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For the crystals of any one substance the angles between corresponding faces are constant. This law is known as the law of constancy of interfacial angles. It corresponds to the law of definite proportions in chemistry.

The proportions in which two elements combine determines the atomic weight of the elements. In an analogous way the intercepts, which are determined by certain constant interfacial angles, establish the axial ratio which, like the atomic weight, is a constant.

Crystal measurement corresponds to quantitative analysis in chemistry. Exact measurements establish the axial ratio of a crystal just as exact analyses establish the atomic weight of an element.

Two chemical elements A and B unite not only to form the compound AB but also the compounds AB, A,B,, A,B, etc. This fact is known as the law of multiple proportions. These proportions for most chemical compounds are usually simple but in many compounds, especially those containing silicon or carbon, they are often far from simple. Among silicate minerals we have such compounds as Mg:Al,S1,027 and H20MgAl,Si015. Among organic compounds we have C&H 22, C,,H.22NO3, C2H 6014, and many others with fifty or more carbon atoms in the molecule. In spite of these complex formulæ all chemists accept the law of multiple proportions as an established fact. Without it chemistry would scarcely deserve to be called a science. The law of rational indices in crystallography corresponds to the law of multiple proportions in chemistry. The same difficulties are encountered in crystal measurement as in quantitative analysis. That is, there are certain errors which usually render it impossible to prove absolutely the law of rational indices or the law of multiple proportions.20 According to Jaquet the formula of hemoglobin (of the dog) is C., H 203 N195S,Fe0.218. This formula can hardly be regarded as established. It may be a little different but it is very probable that these elements unite in definite proportions. This is exactly analogous to vicinal faces such as (251-250-250) observed on alumn by Miers.

The law of multiple proportions was deduced by Dalton from his atomic theory before there were accurate analyses to prove it,

Organic chemistry has an advantage over inorganic chemistry in that the formulæ may usually be determined by the method of formation.

just as the law of rational indices was deduced by Haüy from his theory of crystal structure. If chemical compounds are made up of atoms they must necessarily unite in definite proportions. This it will be recalled is precisely analogous to the argument used for proof of the rationality of the indices. If crystals are made up of particles or molecules, the crystal faces necessarily have rational indices.

Two or more given elements do not unite in all possible proportions but in a comparatively few, usually simple, proportions which we explain by the term valence. There are but two oxids of mercury Hg,O, and Hg which we explain by saying that the valence of mercury is one and two. This is analogous to the limitation imposed by the law of complication of Goldschmidt or the law of maximum reticulate density of Bravais.

To complete the analogy between the laws and theories of crystallography and chemistry let us consider the periodic law and its analogue. Mendeléef, the Russian chemist, predicted the existence of several chemical elements, scandium and gallium, which he called ekaboron and eka-aluminum, before they were discovered. Not less remarkable was the deduction by Hessel, a German mathematician, of the thirty-two possible types of symmetry in crystals, assuming 2-, 3-, 4-, and 6-fold symmetry-axes, in 1830, at a time when only about half of them were known. Of the thirty-two possible types of symmetry, only one remains to be found.


Judging from various text-books and articles a difference of opinion exists as to the exact meaning of the law of rational indices. Some authors limit the indices to simple numbers while others admit that occasionally the indices are large numbers. Unfortunately this question can not be decided by direct measurement of the angles on account of errors in measurement. As crystals possess axes of only 2-, 3-, 4-, and 6-fold symmetry they must consist of regularly arranged molecules, or particles of some sort, whatever their nature may be. Crystal faces, then, necessarily have rational indices. The indices are usually small numbers but may also be

complex, the complexity in general increasing with the rarity of the face. The structure theory of Bravais offers a satisfactory explanation of the abundance of faces with simple indices and the rarity of faces with complex indices. There is a remarkable analogy between the fundamental laws of chemistry and crystallography. STANFORD UNIVERSITY, CALIFORNIA,

Feb., 1912.

PROC. AMER. PHIL. SOC., L1. 204 E, PRINTED JUNE 5, 1912.


By T. J. J. SEE.

(Read April 19, 1912.)

(Plates VIII (bis) AND IX.)

1. INTRODUCTORY REMARKS. More than a century and a quarter have elapsed since it was confidently announced by Sir William Herschel that sidereal systems made up of thousands of stars exhibit the effects of a clustering power which is everywhere moulding these systems into symmetrical figures, as if by the continued action of central forces (Phil. Trans., 1785, p. 255, and 1789, pp. 218–226). In support of this view he cited especially the figures of the planetary nebulæ, and the globular clusters, as well as the more expanded and irregular swarms and clouds of stars visible to the naked eye along the course of the Milky Way, which thus appears to traverse the heavens as a clustering stream. And yet notwithstanding the early date of this announcement and the unrivaled eminence of Herschel, it is only very recently that astronomers have begun to consider the origin of sidereal systems of the highest order.

The historical difficulty of solving the problem of n-bodies, when n exceeds 2, which dates from the establishment of the law of universal gravitation by Newton in 1687, will sufficiently account for the restriction of the researches of mathematicians to the planetary system, where the central masses always are very predominant, the orbits almost circular and nearly in a common plane, and to other simple systems such as the double and multiple stars: but owing to the general prevalence of the clustering tendency pointed out by Herschel and now found to be at work throughout the sidereal uni

verse, it becomes necessary for the modern investigator to consider also the higher orders of sidereal systems, including those made up of thousands and even millions of stars. It is only by such a comprehensive view of nature, which embraces and unites all types of systems under one common principle, that we may hope to establish the most general laws governing the evolution of the sidereal universe.

Accordingly, although the strict mathematical treatment of the great historical problem of n-bodies is but little advanced by the recent researches of geometers, yet if we could arrive at the general secular tendency in nature, from the observational study of the phenomena presented by highly complex systems of stars, operating under known laws of attractive and repulsive forces, the former for gathering the matter into large masses, the latter for redistributing it in the form of fine dust, the result of such an investigation would guide us towards a grasp of problems too complex for rigorous treatment by any known method of analysis.

Now it happens that in the second volume of the “Researches on the Evolution of the Stellar System,” 1910, the writer was able to establish great generality in the processes of cosmogony, and to show that the universal tendency in nature is for the large bodies to drift towards the most powerful centers of attraction, while the only throwing off of masses that ever takes place is that of small particles expelled from the stars under the action of repulsive forces and driven away for the formation of new nebulæ. The repulsive forces thus operate to counteract the clustering tendency noticed by the elder Herschel, and so clearly foreseen by Newton as an inevitable effect of universal gravitation upon the motions of the solar system that he believed the intervention of the Deity eventually would become necessary for the restoration of the order of the world (cf. Newton's " Letters to Bentley,” Brewster's “Life of Newton," Vol. II., and Chapter XVII., and Appendix X).

But whilst the argument developed in the second volume of my “Researches "gives unexpected simplicity, uniformity and continuity to the processes of cosmogony, there has not yet been developed, so far as I know, any precise investigation of the attractive forces operating in globular clusters, which might disclose the nature of the

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