophical
Magazine, June 1833, p. 420.
- The interior cone emerges as a cylinder.
- This I suppose to refer to the Paper
read by Mac Cullagh before the Royal Irish Academy on the 24th June, 1833.
Geometrical Properties, &c., referred to below
as subsequently printed in the xvii.th vol., part ii., of the
Royal Irish Academy.
- See letter from W. R. Hamilton to Lloyd,
dated September 2, 1833.
- On a General Method of expressing the
Paths of Light and of the Planets by the Coefficients of a
Characteristic Function.
Links:
D.R. Wilkins
(dwilkins@maths.tcd.ie)
School of Mathematics
Trinity College, Dublin