Law of symmetry. Relation to chemical composi tion. Criticisms of Haiiy s system. 674 not parallel to the edges, but took the angles as its point of departure, and thus was parallel to a diagonal. In the case above supposed the decrease took place by two ranges in breadth for one in height or thickness, but other less simple ratios might be supposed, as of two in breadth to. three in height, and to these the name mixed decrements were given. There were other possible modes of decrease also distinguished, to which it is needless now to refer. But by these and other modes of procedure Haiiy showed how the various secondary crystals could arise from his assumed primitive forms or molecules. The great advance secured by this theory of Haiiy s was the firm establishment of the idea that the forms of crystals were not irregular or capricious, but definite and based on fixed and ascertainable laws. Hence he showed that, whilst certain secondary forms may be deduced from a given nucleus, there are other forms that cannot occur. Further he pointed out what he named " the law of symmetry," in consequence of which, when any change of a crystal form took place by its combination with other forms, all similar parts angles, edges, faces were modified in the same way at the same time. All these changes too, he said, could be indicated by rational coefficients or commensur able numbers. A not less important principle, which Haiiy endeavoured to establish, was the intimate relation of the crystalline form to the chemical composition of minerals, so that even prior to analysis the real diversity of species formerly con joined might be inferred from differences in the angles. As an example of this may be mentioned his discovery of the difference of the angles in crystals classed together as "heavy spar," a difference only explained when Vauquelin showed that those with the larger angle from Sicily con tained the new earth strontia, discovered by Klaproth, instead of the baryta found in those from Derbyshire. The modifications which this view has had to undergo from wider observations will be noticed afterwards, but even its enunciation by Haiiy formed a great stimulus to research both as to the forms and the composition of minerals. Taken in connection with the perspicuous and elegant style of his work, its clear arrangement and full illustration by figures, its influence on the progress of the science may be readily understood. Many deficiencies in his system are now easily seen, and some of the most fatal were soon brought to light by the very stimulus his works gave to the science. Thus one of the first to criticise the system was Weiss, who translated Haiiy s work into German in 1804.. He not only pointed out that the primitive forms erred both in excess and defect, but struck deeper at the theory by showing that the integrant molecules might better be entirely laid aside. They were not wanted to explain the observed facts, and the so-called planes built up of them would not reflect the light. Bernhardi, a medical man in Erfurt, attacked the theory from other points of view. Thus he objected to the prisms which Haiiy had chosen as primitive forms that their dimensions could not be deter mined from themselves, their height depending on another form, and therefore that octahedrons or double pyramids were preferable. Then he showed that various crystals were more readily explained from other forms than those taken as their primaries by Haiiy, and that in the regular formsit wasquite indifferent whether the cube or regularocta- hedron was cho sen, whilst among the irregular forms other divisions might be established, more conformable to nature. It is needless to specify further criticisms on Haiiy s theory, as its very merits soon led to its being replaced by more profound views. Thus the importance it ascribed to the angles of the faces and cleavages of crystals for the true determination of minerals formed a strong motive for their more accurate determination. The discovery also of the reflecting goniometer in 1809 by Wollastcn (born 1766, died 1 829) enabled this to be done with a degree of accuracy previously impossible. The writings of Dr Wollaston himself, of Mr Brooke, and especially the Introduction to Mineralogy (1816) of William Phillips (born 1773, died 1828) were specially rich in material of this kind. The influence of this accumulation of facts was shown less in the correction of Haiiy s data than in the necessity it involved of some new and more workable theory for con necting the facts than that adopted by the French mineralogist. For this science is chiefly indebted to Weiss, already mentioned as the translator and critic of Haiiy s great work. Born at Leipsic in 1780, and educated in its university, where he began to teach in 1803, he inaugurated his appointment as ordinary professor of physics in 1808 by the publication the following year of a dissertation, De indagando formarum crystallinarum charactere geometrico prineipali. In this he pointed out for the first time the im portance of the axes of crystals, to which, however, Haiiy had referred. " The axis," he says, " is truly the line governing every figure (pmnis figurce dominatrix) round which the whole is uniformly disposed. All the parts look to it, and by it they are bound together as by a common chain and mutual contact." But the axes are not mere geometric lines physically dead and powerless. It is in reference to them that the forces work which have formed the crystals. Hence the importance of the inclination of the faces to the axes as characterizing forms, and the simpler numbers by which the relations of these faces might be expressed. He further points out various distinctions in the forms of crystals, in which his followers have traced the germs of the systems of crystallization he subsequently established. This was done in his memoir, " Uebersichtliche Darstellung der verschiedenen naturlichen Abtheilungen der Krystallisa- tionssysteme," published in 1815 in the Transactions of the Academy of Berlin, to which city he had been transferred in 1820. In this memoir the terms regular system, four-membered system, two-and-two-membered system, and others afterwards used first appear. In other memoirs in the same series, of which the more important were those on the crystallization of felspar, epidote, gypsum, and quartz, his views were more fully developed. Along with these views of the general relations of crystals Weiss also introduced important improvements in the mode of designating the faces of crystals, so as to render it more easy to calculate their angles. Haiiy had already done this in conformity to his theory of decrements, but the expressions were complex and the numbers large. But here as elsewhere, Weiss says, the mechanical atomistic views by which Haiiy was led must be laid aside, in order to allow the ascertained knowledge of the mathematical laws and relations of crystalline structure to come out purely. Leaving out of view, therefore, the supposed primitive forms, and looking only to v/hat was above and beyond them, Weiss referred all to the essential relations of the axes or the co-ordinates of the faces, and thus gave at once far more precision and simplicity to the symbols, and facilitated the necessary calculations. It often happens in periods of intellectual activity that several inquirers are engaged on the same subject, and, following it out in similar directions, come to results that i.iore or less coincide. Such seems to have been so far true in regard to crystallography, and these discoveries of Weiss have been claimed for Mohs. Born in the Hartz in 1773, he studied at Halle, turning his attention specially to mining. In 1812 he became professor in Griitz, and in 1818 succeeded Werner in Freiberg, which a few years later he left for Vienna, where he taught with great success.
He died in 1839. The dates of their publications leave no