1902 Encyclopedia > Crystallography

Crystallography




CRYSTALLOGRAPHY. When water containing saline matter in solution is allowed to evaporate slowly, the salt it contains is thrown down in bodies of peculiar forms, bounded by smooth, even surfaces meeting in straight lines. Fused metals consolidating in certain favourable conditions appear as similar bodies. And in nature also, in cracks or fissures of the rocks, or imbedded in their mass, minerals resembliug these in form are frequently found. These regular polyhedric, or many-sided Definition bodies, whether natural or artificial, are named crystals, u* Cryata1*' and the science naming and describing the forms they, assume, and pointing out the relations that exist among them, is termed crystallography. In a theoretical point of view this science may.be regarded as a branch of mathe-matics, and might be studied independent altogether of the fact of any material bodies existing in the forms described. Practically, however, its chief interest and value is as a mean of distinguishing many salts, ores, and other substances, either formed artificially or, more especially, occurring naturally as minerals. At present no particular system of crystallography has found general acceptance, and referring for the details of the one adopted in the description of mineral species to MINERALOGY, we propose in this place to give an account of the history of the science, pointing out the more remarkable steps in its progress, the chief general results attained, and some of the best works from which further information in regard to it may be obtained.

The term crystal, found in most modern European Origin of languages, is derived from the Greek woid _______, name-meaning ice or frozen water, and subsequently tiansferred to pure transparent stones cut into seals, and, as was thought, only produced in the extreme cold of the lofty passes of the Alps. Pliny, who notices this rock-crystal in his Natural History (book xxxvii.), points out clearly enough the hexagonal form of the crystals, remaiking that it is not easy to give any reason why they grow in this foim, more especially as the points have not the some appearance (eo magis epiod neque mucronibus cctdim species est), and the polish of the sides is such that no art can equal it. The forms of other minerals are also noticed by him, but the term crystal still had regard to the ice-like transparency and purity of the stone, a icference entirely lost in the modern scientific use of the word.

It is not wonderful that these bodies, often so remarkable for the beauty of their forms, colours, and other physical properties, attracted considerable attention even in the so-called dark ages. But these notices rather amuse us by their quaint absurdity, as we should now regard it, than throw light on the progress of the science. Thus Albeitus I Magnus in the middle of the 13th century tells how the cold in the lofty mountains makes the ice so dry that it congeals into crystal (ex illo sieco coagulat glaciem in crystallum). Agricola, three centuries later, knew little more, though affirming that crystallus was not ice but rather succus frigore densatus. Still he indicates some simple forms of crystals, and notes the fissile structure of some stones, as the lapis specularis (probably gypsum or mica), a property which as cleavage soon exercised much influence. Nicolaus Steno, the Dane, born in 1638 at Copenhagen, but for a time resident at Florence, amidst his well-known studies in anatomy, found leisure also to speculate on questions concerning the structure of the earth and the nature of gems and precious stones. As his treatise De solido intra solidum naturaliter contento, published in 1669, anticipated in geology some modern speculations and theories subsequently confirmed, so it also contains the germs of important facts in crystallography. It was still the wondrous rock-crystal with the polished sides of the middle prism and the terminal points of the pyramids, joined by the central axis of the crystal, that formed the starting-point of his speculations, and led him to introduce some new notions and terms into the science. How these crystals originated was doubtful, but they evidently grew, not from within like plants, but from without, by the addition of new layers of minute particles carried to the crystal by a fluid, and laid down specially at the ends, as shown by the fine striee that are never wantiug on the middle planes. His rejection of extreme cold as the causa efficiens, for something similar to magnetic power, is again a suggestive idea, and not less his conclusion that crystals therefore were not formed only at the first beginning of things, but continue to grow even at the present day. Still more important as a step in the progress of the science would be his assertion that the number and length of the sides in the plane of the axis may vary widely without change in the angles (in piano axis, lalerum et numerum et longitudinem varie mutari non mutatis angulis), could we regard it as having a wider application than to the case in hand. It was perhaps more a deduction from the mathe-matical form of the body than a generalization from observed facts. But some of his other descriptions show great powers of observation, and in his notice of the cleavage of calcspar, and its division into other rhomboidal bodies, we have again a fact that in other hands was to bear important fruits.

Erasmus Bartholinus, another Dane (born 1625, died 1698), made known, in his Experimenta Crystalli Islandici disdiaclastici (1670), another property of the same mineral, very remarkable in itself and its results to science. This was the double refraction of the beautifully transparent variety sent from the Rbdefiord in Iceland to Copenhagen. In the same tract, it may be mentioned in passing, occurs the first reference to the blowpipe as a mean of applying heat to minerals. But the optical fact, turning the attention of mathematicians to crystals, had more direct influence on our science. The celebrated Huyghens described the same miranda refractio, and pointed out its laws; he also measured the angles of the rhomboids with a close approximation to truth, and remarked the occurrence of a less distinct double refraction in quartz—in crystallo duplex esset refractio. He likewise observed the peculiar cleavage of calcspar, which he tried to explain by building up the crystals of spheroids. Leeuwenhoek also, in his Arcana Natures (1695), mentions cleavage in gypsum and Muscovy glass, and tried to estimate the thickness of the laminae, which Newton in his Optics in 1706 showed could be calculated from his doctrine of the colour of thin plates. In the same work Newton gives an account of the double refraction of Iceland spar and the laws it follows, and, observing the changes to which the rays were subject, asks if these rays of light may not have different sides, with different properties—the first anticipation of the polarization of light, so important in this science. Returning to Leeuwenhoek, we find him showing salts of various forms, growing up in solutions under his microscopes. About this time too Guglielmini in his treatise on these bodies, Be Salibus Dissertatio Epistolaris (1707), tried to prove that they could all be divided into molecules of a few regular forms, and affirms as a consequence that the inclination of the planes and angles is always constant. At a somewhat earlier period the celebrated Robert Boyle had published a treatise on precious stones, in which he describes many properties of crystals and their peculiar forms which he compares to those of salts. He also pointed out the crystallization of bismuth from fusion by heat—a fact often overlooked by later observers.

The attention of men of science was now thoroughly directed to the forms and origin of these bodies, and many curious observations might be collected from the writings of De la Hire, Woodward, Cappeller, Henckel, and others. But we pass on to Linnaeus, whose Systema Naturte formed, Lmmet in this as in other departments of natural science, the com-mencement of a new period in its history. In his first edition in 1736 he gave a classification which, as he says himself, though far from perfect and often blamed, had enabled others mounted on his shoulders to see wider. Some of these successors he enumerates in the twelfth edition of his /System (1768)—among them WTallerius, Swab, and Cronstedt. He admits in the preface that he had laid aside the study of stones in which he once delighted, and therefore could not boast of his knowledge of lithology. Lithologia mihi cristas non eriget, lapides enim quos quondam in deliciis liabui, tradita demnm aliis dis-ciplina, seposui, are his characteristic words Still there is much that was important in his work. Thus he distinguishes figured stones from those that are amorphous, and notes the difference of the tessellata or cubical from the prisma or long columnar and the pyramis or pointed forms. Then he figures rudely, it may be, and describes some forty common forms of crystals, and gives examples of minerals in which they occur. His table of " Afiinilates crystalloruin" is even more suggestive, and could scarcely fail, if followed out, to lead to further advances. The use he made of these forms as important characters in describing and classifying minerals was well calculated to promote their study. Even the fact that he cut out models in wood of the forms he saw, shows in what a truly practical maimer he regarded the subject. His notions regarding the formation of crystals were, however, very imperfect. Salt, he affirms, is the only known natural cause of crystallization, and consequently the forms of the crystals of other substances were determined by the salts in union with them. This is the more remarkable, as he refers to an anonymous author in his own country, to whom he applies the words of Isaac, vox Sioabii, manus Cronstedti, as refuting this theory from the fact that crystals of metals were produced by fusion.

The advanced character of these views of Linnaeus appears more strongly when we contrast them with those of his great rival Buffon. According to him crystals are only a result of organization, so that the prisms of rock crystal, the rhombs of calcspar, the cubes of sea salt, the needles of nitre, and others are produced by the motions of organic molecules, and specially of those derived from the remains of animals and plants found in calcareous masses, and in the layer of vegetable earth covering the surface of the globe. Hence he takes no note of crystallization among the characters of minerals. Very different was the influence of Linnaeus on Rome Delisle (born 1736, died 1790) Delisla. whose Essai de CmtaZfop-orp/iie appeared first in 1772, and in an enlarged form in 1783. Working in the spirit of his master, he formed a large collection of mineral crystals which he examined with great care, comparing the forms of the faces and measuring the angles. In doing this he soon found that the same mineral assumed various forms,-— calcspar, for instance, sometimes that of a six-sided prism, at others of a rhomboid, and fluor-spar in some cases form-ing cubes, in others octahedrons. In trying to explain this fact, he assumed that in each species there was a certain original form, generally the most simple he could find, from which all the others might be derived when cut in a parti-cular manner. Thus by cutting off the angles of the cube, it may be converted into an octahedron. Werner in his treatise On the External Character of Minerals had used the terms truncation, bevelling, acumination (Abstumpfung, Zuscharfung, Zuspitzung) for similar changes on the fundamental forms, but Delisle probably had no knowledge of this fact, and in other respects could borrow little from Werner, who in crystallography scarce went beyond Linnaeus. The progress Delisle made in the ten years between his first and second work (Cristallographie on Description des formes propire a, tous les corps du regne mineral, 1783) is truly remarkable. He now affirms in clear and distinct terms "that amidst all the innumerable variations of which the primitive form of a salt or crystal is susceptible, there is one thing that never varies and remains constantly the same in each species,—that is, the angle of incidence, or the respective inclination of the faces to each other." Hence these angles are truly characteristic of each species, but only of the primitive forms, from which others, which he names secondary, are derived by various modifications. Of these principal primitive forms he assumes six; but these are less skilfully chosen and, as now seen, not always truly distinct. But many defects were compensated for by the great labour he expended on figuring and his superior accuracy in measuring crystals. This he was able to secure by the use of the goniometer recently invented by Carangeau, the new instrument, as it were, transforming the science. Then his observation of twin crystals, or macles, as he named them,—which he showed were characterized by their re-entering angles as made up of two crystals, or two halves of one crystal, in a reversed position,—was also a noteworthy step. How much he accomplished may be judged from the fact that he gives figures of more than 500 regular forms, in place of the forty described by Linnaeus. He had probably carried his system as far as it could go, and not merely familiarized the forms of crystals to mineralogists, but also suggested the possible connections that might exist among them.

Delisle seems to have assigned little value to cleavage, and in his preface speaks contemptuously of the crystal-loclastes (brise-cristaux) as innovators in the science. But Beitpnan. even earlier, in 1773, Bergman, the well-known Swedish chemist, had shown its importance, and used this peculiar structure to explain the relations of the different forms of crystals observed in the same mineral. Starting from the rhombohedron of calcspar, he placed it with the chief axis upright, and then building up other similar rhombs on it, formed a six-sided prism with rhombic ends. By stopping at a certain stage, it became a dodecahedron, or body with twelve rhombic faces, which he assumed, not quite accurately, to be the same as that proper to garnet. Again, placing this garnet-form in proper position and adding other rhombs, he showed how it easily changed into another characteristic of the hyacinth (in aliam facile migrat), whilst by other changes different crystals were produced. But he did not proceed far in the direction thus indicated, and deeper views, with more accurate facts and measure-ments, were required before this could be done.





These were found in the works of Bene Just Haiiy (born 1743, died 1822), who seems to have been led almost by accident to his theory. Curiously it is still the same mineral that with him, as with so many of his predecessors, forms the starting-point. When looking over the cabinet of Citizen Defrance a hexahedral prism of calcspar was accidentally broken from a group to which it belonged, and given him in a present. This crystal showed at the base, where it had been detached, a broken corner with thp peculiar brilliant lustre, "poli de la Nature," of the cleavage faces. Haiiy's attention was arrested by the fact, and he tried to obtain similar faces on other corners, but he only succeeded on the three alternate edges at each end of the prism. Continuing the process further, he found that he could remove slice after slice, till no vestige of the original prism was left, but in place of it a rhomboid per-fectly similar to the Iceland spar and lying in the middle of the prism. The fact struck him with surprise, mingled with the hope that it was not isolated, and this, he says, served to " develop my ideas regarding the structure of crystals, and has been, as it were, the key of the theory " (et a été comme la clef de la théoi-ie). Following it out on differently formed crystals of this mineral he found they could all be reduced to a similar internal nucleus. But wdien the mineral was distinct the nucleus had a different form. Thus in fluor-spar the nucleus was an octahedron ; in heavy spar a right prism with rhombic bases ; in galena, or sulphate of lead, a cube ; and so of other substances. In each also these forms were constant, relative to the entire species, so that its angles were subject to no appreciable variation. Even where crystals cannot be thus mechanically divided, Haiiy stated that theory aided by certain indica-tions might serve to discover the primitive form.

On these and other similar facts, Haiiy erected his celebrated theory of the structure of crystals. In each mineral theory or there exists what he calls its integrant molecules,—solid bodies incapable of further division and of invariable form, with faces parallel to the natural joints indicated by the mechanical division of the crystals, and with angles and dimensions given by calculation and observation combined. These molecules are marked in different species by distinct and determinate forms, except in a few regular bodies, such as the cube, which do not admit of variations. From these primitive or integrant molecules all the various crystals found in each species are built up according to certain definite laws, and thus the secondary crystals, as he names them, are produced. Of primitive forms only six were known from observation. These were the parallelopiped, the octahedron, the tetrahedron, the regular hexahedral prism, the dodecahedron with equal and similar rhombic faces, and the dodecahedron with triangular faces, consist-ing of two regular six-sided pyramids joined base to base. In order to produce those secondary crystals which covered over the primitive form, so as to disguise it in so many different ways, he supposed the enveloping matter to be made up of a series of lamina?, each decreasing in extent either equally in all directions, or only at certain parts. This decrease takes place by the regular subtraction of one or several ranges of integrant molecules in each successive layer ; and theory, determining by calculation the number of these ranges, can represent all the known results of crys-tallization, and even anticipate discoveries, and indicate hypothetical forms which may one day reward the research of naturalists. He thus claims for his theory that greatest proof of its truth and value which a scientific theory can present,—the power to anticipate observation and to foretell future discoveries. As an example of this process Haiiy showed how by applying successive layers of integrant molecules, each less by one row all round, to the faces of the primitive cube, a rhombic dodecahedron was necessarily formed. In other cases he assumed that the decrease was 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 Hatty'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 Law of he pointed out what he named " the law of symmetry," in symmetry, 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 chemical 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. C™orsms Thus one of the first to criticise the system was Weiss, ysi^ys 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 forms it wasquite indifferent whether the cube or regularocta-hedron was chosen, 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 aseribed 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 1829) 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 Weiss, mentioned as the translater 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 principali. 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 (ornnis figures 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 natürlichen 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, tvvo-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. Haüy 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 Haüy 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 what 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 more 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 Grätz, 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 doubt that Weiss preceded him in promulgating these new views, but also show that Mohs wrought them out in a more systematic form, and made them more generally known. In 1820 he published his Charakteristih des naturhistorischen Mineralsystem.es, followed in 1822 by his Grundriss der Mineralogie. Both these treatises were translated into English, the second by the well-known Haidinger, then residing in Edinburgh. The clearness and precision with which he marked out and defined the various terms and new ideas required, and followed out the laws regulating combinations, had a great effect in giving a wider currency to his writings. The thorough mode in which he traced out the series of forms in the systems and explained these also added to their popularity. Professor Jameson too gave it a higher authority and wider accept-ance, describing it " as eminently distinguished by its originality and simplicity." Its success was further promoted by the remarkable discovery made about the Brewster, same time by Sir David (then Dr) Brewster. In connection with his observations on the polarization of light, this dis-tinguished optician had endeavoured to point out the con-nection between Haiiy's nuclei or primitive forms of crystals and the number of their axes of double refraction, and even shown that Haiiy had in some cases chosen erroneous forms, as they did not agree with their optical characters. The appearance of Mohs's views threw unex-pected light on the fact, as his system of crystallography harmonized in a most remarkable manner with the arrange-ment proposed on optical grounds. In reality, as now well known, all minerals crystallizing in the regular system of Weiss and Mohs with equal and uniform axes show only _single refraction ; those belonging to the two and one axial and three and one axial systems of Weiss, the pyramidal oand rhombohedral of Mohs, have double refraction with only one optical axis ; whilst those in the three other systems show double refraction and two optical axes. As Whewell has well remarked, " Sir D. Brewster's optical experiments must have led to a classification of crystals into the above systems, or something nearly equivalent, even if the crystals had not been so arranged by attention to their forms."

The establishment of this system, whether due to Weiss
progress. or Mohs, or in part to both, gave to crystallography as a
pure science essentially its present form. Taken in
oconnection with the law that the indices marking the
'relative dimensions of the parameters are always rational
numbers, and seldom large, with the symmetry of forms,
and the grouping of the faces in zones, we have the leading
principles on which it depends. The subsequent progress
of the science has been rather directed to working out and
completing the structure, and showing the mutual relations
of its essential principles, than to modifying the foundations
on which it rests. These researches have taken two chief
directions, the one explaining the geometrical properties of
crystals, and the systems under which in consequence of
these properties they necessarily fall to be classed, while
the second has regard to the physical properties of crystals,
that is, of the various bodies, especially the native minerals,
assuming these forms. Before noticing these we must refer
to another point in which Haiiy's views were also about the
same time remarkably modified and extended,
j Haiiy, we have seen, maintained that a very close connec-
morpbism ti°n always existed between the crystalline character and the chemical composition of minerals, so that from diversity in the angular measurement of two crystals we might infer a difference in their chemical composition, or the reverse. More accurate analyses soon showed that this law had not that universal application which Haiiy assumed, and even in 1815 Fuchs had pointed out that certain elements were what he named vicarious, so that in compounds a certain amount of one could replace so much of some other. The remarkable theories and researches of Berzelius soon rendered some change in this respect inevitable, and it was carried out by the discovery of isomorphism by his pupil Mitscherlich in 1822. The subject, however, belongs less to crystallography than to chemistry or mineralogy, and we can only mention the general principle. Mitscherlich showed that there are certain substances which crystallize in forms closely resembling each other, and with the corresponding angles only differing by one or two degrees, or even less. Thus the carbonates of iron and manganese, or lime and magnesia, agree nearly in form and dimensions. Such substances were named isomorphous, and were found to have the tendency to replace or be substituted for each other in compound bodies, with very slight modification of the forms or angles of the crystals. Though at first denied by Haiiy and his followers, this truth is now fully established, and has had vast influence in the determination and classification of minerals. As modifying the same conclusion of Haiiy, but in an opposite direction, we must also mention Mitscherlich's further discovery of dimorphism, according to which the same element (as sulphur), or the same compound (as carbonate of lime), when crystallizing under different conditions, especially as regards tempera-ture, may assume two distinct forms of crystals belonging even to different systems. Instances are even known of trimorphism and polymorphism, in which the same sub-stance may occur in three more or forms of crystalliza-tion.

The mode of formation of crystals, and the powers that are active in their formation, were, as we have seen, favourite subjects of speculation with the earlier writers on known crystallography, and are closely connected with the chemical composition of minerals to which we have just referred. This subject continues to attract many inquirers, and has given occasion to some remarkable speculations ; but it can hardly be affirmed that much progress has been made in this direction. Crystals may still be seen, as in the time of Leeuwenhoek, springing out of solutions under the microscope, and continuingto increase in size, but the powers that are active escape our notice, and we are still left almost in the same region of speculation as our predeces-sors. Such discussions, in truth, concern rather the general constitution of matter than the special corner whose history we have been following, so that the words of Brewster still hold true :—" In whatever way crystallographers shall succeed in accounting for the various secondary forms of crystals, they are then only on the threshold of their subject. The real constitution of crystals would be still i unknown ; and though the examination of these bodies has been pretty diligently pursued, we can at this moment form no adequate idea of the complex and beautiful organization of these apparently simple bodies."

Returning to the more special subject of pure or Crystallo-geometric crystallography, one great object of recent symi^is. inquiry has been to discover some method of designating the forms or faces of crystals by numbers or symbols, that would at once point out their general relations to each other, and facilitate the calculation of their angles so as to check or control observation. Haiiy had already attempted to do this in his great work, by means of his theory of decrements, but his materials were still too imperfect, and his symbols are often very complex. Still the weight of his name retains great influence in France, where a system founded on his, but modified by the more recent views, prevails. It is generally associated with the name of Armand Levy (born 1794, died 1841), who in 1837 published an important work on Mr Heuland's collection (Description d'une collection des Minéraux formée par M. H. Heuland) illustrated by numerous plates of crystals. He 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 Dufrénoy in his Traité de Minéralogie (Paris, 1844-56), and by Des Cloizeaux in his 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" (Beiträge zur Krystallonomie) in 1823, showed how crystals might be represented not so much by the faces as by their normals, that is, by lines drawn from the centre of the system vertical to the faces. Cleavage, he says, and the reflection of light, etc., all indicate a force acting vertical to the faces, or in the normal. He further brought clearly out the 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 Tübingen, another pupil of Weiss, made known a similar method in 1835, 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 Grundriss der bestimmenden und rechnenden Krystallographie, 1873. The truest representa-tive of Weiss, however, is generally regarded as Gustaf Hose, 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 1825 ; and is further explained in Mohs's Anfangs-gründe der Naturgeschichte des Mineralreichs (1832, 2d edi-tion by Zippe, 1839). Haidinger, besides many memoirs, has also published a separate work in which the method is fully explained (Handbuch der bestimmenden Mineralogie, 1845). Put 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 Lehrbuch der reinen und angewandten Krystallographie appeared in 1830 ; his Anfangsgründe der Krystallographie in 1841, 2d edition, 1855, and his Theoretische Krystallo-graphie in 1856. His Elemente der Mineralogie, first published in 1846, 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 lie 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 Professor 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-books, among which may be mentioned the Lehrbuch dei 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 the 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 Levy, so that the one may be, as it were, translated iuto the other. Similar tables may also be found in Des Cloizeaux's Minéralogie 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 Physical crystallographic characters, proving very distinctly that the ProPertles-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 Brewster 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 l'emploi des propriétés ojjtiques biréfringentes en Minéralogie, 1857, and Sur l'emploi du microscope, in 1864), and in his Manuel de Minéralogie 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 a new impulse was given to the study "by Mr Sorby's memoir "On the Microscopical Structure of Crystals," published in the Journal of the Geologiccd 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 (Mikroslcopische Gesteinsstudien. 1863, Mikroskopische Beschaffenheiten der Mineralien, 1873, <fec), Schrauf (Lehrbuch der physikalischen Minéralogie, 1868), and Rosenbusch LMikroscopische PhysiogranMe der Mineralien, 1873), both for further information on the subject generally, and for lists of the more important recent publications. How valuable it has become may be seen from the fact that these transparent sections, examined between two Nicol's prisms, from the phenomena of the interference of light, readily enable the observer to deter-mine to which of the six systems of crystallization the mineral interposed belongs, and thus to fix one of its most essential characters. In this way the exact composition of many fine-grained crystalline rocks can be determined, and much light thrown on their history.

In regard to the other physical properties of crystals, it must suffice to say that they all indicate a similar close dependence on their geometric character. The same systems shown by their mathematical forms and optic properties reappear in reference to their relations to heat, magnetism, electricity, and other properties. The regular or tesseral minerals, with simple refraction of light, are shown by Senarmont's researches also to conduct heat uniformly in all directions, and their magnetic and electric peculiarities are similar. The tetragonal and hexagonal crystals with one chief axis, as they show double refraction of light with a single optic axis, have also analogous modes of conducting heat, of expanding under its influence, and of transmitting magnetism and electricity. And again, the three other systems with unequal axes, as they show two optic axes, exhibit also corresponding peculiarities in respect to the other properties mentioned. In this we have a remarkable instance of connection of the various physical sciences, and a strong proof of the sound basis that crystallography attained by the discoveries of Weiss and Mohs. A great deal has been recently done in improving the instruments employed in determining the forms and dimensions of crystals. Though for first and rough approximations to the angles the early form of the hand goniometer may still be used, even the reflecting goniometer of Wollaston no longer meets the requirements of modern accuracy. In 1820 the Eoyal Academy of Sciences at Berlin offered a prize for the best methods of measuring these angles, which was gained by Dr Kupfter. Malus added a telescope to Wollaston's goniometer, and other methods of increasing its accuracy have been proposed, as by Babinet and Mitscherlich. Frankenheim, Haidinger, and others have also endeavoured to perfect the methods or means of observation which now, as tested by comparison with the results of calculation, seem fully adequate to the w7ants at least of determinative mineralogy.

The forms of crystals that occur in native minerals are described more or less fully in several of the works already mentioned. The Atlas der Krystal-Formen des Mineralreiches of Dr A. Schrauf, 1865-1873, is intended to form a complete collection of all the forms observed in the mineral kingdom,—estimated as exceeding 10,000 in number, described in very many separate treatises and memoirs, and every day becoming more numerous. The arrange-ment is alphabetical, and the work, from its accuracy and rich material, is highly valuable but still unfinished. The forms of salts and artificial crystals are described in Rammelsberg's Handbuch der krystallographische Chcmic, 1855, and supplement, 1857, and many of them also in Groth's Kryslallographie.

The history of crystallography is related in C. M. Marx's Geschichte der Kryslallkunde, 1825 ; WheweU's History of the Inductive Sciences, vol. iii. ; Von Kobell's Geschichte der Mineralogie, 1864; Kenngott's Ucbcrsicht der Resullate der Mineralogische Forschungen, 1844-1865 ; Quenstedt's Grundriss, 1873, &c., from which further information may be obtained. (J. N1.)



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