Part 6 (2/2)

(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_ = 1:2340:1233.

Crystals of gypsum have _a_:_b_:_c_ = 1:0413:0691.

Crystals of tin-stone have _a_:_b_:_c_ = 1:1:06724.

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 circ.u.mscribed 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 cla.s.s 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 quant.i.ties 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 circ.u.mstance 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 Hauy to Miller, and yet now we can see the great fundamental ideas s.h.i.+ning through the obscurity from the first!

What we now call the parameters of a crystal were to Hauy 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 a.s.sistance is pa.s.sed. 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 ma.s.s 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 a.s.sociates 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 n.o.ble knights were Faraday, Graham, and Miller.

VII.

WILLIAM BARTON ROGERS.

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