076 CRYSTALLOGRAPHY assumes six prisms or parallelopipeds as primary forms, and designates the faces, angles, and edges by letters, as was done by Haiiy. This system is adopted and explained by Dufrenoy in his Traite de Mineralogie (Paris, 1844-56), and by Des Cloizeaux in Ms Manuel (Paris, 1862-74.) In Germany also various methods have appeared. Weiss himself only published special papers, but his views have been wrought out by several of his many followers. Thus one of his favourite pupils, F. E. Neumann, in his " Contri butions" (Beitrdje zur Krystallonomie) in 1823, showed how crystals might ba represented not so much by the faces as by their normals, that is, by lines drawn from the centre of tli3 system vertical to the faces. Cleavage, he says, and the reflection of light, &c., all indicate a force acting vertical to the faces, or in the normal. He further brought clearly out ths arrangement of the faces in zones, and showed how they could be represented to the eye either by lines on a plain surface, or by great circles on the circumscribing sphere. Quenstedt of Tubingen, another pupil of Weiss, made known a similar method in 1 835, which he has since illustrated in his Methode der Krystallo graphie in 1840, in his Mineralogie, 1855 (2d ed. 1863), and more fully in his Grandriss der bestimmenden und rechnenden Krystallographie, 1873. The truest representa tive of Weiss, however, is generally regarded as Gustaf Rose, who laid the foundation of his reputation by his ac count of the "crystallization of sphene and titanite" in 1820. His Elemente der Krystallographie first appeared at Berlin in 1833, and in a third edition in 1873. Mohs s method was expounded in his works already noticed, and became better known in Britain by Haidinger s translation of his Treatise on Mineralogy, published at Edin burgh in 1 825 ; and is further explained in Mohs s Anfangs- griinde der Naturgeschichte des MineralreicJts (1832, 2d edi tion by Zippe, 1839). Haidinger, besides many memoirs, has also published a separate work in which the method is fully explained (llandbuch der bestimmenden Mineralogie, 1845). But wider success and more general adoption has attended the method of Dr Carl Naumann, in which the faces are represented by means of their co-ordinates, and thus in an easily understood form. Born in 1797, Naumann began his studies under Werner, and completed them under Mohs, and has been regarded as carrying out the system of his teacher, whilst trying to mediate between him and Weiss. His LeJirbuch der reinen und angewandten Krystallographie appeared in 1830 ; his Anfangsgriinde der Krystallographie in 1841, 2d edition, 1855, and his Theoretische Krystallo graphie in 1856. His Elemente der Mineralogie, first published in 1816, and of which a ninth edition appeared in 1874, has still further extended his method and nomen clature. His system, occasionally in slightly altered form, has wide prevalence in Germany, and has been introduced into this country in Nicol s Mineralogy, 1849, and in the article on MINERALOGY in the eighth edition of the present work. Dana in his Mineralogy, 1854, has given it wide currency in America ; he has endeavoured to simplify the mode of representing the faces. Another method, which in Germany in great measure divides the field with Naumann s, may be said to "have had its origin in Britain. In 1825 Dr. Whewell published in the Philosophical Transactions a memoir on " A General Method of Calculating the Angles of Crystals," in which he referred only to Haiiy s views, and in 1826 another " On the Classification of Crystalline Combinations, " founded on the methods of Weiss and Mohs, especially of the latter, with which he had in the meantime become acquainted. The author himself states that his method had little value as a method of calculating the angles of crystals. But in 1839 Profsssor Miller of Cambridge, partly adopting his views, and partly aiding himself by the suggestions of Neumann and of Grassmann, who, without any knowledge of what his predecessor had done, had re-invented the method of representing the position of the faces of crystals by corres ponding points on the surface of a circumscribing sphere, brought out his Treatise on Crystallography. In his edition of Phillips s Mineralogy, 1852, the same system was also employed. In Germany this system has found many followers, and is used in several of the best text-bookj, among which may be mentioned the Lehrbuch der Krystallographie of Karsten, 1861, and the works with the same title of Von Lang, 1866, and Dr A. Schrauf, 1866. The relative merit of the methods mentioned cannot be discussed in this place. The system of Naumann is. perhaps, the one now most generally prevalent, and most easily understood by beginners, as giving the most graphic picture of the various forms and their combinations. Miller s system, on the other hand, is regarded as better adapted for the various calculations needed in the higher portions of the science, and is therefore often preferred by those who make a special study of tho subject. How closely their merits are balanced is shown by the fact that Groth, in his recent valuable work, Physikalische Krys tallographie, Leipsic, 1876, whilst preferring Naumann s, deems it necessary to explain Miller s also to his readers, and to give a comparative table of the symbols employed by Naumann, Miller, and Le vy, so that the one may be, as it were, translated into the other. Similar tables may also be found in Des Cloizeaux s Mineralogie and in Schrauf s Atlas. Many very interesting facts have also been recently ascertained, showing the intimate relation that exists between the various physical properties of crystals and their p crystallographic characters, proving very distinctly that the r systems of crystals are not mere artificial arrangements of speculative men, but have a real foundation in the structure of the bodies observed. We saw already how Brewstcr proved this connection in reference to their optical properties. He continued his researches on this subject for many years, and it was also pursued by many of his contemporaries, among whom Biot, Sir John Herschel, and Haidinger may be named. More recently the stauroscope, invented by Von Kobell, and the polarizing microscope of Norremberg have proved valuable aids in investigating these properties. In France M. des Cloizeaux has specially directed attention to the optical properties of crystals and their value in mineralogy (De Vemploi des proprittcs optiques birefringentes en Mineralogie, 1857, and Sur Vemploi du microscope, in 1864), and in his Manuel de Mineralogie records many remarkable observations made both by him self and others. In Britain and in Germany these investigations have recently been conjoined with the examination with the microscope of thin slices of minerals and rocks. The method of preparing such transparent sections was first described by William Nicol of Edinburgh, to whom is also due the discovery in 1828 of the peculiar prisms of Iceland or calcareous spar which are now known by his name, and which form an almost indispensable part of apparatus for such researches. It is scarcely possible to avoid noticing the important influence which this one mine ral with its marked properties has had on the progress of the science whose history we are describing. In this country n new impulse was given to the study by Mr Sorby s memoir " On the Microscopical Structure of Crystals," published in the Journal of the Geological Society in 1848. In Germany the workers in this field are so numerous that we cannot specialize individuals, but shall only refer to the works of Zirkel (Mikroskopische G esteinssludien, 1863, Mikroskopische Bescha/enheiten der Mineralien, 1873, <fec.), Schrauf (Lehrbuch der physikalischen Mineralogie, 1868),
and Rosenbusch (Mikroscopische Physiographic der