OF all the physical properties of bodies, there is none so fixed, and in every way so remarkable, as this;—that the same chemical compound always assumes, with the utmost precision, the same geometrical form. This identity, however, is not immediately obvious; it is often obscured by various mixtures and imperfections in the substance; and even when it is complete, it is not immediately recognized by a common eye, since it consists, not in the equality of the sides or faces of the figures, but in the equality of their angles. Hence it is not surprising that the constancy of form was not detected by the early observers. Pliny says,1 “Why crystal is generated in a hexagonal form, it is difficult to assign a reason; and the more so, since, while its faces are smoother than any art can make them, the pyramidal points are not all of the same kind.” The quartz crystals of the Alps, to which he refers, are, in some specimens, very regular, while in others, one side of the pyramid becomes much the largest; yet the angles remain constantly the same. But when the whole shape varied so much, the angles also seemed to vary. Thus Conrad Gessner, a very learned naturalist, who, in 1564, published at Zurich his work, De rerum Fossilium, Lapidum et Gemmarum maxime, Figuris, says,2 “One crystal differs from another in its angles, and consequently in its figure.” And Cæsalpinus, who, as we shall find, did so much in establishing fixed characters in botany, was led by some of his general views to disbelieve the fixity of the form of crystals. In his work De Metallicis, published at Nuremberg in 1602, he says,3 “To ascribe to inanimate bodies a definite form, does not appear consentaneous to reason; for it is the office of organization to produce a definite form;” 317 an opinion very natural in one who had been immersed in the study of the general analogies of the forms of plants. But though this is excusable in Cæsalpinus, the rejection of this definiteness of form a hundred years later, when its existence had been proved, and its laws developed by numerous observers, cannot be ascribed to anything but strong prejudice; yet this was the course taken by no less a person than Buffon. “The form of crystallization,” says he,4 “is not a constant character, but is more equivocal and more variable than any other of the characters by which minerals are to be distinguished.” And accordingly, he makes no use of this most important feature in his history of minerals. This strange perverseness may perhaps be ascribed to the dislike which Buffon is said to have entertained for Linnæus, who had made crystalline form a leading character of minerals.
It is not necessary to mark all the minute steps by which mineralogists were gradually led to see clearly the nature and laws of the fixity of crystalline forms. These forms were at first noticed in that substance which is peculiarly called rock-crystal or quartz; and afterwards in various stones and gems, in salts obtained from various solutions, and in snow. But those who observed the remarkable regular figures which these substances assume, were at first impelled onwards in their speculations by the natural tendency of the human mind to generalize and guess, rather than to examine and measure. They attempted to snatch at once the general laws of geometrical regularity of these occurrences, or to connect them with some doctrine concerning formative causes. Thus Kepler,5 in his Harmonics of the World, asserts a “formatrix facultas, which has its seat in the entrails of the earth, and, after the manner of a pregnant woman, expresses the five regular geometrical solids in the forms of gems.” But Philosophers, in the course of time, came to build more upon observation, and less upon abstract reasonings. Nicolas Steno, a Dane, published, in 1669, a dissertation De Solido intra Solidum Naturaliter contento, in which he says,6 that though the sides of the hexagonal crystal may vary, the angles are not changed. And Dominic Gulielmini, in a Dissertation on Salts, published in 1707, says,7 in a true inductive spirit, “Nature does not employ all figures, but only certain ones of those which are possible; and of these, the determination is not to be fetched from the brain, or proved à priori, but obtained by experiments and observations.” And 318 he speaks8 with entire decision on this subject: “Nevertheless since there is here a principle of crystallization, the inclination of the planes and of the angles is always constant.” He even anticipates, very nearly, the views of later crystallographers as to the mode in which crystals are formed from elementary molecules. From this time, many persons labored and speculated on this subject; as Cappeller, whose Prodromus Crystallographiæ appeared at Lucern in 1723; Bourguet, who published Lettres Philosophiques sur la Formation de Sels et de Cristaux, at Amsterdam, in 1792; and Henckel, the “Physicus” of the Elector of Saxony, whose Pyritologia came forth in 1725. In this last work we have an example of the description of the various forms of special classes of minerals, (iron pyrites, copper pyrites, and arsenic pyrites;) and an example of the enthusiasm which this apparently dry and laborious study can excite: “Neither tongue nor stone,” he exclaims,9 “can express the satisfaction which I received on setting eyes upon this sinter covered with galena; and thus it constantly happens, that one must have more pleasure in what seems worthless rubbish, than in the purest and most precious ores, if we know aught of minerals.”
Still, however, Henckel10 disclaims the intention of arranging minerals according to their mathematical forms; and this, which may be considered as the first decided step in the formation of crystallographic mineralogy, appears to have been first attempted by Linnæus. In this attempt, however, he was by no means happy; nor does he himself appear to have been satisfied. He begins his preface by saying, “Lithology is not what I plume myself upon.” (Lithologia mihi cristas non eriget.) Though his sagacity, as a natural historian, led him to see that crystalline form was one of the most definite, and therefore most important, characters of minerals, he failed in profiting by this thought, because, in applying it, he did not employ the light of geometry, but was regulated by what appeared to him resemblances, arbitrarily selected, and often delusive.11 Thus he derived the form of pyrites from that of vitriol;12 and brought together alum and diamond on account of their common octohedral form. But he had the great merit of animating to this study one to whom, more perhaps than to any other person, it owes its subsequent progress; I mean Romé de Lisle. “Instructed,” this writer says, in his preface to his Essais de Crystallographie, “by the works of the celebrated Von Linnée, how 319 greatly the study of the angular form of crystals might become interesting, and fitted to extend the sphere of our mineralogical knowledge, I have followed them in all their metamorphoses with the most scrupulous attention.” The views of Linnæus, as to the importance of this character, had indeed been adopted by several others; as John Hill, the King’s gardener at Kew, who, in 1777, published his Spathogenesia; and Grignon, who, in 1775, says, “These crystallizations may give the means of finding a new theory of the generation of crystalline gems.”
The circumstance which threw so much difficulty in the way of those who tried to follow out his thought was, that in consequence of the apparent irregularity of crystals, arising from the extension or contraction of particular sides of the figure, each kind of substance may really appear under many different forms, connected with each other by certain geometrical relations. These may be conceived by considering a certain fundamental form to be cut into new forms in particular ways. Thus if we take a cube, and cut off all the eight corners, till the original faces disappear, we make it an octohedron; and if we stop short of this, we have a figure of fourteen faces, which has been called a cubo-octohedron. The first person who appears distinctly to have conceived this truncation of angles and edges, and to have introduced the word, is Démeste;13 although Wallerius14 had already said, in speaking of the various crystalline forms of calcspar, “I conceive it would be better not to attend to all differences, lest we be overwhelmed by the number.” And Werner, in his celebrated work On the External Characters of Minerals,15 had formally spoken of truncation, acuation, and acumination, or replacement by a plane, an edge, a point respectively, (abstumpfung, zuschärfung, zuspitzung,) as ways in which the forms of crystals are modified and often disguised. He applied this process in particular to show the connexion of the various forms which are related to the cube. But still the extension of the process to the whole range of minerals and other crystalline bodies, was due to Romé de Lisle.
WE have already seen that, before 1780, several mineralogists had recognized the constancy of the angles of crystals, and had seen (as Démeste and Werner,) that the forms were subject to modifications of a definite kind. But neither of these two thoughts was so apprehended and so developed, as to supersede the occasion for a discoverer who should put forward these principles as what they really were, the materials of a new and complete science. The merit of this step belongs jointly to Romé de Lisle and to Haüy. The former of these two men had already, in 1772, published an Essai de Crystallographie, in which he had described a number of crystals. But in this work his views are still rude and vague; he does not establish any connected sequence of transitions in each kind of substance, and lays little or no stress on the angles. But in 1783, his ideas16 had reached a maturity which, by comparison, excites our admiration. In this he asserts, in the most distinct manner, the invariability of the angles of crystals of each kind, under all the changes of relative dimension which the faces may undergo;17 and he points out that this invariability applies only to the primitive forms, from each of which many secondary forms are derived by various changes.18 Thus we cannot deny him the merit of having taken steady hold on both the handles of this discovery, though something still remained for another to do. Romé pursues his general ideas into detail with great labor and skill. He gives drawings of more than five hundred regular forms (in his first work he had inserted only one hundred and ten; Linnæus only knew forty); and assigns them to their proper substances; for instance, thirty to calcspar, and sixteen to felspar. He also invented and used a goniometer. We cannot doubt that he would have been 321 looked upon as a great discoverer, if his fame had not been dimmed by the more brilliant success of his contemporary Haüy.
Réné-Just Haüy is rightly looked upon as the founder of the modern school of crystallography; for all those who have, since him, pursued the study with success, have taken his views for their basis. Besides publishing a system of crystallography and of mineralogy, far more complete than any which had yet appeared, the peculiar steps in the advance which belong to him are, the discovery of the importance of cleavage, and the consequent expression of the laws of derivation of secondary from primary forms, by means of the decrements of the successive layers of integrant molecules.
The latter of these discoveries had already been, in some measure, anticipated by Bergman, who had, in 1773, conceived a hexagonal prism to be built up by the juxtaposition of solid rhombs on the planes of a rhombic nucleus.19 It is not clear20 whether Haüy was acquainted with Bergman’s Memoir, at the time when the cleavage of a hexagonal prism of calcspar, accidentally obtained, led him to the same conception of its structure. But however this might be, he had the indisputable credit of following out this conception with all the vigor of originality, and with the most laborious and persevering earnestness; indeed he made it the business of his life. The hypothesis of a solid, built up of small solids, had this peculiar advantage in reference to crystallography; it rendered a reason of this curious fact;—that a certain series of forms occur in crystals of the same kind, while other forms, apparently intermediate between those which actually occur, are rigorously excluded. The doctrine of decrements explained this; for by placing a number of regularly-decreasing rows of equal solids, as, for instance, of bricks, upon one another, we might form a regular equal-sided triangle, as the gable of a house; and if the breadth of the gable were one hundred bricks, the height of the triangle might be one hundred, or fifty, or twenty-five; but it would be found that if the height were an intermediate number, as fifty-seven, or forty-three, the edge of the wall would become irregular; and such irregularity is assumed to be inadmissible in the regular structure of crystals. Thus this mode of conceiving crystals allows of certain definite secondary forms, and no others.
The mathematical deduction of the dimensions and proportions 322 of these secondary forms;—the invention of a notation to express them;—the examination of the whole mineral kingdom in accordance with these views;—the production of a work21 in which they are explained with singular clearness and vivacity;—are services by which Haüy richly earned the admiration which has been bestowed upon him. The wonderful copiousness and variety of the forms and laws to which he was led, thoroughly exercised and nourished the spirit of deduction and calculation which his discoveries excited in him. The reader may form some conception of the extent of his labors, by being told—that the mere geometrical propositions which he found it necessary to premise to his special descriptions, occupy a volume and a half of his work;—that his diagrams are nearly a thousand in number;—that in one single substance (calcspar) he has described forty-seven varieties of form;—and that he has described one kind of crystal (called by him fer sulfuré parallélique) which has one hundred and thirty-four faces.
In the course of a long life, he examined, with considerable care, all the forms he could procure of all kinds of mineral; and the interpretation which he gave of the laws of those forms was, in many cases, fixed, by means of a name applied to the mineral in which the form occurred; thus, he introduced such names as équiaxe, métastatique, unibinaire, perihexahèdre, bisalterne, and others. It is not now desirable to apply separate names to the different forms of the same mineral species, but these terms answered the purpose, at the time, of making the subjects of study more definite. A symbolical notation is the more convenient mode of designating such forms, and such a notation Haüy invented; but the symbols devised by him had many inconveniences, and have since been superseded by the systems of other crystallographers.
Another of Haüy’s leading merits was, as we have already intimated, to have shown, more clearly than his predecessors had done, that the crystalline angles of substances are a criterion of the substances; and that this is peculiarly true of the angles of cleavage;—that is, the angles of those edges which are obtained by cleaving a crystal in two different directions;—a mode of division which the structure of many kinds of crystals allowed him to execute in the most complete manner. As an instance of the employment of this criterion, I may mention his separation of the sulphates of baryta and strontia, which had 323 previously been confounded. Among crystals which in the collections were ranked together as “heavy spar,” and which were so perfect as to admit of accurate measurement, he found that those which were brought from Sicily, and those of Derbyshire, differed in their cleavage angle by three degrees and a half. “I could not suppose,” he says,22 “that this difference was the effect of any law of decrement; for it would have been necessary to suppose so rapid and complex a law, that such an hypothesis might have been justly regarded as an abuse of the theory.” He was, therefore, in great perplexity. But a little while previous to this, Klaproth had discovered that there is an earth which, though in many respects it resembles baryta, is different from it in other respects; and this earth, from the place where it was found (in Scotland), had been named Strontia. The French chemists had ascertained that the two earths had, in some cases, been mixed or confounded; and Vauquelin, on examining the Sicilian crystals, found that their base was strontia, and not, as in the Derbyshire ones, baryta. The riddle was now read; all the crystals with the larger angle belong to the one, all those with the smaller, to the other, of these two sulphates; and crystallometry was clearly recognized as an authorized test of the difference of substances which nearly resemble each other.
Enough has been said, probably, to enable the reader to judge how much each of the two persons, now under review, contributed to crystallography. It would be unwise to compare such contributions to science with the great discoveries of astronomy and chemistry; and we have seen how nearly the predecessors of Romé and Haüy had reached the point of knowledge on which these two crystallographers took their stand. But yet it is impossible not to allow, that in these discoveries, which thus gave form and substance to the science of crystallography, we have a manifestation of no common sagacity and skill. Here, as in other discoveries, were required ideas and facts;—clearness of geometrical conception which could deal with most complex relations of form; a minute and extensive acquaintance with actual crystals; and the talent and habit of referring these facts to the general ideas. Haüy, in particular, was happily endowed for his task. Without being a great mathematician, he was sufficiently a geometer to solve all the problems which his undertaking demanded; and though the mathematical reasoning might have been made more compendious 324 by one who was more at home in mathematical generalization, probably this could hardly have been done without making the subject less accessible and less attractive to persons moderately disciplined in mathematics. In all his reasonings upon particular cases, Haüy is acute and clear; while his general views appear to be suggested rather by a lively fancy than by a sage inductive spirit: and though he thus misses the character of a great philosopher, the vivacity of style, and felicity and happiness of illustration, which grace his book, and which agree well with the character of an Abbé of the old French monarchy, had a great and useful influence on the progress of the subject.
Unfortunately Romé de Lisle and Haüy were not only rivals, but in some measure enemies. The former might naturally feel some vexation at finding himself, in his later years (he died in 1790), thrown into shade by his more brilliant successor. In reference to Haüy’s use of cleavage, he speaks23 of “innovators in crystallography, who may properly be called crystalloclasts.” Yet he adopted, in great measure, the same views of the formation of crystals by laminæ,24 which Haüy illustrated by the destructive process at which he thus sneers. His sensitiveness was kept alive by the conduct of the Academy of Sciences, which took no notice of him and his labors;25 probably because it was led by Buffon, who disliked Linnæus, and might dislike Romé as his follower; and who, as we have seen, despised crystallography. Haüy revenged himself by rarely mentioning Romé in his works, though it was manifest that his obligations to him were immense; and by recording his errors while he corrected them. More fortunate than his rival, Haüy was, from the first, received with favor and applause. His lectures at Paris were eagerly listened to by persons from all quarters of the world. His views were, in this manner, speedily diffused; and the subject was soon pursued, in various ways, by mathematicians and mineralogists in every country of Europe.
I HAVE not hitherto noticed the imperfections of the crystallographic views and methods of Haüy, because my business in the last section 325 was to mark the permanent additions he made to the science. His system did, however, require completion and rectification in various points; and in speaking of the crystallographers of the subsequent time, who may all be considered as the cultivators of the Hauïan doctrines, we must also consider what they did in correcting them.
The three main points in which this improvement was needed were;—a better determination of the crystalline forms of the special substances;—a more general and less arbitrary method of considering crystalline forms according to their symmetry; and a detection of more general conditions by which the crystalline angle is regulated. The first of these processes may be considered as the natural sequel of the Hauïan epoch: the other two must be treated as separate steps of discovery.
When it appeared that the angle of natural or of cleavage faces could be used to determine the differences of minerals, it became important to measure this angle with accuracy. Haüy’s measurements were found very inaccurate by many succeeding crystallographers: Mohs says26 that they are so generally inaccurate, that no confidence can be placed in them. This was said, of course, according to the more rigorous notions of accuracy to which the establishment of Haüy’s system led. Among the persons who principally labored in ascertaining, with precision, the crystalline angles of minerals, were several Englishmen, especially Wollaston, Phillips, and Brooke. Wollaston, by the invention of his Reflecting Goniometer, placed an entirely new degree of accuracy within the reach of the crystallographer; the angle of two faces being, in this instrument, measured by means of the reflected images of bright objects seen in them, so that the measure is the more accurate the more minute the faces are. In the use of this instrument, no one was more laborious and successful than William Phillips, whose power of apprehending the most complex forms with steadiness and clearness, led Wollaston to say that he had “a geometrical sense.” Phillips published a Treatise on Mineralogy, containing a great collection of such determinations; and Mr. Brooke, a crystallographer of the same exact and careful school, has also published several works of the same kind. The precise measurement of crystalline angles must be the familiar employment of all who study crystallography; and, therefore, any further enumeration of those 326 who have added in this way to the stock of knowledge, would be superfluous.
Nor need I dwell long on those who added to the knowledge which Haüy left, of derived forms. The most remarkable work of this kind was that of Count Bournon, who published a work on a single mineral (calcspar) in three quarto volumes.27 He has here given representations of seven hundred forms of crystals, of which, however, only fifty-six are essentially different. From this example the reader may judge what a length of time, and what a number of observers and calculators, were requisite to exhaust the subject.
If the calculations, thus occasioned, had been conducted upon the basis of Haüy’s system, without any further generalization, they would have belonged to that process, the natural sequel of inductive discoveries, which we call deduction; and would have needed only a very brief notice here. But some additional steps were made in the upward road to scientific truth, and of these we must now give an account.
IN Haüy’s views, as generally happens in new systems, however true, there was involved something that was arbitrary, something that was false or doubtful, something that was unnecessarily limited. The principal points of this kind were;—his having made the laws of crystalline derivation depend so much upon cleavage;—his having assumed an atomic constitution of bodies as an essential part of his system; and his having taken a set of primary forms, which, being selected by no general view, were partly superfluous, and partly defective.
How far evidence, such as has been referred to by various philosophers, has proved, or can prove, that bodies are constituted of indivisible atoms, will be more fully examined in the work which treats of the Philosophy of this subject. There can be little doubt that the 327 portion of Haüy’s doctrine which most riveted popular attention and applause, was his dissection of crystals, in a manner which was supposed to lead actually to their ultimate material elements. Yet it is clear, that since the solids given by cleavage are, in many cases, such as cannot make up a solid space, the primary conception of a necessary geometrical identity between the results of division and the elements of composition, which is the sole foundation of the supposition that crystallography points out the actual elements, disappears on being scrutinized: and when Haüy, pressed by this difficulty, as in the case of fluor-spar, put his integrant octohedral molecules together, touching by the edges only, his method became an empty geometrical diagram, with no physical meaning.
The real fact, divested of the hypothesis which was contained in the fiction of decrements, was, that when the relation of the derivative to the primary faces is expressed by means of numerical indices, these numbers are integers, and generally very small ones; and this was the form which the law gradually assumed, as the method of derivation was made more general and simple by Weiss and others.
“When, in 1809, I published my Dissertation,” says Weiss,28 “I shared the common opinion as to the necessity of the assumption and the reality of the existence of a primitive form, at least in a sense not very different from the usual sense of the expression. While I sought,” he adds, referring to certain doctrines of general philosophy which he and others entertained, “a dynamical ground for this, instead of the untenable atomistic view, I found that, out of my primitive forms, there was gradually unfolded to my hands, that which really governs them, and is not affected by their casual fluctuations, the fundamental relations of those Dimensions according to which a multiplicity of internal oppositions, necessarily and mutually interdependent, are developed in the mass, each having its own polarity; so that the crystalline character is co-extensive with these polarities.”
The “Dimensions” of which Weiss here speaks, are the Axes of Symmetry of the crystal; that is, those lines in reference to which, every face is accompanied by other faces, having like positions and properties. Thus a rhomb, or more properly a rhombohedron,29 of 328 calcspar may be placed with one of its obtuse corners uppermost, so that all the three faces which meet there are equally inclined to the vertical line. In this position, every derivative face, which is obtained by any modification of the faces or edges of the rhombohedron, implies either three or six such derivative faces; for no one of the three upper faces of the rhombohedron has any character or property different from the other two; and, therefore, there is no reason for the existence of a derivative from one of these primitive faces, which does not equally hold for the other primitive faces. Hence the derivative forms will, in all cases, contain none but faces connected by this kind of correspondence. The axis thus made vertical will be an Axis of Symmetry, and the crystal will consist of three divisions, ranged round this axis, and exactly resembling each other. According to Weiss’s nomenclature, such a crystal is “three-and-three-membered.”
But this is only one of the kinds of symmetry which crystalline forms may exhibit. They may have three axes of complete and equal symmetry at right angles to each other, as the cube and the regular octohedron;—or, two axes of equal symmetry, perpendicular to each other and to a third axis, which is not affected with the same symmetry with which they are; such a figure is a square pyramid;—or they may have three rectangular axes, all of unequal symmetry, the modifications referring to each axis separately from the other two.
These are essential and necessary distinctions of crystalline form; and the introduction of a classification of forms founded on such relations, or, as they were called, Systems of Crystallization, was a great improvement upon the divisions of the earlier crystallographers, for those divisions were separated according to certain arbitrarily-assumed primary forms. Thus Romé de Lisle’s fundamental forms were, the tetrahedron, the cube, the octohedron, the rhombic prism, the rhombic octohedron, the dodecahedron with triangular faces: Haüy’s primary forms are the cube, the rhombohedron, the oblique rhombic prism, the right rhombic prism, the rhombic dodecahedron, the regular octohedron, tetrahedron, and six-sided prism, and the bipyramidal dodecahedron. This division, as I have already said, errs both by excess and defect, for some of these primary forms might be made derivatives from others; and no solid reason could be assigned why they were not. Thus the cube may be derived from the tetrahedron, by truncating the edges; and the rhombic dodecahedron again from the cube, by truncating its edges; while the square pyramid could not be legitimately identified with the derivative of any of these forms; for if we were to 329 derive it from the rhombic prism, why should the acute angles always suffer decrements corresponding in a certain way to those of the obtuse angles, as they must do in order to give rise to a square pyramid?
The introduction of the method of reference to Systems of Crystallization has been a subject of controversy, some ascribing this valuable step to Weiss, and some to Mohs.30 It appears, I think, on the whole, that Weiss first published works in which the method is employed; but that Mohs, by applying it to all the known species of minerals, has had the merit of making it the basis of real crystallography. Weiss, in 1809, published a Dissertation On the mode of investigating the principal geometrical character of crystalline forms, in which he says,31 “No part, line, or quantity, is so important as the axis; no consideration is more essential or of a higher order than the relation of a crystalline plane to the axis;” and again, “An axis is any line governing the figure, about which all parts are similarly disposed, and with reference to which they correspond mutually.” This he soon followed out by examination of some difficult cases, as Felspar and Epidote. In the Memoirs of the Berlin Academy,32 for 1814–15, he published An Exhibition of the natural Divisions of Systems of Crystallization. In this Memoir, his divisions are as follows:—The regular system, the four-membered, the two-and-two-membered, the three-and-three-membered, and some others of inferior degrees of symmetry. These divisions are by Mohs (Outlines of Mineralogy, 1822), termed the tessular, pyramidal, prismatic, and rhombohedral systems respectively. Hausmann, in his Investigations concerning the Forms of Inanimate Nature,33 makes a nearly corresponding arrangement;—the isometric, monodimetric, trimetric, and monotrimetic; and one or other of these sets of terms have been adopted by most succeeding writers.
In order to make the distinctions more apparent, I have purposely omitted to speak of the systems which arise when the prismatic system loses some part of its symmetry;—when it has only half or a quarter its complete number of faces;—or, according to Mohs’s phraseology, when it is hemihedral or tetartohedral. Such systems are represented by the singly-oblique or doubly-oblique prism; they are termed by Weiss two-and-one-membered, and one-and-one-membered; by other writers, Monoklinometric, and Triklinometric Systems. There are also other 330 peculiarities of Symmetry, such, for instance, as that of the plagihedral faces of quartz, and other minerals.
The introduction of an arrangement of crystalline forms into systems, according to their degree of symmetry, was a step which was rather founded on a distinct and comprehensive perception of mathematical relations, than on an acquaintance with experimental facts, beyond what earlier mineralogists had possessed. This arrangement was, however, remarkably confirmed by some of the properties of minerals which attracted notice about the time now spoken of, as we shall see in the next chapter.
~Additional material in the 3rd edition.~
DIFFUSION of the Distinction of Systems.—The distinction of systems of crystallization was so far founded on obviously true views, that it was speedily adopted by most mineralogists. I need not dwell on the steps by which this took place. Mr. Haidinger’s translation of Mohs was a principal occasion of its introduction in England. As an indication of dates, bearing on this subject, perhaps I may be allowed to notice, that there appeared in the Philosophical Transactions for 1825, A General Method of Calculating the Angles of Crystals, which I had written, and in which I referred only to Haüy’s views; but that in 1826,34 I published a Memoir On the Classification of Crystalline Combinations, founded on the methods of Weiss and Mohs, especially the latter; with which I had in the mean time become acquainted, and which appeared to me to contain their own evidence and recommendation. General methods, such as was attempted in the Memoir just quoted, are part of that process in the history of sciences, by which, when the principles are once established, the mathematical operation of deducing their consequences is made more and more general and symmetrical: which we have seen already exemplified in the history of celestial mechanics after the time of Newton. It does not enter into our plan, to dwell upon the various steps in this way 331 made by Levy, Naumann, Grassmann, Kupffer, Hessel, and by Professor Miller among ourselves. I may notice that one great improvement was, the method introduced by Monteiro and Levy, of determining the laws of derivation of forces by means of the parallelisms of edges; which was afterwards extended so that faces were considered as belonging to zones. Nor need I attempt to enumerate (what indeed it would be difficult to describe in words) the various methods of notation by which it has been proposed to represent the faces of crystals, and to facilitate the calculations which have reference to them.