_Reprinted from the "Proceedings of the American Academy of
Arts and Sciences," Vol. XVI, May 24, 1881._
William Hallowes Miller, who was elected Foreign Honorary Member of this
Academy in the place of C. F. Naumann, May 26, 1874, died at his
residence in Cambridge, England, on the 20th of May, 1880, at the age of
seventy-nine, having been born at Velindre, in Wales, April 5, 1801. His
life was singularly uneventful, even for a scholar. Graduating with
mathematical honors at Cambridge in 1826, he became a fellow of his
college (St. John's) in 1829, and was elected Professor of Mineralogy in
the University in 1832. Under the influence of the calm and elegant
associations of this ancient English university, Miller passed a long
and tranquil life--crowded with useful labors, honored by the respect
and love of his associates, and blessed by congenial family ties. This
quiet student-life was exactly suited to his nature, which shunned the
bustle and unrest of our modern world. For relaxation, even, he loved to
seek the retired valleys of the Eastern Alps; and the description which
he once gave to the writer, of himself sitting at the side of his wife
amid the grand scenery, intent on developing crystallographic formulae,
while the accomplished artist traced the magnificent outlines of the
Dolomite mountains, was a beautiful idyl of science.
Miller's activities, however, were not confined to the University. In
1838 he became a Fellow of the Royal Society, and in 1856 he was
appointed its Foreign Secretary--a post for which he was eminently
fitted, and which he filled for many years. In 1843 he was selected one
of a committee to superintend the construction of the new Parliamentary
standards of length and weight, to replace those which had been lost in
the fire which consumed the Houses of Parliament in 1834, and to
Professor Miller was confided the construction of the new standard of
weight. His work on this important committee, described in an extended
paper published in the "Philosophical Transactions" for 1856, was a
model of conscientious investigation and scientific accuracy. Professor
Miller was subsequently a member of a new Royal Commission for
"examining into and reporting on the state of the secondary standards,
and for considering every question which could affect the primary,
secondary, and local standards"; and in 1870 he was appointed a member
of the "Commission Internationale du Metre." His services on this
commission were of great value, and it has been said that "there was no
member whose opinions had greater weight in influencing a decision upon
any intricate and delicate question."
Valuable, however, as were Professor Miller's public services on these
various commissions, his chief work was at the University. His teacher,
Dr. William Whewell--afterward the Master of Trinity College--was his
immediate predecessor in the Professorship of Mineralogy at Cambridge.
This great scholar, whose encyclopaedic mind could not long be confined
in so narrow a field, held the professorship only four years; but during
this period he devoted himself with his usual enthusiasm to the study of
crystallography, and he accomplished a most important work in attracting
to the same study young Miller, who brought his mathematical training to
its elucidation. It was the privilege of Professor Miller to accomplish
a unique work, for the like of which a more advanced science, with its
multiplicity of details, will offer few opportunities.
The foundations of crystallography had been laid long before Miller's
time. Hauey is usually regarded as the founder of the science; for he
first discovered the importance of cleavage, and classed the known facts
under a definite system. Taking cleavage as his guide, and assuming that
the forms of cleavage were not only the _primitive forms_ of crystals as
a whole, but also the forms of their _integrant molecules_, he
endeavored to show that all secondary forms might be derived from a few
primary forms, regarded as elements of nature, by means of _decrements_
of molecules at their edges. In like manner he showed that all the forms
of a given mineral, like fluor-spar or calcite, might be built up from
the integrant molecules by skillfully placing together the primitive
forms. Hauey's dissection of crystals, in a manner which appeared to lead
to their ultimate crystalline elements, gained for his system great
popular attention and applause. The system was developed with great
perspicuity and completeness in a work remarkable for the vivacity of
its style and the felicity of its illustration. Moreover, a simple
mathematical expression was given to the system, and the notation which
Hauey invented to express the relation of the secondary to the primary
forms, as modified and improved by Levy, is still used by the French
mineralogists.
The system of Hauey, however, was highly artificial, and only prepared
the way for a simpler and more general expression of the facts. The
German crystallographer, Weiss, seems to be the first to have
recognized the truth that the decrements of Hauey were merely a
mechanical mode of representing the fact that all the secondary faces of
a crystal make intercepts on the edges of the primitive form which are
simple multiples of each other; and, this general conception once
gained, it was soon seen that these ratios could be as simply measured
on the axes of symmetry of the crystal as on the edges of the
fundamental forms; and, moreover, that, when crystal forms are viewed in
their relation to these axes, a more general law becomes evident, and
the artificial distinction between primary and secondary forms
disappears.
Thus became slowly evolved the conception of a crystal as a group of
similar planes symmetrically disposed around certain definite and
obvious systems of axes, and so placed that the intercepts, or
parameters, on these axes bore to each other a simple numerical ratio.
Representing by _a_:_b_:_c_ the ratio of the intercepts of a plane
on the three axes of a crystal of a given substance, then the intercepts
of every other plane of this, or of any other crystal of the same
substance, conform to the general proportion _m_._a_:_n_._b_:_p_._c_,
in which _m_, _n_, _p_ are three simple whole numbers. This simple
notation, devised by Weiss, expressed the fundamental law of
crystallography; and the conception of a crystal as a system of planes,
symmetrically distributed according to this law, was a great advance
beyond the decrements of Hauey, an advance not unlike that of astronomy
from the system of vortices to the law of gravitation. Yet, as the
mechanism of vortices was a natural prelude to the law of Newton, so the
decrements of Hauey prepared the way for the wider views of the German
crystallographers.
Whether Weiss or Mohs contributed most to advance crystallography to its
more philosophical stage, it is not important here to inquire. Each of
these eminent scholars did an important work in developing and diffusing
the larger ideas, and in showing by their investigations that the facts
of nature corresponded to the new conceptions. But to Carl Friedrich
Naumann, Professor at the time in the "Bergakademie zu Freiberg,"
belongs the merit of first developing a complete system of theoretical
crystallography based on the laws of symmetry and axial ratios. His
"Lehrbuch der reinen und angewandten Krystallographie," published in two
volumes at Leipzig in 1830, was a remarkable production, and seemed to
grasp the whole theory of the external forms of crystals. Naumann used
the obvious and direct methods of analytical geometry to express the
quantitative relations between the parts of a crystal; and, although his
methods are often unnecessarily prolix and his notation awkward, his
formulae are well adapted to calculation, and easily intelligible to
persons moderately disciplined in mathematics.
But, however comprehensive and perfect in its details, the system of
Naumann was cumbrous, and lacked elegance of mathematical form. This
arose chiefly from the fact that the old methods of analytical geometry
were unsuited to the problems of crystallography; but it resulted also
from a habit of the German mind to dwell on details and give importance
to systems of classification. To Naumann the six crystalline systems
were as much realities of nature as were the forms of the integrant
molecules to Hauey, and he failed to grasp the larger thought which
includes all partial systems in one comprehensive plan.
Our late colleague, Professor Miller, on the other hand, had that power
of mathematical generalization which enabled him to properly subordinate
the parts to the whole, and to develop a system of mathematical
crystallography of such simplicity and beauty of form that it leaves
little to be desired. This was the great work of his life, and a work
worthy of the university which had produced the "Principia." It was
published in 1839, under the title, "A Treatise on Crystallography"; and
in 1863 the substance of the work was reproduced in a more perfect form,
still more condensed and generalized, in a thin volume of only
eighty-six pages, which the author modestly called, "A Tract on
Crystallography."
Miller began his study of crystallography with the same materials as
Naumann; but, in addition, he adopted the beautiful method of Franz
Ernst Neumann of referring the faces of a crystal to the surface of a
circumscribed sphere by means of radii drawn perpendicular to the faces.
The points where the radii meet the spherical surface are the poles of
the faces, and the arcs of great circles connecting these poles may
obviously be used as a measure of the angles between the crystal faces.
This invention of Neumann's was the germ of Miller's system of
crystallography, for it enabled the English mathematician to apply the
elegant and compendious methods of spherical trigonometry to the
solution of crystallographic problems; and Professor Miller always
expressed his great indebtedness to Neumann, not only for this simple
mode of defining the position of the faces of a crystal, but also for
his method of representing the relative position of the poles of the
faces on a plane surface by a beautiful application of the methods of
stereo-graphic and gnomonic projection. This method of representing a
crystal shows very clearly the relations of the parts, and was
undoubtedly of great aid to Miller in assisting him to generalize his
deductions.
From the outset, Professor Miller apprehended more clearly than any
previous writer the all-embracing scope of the great law of
crystallography. He opens his treatise with its enunciation, and, from
this law as the fundamental principle of the subject, the whole of his
system of crystallography is logically developed. Beyond this, all that
is peculiar to Miller's system is involved in two or three general
theorems. The rest of his treatise consists of deductions from these
principles and their application to particular cases.
One of the most important of these principles, and one which in the
treatise is involved in the enunciation of the fundamental law of
crystallography, is in its essence nothing but an analytical device.
As we have already stated, Weiss had shown that, if _a_:_b_:_c_
represent the ratio of the intercepts of any plane of a crystal on the
three axes _x_, _y_, and _z_, respectively, the intercepts of any other
possible plane must satisfy the proportion--
_A_:_B_:_C_ = _m_._a_:_n_._b_:_p_._c_,
in which _m_, _n_, and _p_ are simple whole numbers. The irrational
values _a_, _b_, and _c_ are fundamental magnitudes for every
crystalline substance;[G] and Miller called these relative magnitudes
the parameters of the crystals, while he called the whole numbers, _m_,
_n_, and _p_, the indices of the respective planes. But, instead of
writing the proportion which expresses the law of crystallography as
above, he gave to it a slightly different form, thus:
_A_:_B_:_C_ = (1/_h_)._a_:(1/_k_)._b_:(1/_l_)._c_,
and used in his system for the indices of a plane the values
_h_:_k_:_l_, which are also in the ratio of whole numbers, and usually
of simpler whole numbers than _m_:_n_:_p_. This seems a small
difference; for _h_ _k_ _l_ in the last proportion are obviously the
reciprocals of _m_ _n_ _p_ in the first; but the difference, small as it
is, causes a wonderful simplification of the formulae which express the
relations between the parts of a crystal. From the last proportion we
derive at once
(1/_h_).(_a_/_A_) = (1/_k_).(_b_/_B_) = (1/_l_).(_c_/_C_),
which is the form in which Miller stated his fundamental law.
[G] For example, the native crystals
of sulphur have _a_:_b_:_c_ = 1:2.340:1.233.
Crystals of gypsum have _a_:_b_:_c_ = 1:0.413:0.691.
Crystals of tin-stone have _a_:_b_:_c_ = 1:1:0.6724.
And crystals of common salt have _a_:_b_:_c_ = 1:1:1.
If _P_ represents the "pole" of a face whose "indices" are _h_ _k_ _l_,
that is, represents the point where the radius drawn normal to the face
meets the surface of the sphere circumscribed around the crystal (the
sphere of projection, as it is called), and if _X_, _Y_, _Z_ represent
the points where the axes of the crystal meet the same spherical
surface,[H] then it is evident that _X Y_, _X Z_, and _Y Z_ are the
arcs of great circles, which measure the inclination of the axes to each
other, and that _P X_, _P Y_, and _P Z_ are arcs of other great circles,
which measure the inclination of the plane (_h_ _k_ _l_) on planes
normal to the respective axes; and, also, that these several arcs form
the sides of spherical triangles thus drawn on the sphere of projection.
Now, it is very easily shown that
(_a_/_h_).cos _P X_ = (_b_/_k_).cos _P Y_ = (_c_/_l_).cos _P Z_;
and by means of this theorem we are able to reduce a great many problems
of crystallography to the solution of spherical triangles.
[H] The origin of the axes is always taken as the center of the
sphere of projection.
Another very large class of problems in crystallography is based on the
relation of faces in a zone; that is, of faces which are all parallel
to one line called the zone axis, and whose mutual intersections,
therefore, are all parallel to each other. If, now, _h_ _k_ _l_ and
_p_ _q_ _r_ are the indices of any two planes of a zone (not parallel to
each other), any other plane in the same zone must fulfill the condition
expressed by the simple equation
u._u_ + v._v_ + w._w_ = _o_,
where _u_ _v_ and _w_ are the indices of the third plane, and u v w
have the values
u = _k_._r_ - _l_._q_
v = _l_._p_ - _h_._r_
w = _h_._q_ - _k_._p_.
Since _h_ _k_ _l_ and _p_ _q_ _r_ are whole numbers, it is evident that
u v w must also be whole numbers, and these quantities are called the
indices of the zone. The three whole numbers which are the indices of a
plane when written in succession serve as a very convenient symbol of
that plane, and represent to the crystallographer all its relations; and
in like manner Miller used the indices of a zone inclosed in brackets as
the symbol of that zone. Thus 123, 531, 010 are symbols of planes, and
[111], [213], [001] symbols of zones.
An additional theorem enables us to calculate the symbols of a fourth
plane in a zone when the angular distances between the four planes and
the symbols of three of them are known, but this problem can not be made
intelligible with a few words.
The few propositions to which we have referred involve all that is
essential and peculiar to the system of Professor Miller. These given,
and the rest could be at once developed by any scholar who was familiar
with the facts of crystallography; and the circumstance that its
essential features can be so briefly stated is sufficient to show how
exceedingly simple the system is. At the same time, it is wonderfully
comprehensive, and the student who has mastered it feels that it
presents to him in one grand view the entire scheme of crystal forms,
and that it greatly helps him to comprehend the scheme as a whole, and
not simply as the sum of certain distinct parts. So felt Professor
Miller himself; and, while he regarded the six systems of crystals of
the German crystallographers as natural divisions of the field, he
considered that they were bounded by artificial lines which have no
deeper significance than the boundary lines on a map. How great the
unfolding of the science from Hauey to Miller, and yet now we can see the
great fundamental ideas shining through the obscurity from the first!
What we now call the parameters of a crystal were to Hauey the
fundamental dimensions of his "integrant molecules," our indices were
his "decrements," and our conceptions of symmetry his "fundamental
forms." There has been nothing peculiar, however, in the growth of
crystallography. This growth has followed the usual order of science,
and here as elsewhere the early, gross, material conceptions have been
the stepping-stones by which men rose to higher things. In sciences like
chemistry, which are obviously still in the earlier stages of their
development, it would be well if students would bear in mind this truth
of history, and not attach undue importance to structural formulae and
similar mechanical devices, which, although useful for aiding the
memory, are simply hindrances to progress as soon as the necessity of
such assistance is passed. And, when the life of a great master of
science has ended, it is well to look back over the road he has
traveled, and, while we take courage in his success, consider well the
lesson which his experience has to teach; and, as progress in this
world's knowledge has ever been from the gross to the spiritual, may we
not rejoice as those who have a great hope?
Although the exceeding merit of the "Treatise on Crystallography" casts
into the shade all that was subordinate, we must not omit to mention
that Professor Miller published an early work on hydrostatics, and
numerous shorter papers on mineralogy and physics, which were all
valuable, and constantly contained important additions to knowledge.
Moreover, the "New Edition of Phillips's Mineralogy," which he published
in 1852 in connection with H. J. Brooke, owed its chief value to a mass
of crystallographic observations which he had made with his usual
accuracy and patience during many years, and there tabulated in his
concise manner. As has been said by one of his associates in the Royal
Society, "it is a monument to Miller's name, although he almost expunged
that name from it."[I] It is due to Professor Miller's memory that his
works should be collated, and especially that by a suitable commentary
his "Tract on Crystallography" should be made accessible to the great
body of the students of physical science, who have not, as a rule, the
ability or training which enables them to apprehend a generalization
when solely expressed in mathematical terms. The very merits of
Professor Miller's book as a scientific work render it very difficult to
the average student, although it only involves the simplest forms of
algebra and trigonometry.
[I] "Obituary Notices from the Proceedings of the Royal Society,"
No. 206, 1880, to which the writer has been indebted for
several biographical details.
Independence, breadth, accuracy, simplicity, humility, courtesy, are
luminous words which express the character of Professor Miller. In his
genial presence the young student felt encouraged to express his
immature thoughts, which were sure to be treated with consideration,
while from a wealth of knowledge the great master made the error evident
by making the truth resplendent. It was the greatest satisfaction to the
inexperienced investigator when his observations had been confirmed by
Professor Miller, and he was never made to feel discouraged when his
mistakes were corrected. The writer of this notice regards it as one of
the great privileges of his youth, and one of the most important
elements of his education, to have been the recipient of the courtesies
and counsel of three great English men of science, who have always been
"his own ideal knights," and these noble knights were Faraday, Graham,
and Miller.