WTH T3RARY , .LIBRARY UNIVERSITY OF CALIFORNIA. Receiued . Accessions No. 4^^^79 Shelf No, m iHBRB TEXT-BOOKS OF SCIENCE ADAPTED FOR THE USE OF ARTISANS AND STUDENTS IN PUBLIC AND SCIENCE SCHOOLS SYSTEMATIC MINERALOGY Text-Books of Science. s d. ABNEY'S PHOTOGRAPHY. 105 Woodcuts 3 6 ANDERSON'S The STRENGTH of MATERIALS and STRUCTURES. 66 Woodcuts 3 6 ARMSTRONG'S ORGANIC CHEMISTRY. 8 Woodcuts 3 6 BALL'S ELEMENTS of ASTRONOMY. 136 Woodcuts 6 BARRY'S RAILWAY APPLIANCES. 207 Woodcuts 3 6 BAUERMAN'S SYSTEMATIC MINERALOGY. 373 Woodcuts 6 o BAUERMAN'S DESCRIPTIVE MINERALOGY. 236 Woodcuts 6 o BLOXAM & HUNTINGTON'S METALS: their PRO- PERTIES and TREATMENT. 130 Woodcuts 5 o GLAZE BROOK & SHAW'S PRACTICAL PHYSICS. 80 Woodcuts 6 o GLAZEBROOK'S PHYSICAL OPTICS. 183 Woodcuts... 6 o GORE'S ART of ELECTRO-METALLURGY. 56 Wood- cuts 6 o GRIFFIN'S ALGEBRA and TRIGONOMETRY 3 6 HOLMES'S The STEAM ENGINE. 212 Woodcuts 6 o JENKIN'S ELECTRICITY and MAGNETISM. 177 Woodcuts 3 6 MAXWELL'S THEORY of HEAT. 41 Woodcuts 3 6 MERRIFIELD'S ARITHMETIC and MENSURATION 3 6 KEY 3 6 MILLER'S INTRODUCTION to the STUDY of INOR- GANIC CHEMISTRY. 72 Woodcuts 3 6 PREECE & SIVEWRIGHT'S TELEGRAPHY. 160 Woodcuts 5 o RUTLEY'S The STUDY of ROCKS. 6 Plates and 88 Woodcuts 4 6 SHELLEY'S WORKSHOP APPLIANCES. 291 Woodcuts 4 6 THOME & BENNETT'S STRUCTURAL and PHYSIO- LOGICAL BOTANY. 600 Woodcuts 6 o THORPE'S QUANTITATIVE CHEMICAL ANALY- SIS. 88 Woodcuts 4 6 THORPE & MUIR'S QUALITATIVE CHEMICAL ANALYSIS. 57 Woodcuts 4 6 TILDEN'S CHEMICAL PHILOSOPHY. 5 Woodcuts... 4 6 (Wi h or without Answers to Problems.) UNWIN'S ELEMENTS OF MACHINE DESIGN. 325 Woodcuts 6 o WATSON'S PLANE and SOLID GEOMETRY 3 6 London : LONGMANS, GREEN, & CO. TEXT-BOOK OF SYSTEMATIC MINERALOGY BY HILARY BAUERMAN, F.G.S. ASSOCIATE OF THE ROYAL SCHOOL OF MINCS SECOND LONDON LONGMANS, GREEN, AND CO. AND NEW YORK : 15 EAST 16* STREET 1889 ights reserved EARTH SCIENCES LIBRARY PRINTED BY SPOTTISWOODE AND CO., NEW-STREET SQUARE LONDON PREFACE. IN preparing the present volume, two main objects have been kept in view : first, that it should form a useful guide to students desirous of acquiring a general knowledge of the subject ; and secondly, that it should serve as an elementary introduction to larger text-books, such as those of Dana, Miller, Descloizeaux,and Schrauf, an acquaintance with which is essential to those who wish to familiarise themselves with the higher branches of the subject For this purpose, the treatment adopted has been as general as possible, the descriptions of the crystalline forms dealing only with their symmetry and general geometrical properties, without enter- ing into the question of the practical calculation and deter- mination of individual examples, which would have increased its bulk beyond admissible proportions. In this part of the text, the methods followed have been mainly those of Groth's admirable treatise on * Physical Crystallography,' except that the plan there adopted of considering the phy- sical structure of crystals before their -geometrical properties has been abandoned in favour of the less logical, though more familiar, one of giving precedence to the latter. The optical properties of crystals have been considered at some- what greater length than is usual in rudimentary books, on account of the great and increasing use made of this branch of investigation. vi Preface. Upon similar utilitarian considerations a mixed system of notation has been adopted in the crystallographic part, the forms being designated in the text by their symbols according to Naumann, while the notation of their faces is by indices on Miller's system. As a matter of personal preference, the latter system would have been adopted exclusively ; but, having regard to the fact that the former is used much more extensively than any other system, both in text-books and in original memoirs, familiarity with its use is very desirable to students. In the hexagonal system, the Bravais-Miller notation by indices on four axes has been adopted, as showing most clearly the relation between it and the tetragonal system. In the chemical portion of the volume the classification followed is that of the second edition of Rammelsberg's 'Handbuch der Mineral-Chemie,' as being the standard modern authority upon the chemistry of minerals. The systematic part having been extended somewhat more than was originally intended, it has been found impos- sible to include physiography, or general descriptive minera- logy, in the same volume, without deviating too widely from the plan of the series. This will therefore be issued as a companion volume. In the preparation of the work, valuable assistance and advice has been received from many friends. In gratefully acknowledging these services, the writer has to mention particularly those rendered by the editor of the series, Mr. Merrifield, who has made several important additions to the text, Mr. R. T. Glazebrook, of Trinity College, Cambridge, and Mr. F. W. Rudler, who has passed the later sheets through the press during the writer's absence abroad. LONDON : January 10, 1881. CONTENTS. CHAPTER PAGE I. PRELIMINARY i II. GENERAL PRINCIPLES OF FORM 6 III. CUBIC SYSTEM 37 IV. HEXAGONAL SYSTEM 73 V. TETRAGONAL SYSTEM 112 VI. RHOMBIC SYSTEM . . . . ... 128 VII. OBLIQUE SYSTEM 145 VIII. TRICLINIC SYSTEM .156 IX. COMPOUND OR MULTIPLE CRYSTALS . . .163 X. MEASUREMENT AND REPRESENTATION OF CRYSTALS 190 XI. PHYSICAL PROPERTIES OF MINERALS -.CLEAVAGE, HARDNESS, SPECIFIC GRAVITY . . . .203 XII. OPTICAL PROPERTIES OF MINERALS . . . . 218 XIII. OPTICAL PROPERTIES OF MINERALS (continued) . 280 XIV. THERMAL AND ELECTRICAL PROPERTIES OF MINERALS 291 XV. CHEMICAL PROPERTIES OF MINERALS . . . . 298 XVI. RELATION OF FORM TO CHEMICAL COMPOSITION . 333 XVII. ASSOCIATION AND DISTRIBUTION OF MINERALS . 345 INDEX. 363 SYSTEMATIC MINERALOGY. CHAPTER I. PRELIMINARY. MINERALOGY is the science that treats of the substances known as Minerals that is, the constituents of the earth considered as they actually occur in nature. The constitution of the solid earth, excluding all con- sideration of its inhabitants that is, of the animals and plants living in the atmosphere may be regarded in many different ways. In the most general view the earth is a spheroid, about five times as heavy as an equal volume of water. Geology, with somewhat more detail, considers the acces- sible portion a shell of some ten to twelve miles thick, as made up of about the same number of different kinds of rock, the inaccessible interior portion being probably not very dissimilar in composition; while Chemistry is con- cerned mainly with the ultimate elementary constituents of the mass, and the properties of these elements as derived from the study of their combinations artificially formed. The position of Mineralogy is intermediate between those of geology and chemistry. With the former it considers the structure of the solid mass of the earth, but in greater detail, resolving the rock masses into a larger number of more B A 2 Systematic Mineralogy. [CHAP. I. exactly defined constituents or minerals ; while, with the latter, it considers the elementary constitution of such substances, restricting its study, however, to such chemical compounds as are actually found in nature. This indi- vidual, natural existence is essential to the idea of a mineral, as distinguishing it from a chemical salt or other artificial preparation. Whether the latter is or is not represented in nature can only be determined by experi- ence, but, speaking generally, it may be said that only the most stable and least soluble compounds, or precisely those that are most difficultly obtainable in the laboratory, are represented in nature, and therefore the chemistry of minerals, though essentially fragmentary, is of no small importance in the general body of chemical knowledge. The qualities essential to the distinction of minerals among themselves are of three kinds namely, form, struc- ture, and chemical composition all of which must be inves- tigated and determined before the specific independence of any mineral can be regarded as properly established. The first of these considers the external form of the substance by methods which are essentially those of descriptive solid geometry, qualified by certain special principles those of symmetry and numerical rationality generalised from the whole body of such observations. This part of the subject is known as Geometrical or Morphological Crystallography. The second quality, that of structure, considers the sub- stance as made up of similar material molecules, whose arrangement is indicated by the physical properties, such as density, cohesion, colour, &c. ; or generally, by their elastic resistance to forces tending to disturb their molecular equi librium, which may or may not vary in different directions. These investigations are essentially part of the work of ex- perimental physicists, but their results, when combined with those of the geometrical crystallographer, collectively form the branch of Physical Crystallography. Lastly, the inves- tigation of the third and most important character, that of CHAP. I.] Definition of Species. 3 elementary composition, is the work of Mineral Chemistry ; and the combination of this with the knowledge derived from the study of form gives rise to the important principle of Isomorphism, or the relation of form to constitution, upon which the most natural and satisfactory systems of classifi- cation are founded. In addition to these three principal heads, information upon subsidiary matters is requisite for the attainment of a complete knowledge of any mineral. These have re- ference to its natural habitat, such as association with other minerals, geographical and geological distribution, and evi- dences of possible changes from the condition of original formation. This last point especially has an important bearing upon the speculative matter of the origin and mode of formation of minerals, which is, or should be, the province of geology proper, as the basis of any reasonable speculation upon the structure of the earth in its largest sense. This branch of the subject is, however, generally spoken of as Chemical Geology. The whole body of knowledge comprised under these several heads, when classified in an orderly manner, forms Descriptive Mineralogy, or Physiography. The number of elements known to the chemist is now between sixty and seventy, all of which are concerned in the production of minerals, though in widely different pro- portions. Some eight or ten are found in the free or un- combined state, and with six or seven hundred combinations of two or more, make up the roll of natural minerals or mineral species. The term species is applied to any substance whose form, structure, and composition are definite, constant, and peculiar to itself, and therefore serve to distinguish it from all other species. By constancy of form and composition in the above definition is not meant that there must be absolute identity in these particulars between different ex- amples of the same substance, but that the variations in B 2 4 Systematic Mineralogy. [CHAP. I. either shall be subject to known laws. Thus, carbonate of calcium in the species Calcite appears in several hundreds of different forms, all of which may, by the application of crystallographic laws, be shown to be derivatives of a single geometrical form the rhombohedron^-and again its composition may differ sensibly in different specimens, but ihese differences are explainable by the law of isomorphism, which shows that one dyad metal may be substituted for another without altering the general molecular constitution. The exact limits to be given to species is to a great extent matter of opinion. Upon purely chemical grounds, all substances combining the same type of molecular consti- tution with analogous forms may be considered as varieties of a single species, without reference to the nature of the elements composing them ; but this is too wide a definition to be of much practical use to the mineralogist. It is therefore customary with such a class of similarly constituted substances to classify them according to their contained metals, giving a different name to each, and to call the whole group by the name of the most prominent species. On the other hand, a variety of any substance marked by some constant peculiarity, whether of form, colour, or other ap- parent property, may often receive a particular name with ad- vantage, even when the distinguishing difference is too slight to allow of the separation on the grounds of systematic form or composition. The ultimate guide is in all cases the con- venience of the observer ; and if the practice of giving names without a previous complete determination of the compo- sition and physical properties be avoided, it is generally better to form new specific names rather than unduly widen the boundaries of older ones. The proportion in which different minerals enter into the composition of the crust of the earth varies very con- siderably, as does also the size of the individual masses of any one. Thus we may find the same substance in particles of microscopic minuteness in some places, and in CHAP. I.] Rocks and Minerals. 5 others in masses measurable by cubic feet or yards, and weighing up to hundreds of tons, or even forming mountain masses, without sensible admixture of other substances. Experience, however, shows that the characteristic proper- ties of any mineral, and especially that of form, are best developed with individuals of a moderate size, as when very minute they become invisible and incapable of exact mea- surement, and when very large the characteristic form is not, as a rule, apparent. Such undefined mineral masses are generally spoken of as Rocks. Quartzite and statuary marbles, for instance, are aggregates of particles of quartz and calcite into masses of a slaty or granular texture in which their proper forms are entirely lost ; while, on the other hand, the crystalline lava called basalt is made up of individuals of the species Labradorite, Augite, Olivine, and Magnetite, all perfectly well denned in form and physical characters, but so minute as to be indistinguishable to the unaided eye, the general effect being that of a uniform, opaque, black substance, separating into masses whose shapes bear no obvious relation to those of their constituent minerals. The distinction between rocks and minerals is, however, one of geological convenience only, and in mineralogy is almost without significance ; the nature of a mineral mass being defined according to its constituents, either as of a single species or an aggregate of two or more, without reference to the size or perfection of the individual components. It often happens that groups of two or more minerals, dis- tinctly separated, pass in the same mass by inappreciable gradations into aggregates in which the individuals are indis- tinguishable by ordinary means, so that it is difficult to say where either condition begins or ends. This difficulty is further increased by the use of the microscope, which very commonly resolves substances apparently uniform into aggre- gates of dissimilar ones, and there is no reason to suppose that the individuality of the constituents ceases when the microscope is no longer able to reveal them. The same 6 Systematic Mineralogy. [CHAP. II. class of observation, however, shows that foreign substances are so commonly included even in the most perfectly deve- loped mineral individuals or crystals, that the condition of homogeneity required by the ordinary definition of a mineral as a homogeneous inorganic substance is seldom, if ever, realised, and therefore this definition can only be accepted as an approximation requiring considerable qualifi- cation in use. The complete discussion of all the subjects indicated in the preceding pages, or even of any one of them, being beyond the scope of an elementary book of limited extent, the space at command will be devoted to a sketch of the principles upon which the methods of determining form, structure, and other elements of classification in minerals are based, without entering into the details of the methods of observing or reduction of observations, for which matters the student is referred to the larger special works as given in the list at the end of the volume. A physiographic sketch of the more important species, classified according to Ber- zelius' and Rammelsberg's method, forms the subject of a companion volume. CHAPTER II. GENERAL PRINCIPLES OF FORM. WITH the exception of water, mercury, and some hydro- carbons which are liquids at ordinary temperatures, minerals are solids, and occur in masses which in some cases are of irregular, and in others of regular, shape. The first of these are called amorphous, and the second crystalline, substances, and any individual mass of the latter kind is a crystal. The branch of mineralogy that treats of the study of such form is called Crystallography. CHAP. II.] Definition of Symmetry. The term c crystal/ derived from the Greek which was applied to ice and transparent quartz or rock- crystal, the latter having been supposed to be produced from water by extreme cold in mountain regions, is ap- plied to natural and artificial substances which, in solidi- fying, from a state whether of solution or fusion, assume definite polyhedral forms, which are constant for the same substance. The leading property of crystals, as distinguished from mere geometrical solids, is the invariability of the angles between corresponding faces in different individuals of the same substance. There is usually a very marked symmetry to be noticed in the arrangement of their plane faces and edges, and occasionally of their points also, although this latter symmetry is not essential, crystallo- graphic symmetry being one of direction and not of position, so that two parallel planes or two parallel lines are not distinguished from one another, and on that account the invariability of the angles is a paramount consideration. The character of the symmetry varies in different groups of crystals, and forms the basis of their classification into systems. If we take any polyhedron and place it upon a looking, glass, the object and its image together constitute a sym- metrical figure, of which the reflecting surface is the plane of symmetry ; and if in any given solid we can find a plane such that, if we were to cut it in half by that plane, and to place against the section a mirror, the reflected image would exactly reproduce the other half (identically, and not reversed, as objects generally are by reflection), the solid is said to be symmetrical about that plane. 1 This is a 1 This may be shown with a model of a cube, painted white, and a plate of red glass, held perpendicularly to one of its faces, when a white image of the part of the face in front will be seen by reflected, and a red one of the hinder part by transmitted, light. If the direction of the plate be parallel to that of a plane of symmetry the two images will 8 Systematic Mineralogy. [CHAP. II. more particular supposition than need be made in crystallo- graphy, in which two parallel planes are not distinguished ; nevertheless it is convenient to use it for purposes of de- scription, in order to escape a vagueness which would other- wise make the description unintelligible. For instance, although figs, i and 2 have exactly the same crystallo- graphic symmetry, it will be more convenient to consider and describe the first, and to regard the second as another example of the same form. They are both symmetrical, as FIG. i. FIG. 2. C regards direction, to the lines A B, CD, but the symmetry of position of fig. i renders it a much more definite thing to talk about and to apply linear measure to. Only it must not be forgotten that this symmetry of position is neither essential nor inherent, but is merely adopted as an aid in forming definite ideas. Symmetry about a line in plane figures corresponds to symmetry about a plane in space. In the latter, symmetry about lines or axes is also observed in many cases, but the discussion of this may be deferred until another character- istic feature of crystals, and that the most important one, as forming the basis of all exact crystallography namely, the principle of rationality has been noticed. The term * Rationality ' will be best understood by using apparently coincide ; but in any other position they will deviate to a greater or less extent. The particular positions of symmetry for the cube are shown in figs, u, 12, 13. CHAP. II.] Principle of Rationality. 9 plane space of two dimensions, instead of actual space of three dimensions, for an illustration. Without considering the exact meaning of axes of refer- . . FIG. 3. ence, let it be assumed that o A, o B, fig. 3, represent two such axes, to which two planes belonging to one crystal, represented by P Q, P' Q', are referred. Then the principle of ra- tionality requires that if PQ" le drawn through P parallel to P' Q', the ratio of o Q to o Q" shall always be rational, or, as it may be more generally stated, . = a simple rational fraction. OQ OQ' Usually, the relation is one of very simple numbers, such as 2 : i, i : 2, 2 : 3, 4 : 5, &c., while it can never be that of an incommensurable surd, such as \/2 or >/5, to unity. This law is an empirical one that is, it expresses the results of observations without explaining their cause but there are no known exceptions. Its geometrical consequences are 1. The exclusion of all but the simpler types of sym- metry about an axis, namely, binary, quaternary, ternary, and senary. 2. The exclusion, from possible crystalline forms, of the Platonic or regular geometrical solids of higher order than the cube or octahedron. The regular dodecahedron and icosahedron involve pentagonal symmetry, and they bring the irrational"valueN/5 into the axial relations. There are numerous examples among natural crystals, notably in iron pyrites, of forms ap- proximating in shape to these that is, they are contained by twelve five-sided or twenty three-sided faces but these are never regular pentagons, or all equilateral triangles. When applied to three dimensions, as in actual crystals, the prin- ciple of rationality requires that if planes of different kinds 10 Systematic Mineralogy. [CHAP. II. occur in the same crystal they must be so related that their intercepts ! upon the axes of reference are in rational pro- portion to one another. That is to say, if one plane meets the three axes in the points pqr, and the other in the points p Q R, the relation expressed by OP ' OQ ' OR will be that of quantities having rational ratios to one another, and usually these will be found to be in low numbers. Axes of symmetry. When a polyhedron is turned about a line so selected that after passing through an aliquot part of a whole revolution its position in space as a whole has not been changed, and there is no apparent difference in shape for the two aspects, it is said to be symmetrical to the line or axis of rotation. The kind of symmetry is denoted by the number of times the positions of symmetry recur in a complete revolution, or between the starting of a marked point on the crystal and its return to the original position. Thus, in a cube, the straight line joining the middle points of two opposite edges is an axis of binary symmetry ; for by turning the solid about such a line through half a revolution, it assumes a position apparently similar to the original one, and the change can only be perceived by observing the alte- ration in the place of a marked point or face with reference to some external object. The cube has also ternary sym- metry about a diagonal line joining opposite points, and quaternary symmetry about a line joining the centres of opposite faces, the original position being apparently re- stored by rotation through one third of a revolution in the former, and one quarter or one right angle in the latter case. Quaternary also includes the lower condition of binary 1 An intercept is the distance intercepted or cut off by a plane upon an axis measured from the origin of the latter. Thus in fig. 3, o Q, o Q', O P, o P', are intercepts upon the axes o B, o A, whose origin is at o. CHAP. II.] Crystallographic Systems. II symmetry. The only other class of symmetry possible in crystals is senary or hexagonal, corresponding to a rotation of one-sixth of a revolution, such as that of a regular hexa- gonal prism about its axis ; this includes ternary symmetry. Quinary symmetry, such as that of a Platonic or regular icosahedron about a diagonal, or of a Platonic dodecahedron about a line joining the centres of opposite faces, is crys- tallographically impossible, as it introduces irrational re- lations. The remarks made on page 8 apply equally in this case : the necessary symmetry being one of direction only, the same symmetry exists about any line parallel to the axis as about the axis itself; but for convenience of description it is best to consider the cases in which there is also symmetry of position, always bearing in mind that this is a mere matter of convenience, and not essential, or affecting the classification. "~ As has already been shown in the case of the cube, a crystal may be symmetrical about more than one axis. If there is binary or quaternary symmetry about two axes at right angles to one another there is a third axis of the same kind at right angles to both. Crystals are classified into systems according to the number and character of their axes of symmetry. Six of such systems are possible, all of which are represented by natural minerals. They are as follows, commencing with those of lowest symmetry : 1. The triclinic system. This is without any axis of symmetry, the faces of any form being only symmetrical to a central point. 2. The oblique system. This has one axis of binary symmetry, and consequently one plane of symmetry. 3. The rhombic system. This has three axes of binary symmetry at right angles to one another. 4. The hexagonal system. This is characterised by one axis of senary or hexagonal symmetry and six of binary symmetry at right angles to the first. It includes the case 1 2 Systematic Mineralogy. [CHAP. II. in which there is an axis of ternary symmetry, or the rhom- bohedral system. 5. The tetragonal system. This has one axis of qua- ternary symmetry at right angles to two of binary symmetry, which are also at right angles to each other. 6. The cubic system. This is specially characterised by three axes of quaternary symmetry at right angles to one another, besides which there are four of ternary and six of binary symmetry, whose positions have been already alluded to on p. 10, and will be more specially noticed sub- sequently. When a system has more than one kind of symmetry, it may be distinguished by the number of its axes of the highest kind, or axes of principal symmetry. Upon this dis- tinction is founded the classification of the systems into the following three groups, which are closely related to their physical properties : 1. Without a principal axis of symmetry. This includes the triclintc, oblique, and rhombic systems, the first being without linear symmetry, while in the second and the third the symmetry is all of the same kind, namely, binary. 2. With one principal axis of symmetry. This includes the hexagonal and tetragonal systems, the principal sym- metry of the first being senary and of the second qua- ternary. 3. With three principal axes of symmetry. This is special to the cubic system, the three axes being those of quaternary symmetry. When a crystal is contained by all the planes or faces ' required by the complete symmetry of the system, each one has a counterpart plane parallel to it, so that their total number is always even and not less than six. These are 1 It is convenient to call the natural surfaces of crystals faces, and those produced artificially, or required in geometrical construction, planes, e.g. planes of symmetry and cleavage planes. CHAP. IL] Hemihedrism. 13 said to be holohedral (full-faced) forms. There are also certain forms in which only one half or one quarter of the full number of faces are present ; these are respectively called hemihedral (half-faced) and tetartohedral (quarter- faced) forms. The selection of these faces may in some instances be made in more ways than one, subject to the condition that the relation of the faces to the axes of sym- metry must be the same as in the holohedral form, or each equivalent axis must cut an equal number of faces namely, one half or one quarter of that which it would do at the same inclination in the full-faced form. Hemihedral forms are not possible in the triclinic system, that being without plane symmetry ; in the oblique system there may be one kind, but it has not been observed in nature ; in the rhombic there may be two kinds, but it is doubtful whether more than one actually exists in nature. In the remaining systems, hexagonal, tetragonal, and cubic, the forms are susceptible of hemihedral development in three ways, but it is only in the tetragonal that all three are actually known to exist, and of these one has only been observed in artificial organic compounds, Tetartohedral crystals are possible in all systems where there are more than two kinds of hemihedrism, that is in the tetragonal, hexagonal, and cubic. As they may be con- sidered as resulting from the successive application of two kinds of hemihedry to holohedral forms, three classes of tetartohedra might be possible in the first two systems, were it not for the circumstance that in either only two out of the three combinations give rise to forms that satisfy the general conditions of symmetry. In the cubic system all three kinds of hemi-hemihedrism produce the same class of form or there is only one kind of tetartohedron. This has not been found in natural minerals, but is characteristic of a group of metallic salts, the nitrates of the lead-barium group. In the tetragonal and hexagonal systems the two possible methods give rise to two different kinds of forms, 14 Systematic Mineralogy. [CHAP. II, neither of which has been observed either in natural or artificial crystals in the former system, but in the latter both kinds are known, one of them being specially characteristic of the commonest mineral constituent of the earth's crust, namely, quartz or rock-crystal. By hemihedral or tetartohedral development a crystal belonging to any system loses a portion of the symmetry which characterises it when possessed of the full number of faces. Thus, in the three kinds of hemihedra possible in the systems with principal axes, one is entirely without plane symmetry, while each of the other two has only one of the two kinds necessary in the full-faced forms. It must there- fore be remembered that in denning systems by their charac- teristic symmetry, such definitions only apply to the holo- hedral forms. There is an essential distinction between the geometrical and the mineralogical idea of hemihedrism and tetartohe- drism, which it may be well to notice as early as possible. In the former only those forms are considered as hemihedra and tetartohedra that are geometrically distinguishable from the holohedral forms, while in the latter the presence of any such form in any crystal of a substance is considered as im- parting the same character to all other crystals of the same substance, whether they be geometrically distinguishable from holohedral forms or not. The same kind of symmetry as is displayed in crystals is furnished by an orderly arrangement of points in space, which has the further analogy with crystals of suggesting rational ratios. These arrangements of points may be ex- tremely simple ; indeed all the crystallographic systems may be represented by the points of intersection of three sets of parallel and equidistant planes, as in fig. 4, where we may call a^i = a, did* b, and a\b\ c, and the angles a^fij}^ = a, # j0 4 4 = /3, and a\ci^a b = y. The whole series of points may be regarded as a sort of net in space, of which the strings represent the lines of intersections of these planes, and the CHAP. II.] Reticular Point Systems. 15 knots or nodes the points of intersection. Then, if the equal distances at which one set of planes intersects the other two be taken as a, b, and c respectively, and the angles between the lines as a, /3, FlG . 4 . and y, these quantities will be the characteristics of the system, and symmetrical relations of equality between those charac- teristics will determine a crystal- lographic system. The planes of the system, corresponding to faces of the crystal, will be planes drawn through any three points of the system. The limita- tion to rational ratios is at once suggested by the necessity of taking a whole number of intervals between any two points, in order to satisfy the definition of a plane of the system being one passing through three points of the system. As to the particular modes of symmetry, if /3 y are all different, there is no symmetry. This corresponds to the anorthic or triclinic system. If /3 and y are right angles, a is perpen- dicular to the plane of be, and is an axis of binary sym- metry. We have then the oblique system. If a /3 y are all right angles, we have three axes of binary symmetry, cor- responding to the rhombic system. If, in addition, b = c, a is an axis of quaternary symmetry, and we obtain the tetragonal system ; while if a = b = c we have the cubic system. If, on the other hand, n = /3 = y, without being right angles, and also a = b = c, then there is an axis of ternary symmetry equally inclined to a, b, and c. This represents the rhombohedral system of ternary symmetry which has been mentioned as included in the hexagonal system. It is to be remarked that while these relations between a b c, a /3 y, yield the crystallographic systems in the easiest way, they are sufficient, but not necessary. For instance, if a /3 y are all right angles, it is not necessary that a, b } and c 1 6 Systematic Mineralogy. [CHAP. II. should be all equal in order to give a cubic system : it is sufficient that they should be commensurable. For, let 0=1, = 2, i and ;// >. Dividing this by m n it becomes : : -, whence are deduced h = m n, & = in, and / = n, as the expressions for the conversion of Weiss's into Miller's symbols. Thus: a: a : 4 = (tf 4 J) = (i6 I2 4) = (4 3 *) The above case applies to the cubic system, where the positions of the signs are interchangeable ; in the remaining systems n is always > i, but m is specially restricted to the vertical axis, and may be either greater or less than n or i. Thus : a : f b : -J- c = (3 2 6) ; f a : b : 2 c = (4 6 3), &c. In arranging the indices or other signs fornvng the symbol of any face, care must be taken that the axes are always noted in the same order. Unfortunately, there is no uniformity of practice in this matter, either one of the three axes being considered as the first by different authors. In Miller's system the three axes are indicated by the letters x y z, the first extending right and left, the second from front to back, and the third above and below the centre. Weiss calls the axes a b c, their order being a front and back, b right and left, and c top and bottom. The latter system, as being the oldest and most generally known, is adopted in this volume. The position of the first and second indices in the symbol of any face as noted in the figures must there- fore be transposed to make them correspond with those of Miller's order. 1 Naumanrfs notation. The symbol of any face may be used by implication to indicate the whole of the same kind of faces in a crystal, if the symmetry of the system is known. 1 The difference expressed in words is as follows : The plane whose parameters are positive on all three axes, is, according to Miller, the right front top face, while in Weiss's order it is the front right top one. 22 Systematic Mineralogy. [CHAP II. This is apparent in Weiss's notation by the use of index letters for the different axes ; but not in Miller's, where the symbols are similar for all the systems, and do not of them- selves show the character of the form. Another method modified from that of Weiss, due to the late Dr. C. F Naumann, indicates both form and symmetry in a single symbol by combining the parameter values with certain arbitrary signs, that differ in each system, representing the form. In the cubic system, where the three axes are all rectangular and the parameters equal, the unit form Miller's 1 1 1 or Weiss's a : a : a is the regular octahedron. This is represented by the capital letter O, the initial of octa- hedron. In the other systems, where the forms correspond- ing to the unit values of the parameters are of the kind known as pyramids, the initial P is used as the unit symbol. The derived forms in any system are indicated by the addition of Weiss's co-efficients, according to the number of axes on which the intercepts vary from unity, but the unit letter is never used more than once in any symbol. The general symbol corresponding to h k /, or m a : b : n c, is m P n in the systems with rectangular axes, but in those with one or more oblique axes accents and other arbitrary signs are added to the characteristic letter to show the symmetry of the system. This notation is more extensively used than any other, mainly from the circumstance that the most popular text-book on Mineralogy is written by its author ; ! and in a slightly modified form it is also used in Dana's manual and text-books, which are probably the most abundantly circulated books of their class in the English language. The symbols have the advantage of being short and convenient ; and being essentially arbitrary, when their nature is once understood they cannot be mistaken for any- thing else, and are therefore well suited for descriptive pur- 1 Elemente der Mineralogie. C. F. Naumann. The first edition was published in 1846, and the ninth shortly before the author's death in 1873. A tentn edition, edited by F. Zirkel, appeared in 1877. The same system is followed in the late Professor Nicol's manual. CHAP. II.] Levy's Notation. 23 poses. They are, however, only adapted for indicating forms that is, the whole system of faces corresponding to any particular set of parameters, and not individual planes as the parts of the symbols are arranged in an invariable and arbitrary form. The special method of arranging these for different kinds of symmetry will be considered under the description of the different systems. Levy's notation. In a system with three axes the solid, con- tained by faces, parallel to two of the axes, and intersecting the third at some measurable distance, will either be a cube or some parallelepipedon, and in the hexagonal system it will be an hexagonal prism. In such forms, when exactly developed, the edges will be of the same length as the axes to which they are parallel, and the plane angles of the parallelo- grams forming the faces will have, the characteristic angles of the axes. The system originally invented by the Abbe Haiiy, and subsequently modified by Levy and Descloizeaux, uses reference solids of this kind known as * primitive ' forms, which are essentially the same as the molecular networks of Bravais. The faces in such forms are indicated by the capital letters P M T, their points by vowels a e i 0, and their edges by consonants in an invariable order, from left to right, each primitive form requiring as many letters for its description as it has different parts. The num- ber of these, therefore, is an indication of the degree of symmetry. The symbols of derived faces are compounded of those of the part of the primitive modified, whether an edge or a solid angle, and a series of signs indicating the ratio of the intercepts of the new plane upon the edges, written as exponents. Thus, in the cubic system, the octahedron is written as a\ which means that it intercepts an equal length upon each of the edges measuring from point a where three faces meet. This is the oldest of all the systems of nota- tion, and was at one time almost universally current, but at present it may be considered as restricted to the mineralogists of France, by whom it is generally used. The important works of Descloizeaux and Mallard being written in this Systematic Mineralogy. [CHAP. II. system, a knowledge of its principles will be found useful to the student. 1 Relations of faces. The symbol of any face in a crystal may, by an extension of meaning be considered as typical of the whole form that is, of all faces similarly related to the axes of reference. When this is meant, the symbol is en- closed in brackets, thus {hkl}; but when restricted to an individual face it is put in parentheses, thus (hkl). As will be subsequently seen, the possible number of faces in a form varies with the symmetry of the system, the maximum of forty-eight occurring in the cubic, and the minimum of two that is the face and its counterpart in the triclinic system. There are many instances of crystals contained by the faces of only a single form, especially in the cubic system ; but it is far more common to find them made up of two or more forms grouped in regular order, such compound crystals being known as combinations. There is no limit to the number of forms that may enter into a combination, subject to the conditions that they all have the same degree of symmetry, and are so arranged that all the faces meet in convex angles. Crystals, in which some of the faces meet in concave or re-entering angles, are not uncommon, but these are never simple, being peculiarly arranged groups of two or more, known as twin crystals. The general solution of the problem of the determina- tion of the direction of the edge or line intersection of two dissimilar faces in terms of their parameters is as follows : Let H K L, n' K' L' (fig. 7) 1 A good account of it will be found in Pisani's Traitc de Mine- ral ogie. FIG. 7. CHAP. II.] Relations of Faces. 25 be two such planes, their intersections H K and H K', with the axial plane, A o B, cross in the point y, which will therefore be a point in the required edge as common to both faces ; (3 will be a second point of a similar kind in the plane A o c, and a a third in the plane A o B 1 As the direction of the line of intersection is unchanged by moving either face parallel to itself, the face (h' k I'} if shifted to the position ELF gives the new line L D as the required direction. This, however, is equivalent to multiplying the parameters of the face bv r. These latter will therefore be for the new J O L position , O L , O L OE = OH -, o F = o K' - , and o L o L' OL' that is, the intercept on the third axis, common to both faces, is an original parameter of (h k /) ; the position of the point L is therefore determined. To find the point D, draw in fig. 8, D u parallel to o B, and D v parallel to o A, when the problem takes the form of the determination of the co-ordinates of the point D, the lengths o u and o v, in terms of the parameters, as when these are known, the sides of the parallelogram o u D v, and with them the position of D, are determined. The triangle o K H is similar to u D H, also o F E is similar to u D E, whence follows OK:UD=OH:UH = OH: (OH o u) OF:UD = OE:UE = OE:(O E o u). The first of these ratios gives the equation OK.OH OU.OK = OH.UD, and the second, OF.OE OU. OF = O E. UD ; 1 This demonstration is given in Groth's Physikalische Krystalk- graphic. 26 Systematic Mineralogy. [CHAP. II. whence the two unknown quantities o u and u D are derived as follows : OE. OK. OH OF. OE . OH ou = OE. O K O H. O F OK.O F . OE OK. OH. OF UD = OE. O K OH.O F or, OK OF OU= OH. OE UD = OV= OK. OF O E. OK O F. OH OE OH OE. OK OF. OH Substituting the proper values for o E and o F, these become PL Q H , . PL O L'' O L O I O L OH.^L OH'. _A OH'. OK OK'. OH PL oT' L .OH'-OH oi. O L J.OH'.OK ---.OK. OH o L' o L' OH'. OH O L O H r . O K O K' . O H O H . O H r ^ OK.OL 7 OL.OK 7 0~L/ OK. OH' -OH.OK 7 Q K . O K' OL.OH / -OH.OL r OL ; OK. OH' OH. OK' which are the equations for the co-ordinates of the point D in terms of the parameters. CHAP. II.] Relations of Faces. FIG. 8. If in fig. 8 a length o w = o L be laid off on the negative side of the axis c, and the parallelepiped OUDVWRQS con- structed, the diagonal o Q will also give the direction of the edge between the two faces being parallel to LD. The multiplication of all the para- meters of (Ji k I) by any quan- tity, ;;/, does not affect the direction of the plane or of its intersection with the other plane, so that, if for o H, o K, o L in the preceding formulae the new values m o H, m o K, and m o L be substituted, we obtain o v' = m o v, o v' = m o v, o w = m o w that is, all the sides of the above parallelepiped will be multiplied by the same quantity, whereby the direction of its diagonal is not altered. The same holds good when the expressions for o u, o v, and o w are multiplied by OK. OH' OH. OK' O H . O H'. O K . O K'. O L* whence the following perfectly symmetrical expressions result : i i o u = o v = OW =r OL.OK' OK.OL'' i i O H . O L' "~ O L . O H'' T I UK. OH' OH. OK'' In the application of these equations to the determina- tion of the direction of intersection of two faces, this inter- section is supposed to pass through the origin of the axes, 28 Systematic Mineralogy. [CHAP. II. which gives one point ; the second is found by laying off upon the three axes the values found for the parameters, in the proper directions, positive or negative, according to their signs, which gives. the three sides of the parallelepiped whose diagonal drawn from the origin of the axes is the direction required. If it is required to express o u, o v, o w, by the indices instead of the parameters of the faces, the values of the latter, expressed by the former, must be introduced into the equation, or for the unit parameters a, b, c, OH-J OK=4 OL = ^. OH'=|, OK'.L, OZ/-' If these values are substituted for o u, o v, o w, in the formulae given above, and each expression is brought to one denomination, we obtain ou =^^=. be be o v = = , ow = ac ac hk' -kh' __w ab ~ a when the differences forming the numerators are expressed by the contractions u, v, w. Multiplying these by the pro- duct a b c, which does not alter their relative magnitudes, they become a ?/, b ?', c w, where the sides of the parallelepiped whose diagonal is the required direction are represented by magnitudes depending only on the indices. For their determination the following method is given by Miller, which is easily remembered : Write down the indices one above the other twice, cut CHAP. II.] Principle of Zones. 29 off the first and last columns, multiply the others crosswise, and subtract one product from the other. Thus : "A k k' X / I' X h h> X k k> I 71 kl'-lk', lh' -hi 1 , hk'-kh' = u -=v w, which, like the indices, are rational whole numbers. Zones. Any number of faces parallel to any right line is called a zone. Faces belonging to the same zone, and which consequently intersect in lines which are parallel to one another and to every face of the zone, are said to be tautozonal. If we suppose the faces of a zone to pass all through one point, they will intersect in a right line passing through that point, and the direction of that line is called the zone axis. For instance, in the cube, the front and back and right and left pairs of faces constitute a zone whose edges are all vertical and parallel to the axis ek fh = w 9 the last equation becomes up + v q + w r = o, which shows that the condition required to bring three planes into the same zone depends only on their indices, and is completely independent of the lengths of their axes. The quantities u v w are called the indices of the zone, and to distinguish them from those of a face or form they are written in square brackets, thus \u v w], forming the so- called zone symbol ; the face, if any, to which this symbol belongs is called the zone plane, and is perpendicular to the 1 This condition may be obtained directly from the consideration that three planes through the origin, parallel to the three faces, have for their equations, expressed as in ordinary geometry, ^ + fy +l_ = o,&c.; a b c and the condition that the three planes should be parallel to a line is obtained by eliminating - , y - , and - from the three equations. This a b c gives the determinant above written. CHAP. II.] Determination of Faces by Zones. 31 zone axis. It may be determined from the intersection of any two planes, P and Q, out of those forming the zone, and from it the whole number of possible tautozonal faces may be calculated by substituting for q and r successively all the simple rational numbers o, i, 2, . . . and calculating, in accordance with the above condition, the corresponding values of/. Determination of a face by two zones. As a plane is determined when the positions of two straight lines parallel to it are given, that of the face of a crystal lying in two zones, and therefore parallel to the axes of both, is simi- larly determinable. If the symbols of the zones are [u v w\ and [u' v' w'\, the indices p q r of the face must satisfy the equation of condition in regard to both. Consequently V 1 q + W 1 >=O, whence we derive __^ v w' w v' u v 1 v u' ' wu' ' uw' u v 1 v u 1 ' One of the three indices may, however, be made equal to any number at pleasure ; say, for example, r uv' v u', which gives p = v w 1 w v', q = w u' u w f , as the three indices of the face common to both zones. These indices may be derived from those of the zones by the scheme of cross multiplication and subtraction, in the Systematic Mineralogy. [CHAP. IT. same way that those of a zone are found from those of two of its faces. Thus : w w w' v w' v' w, w u 1 u w,' u v' v u' = / =q =r. By this method the symbols of a face may be found when those of any two zones in which it lies, or of any two faces in each of those zones, are known. Suppose, for example, a face is observed to lie in one zone with the faces (123) and (i i 3), and in another with (o i i) and (i 2 2), the corresponding zone symbols will be X X 3 x x 6 -3, 3~3> i-2=[3o T] 2-2, i-o, o-i=[oi i]; and that of the face common to both 3 i xl I X 3 X I -i, 0-3, 3-0 = (13 3). Any three faces intersecting at any angles may appear in the same zone, but a fourth (or any further number of faces), arbitrarily placed, is not possible according to the principle of rationality, or rather according to the particular develop- ment of it known as the principle of anharmonic ratios. This principle is of such importance that a general demon- stration of it, obligingly furnished by the editor, Mr. Merrifield, is given in the following pages. ANHARMONIC RATIOS. Let A B c D (fig. 9) be any four points on a right line, and let o be an arbitral-}' point through which the right " TT i Anharmonic Ratios of Planes. lines o A, o B, o c, o D are drawn. Also, let fall o P perpen- dicular on AD. Then we have OA.OBSin AOB = AB: OP OB.ODsinBOD = BD.OP Since each side of the equa- tion is a different expres- sion for twice the "~ ~f a triangle. OA.OCSinAOC = AC.OP| OC.ODsinCOD = CD.OPj Dividing out suitably sin A o B . sin. A o c A B . A c sin B o D " sin c o D B D ' c D J which is the anharmonic ratio of the four points or four lines : the second side of this equation shows that the ratio is inde- pendent of the position of the point o, which may therefore be at o'. The first side shows that, the pencil being given, the transversal is immaterial. Thus the anharmonic ratios A B . AC B D ' c D' AC A'B' . A'C' B'D' ' C'D' are all equal, and this whether o A D, O'A D are in the same plane or not. There is more than one anharmonic ratio of four points on a line. They are connected by means of the identity, 34 Systematic Mineralogy. [CHAP. II. which is obtained by the cyclic permutation of the letters B c D in the expression A B . c D. The ratio of any two of the terms in this formula may be taken as the anharmonic ratio of the pencil. A further consequence of this property is, that if we con- sider oand o' as in separate planes, OO'A, OO'B, oo'c, OO'D constitute a pencil of planes such that the anharmonic ratio of any transversal line is invariable, and equal to the corre- sponding sine-ratio of the dihedral angles. If any four points on a line are given, any pencil of planes through the four points will have the anharmonic ratio of the range, and every transversal of any such pencil will again have the same anharmonic ratio. The sine-identity of a pencil is easily written down from the corresponding one of a range by simply inserting the letter at the vertex, and the word sine. Thus AB. CD-I-AC. DB + AD.BC = O, gives sin AOB.sincoD + sinAOC.sin DO B+sin AOD.sin BOC = O, and the anharmonic sine-ratios are sin A o B . sin c o D : sin A o c . sin D o B, &c. N.B. Points , on a right line are a range. Right lines meeting in a point are a pencil. Planes meeting in a right line are also a pencil. ANHARMONIC PROPERTY OF ZONES. The anharmonic proptrty of zones is this : that if we take in space four planes parallel to four of the faces of a zone, and meeting in one line, the anharmcnic ratio of the pencil is rational. It is to be remembered that all the faces of a zone are parallel to a line. Hence all their intersections, two and two, will be in parallel lines. The necessary construction is therefore at once obtained by drawing, through the inter- section of any two faces, planes parallel to the other two. The four iaces of the zone, by the ordinary law of crystal- CHAP. II.] AnJiarmonic Ratios of Zones. 35 lography, cut off from each of the axes intercepts having rational numerical ratios; and what we have to prove is that, as a consequence of this rationality, the corresponding an- harmonic ratio is also rational. Confining our attention to one plane only, let us con- sider four transversal lines meeting the two axes : thus, in the annexed figure, #i^i> #2^2> #3^3> a 4&4i are f ur transversals to ox andojr, so taken that O0 4 = Through b, draw b\a \ parallel to FIG 10 We want to show that if Imn pqr are rational the anhar- monic ratio of the pencil b\ a l a' 2 a' 3 a\ is also rational. This ratio is Now tfl ^3 = 0^3-0^!=^-!^ 00! D 2 36 Systematic Mineralogy. [CHAP. II. ^3^4 = 0^4-0^3= f^- J OflJj y-/) (?-* *), (mq)(nplr) m n^ m q~ r~~ q which is rational, if Imn pqraxe so. The whole of the anharmonic ratios of the zone are con- tained in the ratios two by two of the terms in the identity Since Imn pqr are either integral, or else rational numerical fractions, each term of this identity must be so too, and the ratio of any two terms must be a rational fraction. As a numerical example, let 1=2, w = 3, = 4,/=3, ^ = 5, ?-=8; then the above-written identity becomes and the anharmonic ratios are any one of the following In practice, the dihedral angles will be measured, and the anharmonic ratio to be used will be the sine -ratio. The test of whether we have really got four faces of one crystal is the rationality of this anharmonic sine-ratio, when re- duced to numbers. It will usually be a simple numerical fraction, in low terms. CHAP. III.] Symmetry of Cube. 37 CHAPTER III. CUBIC SYSTEM. 1 THE forms of this system, whose symmetry is completely exhibited in an actual cube, are referred to three principal axes at right angles to each other, whose unit lengths or FIG. ii. FIG. 12. V: - ....... Sr \ parameters are all equal. These are axes of quaternary symmetry, each one passing through the centre, o, and joining the centres of opposite faces perpendicularly : A o A, BOB, coc (fig. n). They lie two by two in planes parallel to the faces which are the principal planes of symmetry. Next in order are the four axes of ternary symmetry OT', OT", ox"', OT"" (fig. 12), making with one another angles of 109 28' 16". These are the four diagonals of the cube, and have no planes of symmetry corresponding to them. Lastly, there are six axes of binary symmetry, the lines jcming the middle points of opposite edges B I to B 6 (fig. 13). They lie by pairs in the principal planes at right angles to each other, at 45 to the principal axes, and are 1 Other names are Tesseral, Regular, Monometric, and Terqua- ternary. 38 Systematic Mineralogy. [CHAP. III. normals to six corresponding planes of symmetry. These latter, which intersect the principal planes singly, at right FIG angles in the binary axes, by pairs at 45 and 135 in the principal axes, and each other by threes in the ternary axes, are the rectan- gular sections of the cube whose sides are the edges and face dia- gonals, or the principal and binary axes, and whose diagonals are the lines joining opposite solid angles, or the ternary axes. The symmetry of the cube is not altered by permuting the order of the principal axes, or by any inversion of sign, and the preservation of these symmetries involves the re- tention of all the others. Taking any face (hkl), or (a : na : ma), the first condition requires that there should be six similar ones due to permutation of indices, or co- efficients, namely hkl, klh, Ihk, Ikh, khl, hlk, or i n m, nm\, m\ n, mm, n\ m, i m n. The second requires that for each of these combinations of letters there should be faces representing each of the following eight permutations of signs : Combining all these together we find that to satisfy full cubic symmetry in the most general form, any face, (hkl), re- quires to be associated with 47 others, constituting the so- called hexakisoctahedron. In its most regular development, with all the faces equally distant from the centre, it is con- tained by 48 plane scalene triangles whose edges all lie in CHAP. III. Hexakisoctahedron. 39 FIG. 14, planes of symmetry. These edges are all three kinds distin- guished in fig. 14 as long, medium, and short (L, M, and s). Those of medium length all lie in the prin- cipal planes of symmetry, so that the sections of these planes are equilateral but not regular octa- gons, while the longer and shorter ones are arranged in alternate pairs in the other six planes of symmetry, the sections of the latter, therefore, are unequal eight-sided figures. The dihe- dral angle between adjacent faces, or the so-called interfacial angles, are always greater than 90 and less than i8o, 1 They are for the particular forms L. {321}- 30f, i5 8 {421}. 402, 162 13', 15', M. 149 oo, 1 54 47', s. 158' 03 The points or solid angles are of three kinds namely, six eight-faced, or formed by the meeting of four long and four medium edges in the poles of the principal axes ; eight six- faced, formed by three long and three short edges in each of the axes of ternary symmetry, which are also known as trigonal interaxes; and twelve four- faced, formed by two short and two medium edges meeting in each of the axes of binary symmetry or rhombic interaxes. Naumann's symbol for this form is m On, signifying that two parameters vary from the unit length, one being m and the other n times greater than that of the third axis. The geometrical construction of the octant, including the 1 That is, no face can be at right angles to a plane of symmetry, or the edges of symmetry are all effective edges of form. 40 Systematic Mineralogy. [CHAP. III. six faces whose indices are all positive of the form {32 i},is given in fig. 15.* FIG. 15. The arrangement of the symbols of the faces for all the positive values of the first parameter in the form {421} is FIG. 16. shown in fig. 1 9, and for those of the whole of the general form {hk 1} in the scheme fig. 16. This supposes the edges to be 1 In this, and figs. 18, 21, 25, and 28, all the parts are similarly lettered to avoid detailed description- CHAP. III.] Icositetrahedron. FIG. 17. projected on the surface of an octahedron, which is then flat- tened out, the octants being numbered from right to left, first above the equatorial or horizontal plane and then in the same order below, the positive pole of the first axis, A, being in the middle. If, in the form hkl, two of the indices are equal, say k = /, one half of the letter permutations are lost, leaving hkk, kkh, khk, which, with the eight permutations of sign, gives a solid of 24 faces, corresponding to two classes of forms, according as the two equal indices are smaller or larger than the third. In the former case, when h>k, the resulting form is of the kind shown in fig. 17, known as an icositetrahedron, or, more properly, trapezoidal icositetrahedron from the shape of the faces which is often contracted to trapezohe- dron. This may be regarded as a hexakisoctahedron, in which the interfacial angle over the long edges is 1 80, or these edges are effective only as edges of symmetry. The actual edges are therefore 24 longer in the principal planes of symmetry, and 24 shorter in the other six planes of symmetry. The solid angles are, six four-faced, each formed by the meeting of four longer edges in the poles of the principal axes ; eight three faced, formed by groups of three shorter edges in the axes of ternary symmetry ; and eight four-faced (two and two edged) or formed by two longer and two shorter edges in the axes of binary symmetry. The values of the angles are, in the commonest kinds, 2 2 and 3 #3, or {211} {311} Over the longer edges . . 131 49' 144 54' Over the shorter edges . 146 27' 129 31' 4 2 Systematic Mineralogy. [CHAP. III. Weiss's symbol for this class of form is a : m a : m a, and Namnann's mOm. 1 The construction of the positive FIG. 18. faces of 2 O 2 is shown in fig. 18, the front half of the same form in fig. 17, and the general arrangement of the symbols of the whole of the faces in fig. 19. FIG. 19. The second case, when the two equal indices are larger than the third, or as the symbol is more conveniently written hhk,'v$> represented by the class of forms known 1 .The largest solid that can be inscribed in a sphere symmetrical to the nine planes of the cubic system is of this class ; but it is not a possible crystallographic one, m having the irrational value 2 -4142 136 or tan 67 30', and these planes do not express its full symmetry. CHAP, ill.] Triakisoctahcdron. 43 as triakisoctahedra (fig. 20). These are contained by 24 isosceles triangles, whose shorter edges make eight three- faced solid angles in the axes of FIG 2o ternary symmetry, and with the longer ones six eight-faced (four are four-edged) solid angles in the poles of the principal axes. They are particular forms of hexakisoctahedra, having the di- hedral angle of the shorter edges 1 80 ; the plane angle between the two medium edges in any quadrant of a principal plane of symmetry is also 180, or they lie in the same line. The effect of this is to make the principal sections ] square or similar to those of the regular octahedron, so that the whole form may be compared to an octahedron having a. low three-faced pyramid superposed upon each of its faces, a property which is indicated by the name. The edges formed by the meeting of these three planes represent by their position the longer edges of the hexakisoctahedron. The interfacial angles of the more important of these forms are Over the longer edges. Over the shorter edges. {221} 2(9 141 03' 152 44' {331} 30 153.28' I 4 2.08'. Weiss's symbol is a : a : m a, and Naumann's m O, signi- fying that one of the parameters is m times the unit or octa- hedral length. The construction for m = 2 for the faces with positive indices is given in fig. 21, the front half of the same form with the faces noted in fig. 20, and the general scheme of notation for the whole form in fig. 22. 1 The principal crystallographic sections are those upon planes containing principal axes of symmetry or form. 44 Systematic Mineralogy. [CHAP. III. FIG. When the three indices are equal, or h = k i = i, there are no permutations of letters, and only the eight sign per- FlG 23 mutations remain, which give the regular octahedron (fig. 23). The eight faces of this form are equilateral triangles, their di- hedral angle is 109 28' 16". The twelve edges are all equal, and lie in the principal planes of symmetry, each one representing two of the medium edges of the hexakisoctahedron. The six solid angles, all four-faced and similar, lie in the poles of the axes of principal symmetry. CHAP, ill.] Octahedron. 45 The axes of ternary symmetry are normal to the faces, and those of binary symmetry normal to the edges. The symbols are {i 1 1}. a : a : a and O. The sections upon the nine planes of symmetry of this solid, when put together, as shown in fig. 24, form a skeleton octahedron, made up of forty- eight trihedral cells, having a common apex in the central point. Models of this kind are useful as showing the relation of the special to the general forms of the system, and their common symmetry; the same cellular ar- rangement is characteristic of all, the difference being in the exter- nal contour of the constituent planes. For the octahedron the three principal sections are the squares described upon the edges, the other six are rhombs having the edges and crystallographic axes for shorter and longer diagonals respectively. When one of the indices becomes zero, or the corre- sponding parameter co-efficient infinity, the change of sign corresponding to that index is lost, so that the number of sign permutations is only four, while the full number of letter permutations (six) are retained as follows : // k o, k o /, o hk, ok/i, &/IQ, h o k with the signs ++, + , !-, attached to each, giving twenty-four faces for the complete form, which is known as a tetrakishexahedron {/i k o} . Weiss's symbol is a : ma : oo # , and Naumann's oo O n. Fig. 25 shows the construction of one-eighth of the form oo O 2, or {201}; fig. 26 one-half of the same; and fig. 27 the general notation of the entire number of faces. The faces are isosceles triangles, whose shorter sides, representing the longer edges of the hexakis- 4 6 Systematic Mineralogy. [CHAP. in. octahedron, meet in four-faced solid angles in the poles of the principal axes, and by threes with three of the longer sides, forming six-faced solid angles in the ternary axes. Each of these longer edges, representing two of the shorter FIG. 25. FIG. 26. . edges of a hexakisoctahedron, is parallel to one of the prin- cipal axes of symmetry, and together they enclose a cube, so that the character of the solid is that of a cube with a low four-sided pyramid on each of its faces. This character is to some extent indicated by the name, and more particularly in the French cube pyramid^ or the German PyramidenwiirfeL The dihedral angles of two of the common varieties are Over the longer edges. Over the shorter edges. {2 IO.} CO O 2 143 08' 126 52' 143 08' 154 09''. CHAP. III.] Rhombic Dodecahedron. 47 When two of the indices are equal, and the third zero, which corresponds to a suppression either of one change of sign in the triakisoctahedron, or of one half of the permuta- tions in the tetrakishexahedron, the resulting form has the symbols h h o, h o h, o h h, with the signs ++, -\ , h , , or twelve faces in all. FIG. 28. FIG. 29. This is the right rhombic dodecahedron, 1 {110}, Weiss's a : a : oo a, and Naumann's GO O. Fig. 28 shows the con- struction from the axes of the octahedron ; fig. 29 the front half of the form ; and fig. 30 the symbols of all the faces. 1 As one index = o and the other two = h, any whole number may be substituted for the latter without changing its character ; or, in other words, the symbol represents a single form, and not a series of forms. The same remark applies to the cube. 48 Systematic Mineralogy. [CHAP. III. Each of the twelve faces is a rhombus whose plane angles are 70 31' 44" and 109 28' 16", and its longer and shorter diagonals are to each other as \/2~: i. The dihe- dral angles of the faces are all 120, the edges which repre- sent the longer ones of the hexakisoctahedron form four- faced solid angles in the poles of the quaternary axes, and three-faced ones in the ternary axes. Each parallel pair of faces is normal to a binary axis, and consequently the whole form is parallel to the six corresponding planes of symmetry, the longer diagonals of the faces are parallel to the edges of an octahedron, and the shorter ones to those of a cube. This is one of the solids having the property of filling up space that is, it will pack together with others of the same size without leaving any hollows. Lastly, when two indices = o, the third h may be i, or any other whole number, leaving only three permutations, h o o, o^o, o o h, with + and signs to the index h in each, or six in all. This is the cube (fig. 31), {100} (a:a:ooa) or co O GO, Con- FIG. 31. x . ' sidered as a special case of a hexa- kisoctahedron it is that having only the shorter edges effective as edges of form, their dihedral angle being 90, the longer edges are represented by the diagonals of the faces and the medium ones by lines parallel to the edges passing through the centre of the faces. The principal planes of symmetry correspond to the axial planes, which are also parallel to the faces ; this form, therefore, like the rhombic dodecahedron, is sym- metrical to its own faces, or is contained by its principal planes of symmetry. The above include all the possible cases of holohedral cubic forms, as will be seen by the following enumeration, in which a different order from that adopted in the descrip- tion is followed : CHAP, in.] Naumanri's Diagram. 49 1. hkl. Hexakisoctahedra. Indices all different, the least > o. 2. hko. Tetrakishexahedra. Indices all different, the least o. 3. hkk. Icositetrahedra. Two indices equal, less than the third, and > o. 4. h o o. Cube. Two indices equal, less than the third, and = o. 5. hhk. Triakisoctahedra. Two indices equal, and greater than the third, which is > o. 6. h h o. Rhombic dodecahedron. Two indices equal, and greater than the third, which is = o. 7. h h h. Octahedron. All three indices equal. These relations may also be shown graphically by the diagram fig. 32, using Naumann's notation. The symbol of the unit form, O, is placed at FIG the summit of a triangle, the left- hand side contains the forms ob- / \ tained by varying one of the / \ unit axes, or the triakisoctahedra, / \ m O, which pass downwards into "/^^ ^*\ the rhombic dodecahedron oo O, / m V n \ when ;;/ = oo ; the right side con- / \ tains the different icositetrahedra, a- d7t -o* m O m similarly passing downwards into the cube oo o GO ; when m = oo, and the base contains forms intermediate between the cube and rhombic dodeca- hedron, or the tetrakishexahedra oo O n. The general symbol of the hexakisoctahedra, mOn, occupies the centre, and the guide lines connecting it with the sides indicate a passage to the triakisoctahedra when n = i, to the icosi- tetrahedra when n = m, and to the tetrakishexahedra when m = oo . Hemihcdral forms of the cubic system. These are ob- tained from the holohedral forms by the symmetrical sup- Systematic Mineralogy. [CHAP. III. pression of one-half of their faces, which may be done in different ways. The relations of these to each other is best shown by considering first those derived from the general form {h k 1} of which all the others are particular cases. In this there are, as already shown, six permutations of letters and eight of signs, giving the forty-eight faces. But these divide symmetrically into four groups namely, the letter permutations into the two following cyclical groups : Direct hkl, klh, lhk\ Inverse Ik h t k h l*h Ik ; and the signs into two groups of four : Direct . . + + +,+--,_ + _,_- + ; Inverse. . , + +, H !-,+ + ; the full arrangement being represented in the following table given by Miller : A \ hkl klh Ihk Ikh khl hlk hkl kilt Ihk Ikh khl hlk ~hkl kill Ihlz Ikh khl Tilk ~hkl ~klh Ihk l~kh khl hlk C D hkJ kill llik nil kill iri~k Tiki 'klh Ih k Ikh khl hik hkl klh Ihk Ikh khl hlk hkl klh Ihk Ikh khl hl~k There are three ways in which the four parts of the above table, distinguished by the letters A B c D, can be grouped so as to divide the whole into two symmetrical parts, namely : Chequerwise, or A with D and B with c ; Into right and left halves, or A with c and B with D ; Into upper and lower halves, or A with B and c with D. CHAP, in.] Plagihedral Hemihedrism. 5 1 The first of these, corresponding to the two orders 1. Direct letters and signs + inverse letters and signs (A + D), 2. Direct letters and inverse signs + inverse letters with direct signs (B + c) gives the forms represented in figs. 33 and 35, the former being derived from the extension of the white faces, and the latter of the shaded ones, in the hexakisoctahedron {321} (fig. 34). It corresponds to an extension of alternate faces FIG. 33- FIG. 34. FIG. 35. i23 of the holohedral form, so that all the edges of the latter are obliterated, the only points common to the original and derived forms being the ends of the principal and ternary axes ; and as in the form {/i k 1} any face is inclined to each of the three planes of symmetry in which its edges lie, at less than a right angle, it will by extension lose its symmetry to all of them, and the resulting forms will be plagihedral (skew- faced) or asymmetric, which names are used to indicate this particular kind of hemihedrism. There is, however, no change in the axes of symmetry, which are of the same number and kind as in the holohedral forms. Geometri- cally, their faces are irregular pentagons, which are so arranged that the >wo correlated (direct and inverse or positive and negative *) hemihedra, derived from the same holohedral form, cannot be superposed or made to corre- 1 Either may be considered as the positive or negative one ; but the choice, when made, applies to all similar forms in the same substance. Generally that containing the face h k I is taken as positive. E 2 53 Systematic Mineralogy. [CHAP. in. spond with each other by rotation. It will be seen that the edges forming the four-faced solid angles at the extremities of the principal axes have a right-handed inclination to the vertical and horizontal lines in one form, and a left-handed one in the other, which cannot be altered by change of position. Such solids are said to be non-superposable, or enantiomorphous, i.e. permanently right- and left-handed, like gloves. This hemihedrism is not applicable to any of the specialised forms of the hexakisoctahedron, or the other six holohedral forms of the system, for it requires the four groups of signs and letters to be unbroken, which they cease to be if two letters are interchangeable, or if a sign becomes ambiguous. This may be seen in another way by considering each of these six forms as a hexakisoctahedron, in which the dihedral angle over one or more of the three kinds of edges is 180, when, supposing one of the faces meeting in such an edge to be removed, the extension of the other will only fill up its place, and the particular shape will be restored. This would be expressed, in ordinary mineralogical language, as follows : The plagihedral hemi- hedral forms of the cubic system are, with the exception of those derivable from hexakisoctahedra, undistinguishable from the holohedral forms. These particular hemihedral forms are not known to exist either in natural or artificial crystals, and therefore no special class of symbols are re- quired for them. They may be indicated by a {hkl} a [Ikh], the prefix a representing asymmetric. Parallel hemihedrism. The second kind of hemi- hedrism represented by the division of the table on page 50 into right- and left-hand halves, which corresponds to the ap plication of both kinds of sign permutations to each group of letter permutations taken separately, gives rise to hemihedral forms with parallel faces. For the hexakis- octahedron these are shown in figs. 36 and 38, the former being derived from the white, and the latter from the shaded faces in fig. 37. These are twenty- four faced solids, known CHAP. III.] Parallel Hemihcdrism. 53 as dyakisdodecahedra ; the faces are trapezoidal with two equal and two unequal sides. In each of the principal sections half the edges represent those of the holohedral form extended, the other half being replaced by shorter FIG. 36. FIG. 37. FIG. 38. ones, which are less steeply inclined to the principal axis in which they meet than the former. The remaining edges make three-faced solid angles in each of the axes of ternary symmetry, having no symmetrical relation to those of the holohedral form. The general result of this is, that the symmetry to the binary axes is lost and that to the principal axes reduced from quatemary to binary, while that to the ternary axes is unchanged, and only the three principal sections remain planes of symmetry. The general symbols for these forms are, for fig. 36 TT {Ikh}, and for fig. 37 *{hkl}. Naumann's are and prefix TT and the square brackets respectively indicating parallel hemihedrism. The positive and negative forms derived from the same hexakisoctahedron are superposable, or either one may be brought in the position of the other by a quarter revolution about a principal axis. The dihedral angles in the particular case given ^,{32 1} are : Over the longer edges in the principal sections . shorter edges in the principal sections . unsymmetrical edges in the principal sections 150 o' 115 23' 141 47' 54 Systematic Mineralogy. [CHAP. III. The tetrakishexahedron, when similarly developed, pro- duces fig. 39 from the white faces, and fig. 41 from the shaded ones, in {2 i o} fig. 40. These are pentagonal dode- cahedra with irregular faces, one of the sides being promi- nently longer ! than the other four, which are equal to each FIG. 39. FIG. 40. FIG. 41. other. These longer sides are the only effective edges in the principal sections, and as they are parallel to the principal axes the partial cubic symmetry is at once ap- parent. The three-faced solid angles mark the symmetry to the ternary axes, and that to the binary axes is wanting, as in the preceding instance. The symbols are : 7r{o/*}and7r{/z/fco}or 2 and - J For the particular case given, TT [2 i o} , the dihedral angles are Over the longer e^ges . Over the shorter edges . 126 52' "3 35' The dihedral angle over the edges in the principal sections of the hexakisoctahedron increases as the value of m in its symbol is increased, becoming 1 80 when m = oo, or when it becomes a hexakistetrahedron ; the same rela- tion holds good with its hemihedral form, the dyakisdode- cahedron, which under similar conditions passes into a. 1 Or shorter when n approximates to i. The regular or Platonic' dodecahedron is an impossible foim, as the indices of the hexakis- tetrahedron producing it are V^+ i. 2, o, which are not admissible on account of irrationality. IT {850; is very near to it. CHAP. III.] Inclined Hemihedrism. 55 pentagonal dodecahedron. In the same way, when n is the symbol of the hexakistetrahedron, and its hemihedral form becomes oo, they pass into the cube. This mode of hemihedrism is not applicable to the other five holohedral forms, or rather it does not produce geo- metrically different forms from them. Inclined hemihedrism. When the faces are so selected that the whole of the letter permutations with only one-half of the sign permutations appear in a hemihedral form, the latter is said to be hemihedral with inclined faces. This corresponds to the division of the table on page 50 into upper and lower halves, the former containing the symbols with an odd number one or three of positive indices, and the latter those with one or three negative indices, or, in other words, a face and its counterpart can never appear on the same form. Geometrically, this signifies the extension of all the faces in alternate octants, or the suppression of those in opposite and adjacent octants of the holohedral form, as shown in fig. 42 for the white, and in fig. 44 for the FIG. 42. FIG. 43. FIG. 44. shaded faces of {3 2 1} fig. 43. This solid is called a hexakis- tetrahedron, from its resemblance to a tetrahedron enclosed by four groups of six-faced pyramids. From the construc- tion it will be easily seen that the edges forming the six- faced solid angles are the same as the longer and shorter edges of the holohedral form, and preserve their character- istic inclination, but the new ones formed by the meeting of the extended faces have no symmetrical relations to the 56 Systematic Mineralogy. [CHAP. III. original form. The highest symmetry is therefore ternary about the normals of the octahedron, that about the prin- cipal axes is reduced from quaternary to binary, and the normals to the rhombic dodecahedron are not axes of symmetry. As a consequence, the symmetry to the faces of the cube is lost, while that to the faces of the rhombic dodecahedron remains. 1 The symbols are : K- {hkl} and K {h~k7} or ^^- and - ^A The dihedral angles of the particular form given, K {32 i}, will be the same as those over the longer and shorter edges in {321} ; those over the special hemihedral edges are 120 55'. The inclined hemihedral forms of the icositetrahedron are called triakistetrahedra, fig. 45 being that derived from the white, and fig. 47 from the shaded faces of fig. 46. FIG. 45. FIG. 46. FIG. 47. In this the new edges formed by the extended faces enclose a regular tetrahedron, upon each of whose faces a trian- gular pyramid formed by the original faces is superposed. The symbols are : K {h k k] and K {/& k k] , or -y and . For the particular case, K- {2 1 1} , the dihedral angles over the 1 As these relations, which hold good for all inclined hemihedral forms, are not readily seen in perspective figures, it will be well for the student to study them upon a model. The regular tetrahedron is the most convenient form for this purpose. CHAP. III.] Inclined Hemihedrism. 57 tetrahedral edges are 109 28', or the supplements of those of the regular tetrahedron; those over the shorter edges have the same value as in {211}. The inclined hemihedral forms of the triakisoctahedron are called deltoid dodecahedra (figs. 48 and 50), of which the first is derived from the white, and the second from the shaded faces in {221} fig. 49. In these the extended faces FIG. 48. FIG. 49. FIG. 50. form a new three- edged pyramid above the enclosed octa- hedron, but of different proportions to those of the original form, the faces, when regularly developed, being deltoids or four-sided figures, one of whose diagonals joins similar and the other dissimilar angles. The symbols are : K {h h k'\ and K {/z /i k] , or and . The dihedral angles over the hemihedral edges in K {2 2 1} are right angles. The octahedron (fig. 52) by this method produces the FIG. 51. FIG. 52. FIG. 53. two regular tetrahedra (figs. 51 and 53) as its hemihedral forms. Their faces are all equilateral triangles of four Systematic Mineralogy. [CHAP. III. times the area and twice the length of edge of those of the octahedron from which they are derived. The dihedral angles are 70 32', or supplements to those of the octa- hedron ; the edges correspond to diagonals of the faces of a cube described about the axes of the enclosed octahedron. The principal axes join the middle points of opposite edges; the ternary axes are, as in the holohedral form, normals to faces, but on one side only, the opposite sides meeting the extended faces in the solid angles. Neither the cube nor the rhombic dodecahedron can produce forms dissimilar from themselves by any of the three methods of hemihedrism. The former, when con- sidered as a particular form of hexakisoctahedron, has FIG. 54 . FIG. 55- FIG. 56. eight of its faces in the same plane, and therefore the removal of half of these, whether by alternating eighths, as in fig. 54, or by quarters, as in figs. 55, 56, which corre- spond to the three methods of hemihedral selection, can FIG. 57. FIG. 58. FIG. 59. have no effect, as the extension of the remaining parts, whether white or shaded, will restore the original form. In the same way the rhombic dodecahedron in each of its CHAP. III.] Hemihedral Diagrams. 59 faces contains four of those of [h kl} ; and therefore their removal by quarters or halves, as in figs. 57, 58, and 59, and the extension of the remainder, does not change the form. It may therefore be said that there are no hemihedral forms of these solids ; but this is only true as a geometrical proposition, and is at variance with a general principle, deduced from observation, that the forms making up the crystals of any particular substance are all of the same kind i.e. either all holohedral or of the same class of hemi- hedrism. For example, in iron pyrites a very large number of forms are known, the principal ones bting pentagonal- and dyakis-dodecahedra, which often appear alone as well as in combination with the cube and octahedron, but never with a hexakisoctahedron or tetrakishexahedron, or any inclined hemihedral form. It is therefore necessary to consider the cube and rhombic dodecahedron as common to the holo- hedral and hemihedral classes of forms alike, their true character being only determinable by the nature of their combinations. The relation and derivation of the different hemihedral forms may be expressed by their symbols, ac- cording to Naumann, in diagrams similar to that already given for the holohedral form. Fig. 60 gives the scheme for the parallel, and fig. 61 for the inclined, class. In Weiss's notation, parallel hemihedral forms are dis- tinguished by the symbol ^ preceding that of the type face ; and the inclined ones in the same way, with the 60 Systematic Mineralogy. [CHAP. III. addition of a positive or negative sign. Thus the two dyakisdodecahedra, ?r {/ikl} and TT {lk/i}, are respectively \ (a : m a \ n a) and \ (na : ma : a), and the hexakis- tetrahedra, K {hkl} and K [hk 1} + J (a : n a : ma) and ^ (a : n a : m a). Tetartohedral forms. If any hemihedral form of the hexakisoctahedron be subjected to one of the other modes of hemihedry, a new form containing only one-fourth of the full number of faces, but otherwise satisfying the general conditions of systematic symmetry, is obtained. This is said to be tetartohedral. By reference to the table on page 50 it will be easily seen that the result is the same whichever methods of hemihedry be employed, the whole series of symbols being resolved into the four groups, each containing one group of letter and sign permutations, and representing a separate form. Of these, only two are geo- metrically distinct otherwise than by position ; that contain- ing direct letters with inverse signs is superposable to that containing inverse letters with direct signs, or B with c, as are also the groups A and D, where letters and signs are both direct or both inverse, but not otherwise that is, the tetartohedra resulting from the same hemihedron are right- and left-handed to each other. Figs. 62 and 63 represent the forms obtained by the FIG. 62. FIG. 63. application of parallel hemihedrism to the hexakistetra- hedron (fig. 42). The first corresponds to the white faces CHAP. III.] Cubic Tetartohedrism. 61 and the second to the shaded ones, in alternate (the ist, 3rd, 6th, and 8th) octants of the hexakisoctahedron (fig. 34), or to the divisions A and B of the table on page 50. These are called right and left tetartohedral pentagonal dodeca- hedra, their faces, when most regularly developed, being irregular pentagons. The only points in common with the holohedral forms are alternate extremities of the ternary axes, which retain their original positions : those of the principal axes meet in six edges which are oblique to the principal sections. As both kinds of hemihedrism are involved in their production, the symmetry to both series of planes is lost, or the forms are plagihedral, but that to the axes is of the same kind as in the hemihedral forms, or to three binary and four ternary axes. This, therefore, may be regarded as the fundamental axial symmetry of the cubic system, it being the minimum common to all classes of forms, the higher quaternary kind being added in the case of the holohedral and plagihedral hemihedral classes. The inverse tetrakishexahedron gives rise to two similar forms differing only in position, and if the former be con- sidered as positive they will be the negative tetartohedra. The general symbols will be : TTK{kkl} 7TK- {////} ir K {h~k7} 7r K {7&7i} and , m On , m O n 7 T ?* T ' 4 4 _ m O n mO n j F) ~~~ /. 4 4 These are the only geometrically distinct tetartohedral forms, and, although crystal! graphically possible, they are not known to occur independently in either natural or arti- ficial crystals. The actual existence of the condition of tetartohednsm is, however, known from the fact of a few substances appearing in crystals showing both kinds of 62 Systematic Mineralogy. [CHAP. III. hemihedrism a point that will be considered in treating of the combinations of the system. Combinations of the cubic system. Any number of simple forms may appear with their full number of faces in a single crystal, which will then have a sort of composite character, the faces of the constituent forms being, as a rule, recog- nisable not by their shapes, but by their positions. Such crystals are termed combinations. That the shapes of the faces must be altered, even with the most exact regularity of position, will be apparent when it is considered that the simple forms described about any definite lengths of axes are all exterior to the octahedron, or enclosed by the cube, having only points or lines in common, and therefore no combination is possible between them in this condition. If, however, the size of any one of them be altered rela- tively to another by shifting its faces parallel to the original positions, either nearer to the centre in the case of an exterior, or further from- it in an interior form, which does not alter its crystallographic significance, a new solid will be produced, with edges and solid angles either wholly or FIG. 64. FIG. 65. partly different from those of the component forms. For instance, fig. 64 represents the complete interpenetration of a cube by an octahedron, their principal axes being common as regards position, but differing in length ; the points of the latter form project as four-faced pyramids above the CHAP. III.] Cube and Octahedron. faces of the former, and similarly the solid angles of the cube form three-faced pyramids above the faces of the octahedron. If these projecting portions be removed, which is necessary to produce convex angles, the form is reduced to fig. 65, which may be considered as an octahedron with its solid angles cut off or truncated l by the faces of a cube. Fig. 66 is another example in which the faces of the two forms are evenly balanced, so that it may be equally well FIG. 67. FIG. 66. 100 regarded as a cube modified by an octahedron, or the reverse ; while in fig. 67 the cube is the principal or domi- nant form, the octahedron being only represented by a small triangular plane in each of the corners. In neither of these are the characteristic shapes of the faces seen in the simple forms apparent, but they are nevertheless easily recognisable from their constant position, and the paral- lelism of the new edges of combination. As a rule, special names are not applied to combinations, but they are described by their symbols, that of the most prominent form being placed first. Thus, fig. 65 is indi- cated by the symbols O . oo <9oo, and fig. 67 as oo <9oo . O, while either order applies equally well to fig. 66. The faces of the octahedron when in combination with the rhombic dodecahedron appear as triangular planes 1 An edge replaced by a single plane making equal angles with the adjacent faces is truncated ; when the replacement is by two planes it is bevelled. Similarly, a solid angle is truncated by a single plane, bevelled by two, and acuminated or blunted by three or more. 6 4 Systematic Mineralogy. [CHAP. III. truncating the three-faced solid angles of the latter, as in fig. 68. When the octahedron is the dominant form, the faces of the rhombic dodecahedron truncate its edges as in FIG. 68. FIG. 69. .fig. 69. The cube truncates the four-faced solid angles of the rhombic dodecahedron, as in fig. 70, and conversely the latter truncates the edges of the former, as in fig. 71. FIG. FIG. 71. Examples of these combinations are of very common occur- rence in nature, especially in magnetite, galena, and fluor- spar. The faces of the icositetrahedron, 2 O 2 or {2 i i}, lie in zones whose axes are parallel to the edges of the octahedron and the plane diagonals of the cube, and therefore form blunt four-faced pyramids upon the solid angles of the former (fig. 72), and three-faced ones upon the latter (fig. 73). CHAP. III.] Combinations of Icositetrahedron. In a combination of these three forms, when the first pre- dominates, as in fig. 74, the cube truncates its four-faced, and the octahedron its three-faced, solid angles. The combinations of icositetrahedra with the rhombic dodecahedron vary very considerably in appearance, ac- FIG. 72. FIG. 73. cording to the value of m or th.e relation of h to k in their symbols. The form most commonly observed, which is 2 O 2 or {211}, truncates the whole of the edges of {i 10} ; as the two forms are tautozonal (fig. 75). This combination FIG. 74. FIG. 75. gives a good example of the determination of the symbols of a form from those of its zones. If the edges of the cube and octahedron be added, as shown by the dotted lines, it will be apparent that any face such as the upper one, marked #, 66 Systematic Mineralogy. [CHAP. III. lies in two zones, the first containing i o i and o i i or [i i i] and the second oo i and i i i or [i i o]. From these sym- bols we get, by the method given on page 32, a symbol (1 1 2) or the counterpart of that of the particular face, and similarly (2 11) for the second face, marked a. 'In such a case, therefore, no special calculation is required for the determination of the form, when the parallelism of the edges is exhibited, by measurement. When m > 2 or h : k > 2 : i, the faces of the icositetra- hedron appear as deltoids in groups of four upon the four- FIG. 76. FIG. 77. FIG. 78. faced solid angles of the rhombic dodecahedron, as shown for the combination GO O. 3 #3, or{i i o}, {31 1} in fig. 76. When m < 2 or h : k < 2 : i, similar planes appear in groups of three upon the three-faced solid angles, as in the combination oo O. | 6>f or {i i o}, {3 2 2} (fig. 77). The triakisoctahedron, having its longer edges in common with the octahedron, bevels the edges of the 'latter and conversely the octahe- dron truncates the three-faced solid angles of the former, both of which relations are apparent in fig. 78, con- taining the forms O. 2 O or {i i i}, {221}. In combination with the rhombic dodecahedron a CHAP. III.] Combinations of Triakisoctahedron. 67 triakisoctahedron forms obtuse pyramids blunting the three- faced solid angles (fig. 79), and with the cube groups of three deltoid planes upon the solid angles (fig. 80). FIG. 79. Fin. 80. The triakisoctahedron, in combination with the icosi- tetrahedron, modifies the edges lying in the dodecahedral planes of symmetry (the short edges of h k /). In the par- ticular case represented in fig. Si, FIG. 81. 2 6>2, | O, or {JIT}, {332} the two forms are tautozonal, and therefore the new edges are parallel to the original ones. When m is greater than f , or the form is nearer to a rhombic dodecahedron, the re- placing faces appear as very acute jriangles whose summits meet in the ternary axes. The tetrakishexahedron having its longer edges in common with the cube, will in com- bination bevel these edges, and conversely the cube will truncate the four-faced solid angles of the first form, as shown in fig. 82, representing the combination oo O oo . QO (92 {100}, or {210}, which is commonly observed in fluorspar. The same forms being also tautozonal to the rhom- bic dodecahedron, the latter will have its four-faced solid F 2 68 Systematic Mineralogy. [CHAP. III. angles blunted by the faces of GO O 2 (fig. 83). If the cube were to be added its faces would truncate these new solid angles, and any more obtuse tetrakishexahedron would bevel the edges between {2 i o} and {i oo}, and any more acute one those between {2 i 0} and {i i o}. FIG. 82. FIG. ^ \ 201 /\ zlo 100 210 120 / *& \ FIG. 84, The tetrakishexahedron and octahedron are not tauto- zonal forms, and therefore in combination their faces assume irregular shapes, as seen in fig. 84. By comparing this with figs. 65 and 69 it will be seen that these shapes will vary with the value of m or the inequality of h and k in the symbols of the tetrakishexa- hedron. As these values diminish, the octahedral faces become more nearly triangular, until when m = i or h = k they are equilateral tri- angles, as in fig. 69 ; and in the reverse direction, when m = oo or h : k = i : o, they are equiangular hexagons of 120, as in fig. 65. The first of the cases, how- ever, is the condition of a rhombic dodecahedron, and the second that of a cube, which are the respective limiting forms of the tetrakishexahedron in these directions. The acute angles of the deltoid planes in fig. 84 wijl alter in a cor- responding manner to a minimum of o in fig. 69, and to a CHAP. III.] Combinations of Hexakisoctahedron. 69 maximum of 96 in fig. 65. By comparing analogous combinations between limiting forms, their relations may often be more readily appreciated than by the most elabo- rate verbal explanations ; and therefore the construction of forms with different parameters to those given in the figures may be recommended to the learner as a useful exercise. The combinations of the tetrakishexahedron with the other twenty-four-faced forms vary very considerably in appearance, accordingly as different values are assigned to the variable parameters in their symbols, and to illustrate them properly a larger number of figures would be required than can be given here. One special case deserves men- tion namely, that by regular truncation of the edges lying in the principal sections, the icositetrahedron {2 i 1} may be changed into the tetrakishexahedron {2 i o}. The hexakisoctahedron appears in combination with the octahedron in groups of eight triangular faces, blunting the FIG. 85. FIG. 86. 231 010 315 solid angles (fig. 85), and in similar groups of six faces upon the solid angles of the cube (fig. 86). From these it will also be apparent that the octahedron truncates the six-faced, and the cube the eight-faced, solid angles of the hexakisocta- hedron. In combination with the rhombic dodecahedron, the hexakisoctahedron may appear either as bevelling the edges or as modifying the three- or four-faced solid angles. The Systematic Mineralogy. [CHAP. III. first case is that in which the forms are tautozonal, repre- sented in fig. 87, for {i i o}, {3 2 1} oroo (^,3 O f corresponds to the condition h k + / in \hkl\ or m n = m + n, or ;/ = n. = in m O n. The known forms of m i m + i this character are {3 2 1} . (4 3 1} and {64. 63. i}, the first being that most frequently observed in nature. When /*< + /. n < or m n < m + n. m i as in {543} . {43 2} or {15.11.7}, the replacement is by six faces upon the three-faced solid angles of {no} ; and lastly, when h > k + l } n> m m i or m 11 > m + , as in {42 1} . {73 1}, &c., the replacement is by eight faces upon the four-faced solid angles of {i i o}. By far the larger FIG. 87. FIG. 88. 01 number of known hexakisoctahedra are of the latter kind, but they are not easily recognised, as they only occur in very subordinate combinations in crystals containing nume- rous other forms. For such complex combinations the reader is referred to Schrauf s Atlas, and the larger treatises on Mineralogy. Hemihedral combinations. Among the simpler and more important cases, that of the cube and pentagonal dode- cahedron is seen in fig. 88. This differs from fig. 82, CHAP. III.] Hemihedml Combinations. 71 by the omission of the alternate bevelling planes in each zone, or the faces of TT {2 i o} truncate the edges of {100} unsymmetrically, being unequally inclined to adjacent faces ; but this inequality diminishes with that between h and k or k and /, or as we approach the cube and rhombic dode- cahedron respectively. This class of combination is very characteristic of the pyrites group of minerals. In fig. 89, the pentagonal dodecahedron inverse to that in fig. 88 modifies the solid angles of the octahedron symmetrically. If these planes be extended to the complete obliteration of the octahedral edges, a solid is obtained with twenty faces which are very nearly equilateral triangles, and approxi- mating in appearance to the regular icosahedron, which, however, we have seen, is not possible in crystallography. 1 The dyakisdodecahedron appears as an obtuse three- faced pyramid upon the ternary solid angles of the pentagonal FIG. 89. FIG. 90. dodecahedron, and has its own solid angles of the same kind truncated by the octahedron, as in fig. 90. It also forms groups of three irregular four-sided planes upon the solid angles of the cube, as in fig. 91. In the inclined hemihedral forms, the cube truncates the edges of a tetrahedron (fig. 91), and has half its own solid angles, or one in each pair joined by a ternary axis truncated by the faces of the latter (fig. 93). Two opposite tetrahedra in combination appear as in fig. 94, the faces of one being 1 1 1 I \ ir |8 5 oj or 0. | [oo f ] is a very close approximation, possible but not actually observed. Systematic Mineralogy. [CHAP, III. prominently larger than the other. It will be easily seen that when they are equally developed the combination will FIG. 91. FIG. 92. FIG. 93. be an octahedron, and therefore indistinguishable from a holohedral form. When this does occur, the hemihedral nature of a substance is often apparent from some physical dissimilarity in the two classes of faces, such as one set being more brilliant than the other, or striated when the others are smooth, and therefore indicating that the crystal is not a true octahedron. The rhombic dodecahedron forms a low three-faced pyramid upon each of the solid angles of a tetrahedron, as in fig. 95. FIG. 94. FIG. 95. The triakistetrahedron bevels the edges of the tetra- hedron of the same direction as in the combination K {2 i 1} K {111} (fig. 96), which is very characteristic of the antimo- nial copper ore known as fahlerz. CHAP. IV.] Tetartohcdral Combinations. 73 Fig. 97 represents one of the few known cases of cubic tetartohedrism, as evidenced by the occurrence of inclined FIG. 97. FIG. 96. and parallel hemihedral forms in the same combination. It is not a natural substance, being a crystal of an artificial salt chlorate of sodium. The same class of development characterises the crystals of the nitrates of lead, strontium, and barium, an actual tetartohedron +5(5 f ) r, or /,- u {3 5 1} , having been determined by Lewis in the latter salt. CHAPTER IV. HEXAGONAL SYSTEM. THE symmetry characteristic of the most completely de- veloped forms in this system is seen in the regular hexagonal prism with parallel end faces (fig. 98). This has binary sym- metry about six axes, making angles of 30 and 150 to each other, the strong lines 0,, a 2 , # 3 , and the dotted ones alternat- ing with the same plane, and senary or hexagonal about a seventh axis, which is normal to the other six, and indicated by the vertical line c. These axes correspond to seven planes of symmetry : a principal one, or that containing the six binary 74 Systematic Mineralogy. [CHAP. IV. or lateral axes, normal to the principal or vertical axis, and six others corresponding to the diametral sections of the prism upon each of the lateral and the vertical axes. Three of these, or those parallel to the faces of the prism, are dis- tinguished as lateral axial planes, and the alternate ones containing the dotted lateral axes as lateral interaxial planes. The geometrical relations of faces of the above kind may be expressed in four different ways, each of which has been adopted as the basis of a system of notation. These are : 1. Weiss 's system. 1 With four reference axes, three lateral at 60 and 120 to each other, and perpendicular to a fourth or principal axis, taken in the order of fig. 98, or a l9 a%, 3 , c. 2. Schraufs system. If in fig. 98 the faces meeting the right and left lateral axis a 2 be produced until they meet both in front and behind, the result is a four-faced prism upon a rhombic base, whose diagonals are at right angles, and in the ratio of i : <\/3, or that of the side of an equi- lateral triangle to its altitude. This therefore gives a method whereby hexagonal forms may be referred to three indepen- dent axes all at right angles to each other, as in the rhombic system ; if the constant relation of i : \/3 be assumed for the lateral axes. This method is adopted by Schrauf, who calls it the orthohexagonal system ; by it the prism in fig. 98 is not a simple form, but a combination of the unit rhombic prism {i i o}, and a form containing two faces parallel to the right and left diametral plane, or {i oo}. 3. Miller's system. If a cube be placed with a ternary axis upright, the three edges meeting in either of the poles of that axis will be parallel and opposite to the three at the other pole, so that if lines parallel to them be drawn through the centre a system of three axes will be obtained, inclined to the vertical, but making equal angles with each other, and having equal parameters. In this particular case these angles will be right angles, but the same relation holds good for any analogous form contained by six equal rhombic faces, 1 Weiss calls the system sechsgliedrig, or six-membered. CHAP. IV.] Hexagonal Notation. 75 rhombohedron, whose solid angles are formed by edges meet- ing at a greater or less angle than 90, which inclination will also be characteristic of their axes. The relation of such a system of axes to the unit faces is that of the legs of a table formed by three sticks, tied together in the middle, to the table top and the ground respectively the first being the face i i i, and the second i 1 1. This prin- ciple is adopted in Miller's rhombohedral notation. 4. Bravais-Miller system. The three semi-axes, having like signs in Miller's system, when represented by their ortho- gonal projections, are resolved into a common vertical and three horizontal lines, the latter making angles of 120 with each other. This brings us back to the system of four axes with the difference that the positive and negative semi-axes alternate with, instead of succeeding, each other at 60 in the horizontal plane ; and in this way we obtain the hexa- gonal notation of Bravais as adapted to Miller's system. It has the advantage of maintaining the relation between the notation by indices and that by parameter-coefficients subsisting in the other systems, as well as of expressing the physical symmetry of the forms more readily than the other- wise preferable rhombohedral notation of Miller. The pro- perties of any face are determined by three indices, one referring to the independent vertical axis and the others to two of the lateral axes ; but to determine its position in the form, a fourth index, referring to the third lateral axis, is required, giving a general symbol of the form {// k I i} , in which the posi- tion of the last letter, referring to the vertical axis, is invariable, while the other three are interchangeable with positive and negative signs, subject to the condition that their algebraical sum is always equal to zero, or h + k + / = o. This property will be apparent from fig. 99, where Ox, Oy, and O z represent the distances Systematic Mineralogy. [CHAP. IV. at which a right line cuts three axes, making angles of 60 to each other conjointly at O. For, A Oxz + A O zy + A Oyx = o, or, Ox. Ozsm.xOz + Oz. Oy.sin. z Oy+ Oy Ox sin. yOx = o; but since xOz = 6o, zOy = 6o,yOx = 120* the sines are all equal. Dividing them out, and also dividing by the product O x. Oy. O z, we obtain which quantities are represented by the three indices in the order given above. Further, if we consider the axes x andj> as positive, z will be negative, and vice versa \ and as the intercept of the latter is the shortest, it will have the largest FIG. TOO. of the three indices, but with the opposite sign to that of the other two ; so that the general symbol becomes [hkli], in CHAP. IV.] Hexagonal Notation. 77 which the first two indices are independent, the third being equal to their sum, but with the opposite sign. The number of permutations of letters and signs satisfying these con- ditions is twelve, where order is shown by the twelve divisions numbered like a clock-face, making up the interior polygon in the diagram, fig. 100. They represent the hori- zontal projection of the faces corresponding to any value of /, and when this is greater than o there will be twelve similar faces below the horizontal plane having / in their symbols, the whole making up the twenty-four-faced solid known as a dihexagonal pyramid, which is the general representative form of the system. These symbols are given in full on page 88. In Weiss's system, the lateral axes are noted as a l a 2 a^ their positions being also shown in fig. 100. According to this, the symbols determining a face, those of the two inde- pendent lateral and the vertical axes, are a\na\m as P or {o i i 1} , the outermost will be 2 P or {o 2 2 1} , and the innermost \ P or {0112}; if the latter is considered as P the others will be 2 P and 4 P or {0441}; and lastly, if the outermost is the unit, the others will be \ P and J P or {0114} respectively. When m = oo or / = o the basal angles become 180, and as those of the polar edges change in the reverse order they diminish until they correspond to the plane angle of the base ; or, in other words, the pyramid becomes a prism. The forms of this class corresponding to the three kinds of pyramids are represented in figs. 108, 109, and no. The first of these, derived from fig. 101, is a dihexagonal prism contained by twelve similar rectangular faces meeting in edges parallel to the vertical axes at angles which are alternately greater and less than 150 (that being the angle of a regular dodecagon, which is excluded by the CHAP. IV.] Hexagonal Prism. irrationality of its parameters). The symbols, as will be apparent from the derivation, are {h k T6\ , n _ t a : n a : a : cc r, and oo Pn. Fig. 109 is the hexagonal prism of the first order, con- tained by six faces meeting at 120 in the lateral axial FIG. iog. FIG. no. 1010 oaio 1120 planes, or its base is a regular hexagon similar in position to that of the unit-pyramid. It is represented by {o 1 1 o}, oo a : a : --a : oo ? and P2. As all the faces of prisms lie in single zones which are unlimited in the direction of their axes, they cannot of them- selves form complete crystals, but can only appear in com- bination. These are said to be open forms, and are common in all except the cubic systems, where the simple forms are necessarily closed. When m is less than i, the basal edges of the pyramid become sharper and the polar ones blunter than those of G 2 84 Systematic Mineralogy. [CHAP. IV. the unit form ; and when m = o the angles of the former are o and of the latter 180, or the whole of the faces meeting the vertical axes at either extremity fall into the same surface, and the form is reduced to the single unlimited plane containing the lateral axes. When, however, this is shifted from the central position, it will intercept some length upon the vertical axis, and require a corresponding parallel face on the opposite side of the origin ; the com- plete form is therefore represented by two faces parallel to the basal section. This, known as the basal or terminal pinakoid) is another form only possible in combination, having no proper geometrical form, its shape being con- ditioned by the edges formed in combination. For instance, in fig. 1 08, it is dihexagonal, in fig. 109 hexagonal of the first order, and in fig. no of the second, these differences being obviously due to the different prisms with which it is combined. The symbols are : {oooi},oo# : 000 : a : c, and oP. The seven classes of forms described above namely, the three pyramids, with their corresponding prisms and the terminal pinakoid, are the only kinds possible with full hexagonal symmetry to seven axes, as will easily be seen by m giving special values to the indices in the general symbol hkli, and working P Pn P2 them out subject to the condition h + k + / = o. Their relation to each mP...mPn...mP2 other is best seen j n Naumann's dia- p fj p gram, formed byarranging their symbols as in the margin. The first vertical line contains the hexagonal pyramids of the first order ; the second, the dihexagonal pyramids; and the third, the pyramids of the second order. The unit forms of these three classes are in the third horizontal line; their obtuse modifications CHAP. IV.] Hexagonal Combinations. 85 having ;;/ < i, or generally a proper fraction expressed as in the second ; the fourth line contains the more acute m pyramids having ; > i ; the fifth, the corresponding infinitely acute form or prisms, and the top line the common limit of obtuse form, or the terminal pinakoid. Any one of these forms lies in the same zone with any other in the same line, whether horizontal or vertical. In the use of this scheme it must always be remembered that the rational coefficient m applies not to the natural unit number i, but to the funda- mental arbitrary ratio a : c. From what has been said con- cerning the properties of pyramids of varying altitudes, it will be easily seen that the basal angles vary more rapidly than those over the polar edges, as they may range from o to 180, while the latter can only vary between 120 and 180. In the description of hexagonal minerals, therefore, the angle of the basal edge is usually given as that characteristic of the species, and determining the ratios a : c most accurately. FIG. in. FIG. 112. In combinations of hexagonal forms of the same order, the more acute, or those with the highest m or lowest /", bevel the basal edges of the more obtuse, and, conversely, the latter acuminate or blunt the polar summits of the former. The 86 Systematic Mineralogy. [CHAP. IV. steepest form of any series, the prism, and the flattest, the terminal pinakoid, respectively truncate the basal edges and polar summits of any pyramid, as in fig. m, containing ccP, $P,P, o/> or {oiio}, {0331}, {oi7o}, {oooi}. FIG. 113. FIG. 114. In combinations of forms of different orders, having m in common, or of the same altitude upon different bases, FIG. 115. FIG. 116. m P2 truncates the polar edges of m P, and the latter bevels the basal solid angles of the former, as in figs. 112, 113, while m /'truncates the edges of mPn lying in the lateral CHAP. IV.] Hexagonal Combinations. FIG. 117. interaxial planes (figs. H4, 1 115), and m P 2, those in the lateral axial planes, the relations of the three classes of prisms being similar to those of the corresponding pyramids. In combinations of forms in which both m and n, or base and altitude, are different, mPn, when combined with a more acute hexagonal pyra- mid of either order, appears as an obtuse twelve-faced point upon the polar summit of the latter, as seen in plan (fig. 1 1 6), for P- Pn. A pyramid of the second order upon a more acute form of the first forms an an- alogous six-faced point (fig. 117), the plan of n P, P2. In the reverse condition, when the pyramid of the second FIG. 118. FIG. 119. order is the more acute form, its faces modify the solid angles in the basal section of that of the first order, the 1 This is noted as PP\, or {o I i 1} {12 3" 3} . 88 Systematic Mineralogy. [CHAP. IV. aspect of such combinations varying with the differences in the parameters of the two forms. For instance, in fig. 118 the faces of 2Pz or {1121} make new edges, which are parallel to the polar edges of P or {o 1 1 1} ; while in fig. 119 the combination edges are parallel to the polar ones of the replacing form J f P 2, or {2 2 4 3} . Lastly, when m = oo, the basal solid angles of the less acute form are truncated, or, in other words, the prism of either order truncates the basal solid angles of the pyramid of the other, as in fig. 120, which is noted as P, aoP 2, o P, but would do equally well for P2, GO P o P, if the order of the first two con- stituents be supposed to be re- versed. The above include all the simpler cases of holohedral hexagonal combinations, but they are not very commonly met with in nature, there being but few species in which full hexagonal symmetry pre- vails, and these are generally remarkable for the large number of forms present, the most of which are, however, as a rule, very subordinate to the dominant form, gene- rally a prism. A few examples of such combinations, which are usually best represented in horizontal pro- jections, will be found in the volume on 'Descriptive Mineralogy.' Hemihedral hexagonal forms. The faces of the di- hexagonal pyramid divide symmetrically into four groups of six, which, as the corresponding faces above and below the basal section differ only in the sign of their fourth index, may be represented by the four triads indicated by dif- ferent shadings in the horizontal projection (fig. 102), cor- responding to the following table, in which the faces are 1 In other words P truncates the polar edges of \ P2, a relation which is very commonly observed in natural crystals. CHAP. IV.] Hexagonal Hemihedrism. numbered in order from left to right, commencing on the right side of the positive semi-axis h : A B C D i. IJiki v. klJii ix. hkli ii. khli vi. Ik hi x. hlki in. hkli vn. Ihki XI. //"/*/ iv. 7i 1 % i vin. ///// XII. ///h E F G H XTII. l7i%i xvii. klhl xxi. hkll xiv. khll xviii. Ik hi xxn. hlkl xv. hkll XIX. //2/* XXIII. //$* xvi. hlkl . xx. khll XXIV. /I^J This may be halved symmetrically in the three following ways, each corresponding to a possible case of hemihedrism : A C F H A C E G and and and B D F H In the first case, one form contains uneven numbered faces above alternating with even numbered ones below, and the other even ones above and uneven below. This is known as plagihedral or trapezohedral hemihedrism ; in the second, or rhombohedral hemihedrism^ the grouping is by alternate pairs of faces above and below the basal section, or the ist, Systematic Mineralogy. [CHAP. IV. 3rd, and 5th, with the 8th, loth, and i2th pairs ; and the 2nd, 4th, and 6th with the 7th, 9th, and nth pairs. In the third, m pyramidal hemihedrism, the grouping is symmetrical to the base, one form containing all even and the other all odd numbered faces. Of the two most obvious methods of dividing the table into upper and lower, and right and left halves, the first is excluded by not satisfying the conditions of symmetry, giving forms having the indices of the vertical axis either all posi- tive or all negative. The second, a one-sided distribution of faces, has been called trigonotype hemihedrism by some writers, while others say that it does not possess true hemi- hedrism. This is possibly rather a question of terms than of fact. It need not be discussed, as the form does not occur in minerals. Trapezohedral hemihedrism. The dihexagonal pyramid, when divided hemihedrally in the manner shown in fig. 122, corresponding to the first case, as defined above, gives rise to the two hemihedral forms (figs. 121 and 123), the first FIG. 121. FIG. 122. FIG. 123. from the white, and the second from the shaded faces of fig. 122. Fig. 124 is a horizontal of projection of fig. 123 ; the faces are numbered according to the table on page 89, the edges below the horizontal plane being shown in dotted lines. These known as hexagonal trapezohedra are con- CHAP, iv.] Trapezofodral Hemihedrism. tained by twelve trapeziform faces meeting in twelve similar polar and twelve dissimilar middle edges, the latter being alternately longer and shorter. Like FIG. 124. the plagihedral hemihedra of the cubic \i. system, they have the same number and kind of axes of symmetry as the holohedral forms, but no planes of sym- metry, and are therefore non-super- posable. As a face of a hexagonal pyramid of either order or a dihexagonal prism contains two, that of a hexagonal prism four, and the ter- minal pinakoid six, faces of the general form {hkli}, it is clear that none of these holohedral forms will be geometri- cally changed by this kind of hemihedry, which is therefore only effective in producing new forms in the dihexagcnal pyramid. No examples of crystals of this kind of hemihe- drism, whether of natural or artificial origin, are known, so that as yet they only represent a geometrical possibility. Jt may be represented by the symbols : na:-a: and mPn mPn L Rhombohedral hemihedrism. The second method of hemi- hedral division of the dihexagonal pyramid, that by alternate pairs effaces above and below the base, as in fig. 126, pro- duces from the white faces fig. 125, and from the shaded ones fig. 127. The corresponding horizontal projections are seen in figs. 128 and 129. A form of this class, known as a 1 The hemihedrism is here shown by the prefix \ to the holohedral symbol without special indication of the particular case meant, and is therefore general for all kinds. 9 2 Systematic Mineralogy. [CHAP. IV. scalenohedron, a contraction of scalene dodecahedron, is con- tained by twelve faces, all similar scalene triangles meeting FIG. 125. FIG. 126. FIG. 127. in two kinds of polar edges, six longer or more obtuse (x), six shorter or more acute (Y), and six middle edges (z), lying in zigzag order about the basal section. From the derivation of this form it will be apparent that FIG. 128. FIG. 129. the longer polar edges have the same character as in the holohedral form, and that new shorter ones (Y) lie in the same transverse sections with them that is, in the lateral CHAP, iv.] Rhombohedral Hemihedrism. 93 interaxial planes, which are therefore planes of symmetry, while the lateral axial planes are not ; and from the obliquity of the middle edges (z) to the original basal section, the latter cannot be a plane of symmetry. The lateral crystallo- graphic axes being normals to the interaxial planes, they will be axes of binary symmetry, and as these planes also inter- sect in the vertical axes, the latter will also be an axis of symmetry, but reduced from hexagonal to ternary. We therefore have ternary symmetry about the principal axis, binary about the lateral crystallographic axes, and three planes of symmetry inclined at 60 to each other as charac- teristics of this class of hemihedrism. The two scalenohedra derivable from the same di- hexagonal pyramid are superposable that is, either may be brought into coincidence with the other by rotation through 60 or T 80 about the vertical axis. They are distinguished as positive and negative, or direct and inverse forms, accord- ing to position. The choice of position is, however, arbi- trary, and usually depends upon structural peculiarities. The general symbols of the scalenohedron are: {**//> and The same selection, applied to the hexagonal pyramid of the first order, as in fig. 131, produces the two hemihedral forms, figs. 130 and 132 the former from the white and the latter from the shaded faces. Figs. 133 and 134 are the corresponding horizontal projections. A form of this class, known as a rhombohedron, a contraction for rhombic hexahedron, is contained by six faces, all equal rhombs, meeting in three polar edges at either end of the vertical axis, and six middle edges in zy order about the basal section. The polar diagonals of the faces meeting the vertical axis, as shown in the dotted line in fig. 130, represent the obtuse edges (x) of the scalenohedron, as will be readily seen when it is remembered that the hexagonal pyramid is 94 Systematic Mineralogy. [CHAP. IV. that particular dihexagonal pyramid whose faces meet at 1 80 in the obtuse polar edges. The symmetrical relations FIG. 130. FIG. 131. FIG. 132. are therefore in every respect similar to those of the scaleno- hedron. The rhombohedron, being the most important form FIG. 133. FIG. 134. of its class, and even of its system, is considered as cha- racteristic of this kind of hemihedrism, which is therefore called rhombohedral instead of scalenohedral. The symbols of the two rhombohedra derived from the same (unit) form are : P P K-{oin},K{ioIi},and + - - - The dihedral angles over the two kinds of edges in the rhombohedron are mutually supplemental, or one kind is as much above as the other is below 90. When the larger angle is in the polar edges, the rhombohedron is obtuse^ but CHAP. IV.] Rhombohedra. 95 when it is in the middle edges it is acute. The form occu- pying the middle position, or having the angles of both polar and middle edges right angles is the cube, which is a possible rhombohedron, deriving from the hexagonal pyra- mid, havings : c= i : 1*2247 ; and although it is not known to exist in nature, there are several forms very nearly approaching it. This is an example of what are called limiting forms, where the same geometrical solid may arise in two systems. In this case it is obviously possible, from the circumstance that the ter-quaternary symmetry of the cube includes the lower ter-binary kind of the rhombohedron, and the real test of the nature of the form is to be looked for not in the presence of the lower, but the absence of the higher symmetry, which is usually apparent in the character of its combinations ; but assuming it to appear as a simple form, its true nature could only be determined by optical or other investigation of the structural peculiarities of the substance. The symbols of the two positive and negative, or direct and inverse, rhombohedra, originating from the same hexa- gonal pyramid, where the latter is a unit form : P P r {01 1 1}, < {101 1}, and +-. - ; or, generally, K-{o:7/}, *{ioi/},and -\~- Like the scalenohedra they are superposable, either one being brought into coincidence with the other by rotation through 60 or 180 about the vertical axis. From their derivation it will be apparent that in co.nbination they truncate each other's solid angles obliquely, as in fig. 135 ; and when the two are exactly balanced, the hexagonal pyramid (fig. 131) is reproduced. A rhombohedron of either position and any length of vertical axis has its polar edges truncated by the faces of 9 6 Systematic Mineralogy. [CHAP. IV. another of the same series of the opposite position, whose height is one half or breadth of base twice that of the first, FIG. 135. as in fig. 136. This case is commonly observed in calcite, p whose principal rhombohedron + _ has its polar edges replaced by faces of !_, and similarly modifies those of ?_ , and in the same way H is modified by +1-5 2 22 - 2 _^by +4/, and so forth. 2 2 By varying the value of m in the symbol of a rhombo- hedron, a series of other forms of greater or less altitude upon the same base are obtained in the same way as with the pyramid. As m decreases, the angles of the middle edges, as well as their jobliquity to the basal section, diminish as those of the polar edges increase^ the latter becoming 180 when m = o, producing the basal pinakoid. In the opposite direction the middle edges become more obtuse and increase their inclination to the basal plane ; and when m = oo they fall into the same vertical lines with the polar edges, or the hexagonal prism GO Pis produced. The limits of the series of the rhombohedron are therefore the same as those of the hexagonal pyramid of the first order upon the same base. The same kind of relation subsists in one direction CHAP. IV.] Rhomboliedra and Scalenohedra. 97 between the scalenohedron and its holohedral form, and therefore, by increasing the value of m in , more acute forms upon the same base may be obtained up to a dihexa- gonal prism P n ; but when m is diminished, the obtuse dihedral angle of the edges (x) increases more rapidly than that of the acute ones (Y) (fig. 125), and becomes 180 when the latter has still a measurable inclination, while that of the middle ones is unchanged. The twelve faces at either end are therefore changed into six, producing a rhombo- hedron, when the value of m is still far in excess of o ; or the inferior limit of the series of the scalenohedron is not the basal pinakoid, but a rhombohedron whose symbol is m(2-n) p n , when that of the scalenohedron is . As , these forms have their middle edges in common, the rhom- bohedron is completely enclosed in the scalenohedron, as in fig. 137. This is known as the rhombohedron of the middle edges. 98 Systematic Mineralogy. [CHAP. IV. There are two other rhombohedra included in any scalenohedron, each having its polar edges parallel to one or other kind of the same edges in the latter form. The first of these, or rhombohedron of the shorter polar edges (fig. 138), is similar in direction to the scalenohedron, and is m(2n i)p represented by the symbol n , while the second, or 2~~ rhombohedron of the longer polar edges (fig. 139), which i the most acute one that can be included, is inverse in m(n+ i) direction, having the symbol n _ ' when 1 - is 2 positive, and vice versa. In this the length of the vertical axis is the sum of those of the other two, as will be seen by comparing the factors measuring this dimension in the three symbols, as n + i = (2 n i) + (2 n). The special values of these three kinds of rhombohedra for the most commonly observed scalenohedra are as follows : Scalenohedra . 3^ *Zf ,5^1 6P $ 2 ' 2 ' 2 ' ~^T Rhombohedra of middle edges . ^, 2 , 3** 9 Rhombohedraofshorterpolaredges 4 ~, SJ? 6_P n_ 222 2 Rhombohedra of longer polar edges Ll! L?, _ 9 J? _ i In the combinations of these four correlated forms, the most obtuse one, the rhombohedron of the middle edges, modifies the polar summits of the scalenohedron, producing blunt three-faced points, the new edges being parallel to the original middle edges, as in fig. 140, which represents the CHAP. IV.] Scalenohedral Combinations. 99 common scalenohedron of calcite, reduced by cleaving away its points, the faces produced by cleavage being those of the unit rhombohedron of the species. The rhombohedron of the shorter polar edges truncates the longer ones of the scalenohedron of the same direction ; and that of the longer FIG. 140. FIG. 141. FIG. 142. polar edges the shorter ones of the inverse scalenohedron, as - p in fig. 141, where the faces of r r r will be those of * ., if the scalenohedron is considered as + 3 P ~ The pyramid of the second order, the three classes of prisms, and the basal pinakoid are not changed in appearance by rhombohedral hemihedrism. In combination, the prism of the first order truncates the middle solid angles of rhombohedra and scalenohedra alike ; in the first case the new faces are triangular planes, p as in -.OO.P (fig. 142), and in the second deltoidal, as in -.coP (fig. 143). When, however, the prismatic edges proper, those parallel to the vertical axis, are apparent, both prism and rhombohedron appear as five-sided figures H 2 i oo Systematic Mineralogy. [CHAP. IV. (fig. 144), representing one of the commonest kinds of Calcite crystals, P. 5 The prism of the second order being the common limit of both rhombohedra and scalenohedra, FIG. 143. FIG. 144. FIG. 14 u'oo 1010 OBO it will, in combination, truncate their middle edges, as in (fig. 145), and . oo />2 (fig. 146), both being observed cases in Calcite. The basal pinakoid truncates the polar summits both of FIG. 146. scalenohedra and rhombohedra, producing either six- or three- sided faces ; the latter case is illustrated in fig. 136. When many rhombohedra and scalenohedra of different values are combined together, the result- ing solid is often of great com- plexity, and its proper symmetry is not always easily seen. In such cases projections of the CHAP. IV.] Naumanris Notation. IOI faces upon the plane of the base are very useful, and are generally to be preferred to perspective figures. Naumanrts rhombohedral notation. As by far the larger number of hexagonal minerals are rhombohedrally hemi- hedral, it is generally convenient to adopt the rhombohedron as the unit form of the series rather than the pyramid, which is less common, and as a rule has its two kinds of faces so unequally developed as to be more properly regarded as a combination of two rhombohedra. It is therefore customary p to write the symbol - as + R, which represents the two rhombohedra deducible from any particular value of the ratio a : c. This may also be derived from the angle over a polar edge in a rhombohedron, or from the supplement of that over a middle edge, by the expression -, sm. 60 where r = the measured angle, a the side of a spherical triangle, corresponding to the angle at the vertex of the right-angled plane triangle whose perpendicular and base are the vertical axis, and a lateral interaxis, or c and a 1 respectively, whence = cotan. a. a But as the lateral axes and interaxes a : a' are in the pro- portion of i : -J^3> when a = i, the required length of the vertical axis will be cotan. a. From any rhombohedron + m R by altering the value m, making the vertical axis longer or shorter, a series of forms of the same kind are obtained, ranging from the basal pinakoid o R to the prism oo R, which, when arranged in IO2 Systematic Mineralogy. [CHAP. IV. order, give a series analogous to that of the pyramid on page 83, or VR...+ R...+R... mR ... ~~ m The most obtuse form of scalenohedron being the rhombo- hedron having the same middle edges, the symbol of the latter will serve to indicate any scalenohedron upon the same base, if another sign be added, marking the number of times that its unit vertical axis is lengthened. This class of symbol has the following forms : , m' R ri, or in which m refers to the vertical axis of the rhombohedron of the middle edges, and n to the same axis in the corre- sponding scalenohedron. These lengths are, however, re- lated to each other in the constant ratio of i : , and 2 n therefore the rhombohedral symbol of any scalenohedron whose dihexagonal notation is '- will be * R , 2 n 2 n and conversely, m' R n' will be m' n' P} '- . n 4- i In this notation the rhombohedra enclosed in any scalenohedron, w ./?;/, are i. That of the middle edges, m R ; 2. That of the shorter polar edges, -- (3 n \) R ; and 3. That of the longer polar edges, (3 n 4- i)R; and the table on page 98 becomes Scalenohedra .... ^32^! 3^2 4^ Rhombohedra of middle) ^> 2 ^ g edges ) Rhombohedra of shorter) r, ~ ^^ ~ polar edges . . . ) Rhombohedra of longer) _/,^_ 7 ^ -g -u polar edges . . . ) CHAP. IV.] Pyramidal Hemihedrism. 103 Pyramidal hemihedrism. This, the third case on page 89, corresponds to the extension of alternate faces in the dihexagonal pyramid, to the obliteration of the adjacent ones on the same side of the basal section, half the original edges in that plane being retained, but no others. The result is a regular hexagonal pyramid, indistinguishable geo- metrically from the holohedral forms, but differing from them in the position of the polar edges, which do not lie in FIG. 147. FIG. 148. FIG. 149. the same planes with the lateral axes or interaxes, but in some intermediate unsymmetrical position. Fig. 147 is the form produced from the left-hand (white or even-numbered) FIG. 151. faces in fig. 148, fig. 149 its horizontal projection, and fig. 150 that of the right-hand form, from which it will be seen that in the first case the greatest length of the basal edges is to the left, and in the second to the right of the 104 Systematic Mineralogy. [CHAP. IV. lateral axes. These are called hexagonal pyramids of the third order ; they have one axis and one plane of hexagonal symmetry, but the original binary symmetry is completely lost. By a similar kind of derivation, prisms of the third order are derived from the dihexagonal prism, but none of the remaining holohedral forms are geometrically affected by this class of hemihedrism. The direct and inverse forms are superposable. The symbols are : for the pyramids and prisms respectively. These forms are only known in combination. Fig. 151 is an example in [~ pz~\ 2 ? . The last form appears only on the right-hand basal solid angles of />, or in the corresponding diagonal zones between P and oo P. Tetartohedral hexagonal forms. Supposing a dihexa- gonal pyramid to be subjected to two different kinds of hemihedrism, the resulting form will have only one-fourth of the full number of faces, or will contain two out of the eight groups in the table on page 89. As three different kinds of hemihedrism are possible, there should be the same number of kinds of tetartohedrism, but of these only two are possible. i. Trapezohedral tetartohedrism : By rhombohedral he- mihedrism the dihexagonal pyramid is resolved into the two scalenohedra containing the groups and - , which, by a further plagihedral development, divide into the four pairs , -, -, 5 i n which, when the faces of the upper H G F E group are even-, those of the lower are odd-numbered, and vice versa. Geometrically this corresponds to taking out the faces of a scalenohedron by pairs adjacent to the alter- CHAP. IV.] Trapezohedral Tetartohedra. 105 nate middle edges, as in fig. 153. The resulting forms (fig. 152 from the white, and fig. 154 from the shaded faces) are called trigonal trapezohedra ; their six faces, which are trapezoids, meet in six polar edges all of the same length, FIG. 152. FIG. 154- and six middle ones alternately longer and shorter, the longer ones being extensions of edges of the scalenohedron. They have the same axes of symmetry as the scaleno- hedron one ternary and three binary but no planes of symmetry, and are therefore not superposable. They are distinguished as right- and left-handed, positive and ne- gative, forms, the latter having reference to the sign of the originating scalenohedron. Thus, if fig. 153 be considered as positive, fig. 154 will be a right-handed and fig. 152 a left-handed positive trapezohedron. The general symbols are, for all four positions, mPn 7 mPn /, mPn or K K' [hkli] if K' be adopted as indicating plagihedral hemihedrism. The first and third and second and fourth of io6 Systematic Mineralogy. [CHAP. IV. these that is, forms of similar direction and opposite signs -are superposable. The hexagonal pyramid of the second order may be regarded as a special form of scalenohedron, having the FIG. 155. FIG. 156. angles of its obtuse, r, polar edges = 180, and its middle edges horizontal ; and therefore if half its faces be taken out by alternate pairs above and below, having a middle edge in common, and the remaining ones be extended, it will satisfy this class of tetartohedrism. The form is a tri- gonal pyramid contained by six isosceles triangles forming a double pyramid, whose base is an equilateral triangle. The symmetry in regard to the axes is therefore the same as in the preceding form : the lateral axial sections as well as the base are planes of symmetry. The derivation of the right-handed form is shown in the horizontal projection (fig. 155), and that of the left-handed one in fig. 156; but there is no distinction required between positive and nega- tive forms, as the first includes both the positive right and negative left trapezohedra, and the second the negative right and positive left ones. The symbols are : K K-' [i 121} and K K 1 {I 2 it}, or - 2 r and ^-^ /. CHAP. IV.] TrapezoJiedral Tetartohedra. 107 These are sometimes written as - - or \ m P 2, signifying that they contain half the faces of the originating form, which, though convenient as indicating the character, are otherwise misleading, as they are not hemihedral forms ; the pyramid of the second order not being susceptible of development into a solid geometrically dis- similar from itself by any of the three possible methods of hemi- drism. & The dihexagonal prism, in the same way, gives rise to two tetartohedral forms called ditri- gonal prisms, contained by three pairs of faces making alternately obtuse and acute angles, the former having the same values as those meeting the lateral axes in the holohedral form, as is seen in the horizontal projection (fig. 157). In combination, they bevel the alternate edges of the unit hexagonal prism. The symbols are : - . oo Pn ccPn , KK' (hklv\ or ?', & The prism of the second order produces two exactly analogous forms known as trigonal prisms, contained by three vertical faces making equal angles with each other, corresponding in position to figs. 155, 156. The symbols // ~ x KK (i 12 o), or 4 4 Pyramids and prisms of the first order do not give any special forms by trapezohedral tetartohedrism. None of the forms of this class ever occur independently or otherwise than in very subordinate combination, and they are almost exclusively confined to one species ; but as that is the most abundant of all minerals, namely quartz, they io8 Systematic Mineralogy. [CHAP. iv. are of considerable interest, fig. 157 a being a characteristic example. It contains R, R> oo P, and two tetartohedra, - - r (s) } and % r (x), which are distinguished as 44 right-handed positive forms on account of their position to the right, and in the case of x, below, the larger rhombohedron R lt considered as direct or positive. In FIG. 157 a. FIG. 157 b. fig. 157 b the same faces occur to the left of R^ and are therefore left-handed positive forms. The trapezohedra are negative when they lie below the faces of the smaller or negative rhombohedron r> or between s and/ 2 in fig. 157 #, or s and / 6 in fig. 157 b, the right- and left-handed character being unchanged. It will be seen from the figures that a face s lies in two diagonal zones of the pyramid or with the faces Ripi and/! r 2 in fig. 157 a, and JS l / 6 , and r^p^ in fig. 157 b^ and that the second of these zones in either case also contains a face, x, of a trapezohedron. No single crystal ever contains both trigonal pyramids, or right- and left- handed trapezohedra of the same direction, as, although a case is known in which both positive and negative trapezo- hedra of the same kind form an apparent scalenohedron, the crystal has been proved by optical tests to be a compound or twin structure. * These, usually known as the rhomb faces in quartz, are remarkable for their brilliancy, whereby they may often be detected in crystals even of microscopic size. CHAP. IV.] Rhombohedral Tetartohedrism. 109 Rhombohedral tetartohedrism. The successive applica- tion of pyramidal and rhombohedral hemihedrism to the dihexagonal pyramid corresponds to the extension of alter- FIG. 158. FIG. 159. nate faces above and below in the scalenohedron, or the groups ^ll and -}-? divide into -, ? -, -, each of which ' G | H E | F G H' E> F contains six faces, either all even- or all odd-numbered. The resulting form is a rhombohedron, whose edges do not lie in any of the principal crystallographic sections, but are oblique to the lateral axes, as shown in the horizontal pro- jections, figs. 158, 159. This is known as a rhombohedron of intermediate position, or of the third order, its symbol being K TT {h k //} , or the whole series, according to Naumann, .rnPnr mPnl mPnr ', mPnl. + .-, + , -., and ; 4 / 4 r 4 ^ 4 r in which positive signs represent the forms derived from the positive scalenohedron, minus signs those from the negative one, and the letters r I indicate whether the right- or the left-hand faces are above or below the middle edges. These, like the ordinary rhombohedra, are all superposable. The hexagonal pyramid of the second order in the same way gives rise to two rhombohedra, whose polar edges lie in the lateral axial planes, or make angles of 30 degrees with those of the ordinary rhombohedron. as shown in figs. i io Systematic Mineralogy. [CHAP. IV. 1 60, 1 6 1. These are said to be of the second order. From the analogy of the preceding, their symbols are : ;; r . and 2 -. 4 / 4 r The dihexagonal prism gives rise to two hexagonal prisms of the third order, oblique to the axes, whose hori- FIG. 161. & zontal projections will be apparent from the contour of the rhombohedron of the same kind previously given, as either one of these will include the two rhombohedra upon the same base originating from the same scalenohedron. These are exactly similar to the same classes of prisms originated by pyramidal hemihedrism. The hexagonal pyramid and rhombohedron of the first order, and both kinds of hexagonal prisms, do not give rise to any special tetartohedral forms of this class. The tetartohedral rhombohedra only occur in combina- tion, and not very frequently. Fig. 161 a is a case observed in an Ilmenite crystal R.&R.2R. J (P2\ The faces of the last form appear only on the left-hand polar edges of R. The impossible method of tetartohedrism is that resulting from plagihedral and pyra- midal hemihedrism, as the first gives forms with even-num- FIG. 161 a. CHAP. IV.] Miller's Notation. 1 1 1 bered faces above and odd below, or vice versa, and the second requires them to be of the same kind above and below. The successive application of the two methods to the dihexagonal pyramid, therefore, leaves only three faces on the same side of the basal section, which is not a sym- metrical crystallographic form. Miller' s rhombohedral notation. In this method, the forms are referred to three axes making equal angles with each other, and having equal parameters. These axes are parallel to the polar edges of the unit rhombohedron of the series, and are therefore, as a rule, oblique to one another. The unit form {i i 1} is_ the basal pinakoid, and contains two faces (111) and (i i i). Their normal is called the axis of the rhombohedron or morphological axis, and corresponds to the principal axis of the hexagonal notation. The unit direct rhombohedron includes the three pairs of faces (i o o), (o i o), (o o i), and the corresponding inverse form those vit'i the indices (221), (121), and (122), which together give the hexagonal pyramid as a combination. The general form h k /, is a direct scalenohedron, and the inverse one with which it combines to form the dihexagonal pyramid, is distinguished as efg, the two being related in the following manner : *=2 (^ + + /) 3^=-+2>&-f2/ /= 2(h + k + /) - 3 = 2k - k+2l or = 2 (h + k + /) - 3 / = 2 h + 2 k - / The unit prism has the faces 211, 121, 112, 211, 121, 112; the prism of the second order, i o i, i~i o, 011, i o i, ii o, 011; and the dihexagonal prism those of the two forms (hko) and (efo). The hemihedral forms are : 1. Asymmetric ahkl, corresponding to the trapezo- hedral tetartohedral forms ; 2. Inclined Khkl. This is the case not recognised as a symmetrical kind of tetartohedrism in the hexagonal nota- 1 1 2 Systematic Mineralogy. [CHAP. v. tion, the faces of the form being all either positive or negative with respect to the principal axis, or the particular class of development subsequently noticed as hemimorphism. 3. Parallel TT hkl, corresponding to rhombohedral tetar- tohedrism in the hexagonal system. For practical purposes, in the calculation and determina- tion of crystals this method is generally preferable to the hexagonal notation, as it dispenses with a fourth index in the symbols ; but for general descriptive purposes it does not so well express the analogy between the hexagonal and tetragonal systems, and it has therefore not been adopted in this work. The student will, however, do well to become acquainted with Miller's notation, as it may probably super- sede the hexagonal form at no very distant date. A simple exposition of it will be found in Gurney's elementary treatise on Crystallography. CHAPTER V. ' TETRAGONAL 1 SYSTEM. THE complete symmetry of this system is contained in an upright prism upon a square base, which has quaternary symmetry about a principal axis parallel to the vertical edges, and binary about four lateral axes, respectively parallel to the sides and diagonals of the base. These cor- respond to five planes of symmetry a basal, or principal plane, and four lateral planes at right angles to the first and at 45 to each other. The reference axes are three, at right angles to each other namely, the vertical or principal axis, c, and two of the four lateral axes, a l9 a 2 those parallel to the diagonals of the base, their order being similar to that of 1 Other names are Pyramidal, Dimetric, Quaternary, Quadratic, and Viergliedig or four-membered. CHAP. V.] Ditetragonal Pyramid. the cubic system. The parameters of the lateral axes are similar, and different from that of the vertical axis, the two being related in the proportion of some arbitrary ratio a : c, proper to the species, which has therefore the same signifi- cation as in the hexagonal system. The general symbol of a face having different intercepts upon the three reference axes, corresponding to different inclinations upon three planes of symmetry, none of which is a right angle, is as in the cubic system (h k I) with the difference that only the first two indices are interchangeable, giving two permutations of letters, hk,kh, and four of signs, + 4-, H , H, , for each value (positive or negative) of /, or sixteen in all as the maximum number of faces possible in a simple tetragonal form. This, known as a ditetragonal pyramid, represented in elevation with its FIG. 162. FIG. 163. symmetrical sections shaded in the manner described on page 77, in fig. 162, and in plan in fig. I63, 1 is a double pyramid, contained by sixteen faces whose dihedral angles in the eight basal edges are all similar, while those in the polar edges, and the corresponding plane angles of the equi- 1 This is noted as I 1 1 4 Systematic Mineralogy. [CHAP. v. FIG. 164. lateral eight- sided base are alternately larger and smaller. The general symbols are : \hkl\y (a \na\ mc\ and mPn. The pyramid, whose base is a regular octagon, and has the angles of its polar edges all equal, is an impossible form, as requiring for n the irrational value tan. 67^, or 2*4142, but for any rational value lower than this, the more obtuse polar edges lie in the interaxial planes, and when n = i or k = h, their angle becomes 180, or the two faces meeting in these planes coincide. This corresponds to the tetragonal or square-based pyramid of the first order, or normal position (figs. 164, 165) [hhl], (a : a : me), and mP } having eight faces meeting at equal angles in the four basal edges and in the eight polar edges which lie in the lateral axial planes, at some other angle whose difference from the first depends upon the disparity in length between the vertical and the lateral axes. The basal angle, -ft corresponds to twice the plane angle between the vertical axis and a lateral interaxis a 1 , and as the length of the latter is to that of the adjacent lateral axis a, inclined to it at 45, as i : i the fundamental V2* ratio a : c for any pyramid, assumed as the unit of the series, may be determined by the expression tan. tan. L- = . and c = -r~- when a =.- i. 2 a' v 2 . = -~ when the measured angle is that over a polar edge = -, the CHAP. V.] Tetragonal Pyramid. 115 fundamental parameter of c is found by computing the side p opposite to that angle in a right-angled spherical triangle described about the pole of the principal axis, when cotan. - = cos./, and cotan./ = c when a= i. When the value of n in the symbol of the ditetragonal FIG. 165. FIG. 166. FIG. 167. pyramid exceeds tan. 67^, the more obtuse polar edges lie in the lateral axial planes, and when n = oo or k = o, the angle becomes 180, or the adjacent faces meeting in them fall into the same plane, giving the tetragonal pyramid of the second order or diagonal position (figs* 166, 167), whose basal ed^es are equal and --= parallel to the lateral crystallo- graphic axes, while its polar edges lie in the lateral interaxial planes. The symbols, as will be apparent from its derivation, are \h o /} , (a : GO a : mc\ and The relations of these three classes of pyramids are similar I 2 u6 Systematic Mineralogy. [CHAP. V. FIG. 168. to those subsisting between the allied forms in the hexagonal system, the difference being in the possible value of n, which ranges from i to GO instead of merely from i to 2. From any pyramid of either kind upon the same base, by multiplying c by any rational quantity m, greater or less than unity, a series of new pyramids of varying altitude is obtained, as in fig. 168, which may be noted as P, 2/> 3/>{iu}, {221}, {3 31}, or $P,$P.P. {113}, {223}, {in}, or P,P,%P, {112} {in} {332}, according as one or other of the three is adopted as the unit form. The basal angle increases with the altitude in these forms, and when m = or / = o it becomes 1 80, or they change to vertical prisms of unlimited height. There are therefore three prisms, one corresponding to each kind of pyramid, namely 1. Ditetragonal prism, {hko}, (a : na : coPn (fig. 169). 2. Tetragonal prism of the first order, {h h o} = {t i o}, (a : a : ooP'~mPn... In combination, forms of the same order will appear in the succession shown in the vertical lines of the above table, the steepest, P....QP. OP m m Pn. ...oojPoc Systematic ", finer alogy. [CHAP. V. the prism, being in the middle, and the flattest, the terminal FIG i?2 pinakoid, at the ends: those of -in- termediate inclination, the pyramids proper, being arranged in regular order towards either of these limiting forms, the steeper truncating the basal edges of the flatter ones, and conversely the latter truncating the polar summits of the former, as in fig. 172, which is noted The pyramid of the second order truncates the polar edges of that of the first order, when both are of the same altitude, or have m in common, as seen in elevation and plan in figs. 173, 174, which also illustrate the converse case of the faces of the pyramid of the first order bevelling the solid FIG. FIG. 174. angles of that of the second order. When the two forms differ in the value of m, if m Pec is steeper it will bevel the basal solid angles of P, as in P, 2 P GO, fig. 175 ; but when m Pec is the flatter form, it will truncate the polar edges of P obliquely, forming four-faced points upon the polar summits, as in P,%P&, fig. 176. A ditetragonal pyramid, mPn, will have its basal solid angles in the lateral interaxial planes bevelled by the faces of a pyramid of the first order of the same altitude, or will bevel the polar edges of the latter, as in P, />3, fig. 177 ; and similarly. the pyramid of the second CHAP. V.] Holohcdral Tetragonal Combinations. 119 order bevels the other basal solid angles, or those in the lateral axial planes of m Pn. A ditetragonal pyramid forms eight-faced: points upon the principal summits of a steeper tetragonal pyramid of either order, as in P%, P$, fig. 178 ; FIG. 175. FIG. 176. when the latter is the obtuser form it reduces the same summits of m Pn from eight- to four-faced ones, as in 3 P$, Pec, fig. 179. This class of combination is of considerable interest, as when a : c approximates to 2 : i, the solid repre- FIG. 177- FIG. 178. sented by 4^2 Por (4 21} {i i i), is almost indistinguish- able from the trapezohedron 221} of the cubic system. This case actually arises in Leucite, where a : c=i : 0-5264, whose crystals, until recently held to be the most typical examples of the particular trapezohedron in question, have been shown to be probably assignable to a tetragonal com- bination of the character of fig. 179. One of the simplest cases of a combination of tetragonal prisms and pyramids is I2O Systematic Mineralogy. [CHAP. V. shown in figs. 180, 181. It contains oo P, oo Poo, P t and is a common form of crystal in Apophyllite. As in the hexagonal system, the number of species with full tetragonal symmetry is comparatively small, but among FIG. 179. FIG. 1 80. these one, namely Idocrase, is remarkable for the large number of forms that are sometimes combined in single crystals. Hemihedral tetragonal forms. The faces of the dihexa- gonal pyramid, when arranged in the following order : A B C D i. hkl v. hkl n. khl vi. khl HI. khl vn. khl IV. hkl viii. hkl E F G H ix. hkl xiii. hkl x. khl xiv. khl xi. khl XV. khl xn. hkl xvi. hkl where the Roman numeration corresponds to that in fig. 182, may be symmetrically halved in three ways, giving rise to three kinds of hemihedral forms analogous to those of the hexagonal system, the only difference being that each group contains two instead of three faces. CHAP. V.] Tetragonal Hemihedrism. 121 Trapezohedral hemihedrism. The first method of selec- tion, that by alternate groups both above and below, or the FIG. 182. arrangements - c and , the former containing faces which are all odd-numbered above, and all even-numbered FIG. 183. FIG. 184. FIG. 185. below, while in the latter the order is reversed, gives the forms figs. 183, 184, the first originating from the white, and the second from the shaded, faces of fig. 1 84. These, known 122 Systematic Mineralogy. [CHAP. V. as tetragonal trapezohedra, have eight faces, meeting four equal polar edges, at either end of the principal axis, and eight unequal, alternately longer and shorter, middle ones in zigzag order about the basal section. From these figures and their horizontal projections (figs. 1 86, 187), it will be apparent ist, that as none of these edges lie in planes of symmetry, the forms are plagihedral ; and 2nd, that as the principal solid angles are formed by the meeting of four similar edges, and the lateral axes and inter- axes bisect the middle edges, the number and kinds of axes of symmetry are the same as in the holohedral form, namely, one principal or quaternary, and four binary. The symbols are : a \hkl} {/*/}, and m J*JL r No other holohedral form than the ditetragonal pyramid gives hemihedra of this class distinguishable from itself, and it is not known whether this particular kind of hemihedrism actually occurs or not. Its existence has been inferred upon physical grounds in a few salts of organic bases, the most pronounced example being a sulphate of strychnine, but no- actual plagihedra of the above kind have as yet been ob- served with certainty. Sphenoidal hemihedrism. The second method of selec- tion, that by alternate pairs of faces both above and below, gives the arrangements and , the first contain- G I H E J F CHAP. V.] Tetragonal Scdlenohedra. 123 ing odd-numbered (ist and 3rd) pairs above, and even-num- bered ones (6th and 8th) below ; and the second the 2nd and 4th pairs above, and the 5th and 7 th below, correspond- ing to the rhombohedral selection of the hexagonal system. When applied to the ditetragonal pyramid it produces the forms figs. 1 88, 190, the former from the white, and the latter from the shaded, faces of fig. 189. These, known as tetra- gonal scalenohedra, are contained by eight faces meeting FIG. 188. FIG. 189. FIG. 190. in three prominently dissimilar kinds of edges. Each of the principal solid angles is formed by two longer and two shorter polar edges ; the former are those of the holohedral form, and have the same dihedral angles ; the third kind, or middle edges, those of the zigzag middle belt, represent alternate middle edges of the tetragonal trapezohedron, and are bisected by the lateral crystallographic axes. On the same side of the base the longer and shorter polar edges lie alternately in planes at right angles to each other ; but on opposite sides they are in the same planes, that is, in the lateral interaxial sections, which are therefore the only planes of symmetry. The axial symmetry is binary about the three crystallographic axes. 1 24 Systematic Mineralogy. The symbols are : [CHAP. V. K {hkl} , and + mPn m P -n : the direct and inverse form of the same origin being super- posable by rotation through 90 about the principal axis. The tetragonal pyramid of the first order in the same way gives rise to figs. 191, 193,' the first from the white, and the second from the shaded, faces of fig. 192. These, known FIG. 191. FIG. 192. FIG. as tetragonal sphenoids, from their wedge-like appearance, are obviously only special cases of tetragonal scaienohedra, having the obtuse polar angles = 180 ; but as they are more frequently met with than the latter forms they are con- sidered as most characteristic of this kind of hemihedrism, which is therefore called sphenoidal. The shorter or hori- zontal edges represent the shorter polar edges of the sca- lenohedron, and the longer ones, which are parallel to the ; These, as well as figs. 188-190, are on a smaller scale than the holohedral form. CHAP. V.] Tetragonal Sphenoids. 125 axes of the diagonal zone in the pyramid, the middle edges in the same form. The symbols are : c {khl} {/*/$/}, and +--^ - m -~. Like the scalenohedra, they are superposable, and have the same planes and axes of symmetry. The remaining holohedral forms are not changed by this hemihedrism. The wedge-like character is not apparent in the forms generally observed, as they mostly originate from pyramids which are either very obtuse or approximate to a regular octahedron, in which latter case the sphenoid is very similar to a regular tetrahedron. This is seen in copper pyrites, which is the only characteristic example of this hemihedrism among minerals, the combination of the two P P sphenoids - - (fig. 194), being very like a slightly dis- torted regular octahedron. This, however, is only one out of many sphenoids found in the same mineral. The flatter ones have the more characteristic combinations shown in fig. 195, where the faces of one sphenoid truncate the edges between oo P and o P alternately above and below. Neither FIG. 194. FIG. 195. sphenoids nor tetragonal scalenohedra ever occur indepen- dently, and the latter when present truncate the solid angles 126 Systematic Mineralogy. [CHAP. V. between the prism and sphenoid faces obliquely, but the faces are usually very small. Pyramidal hemihedrism. The third method of selection, that by alternate pairs of faces adjacent to the same basal edge, gives the arrangements ^-L5 and ? - E , the first con- EG F I H taining all even-numbered faces, and the second all odd- ones, corresponding to two tetragonal pyramids, which are similar in form to the holohedral ones, but not in position ; the sections upon planes passing through the polar edges lying in one case to the right, and the other to the left of the lateral axial and interaxial planes of symmetry, as seen in the horizontal projections, figs. 196, 197, which also show that the basal edges are half those of the ditetragonal pyramid ; the symmetry is, therefore, to a single plane the base, and quaternary to a single axis the principal one. These forms, which are superposable, are known as tetra- gonal pyramids of the third order, or of intermediate posi- tion, the symbols being : , {hkl} , {^/J.and + The ditetragonal prism in the same way gives two prisms of the third order, whose basal sections are the same as those of the corresponding pyramid. They have the symbols : CHAP. V.] Tetragonal Tetartohedra. 127 The above are the only special geometrical forms pro- duced by this hemihedrism ; they FlG I98 do not occur independently, but only in combinations, and are characteristic of a small but well- defined group of minerals, the Tungstates and Molybdates. One of the simplest examples is given in fig. 198, a crystal of molybdate of lead, in which a tetra- gonal prism of the third order ^{430} or [^ ~^\ trun- cates the basal edges of the unit pyramid obliquely to the right hand of the lateral axes. Tetartohedral tetragonal forms. The faces of the ditetra- gonal pyramid may, by the successive application of two kinds of hemihedral selection, be divided into four symme- trical groups, giving, as in the hexagonal system, two possible classes of tetartohedra, corresponding to the following divisions : Hemihedrism . . i. n. in. iv. Plagihedral and sphenoidal -, -, -, Sphenoidal and pyramidal . , - , -, The faces to which these correspond will be seen in the table on page 120. The forms corresponding to the first of these divisions are sphenoids, differing from those derived by hemihedrism from the tetragonal pyramid of the first order, in the position of their horizontal polar edges, which do not lie in planes of symmetry, but cross each other obliquely, so that the faces are scalene instead of isosceles triangles, and are not sym- metrical to any principal section, while preserving the same axes of symmetry as the sphenoid, the relations in the latter respect being similar to those between the hexagonal rhom- 128 Systematic Mineralogy. [CHAP. vi. bohedra and plagihedral tetartohedra. The pyramid of the second order in the same way becomes a horizontal prism, whose section is the rhomb, having for its diagonals the vertical and a lateral axis. The dketragonal prism gives others of rhombic sections, whose diagonals are in the propor- tion of a : na, and the prism of the second order gives two parallel pairs of faces. By the second method the ditetra- gonal pyramid produces sphenoids of the same geometrical properties as the hemihedral ones, but differently placed with respect to the axes, and which, like the analogous tetarto- hedral rhombohedra, may be said to be of the third order. The pyramid of the second order gives sphenoids of the second order, having their horizontal edges parallel to the lateral axes ; and the ditetragonal prism gives tetragonal prisms of the third order. None of these tetartohedral forms have , as yet been found either in natural or artificial crystals ; but they are interesting as geometrical possibilities, and as showing the complete analogy subsisting between the hexagonal and tetragonal systems in all their modifications. CHAPTER VI. RHOMBIC * SYSTEM. THE forms of this system are referred to three rectangular axes, whose parameters are all different. The three princi- pal sections are planes of symmetry as in the cubic and tetragonal systems, but, on account of the dissimilarity of the parameters, the symmetry about the axes is only binary. These properties are apparent in a vertical prism of definite height upon an oblong rectangular base, whose length, breadth, and depth are all different, which is only symmetrical to its faces and about its edges, and the lengths of the latter are proportional to the parameters. In the 1 Other names are, Orthorhombic, Orthoclinic, Prismatic, Tri- metric, Terbinary, and Zweigliedrig. CHAP. VI.] Rhombic Pyramid. 129 general symbol h k I the order of the indices is invariable, and the different faces are represented by sign permutations only, giving eight as the largest number of faces that can appear in any simple form. The forms represented by h k k and h hh are also of the same geometrical character, and as the crystallographic elements are most readily deduced from these more special kinds, it is convenient to consider them first rather than the general form. The unit of any series is a rhombic pyramid or octahedron, such as fig. 199, which, if represented by {i i 1} a : b : c or P, has the para- meters of OA a ^ O = b = i.o, and OC=c = 8. FIG. 200. The principal sections are all rhombs of different propor- tions, their deviation from a square form increasing with the disproportion of the parameters. It is customary to place the form, so that the longitudinal axis, a, is the shorter diagonal of the basal section and the right and left one, b, the longer, and to call the first the brachy diagonal and the second the macrodiagonal axis, the third vertical axis being noted by c as in the preceding systems. To express the ratios of the parameters, one of them, usually that of the macrodiagonal, is put = i: thus, in fig. 199, a : b : c = 0*6 : 1*0 : 0*8; but if, as some authors prefer, the brachy- K 130 Systematic Mineralogy. [CHAP. VI. FIG. diagonal is considered as the unit a : b : c = i : i '666 : i '333. ! The choice of position in a rhombic pyramid is entirely arbitrary, as there are no peculiarities, either morpho- logical or physical, making any one axis a principal one, so either one may be made vertical or horizontal at pleasure. Thus, turning fig. 199 about a until b is vertical gives fig. 200, when 10 = 075 : i : 1-25; and turning it about b gives the position of fig. 201, where the shortest axis ia vertical and a : b : l {110} oo a : oo b : c a:b: c a : b : c a : b : me a : b : aoc m oP P P mP P m Any pyramid of a principal series will, by successively lengthening its brachy diagonal axis by any rational multiples n. produce new pyramids all of the same height and length on the axes a and c, but increasing in breadth on b as n becomes greater, as seen for 2b and -$b in fig. 203; and when n = oo the form is a rhombic prism, whose edges are hori- zontal and parallel to the axis , the axes a and c being the diagonals of its base. Prisms of this kind are called domes or domas, from their resemblance to house roofs, and the K2 132 Systematic Mineralogy. [CHAP. VI. particular one in question is known as the macrodome, its zone axis being the macrodiagonal. If this is derived from FIG. 202. FIG. 203. the unit pyramid, it will be the unit macro- dome; but as the same development applies equally to any of the pyramids of the prin- cipal series, they will give a corresponding succession of macropyramids and limiting macrodomes, the whole constituting the transverse prismatic or macrodiagonal series represented by Naumann's general symbols mPn,mPao, where the long sign signifies that n applies to the longer lateral axis. This is more directly indicated in Breit- haupt's modification of the symbol, m Pn, which is therefore preferred by some writers, although the first form is most generally used. The order of symbols for the principal types of this series is as follows : {hkl} h>khk {hkl}h>k n< a : nb : c a : nb : c a : nb : me m J-Pn m a : QO b : m m Pn mPn {hoh} {hoi} h>l a : b : c a : & : me m The extension of the brachydiagonal of any unit or CHAP. VI.] Bracky diagonal Series. 133 other pyramid of the principal series as in fig. 204, where that axis is successively made 2 a, 3 a, oo a, while b and c are unchanged produces a third class of pyramids and domes known as the longitudinal or brachydiagonal series, which are distinguished in their symbols from the transverse series by the sign w, indicating the shorter lateral axis, placed over the characteristic POT its coefficient, thus m Pn or m P n. The order of the typical symbols is as follows : {hkl} h2 The prism of the principal series having its lateral axes in the unit proportion a : &, will, by lengthening either of 1 34 Systematic Mineralogy. [CHAP. vi. these axes relatively to the other, give rise to new prisms, whose characteristic angle increases with the value of n. These are called macroprisms, when derived from the varia- tion of ^, and brachyprisms from a, which are respectively represented by Pn and ccPn. When n in either series becomes oo, the angle of the prism is 180, or it is reduced to a single plane parallel to one lateral axis and perpen- dicular to the other, giving the two forms known as the macropinakoid and brachypinakoid ; the former is repre- sented by {i o o} a : oo b : QO ^ or oo Pec, and the latter by {010} oo a : b : oo c, or oo ^oo. The above include all the simple forms possible in the rhombic system, namely, pyramids of eight faces, the only closed forms, prisms of four faces in three positions, and the three pinakoids of two faces. Their relations to each other may be shown by arranging their symbols in a diagram of the kind given in the preceding systems. Here the unit oP oP. oP.....oP oP form Pin the m m : m Pn. centre has the obtuser forms of m : m\ the principal series below and ...Pn Pco the acuter ones : _ above it in the mPca...mPn mP...mPn m ^oo same vertical line. The lines next to right and left contain the series of the macropyramids and brachypyramids respectively, and the first and last lines, the brachydomes and macrodomes. The top horizontal line contains the symbol of the basal pinakoid as the common limit of all the series ; the other pinakoids are at the ends of the lower horizontal line, the intermediate positions being taken by the different prisms. As in the previous system, the forms contained in any line, whether horizontal or vertical, lie in the same zone. CHAP. VI.] N N M N N N M ^^^ KIM N CO Rhombic Notation. 135 vo rl- co N ^ N W C* N N >O N VO M _, _ M CO HH CO si si si <> si si Q *3 KIN KIN Kin KIM -v, K;N KID vO C> co N M 4- N KlN KIN K.N KiN KIN KIN KlI . O O O o O O O ^ 8888888 o '5 o ^ V. Vj Vj Vj Vi Vi *o Si Si Si Si Si Si Si ^a "55 COMCO>-IN^ M 'C "-t>-<^i_irONcoi-<'C ^ & ^^^,r,^ CO &, c^ H-+?1(N(N COIN N CO "ON vo"o\ "CO "cT "cO """ CO CO >O CO co vO CO 8 D JO JO >j^ JO >C >O 3Q_. 8 HwH&iNlw KIN c< co M -S CO co'O co si Si si Si si si Si CO CO HN H^ >H W ^ L, ^ -^- >-^ vT x* ^ ^ ^ ^ 8O HH, oo Poo ; this is drawn to the parameters of Brookite. Another consequence of the absence of diagonal axes of symmetry is the tendency to elongation parallel to one axis, producing solids which are essentially prisms, a peculiarity FIG. 212. FIG. 213. 001 181 101 which is indicated in the name ' prismatic ' applied by Miller to this system; and as the extension may be along either axis indifferently, the same combination in the same substance may appear in many different shapes, according as one or other form predominates. Thus in fig. 210 the three unit prismatic forms (prism and domes) are combined in about CHAP. VI.] Rhombic Combinations. 139 equal dimensions with the three pinakoids ; in fig. 211 the solid is essentially prismatic, the vertical edges being the longest. In fig. 212 the character is longitudinal, prismatic or brachydomatic, the greatest length being parallel to a ; and in fig. 213 it is transverse-prismatic, or macrodomatic, the longest edges being parallel to b. These variations are quite possible, and are to some extent represented in the species Barytes. FIG. 214. FIG. 215. A few more illustrations of the simpler class of combina- tions are given in figs. 213-223, from the closely allied species, Barytes and Anglesite, which are remarkable for their FIG. 216. FIG. 217. \ great variety of forms, these being selected from nearly a hundred described crystals of these minerals. Fig. 214 is the unit macrodome of Barytes, P oo, shortened in the direc- 14 Systematic Mineralogy. [CHAP. VI. tion of its axis by oo P oo, and truncated in its middle edges by oo Pec. Fig. 2 1 5 is similar, with the addition of ^ Pec and o P. Fig. 2 1 6 is Poo, o P, with the middle solid angles trun- cated obliquely by the prism oo P, and the upper and lower FIG. 218. FIG. 219. ozi ones by the brachydomes Pec, 2 Pec, 4 P oo . Fig. 217 contains 2 Pec, 4^j ^, elongated parallel to a, and limited by the prism ecP. In fig. 218, the upper edges be- tween Pec and ecPec are truncated obliquely by the acute brachypyramid 3 P$, and those between ooPand ecPec in front by a more acute macrodome 2 ^oo. Fig. 219 is the FIG. 220. FIG. 221. IK 310 pyramid 2 Pof Anglesite, with its macrodiagonal polar edges truncated by 2 Pec. Fig. 220 is a more acute pyramid, 4 P CHAP. VI.] Rhombic Combinations. 141 with the same brachydome, and the unit macrodome modify- ing the brachydiagonal polar edges. Fig. 221 contains oo P, P, 2.Ajo, oo^oo, and 2 P t the latter modifying the solid angles formed by the meeting of the three prismatic forms. That this is one of the pyramids of the principal series is apparent from the horizontality of its edges of combination with the prism. In fig. 222 the faces of the macrodome 2 fee are so FIG. 222. FIG. 223. proportioned as to form rhombic planes truncating the front solid angles between ccP and P. Fig. 223 is of the same general character, but the basal edges of the prism are FIG. 224. FIG. 225. V 702 10X modified by 2 P ; and the lateral solid angles formed by P, oo P, and ccPcc, are replaced by an acute brachy pyramid 142 Systematic Mineralogy. . VI. FIG. 226. 2P2, whose edges of combination are parallel to the macrodiagonal polar edges of the pyramid. Fig. 224 is the unit macrodome Pec, lengthened parallel to its axis, limited laterally by 2 ^oo, and its middle edges bevelled by J Poo. Figs. 225, 226 are examples of simple combinations of Oli- vine. The first is GO p2, oo Poo, 2 Pec, and the second ecP2, aoPco, Pec, the latter being very commonly observed in crystallised slags obtained in puddling and heating fur- naces. These examples will suffice to show the general character of the com- binations of this system, but they are only of simpler kinds. For those of more complex character the reader is referred to the larger special memoirs and descriptions, especially to Schrauf's atlas of crystalline forms. The fundamental parameters of any rhombic series of crystals, being irrational numbers, they may, when two are nearly equal, produce forms approximating in character to those of the tetragonal system, but the true nature will usually be apparent by their modi- fications. Fig. 221, for instance, would be nearly like the common combination Pec P in Tinstone but for the rhombic faces of 2 Pec. Where the parameters of the axes a and b are related to each other in the proportion of i : \/3. the obtuse angle of the prism will be 120, and as the brachypinakoid truncates its acute angles, the combination of these two forms will be an equal six-sided prism, having all its angles of 120, or geometrically identical with the unit prism of the hexagonal system ; and in like manner any pyramid of the same series combined with a FIG. 227. CHAP. VI.] Rhombic Hemiliedrism. 143 brachydome of twice the height i.e., /> with 2 Pec, or 2 P with 4^00 will produce a regular hexagonal pyramid. Fig. 221 is an example of such a combination in Witherite, which is very similar in appearance to the ordinary form of quartz crystal. Other examples are afforded by Ara- gonite and Bisulphide of Copper, where the prismatic angle approaches very nearly to 120. Such forms may, however, as a rule, be discriminated without much difficulty by the unequal modification of their edges, peculiarities of cleavage, &c., and, when transparent, by their optical properties. The solid formed by the combination of the three pinakoids may also, in some instances, appear very like a cube ; the* best example of this is afforded by Anhydrite, or anhydrous Sul- phate of Calcium. Hemihedral rhombic forms. A rhombic pyramid may, by the omission of alternate faces right and left, above and below the base, give rise to hemihedral forms analogous to the tetartohedral sphenoids of the tetragonal system, as FIG. 228. FIG. 229. FIG. 230. shown in figs. 228 and 230, the first being that derived from the white faces, and the second from the shaded ones in fig. 229. These differ from the tetragonal sphenoid by the inequality of the middle edges, two being obtuse and two acute ; and also their polar edges do not lie at right angles to each other, but cross obliquely, the angle between 144 Systematic Mineralogy. [CHAP. VI. their horizontal projections corresponding to the acute angle of the prism. As in the case of the tetrahedron and tetra- gonal sphenoids, they are not symmetrical to rectangular axial planes ; and these being the only possible planes of symmetry in the system, they are plagihedral, being per- manently right- and left-handed, according to their origin. The symbols are : FIG. 231. and + and - . 2 2 The prismatic forms are not affected geometrically by this kind of hemihedrism, which is not of very frequent occurrence in natural crystals. The principal examples (fig. 231) are found in Sulphate of Magnesium and the isomorphous Sulphate of Zinc. There is another class of hemihedral forms possible in the rhombic system, but only a single example has been demonstrated as existing in an artificially crystallised com- pound. These are analogous to the scaleno- hedral or rhombohedral forms of the pre- ceding systems, being produced by the alternate development of pairs of faces preserving one of the original edges. The result is the production of a prism upon the same base as the pyramid, whose edges are inclined instead of being 110 110 FIG. 232. FIG. 233. vertical or horizontal, the inclination being similar to that of the edges retained. For instance, fig. 233 is the oblique CHAP. VII.] Oblique Symmetry. 145 rhombic prism of this kind produced from the faces i., iv., vi., and vii., in fig. 232 ; its edges are parallel to the more obtuse polar edges in the pyramid, and its faces are sym- metrical to the brachypinakoid alone. The lower front and upper back pairs of faces would produce a similar form, with a forward slope, but symmetrical to the same plane, and in like manner from the extension of alternate pairs of basal and macrodiagonal edges, pairs of similar prismatic forms may be derived, having the same symmetry to one axial plane only. From this latter circumstance, such forms are said to have monosymmetric hemihedrism, a property which they have in common with all other forms of the next system. Formerly several minerals were referred to this type of hemihedrism, but they are now, upon structural considera- tions, placed in the oblique system. CHAPTER VII. OBLIQUE 1 SYSTEM. THE forms of this system are referred to three axes having dissimilar parameters, one being at right angles to the other FIG. 234. c two. If this be considered as the axis of breadth, and placed horizontally, as B B in fig. 234, it will be normal to 1 Other names are clinorhombic, monoclinic, oblique-rhombic, binary, monosymmetric, and zwei und eingliedrig. 146 Systematic Mineralogy. [CHAP. VII. an upright "longitudinal plane containing the other two axes, which may be oblique to each other, and parallel to two others, in each of which it will be at right angles to one of the remaining axes. The solid whose edges are shown by the fine dotted lines, which may be supposed to represent that derived from the unit .parameters, will therefore be rhombic in two of its principal sections, and rhomboidal in the third, as in the three plane projections, figs. 235, 236, 237, which are similarly noted to the preceding one. From these it will be seen _that the rhom- boid ACAC, fig. 237, di- vides the solid symmetrically into right and left halves, whether it be looked at from the front, as in fig. 235, or from above, as in fig. 236 ; while in fig. 237 the division by the other two planes into right and left and upper and lower halves are both unsymmetrical. This property of symmetry to a single longitudinal plane is the most essential character of the system, and there is usually a marked obliquity be- tween the axes in that plane, which as a necessary crystallo- CHAP. VII.] Oblique Hemipyramids. 147 graphic element, in addition to the parameters #, , c, is ex- pressed as the angle /3. When, as is generally done, one of these axes is placed upright, and the third with a forward inclination, as in fig. 234, (3 is considered as the acute angle in front below, or on the negative side of the vertical axis ; the supplemental obtuse angle (180 /3) being on the posi- tive side above, which positions are reversed behind. In this order the axes a and b are as in the preceding system diagonals of a rhombic section ; but the former is inclined, while the latter is horizontal to the third axis c. They are therefore distinguished as clinodiagonal and orthodiagonal axes. In the arrangement of the parameters that of b is con- sidered as unity, but it is not necessarily longer than that of a. The solid under consideration is geometrically an oblique rhombic pyramid or octahedron, but it is not a simple crys- tallographic form, being made up of two dissimilar classes of faces marked by longer or shorter edges in the plane of symmetry, according as they face the obtuse or acute angle of the inclined axis, and either set may occur in combination with or without the other. It is therefore to be considered as contained by two half pyramids or hemipyramids, one having the faces n., in., v., and vni., as in fig. 234, opposite the acute angle, and the other i., iv., vi., VIL, facing the obtuse angle of the axes. For the general forms represented by three dissimilar finite indices, the two groups are hkl ^ hkl\hkl where the stronger letters indicate faces in front of the ortho- diagonal section. Naumann calls these positive and negative hemipyramids, or +/>, and P^ and considers the former as that facing the acute angle of the axes, which conven- tion has the inconvenience of throwing the face whose indices are all positive into the negative form ; but as it is L 2 148 Systematic Mineralogy. [CHAP. VII. that most generally followed, it will be adopted in the follow- ing pages. The notation of the unit forms is therefore : 1 1 1 I 1 1 1 1 1 i 1 1 1 + P = iiiliii 1 1 1 i 1 1 1 Weiss's symbols are (a : b : c) and (a' : b : c) respectively. From the unit hemipyramids P we may, by keeping a and b constant and altering the value of c, obtain other forms, which will be flatter or steeper according as the values assigned are greater or less than unity. These are the hemi- pyramids of the principal series ; the forms are represented by the symbols : -P, ( a:b\~ : A or [hhl] where (h /). When in the forms P the axes b and c are kept constant, and a is lengthened, a new series known as clinodiagonal hemipyramids are obtained, represented by the symbols : + ;P/2, (na \ b \ c\ or {hkl} where (h < k). Here the oblique axis is indicated by the inclined bar in the letter P. The same system of derivation applies, how- ever, to any of the forms of the principal series, P, so that the general symbols of any hemipyramid of the clinodiagonal series are : it m jP/z, (na : b '. me) or {h k 1} A third series, as in the rhombic system, is obtained by varying the length of the orthodiagonal, or axis of symmetry b, a and c being unchanged. These are the hemipyramids of the orthodiagonal series, whose symbols are for the forms derived from Hh P : nh Pn, (a : nb : c\ or {hkh} where (h > k), CHAP. VII.] Oblique Prism. 149 and for those derived from any other of the forms + m P : m Pn, (a : nb : me) or {h k /} where the straight bar and the stem of the P signifies that the orthodiagonal is the axis modified. The geometrical character of these forms will be generally analogous to those similarly produced in the rhombic system, the difference between the two hemipyramids being remembered. When, however, m = GO or / = o, the edge lying in the plane oif symmetry becomes vertical or parallel to c for either hemi- pyramid, and a prism is produced, which is known as the primary vertical prism, with the symbols oo P, (a : b \ c), and {no.} This is only distinguishable from a rhombic prism by the circumstance that the diagonals of the rhomb forming its horizontal section are not the axes b and a, but the orthodiagonal and the horizontal projection of the clino- diagonal, and therefore the fundamental ratio of the axes lying in the basal section cannot, as in the rhombic sys- tem, be determined from a measurement of the angle of the prism alone, a knowledge of the characteristic angle /3 being required in addition. From the clinodiagonal hemi- pyramid m 3?n by a similar method clinodiagonal prisms, oo fn (na : b : co K). The first of these in combination modify the right and left edges of the primary prism, and the second those in the front and back plane, the angles being more obtuse as the value of n increases. In the series of clinodiagonal hemipyramids, Pn> the angle between the edges in the plane of symmetry and the clinodiagonal axis becomes more acute as the value of n is increased, and when it becomes oo, the four faces of either hemipyramid become parallel, forming an inclined rhombic prism, whose edges are parallel to that axis. This is 1 50 Systematic Mineralogy. [CHAP. VII. known as the principal clinodome jPoo = (oo a : b : c)= {01 1} . In like manner a more obtuse hemipyramid + f> gives ~~ m rise to a flatter clinodome -LaPoo = fee a : b : c ) = m \ m J {o k 1} where (k < /), and a more acute one m jP to the steeper form mcc= (cca : b : me) = ( & : c) = {101} . Flatter fonns + L I>>=(a< : <*b : L (h < /) . m \ m Steeper forms + m fee =(&' '. oo b : m c) = (h o /} . (h > I) Negative hemidomes. Principal ' fee = (a : co b : c) {101} . Flatter forms _JL / > OD== /' fl : aol : <} = {Ao/}. (*' 2, and an ortho- prism ccP2. Fig. 241, a combination of a nearly rhombic character observed in Caledonite, is, however, distinguishable CHAP. VII.] Oblique Combinations. 155 by the hemi-orthodome + ^ Pec, which appears under the unit hemi-orthodome + P in front below, but not above. Fig. 242, the common form of Borax crystal, con- tains the three pinakoids, the principal prism, and both prin- cipal hemipyramids, but the oblique character is brought out by the acute hemipyramid 2 P, which occurs only in the positive form. Fig. 243, a crystal of Allanite, contains GO P 9 o P, and + ^oo. Fig. 244, one of the simplest forms FIG. 246. FIG. 247. lio 103 J of Hornblende, contains only the unit prism GO P and clino- dome aPco ; fig. 245 the same, with the acute edges of the prism truncated by the clinopinakoid oo^Poo. Fig. 246 has the edges between oo P and o P modified by the faces of the negative hemipyramid P, together with the clinopinakoid oo Poo. Fig. 247, a crystal of Azurite, elongated parallel to the orthodiagonal axis, contains o P, ^ P oo, oo Pea, + -J aPoo, and + ^P2. These will be sufficient to indicate the general character of the combinations of this system. A few more complex examples will be given in the descrip- tive portion of the work. 156 Systematic Mineralogy. [CHAP. VIII. CHAPTER VIII. TRICLINIC 1 SYSTEM. THE forms of this system are referred to three axes all having different parameters and all oblique to each other. The characteristic elements of crystals belonging to it are therefore, in addition to the lengths of the axes, the three angles between them. This gives forms of the most rudi- mentary character, every face crystallographically possible that is, having a similar face parallel to itself as required by the general conditions of crystallographic symmetry to a centre is a complete form, and may combine with any other having similar or dissimilar indices ; and as no form can have more than two faces, any actual crystal must be a combination of at least three forms. From the obliquity of the axes there can be neither planes nor axes of symmetry, which property is indicated in the name ' asymmetric.' The notation of the axes is similar to that in the rhombic system, when one has been selected as the vertical axis r, the shorter one of the other two is made the brachydiagonal and the longer the macrodiagonal, the latter being so arranged as to slope from left to right. The angle between 1 Other names are anorthic, asymmetric, doubly- oblique, oblique- rhomboidal, and eingliedrig. CHAP. VIII.] Triclinic Symmetry. 157 c and b is called a, that between c and a, (3, as in the oblique system, and that between a and b, y ; these angles being measured upon the positive semi-axes, as shown in per- spective projection fig. 248, and in the orthographic projec- tions upon the three principal sections (figs. 249, 250, and FIG. 249. 251). Each of these latter shows two of the semi-axes in their true lengths and inclinations, the notation being gene- rally similar to that of the rhombic system. From these it will be seen that the principal sections are all rhomboids, FIG. 251. and that the particular solid corresponding to three finite parameters is an oblique rhomboidal pyramid or octahe- dron contained by four 'dissimilar pairs effaces, each of which, therefore, represents a different form. These are known as quarter- or tetarto-pyramids, whose positions are 158 Systematic Mineralogy. [CHAP. VIII. in Naumann's method indicated by the letter P differ- ently accented, according as the face indicated belongs to the right or left, upper or lower octants. The complete notation according to the different systems is as follows : II.(a':b : 0=(fi i)l VIII. (a : V : ^)=(i II)) ' _ VII. (of : V : 0=(i 1 i) III. (*' : ' : c)=(i i i)\ p IV. (a : V : ,)=(il i)| V. (a : b : ^)=(i i i)) VI. (of : b : c')=(i i )) The relations of the observed forms may be developed from the symbols of any of the unit quarter-pyramids in a similar manner to that given for the rhombic and oblique system. Thus, by varying the length of / when c is lengthened, and to ^ P> = ( a : b : - e] = {hhl} where h, the form changes to a pair of planes parallel to the vertical axis, or a hemiprism, which includes both the CHAP. VIII.] Tnclinic Hemiprisms. \ 59 upper and lower quarter-pyramids lying on the same side of the centre. This is indicated by the position of the accents in Naumann's symbol. Thus, from P' and P t is derived the right principal hemiprism : oo P ,'== (a \ b : oo c) = {i i o} , and from 'P and ,P, the left principal hemiprism, /P=(a :b' :oo^) = (ilo}. which, like all other triclinic forms, may appear together in the same combination, or independently of each other, there being no true triclinic prism, but only a prismatic combination of the hemiprisms. By increasing the length of the macrodiagonal or brachydiagonal axis respectively in the principal hemiprisms, the other axes being unchanged, macrodiagonal and brachydiagonal hemiprisms are formed. The symbols of the former are oo jP/ n. GO /Pn = (a \ n b : GO c) (a \ n b' : oo c) = {h ko] {hk o} where h > /, and those of the latter co Pj' n, oo /Pn (n a : b : co c) (n a : b' : GO c) = {h ko] {hk o} where h < k. In the macrodiagonal quarter-pyramids, P' n and 'Pn, when n = GO, the angle between the faces meeting in the front and back .axial plane becomes 180, or two fall into one parallel to the macrodiagonal, producing a hemi-macrodome, 'P' cc (a : GO : <:) = {i o i}, which in combination trun- cates edges parallel to that axis in front of the crystal above the base and below it behind, the correlated form ,P t oo, having the reverse position, or appearing below in front and above behind. Other analogous forms represented by m 'P GO and m t P, co are derived in the same way from the quarter-pyramids m Pn, m 'Pn, &c. Hemiprismatic forms parallel to the axis #, or hemi- brachydomes, are obtained from pairs of the quarter-brachy- 160 Systematic Mineralogy. [CHAP. vill. pyramid series by making n = co in their symbols, when the two faces meeting in the right and left axial plane fall into one. That derived from P' n and t Pn has the symbols f co = (oo a : b : c) {o i 1} which in combination modifies edges parallel to the brachy- diagonal axis above the centre of the crystal to the right and below it to the left, while the correlated form derived from 'Pn and P t n or 'P, oo = (oo a \ b' : c) = {o"i 1} has the opposite position or to the right below and to the left above. As before, the general symbols for the hemi-brachy- domes are m ,P' co = (oo a : b : m c) = {o k /} and : b' : mc = o%!. In these, unlike the other prismatic forms (the hemiprisms and hemi macrodomes), the two quarter-pyramid planes in- cluded in any face have dissimilar signs, or one is a type or positive plane of one form, and the other the negative or counter-plane of another. Hence the accents in Naumann's symbols lie chequerwise, as in naming the forms the front planes are always meant. In the hemi-brachydomes, m f' co and m ' P, GO, when m is made = GO, the faces become parallel to the front and back axial plane, producing the brachypinakoid, which, like that in the rhombic system, has the symbols GO P = (GO a : b : GO c) = {o i o} and similarly m = co in a hemi-macrodome gives rise to the form parallel to the right and left axial plane or the macro- pinakoid co P = (a : cc : GO c) (i o o). The third, or basal pinakoid, is the limiting form of the CHAP. VIII.] Triclinic Combinations, 161 vertical series of quarter-pyramids ri ,P'. when m = o, and is represented by oP= (000 : cob : r)={ooij. This system of development may be represented by the scheme given for the rhombic system on p. 134, if the octants in which the particular quarter-pyramids and hemi- prismatic forms lie, be indicated by properly accentuating the symbol P. The lines on which the symbols are arranged will also have the same significance, that is, those in any horizontal and vertical lines will be in the corresponding zones, except that the axes of the principal zones, instead of being at right angles, will be oblique to each other. The appearance of triclinic combinations is chiefly de- pendent upon the obliquity of the axes. When the three angles differ but slightly from right angles, as in the mineral Cryolite, the crystals have a general resemblance to cubic forms, while, on the other hand, in Axinite and Sulphate of Copper, they are marked by extreme obliquity and apparent want of symmetry. In other species, notably in the felspar group, triclinic crystals occur, which in Albite are closely allied morphologically to those of the analogous species Orthoclase, in the oblique system. In this latter case the resemblance is often so close that the system to which the crystals belong cannot always be determined by considera- tion of forms alone. The combination of the three pina- koid planes, also called the doubly-oblique prism, is the primitive solid of the system according to the French nota- tion, the faces being oblique parallelograms whose sides represent the meshes of the molecular network, each being dissimilar from the other two. The determination of the elements of a triclinic form requires at least five independent observations, and involves calculations which cannot be described in few words. The student is therefore referred for information on this subject to the larger works on determinative mineralogy and prac- tical crystallography. As there is no direct relation between M 1 62 Systematic Mineralogy. [CHAP. vin. form and other physical properties, the choice of position is quite arbitrary, so that there may be and often is consider- able diversity of opinion as to the symbols to be assigned to the faces by different authors. The general characters of the simpler triclinic com- binations will be seen in figs. 252-254. Fig. 252 is one of the most un symmetrical kinds, a crystal of Axinite, 1 contain- ing 'P. P'. \ P'. O P'. 'P' 00. 2 'P' 00. FIG. 252. FIG. 253. Fig. 253, a crystal of Babingtonite, contains oo/oo. oo P. o P. oo ',P\. f' oo . 'P, oo. Fig. 254 is similar, with FIG. 254. FIG. 255. on. llo 110 01 the substitution of the hemiprisms,oo /V. ooi/2. Fig. 255, a crystal of Albite, has a general resemblance to one ol Orthoclase, but the special triclinic character is apparent by the presence of the quarter-pyramid P, only on the edges between oo/oo and ^oo and not on the analogous edges between oo /oo and o P. 1 This is the position adopted by Schrauf. Other authors consider the quaiter-pyramid faces as belonging to the zone of the prism. CHAP. IX.] Hemimorphic Crystals. 163 CHAPTER IX. COMPOUND OR MULTIPLE CRYSTALS. IN demonstrating the geometrical characteristics of the different systems in the preceding chapters, the solids illus- trated have been assumed to be of the most regular cha- racter, every face of the same form being similarly placed in regard to the symmetrical centre or origin of the axes. Such crystals, however, without being absolutely unknown, are of comparative rarity, at any rate in individuals of any size, and in by far the larger number of instances one or more faces of any form may be largely developed, with a corre- sponding reduction or even entire suppression of the re- mainder, as, for example, in the common case of a prism terminated by pyramids or domes, the faces of the latter forms appear only at one end of the prism, because the other forms the surface of attachment to the rock. In such cases the missing faces have to be assumed in reasoning out the character of the completed from the observed form. Besides this, there are other cases in which two or more crystals are united into a mass having a particular regularity of arrangement, the component crystals preserving, to a great extent, their individuality. Such multiple crystals are of two principal kinds known as parallel and twinned groups, but before considering these it is necessary to notice a third special kind of development which is, to some extent, of a compound character. Hemimorphism. There are a few minerals ana artificial products, whose crystals are dissimilarly ended, the faces limiting a prismatic zone at one end of its axis belonging to different forms from those in the corresponding position at the other end. Such crystals are not properly hemihedral, as, although they contain but half the full number of faces possible in their constituent forms, these faces are not, as M 2 164 Systematic Mineralogy. [CHAP. IX. FIG. 256. they should be, uniformly distributed about the axes, but are so grouped that we may have all the faces whose indices are positive to an axis, while the corresponding negative ones are entirely absent, their places being occupied by some totally different form. This arrangement is incom- patible with regular hemihedrism, 1 and it is therefore distinguished by the name of hemimorphism. The most conspicuous examples are afforded by Tourmaline, the Ruby, Silver Ores, and Greenockite in the hexagonal, Struvite and Electric Calamine in the rhombic, and Cane Sugar in the oblique system. Fig. 256 represents a crystal of Tourmaline contained above by R K {i o 1 1} (a), and JR K{O i i 1} (b) ; below, by l?(a) and ^R K {i_p i~2) (c) ; and in the zone of the prism by GO P 2 {112 o} (d\ ccP {o i i o} (*?), and oo/*| {1340} (/). Of the latter three forms the first appears with its full number of faces, and the others with only one-half, as trigonal and di- trigonal prisms respectively. The reason of this is, that the prism of the first order, considered as a rhom- bohedron of infinite altitude, falls into two groups of three faces, one of which belongs to the upper and the other to the lower end of the crystal, either of which may be pre- sent to the exclusion of the other in a hemimorphic group, and the dihex- agonal prism in the same way as an unlimited scalenohedron divides into an upper and a lower ditrigonal prism ; but a face of the hexagonal prism of the second order includes both upper and lower faces of the sca- lenohedron, of which either may be omitted without changing its geometrical character. The occurrence of trigonal prisms of 1 This is, however, distinguished by Miiller as asymmetric hemi- hedrism in the rhombohedral system. CHAP. IX.] HemimorpJdc Crystals. i6 S this kind is therefore evidence of hemimorphic development, even when both ends of the crystal are not available for observation, as is generally the case in the ruby silver ores. In Greenockite the dissimilarity of the ends is much more marked, the crystals showing numerous pyramids at one end of the prism, squared off by the basal plane at the other. Fig. 257, a crystal of Struvite, has the upper faces of Pec {i o i}, combined with the lower ones of ^oo {i o 3} and o P. {o o 1} , which are limited transversely by Pec {o 1 1} , 4^00 {041}, and oo^oo {o i o} ; the latter may be con- FIG. 257. FIG. 258. 101 sidered as common to both sides of the base, while the for- mer two are only represented by their upper faces. Fig. 258 is a crystal of Electric Calamine contained by ccP{i 10}, oo Pec {ioo},ccPcc {o i o}, limited above by 3 -Poo {301}, 3^00 {03 i}, and oP{ooi], and below by the brachy- pyramid 2^2(121}. These are some of the more striking examples of this class of crystals, which, as a rule, are distinguished by the property of pyroelec- tricity, the opposite developing dissimilar polarity when heated. Parallel grouping. In the simplest case of the aggre- gation of two similar crystals, the individuals are so arranged that a line joining their centres is either on the prolongation of a cry stall ographic axis or parallel to it, as in fig. 259, representing two octahedra having a common vertical axis 1 66 Systematic Mineralogy. [CHAP IX. where the surface of contact represented by the shaded plane is obviously equivalent to a face of a cube, and no alteration of character is ef- fected by mere rotation of either crystal through one or more right angles about the line OTO'. The compound nature of such growth is evi- denced by the re-entering angles of the faces adjacent to the plane of contact which will be more apparent as the dis- tance between the centres O 0' is increased. This kind of grouping, often many times repeated, is commonly seen in crystals of alum, and also in native silver and other cubic minerals. If we suppose two cubes to be united in the same way, there will be a mere shifting of the top and bottom faces, giving a cube drawn out in height, but otherwise indistin- guishable from a single crystal. The same re- mark holds good if either cubes or octahedra are in contact parallel to a face of the rhombic do- decahedron, their aspect not being changed by a half turn about the normal to that face. Twin grouping. Fig. 260 represents another method of contact of two octahedra, namely, on a face com- FIG. 260. CHAP. IX.] Twin Structure. 167 mon to both, in which the axes are parallel as long as the par- ticular position is retained, but if one crystal, say the front one, be turned 180 about the line OTO' 9 the result shown in fig. 261 is obtained, where the axes are no longer parallel, the original positive extremities in the movable crystal coinciding with the_negative ones in the fixed one, or A with A' t B with ', and C with C f , while their opposite extremities make large angles with each other, the individual crystals being symmetrical to their common face, the surface of contact, which, however, as we have pre- viously seen, is not a plane of symmetry of the form. This arrangement is known as a twin structure, or twin crystal, the common plane of symmetry is the twin plane^ its normal the twin axis-, and the surface join- ing the two crystals, the plane of contact or com- position. In this, as in many other simple cases, the planes of twinning and com- position coincide, but it is not always so, and the distinc- tion between them must be carefully borne in mind, espe- cially in dealing with the twin forms in the systems of lower symmetry where only the composition face is apparent, and the position of the crystals must often be shifted to arrive at the true twin plane. As a rule, the twin plane may be any actual or possible face of a form proper to the series, other than a plane of symmetry, and it is generally one having low indices, such as 110,100, i i i, &c. If we suppose two octahedra in the position of fig. 261 to be freely pene- trable, and the line O T O' to be shortened until O and O' coincide, we obtain the solid fig. 262, where the faces of 168 Systematic Mineralogy. [CHAP. IX. contact lie in the same planes in front and behind, but all the others meet in re-entering angles, the points and edges of both crystals being fully developed. This is known as a penetration twin, the shaded parts belonging to the inverted, and the white to the direct or fixed crystal, the faces being numbered according to the original positions in the preceding figures. If the individual crystals, instead of being regularly developed, are supposed to be flattened to one half of their FIG. 262. normal thickness upon the twin axis, the group will resemble fig. 263, where there is no penetration, and only those FIG. 264. faces that are parallel to the twin plane appear of their full size. This is exactly what hap- pens when a single crystal is divided by a twin plane pass- ing through the centre as in fig. 264, and one half turned through 1 80, the other re- maining stationary. This is one of the most convenient methods of explaining twin structure, and is that most generally used, the resulting forms are called contact-twins as well as macles and hemitrope crystals. The latter terms, which were formerly CHAP. IX.] Cubic Twin Crystals. 169 in general use, are now mainly confined to the works of French authors. German writers describe twin crystals, zwilling, drilling, vidiins;, &c., according as two, three, or more individuals are apparent in the group. Fig. 265 is a contact twin of two rhombic dodecahedra upon a face of the octahedron. Here there are no re-entering angles, the section upon the twin plane being a regular hexagon. A FIG. 265. FIG. 266. complete penetration twin of the same kind has also been observed in crystals of Sodalite. Fig. 266 is a penetration twin of two cubes, which, being exactly centred, have their points on the twin axis in common. This is a common twin form of Fluorspar, but the FIG. 267. observed crystals are not gene- rally quite regular, so that the projecting portions of the shaded crystal above the faces of the white one are alternately of different sizes instead of being all exactly alike. The greater number of cases of twin crystals among holohedral cubic forms are upon the above type where the twin plane is the face of an octahedron, and this is also seen in inclined hemihedral forms, as in fig. 267, a penetration twin of two Systematic Mineralogy. [CHAP. IX. tetrahedra, two of whose faces, parallel to the twin plane, lie in the same surface at one end of the twin axis, while the other six meet in the same point at the other end. Fig. 268 is a more common case of penetration-twinning of tetrahedra, the twin-plane shown by oblique shading being a face of the cube. When the individuals in such a FIG. 268. FIG. 269. group instead of being simple tetrahedra are unequally developed combinations of both positive and negative ones, the appearance is similar to that of fig. 269, or an octahedron with a V-shaped groove along each of its edges ; and the combination of - oo O (fig. 95), twinned in the same way, resembles a rhombic dodecahedron grooved parallel to the longer diagonals of its faces. In the parallel hemihedral forms one of the most fre- quently observed cases is the penetration twin of two penta- gonal dodecahedra (fig. 270), the twin plane being a face of the rhombic dodecahedron. This is especially characteristic of iron pyrites, and the similarly constituted sulphides and arsenides of nickel and cobalt. The above are the principal kinds of twin-crystals in the cubic system, in their simplest and most regular develop- ment ; other and more complex cases arise when the compo- CHAP. IX.] Cubic Twin Crystals. 171 nent crystals are of different sizes, or the plane of composition is not central, when the groups are often considerably dis- torted. The same structure may also be repeated with three or more individual crystals, producing multiple or polysynthetic FIG. 270. FIG. 271. twin groups. Fig. 271 is an example of a peculiar polysyn- thetic twin of Spinel recently described by Struver. It is made up of six tetrahedral combinations, o to o 6 , the first four being repeated contact twins, on an octahedral face, while the fifth and sixth are parallel to the second and third; and as all their twin axes lie in the same face of a rhombic dodecahedron, whose axis, the line joining the hollow six- faced angles in the centre, is the edge common to all the individuals, there is complete lateral symmetry to that face. In many instances the structure of a twin group may be explained in more than one way, or the twin axis may be exchanged for another line at right angles to itself, rotation about which produces a similar geometrical form, although the position of individual faces may be different. Thus, in fig. 264 the axis O T may be exchanged for a line in the twin plane joining the middle points of opposite edges, and normal to a new plane, cutting the edges at one-half and one-third of their lengths alternately, which has the pro- perties of a face of the icositetrahedron 2 O 2 and gives 172 Systematic Mineralogy. [CHAP. IX. forms exactly similar to figs. 262 and 263, but with this difference, that the faces brought opposite to each other belong to different tetrahedra, instead of to the same one as they do with the octahedron face in the twinning plane. Twin crystals of the hexagonal system. The faces of di- hexagonal pyramids and prisms and hexagonal pyramids are possible twin planes in the holohedral forms of this system, but the only observed groups are twinned upon the latter form, and they are not of very common occurrence. In the rhombohedral hemihedral forms, on the other hand, twin structure is extremely common, the twin plane being most frequently either the face of the same or some other rhom- bohedron or the basal pinakoid, the latter not being one of their planes of symmetry. Fig. 272 is a common twin group of Calcite, in which two rhombohedra of the same FIG. 272. sign are twinned upon a face of the more obtuse rhombohe- dron \ R. The faces in front meet in a re-entering angle, and those below in a parallel salient one, the twin edges in both cases being parallel to the longer diagonals of the faces, while those of the side faces are parallel to the middle edges of the rhombohedron. When this structure is repeated by the addition of a third crystal, as in fig. 273, the middle member CHAP. IX.] Rhombohedral Twin Crystals. 173 of the group R' is often reduced to a thin parallel plate, the third one R" being parallel in position to the first R; and when the number of individuals is much greater, and the intermediate ones are very thin, the group is scarcely dis- tinguishable from a simple crystal, the twin structure being only apparent in the numerous fine striations covering two of the faces parallel to their horizontal diagonals, and the other four parallel to their middle edges. Fig. 2 74 is a contact twin FIG. 274. of the hexagonal prism upon a face of the same rhombohedron - \R, and fig. 275, another having a face of the unit rhom- bohedron as a twin plane, the two crystals making a nearly FIG. 276. FIG. 277- right-angled group, the inclination of the principal axes to each other being 89.o4 / . Fig. 276 is a contact group of the common scalenohedron R 3 of Calcite twinned upon a face of the acute rhombohedron 2 R. 174 Systematic Mineralogy. [CHAP. IX. In the second case, where the basal pinakoid is the twin plane, the axes of the component crystals are parallel. Fig. 277 is the contact twin or hemitrope of a single rhom- FIG. 278. FIG. 279. bohedron, and fig. 278 the same completely developed as a penetration twin of two. Fig. 279 is a hemitrope of the com- mon scalenohedron R 3 of Calcite, also upon the basal plane ; FIG. 280. and fig. 280 a similar twin of the combination GO P. \ R in the same mineral. Here the two prisms are in contact, and their section being a regular hexagon, there are no re- entering angles, but the compound character is apparent from the shape of the prism faces, which are alter- nately rectangular and six-sided, in- stead of all being irregular five-sided figures, as in the single crystal, fig. 144. Fig. 281 is a contact twin of the com- mon combination en p.R. R of Quartz, the faces of the positive rhombohedron in one crystal lying parallel with CHAP. IX.] Tetragonal Twin Crystals. 175 those of the negative one in the other. This is a case of partial penetration, the faces adjacent to the surface of con- tact, which is not the twin plane, meet- FJG 28l ing in re-entering angles; but when, as very frequently happens, there is more complete penetration, these angles are convex, and the group can only be distinguished from a simple crystal by the irregular character of the faces, which rarely have even surfaces, por- tions of one rhombohedron being irregularly distributed through the other in a manner which shows the crystals to be combined, and that their contact is not in a plane surface. A complete description of twin structures of this kind will be found in Descloizeaux's memoir on Quartz. Twin crystals of the tetragonal system. In the type of twin structure most frequently observed in this system, the twin plane is a face of the pyramid of the second order, FIG. 282. FIG. 283. m P. Fig. 282, one of the simplest examples, is a hemi- trope of the tetragonal pyramid in Hausmannite, the lower half being rotated to the left on the lower pJane (oIF). This, when repeated symmetrically upon all four sides of the pyramid, gives the group of five individuals, fig. 283. Systematic Mineralogy. [CHAP. IX. Another very common example of the same kind (fig. 284) occurs in the combination P.aoPof Tinstone, and this, when repeated with a third individual, the middle one being shortened to a parallel plate, gives the bent-kneed or geni- * 28 4- FIG. 285. culated group (fig. 285) whose ends are both in the direct position, the middle one alone being reversed. Fig. 286 is another example of a triple group, very characteristic of the allied species Rutile. Here the ends are bent away from the middle, the twin planes being different faces of Poo. FIG. 286. FIG. 287. In sphenoidal hemihedral forms, the twin plane is commonly a face of a pyramid, and when the proportions of the latter differ but slightly from those of a regular octahedron, the twin groups, whether contact or penetration, are very like tetrahedral twins \r f he cubic system. This is especially CHAP. IX.] Rhombic Twin Crystals. 1 77 observed in Copper Pyrites. In the pyramidal hemihe- dral forms the twin plane is usually a face of the diagonal prism oo Poo, or the twin axis is one of the lateral axes. This in crystals like fig. 287 has the effect of bringing the faces SS' of the pyramids of the third order J (3 P$) into a re-entering or negative solid angle in the basal section of the pyramid of the second order Poo, but when these planes are less completely developed, they usually appear as contrasted diagonal striations upon the faces of Poo. Twin crystals of the rhombic system. Crystals twinned upon the faces of pyramids or prisms are of frequent occur- rence in this system. Some of the most familiar examples FIG. 288. FIG. 289. \ are afforded by the allied species, Aragonite and White-lead Ore. Fig. 288 is a contact twin of Aragonite on the face (I i o) of the prism oo P, and fig. 289 its section on the basal plane, the shading lines being parallel to the brachypinakoid in each individual, which faces meet on one side in a pro- jecting, and on the other in a re-entering, angle. If, as very commonly happens, the hollow space a is filled up by parallel elongation of the two crystals, the group may resemble a single crystal if the development is confined to the zone of the prism. Figs. 2900, 290^ give the same group with a third individual, the twinning being repeated on the same face ; this brings the prism faces in i. and in. into similar position, the middle crystal n. being reduced to a parallel plate. N 178 Systematic Mineralogy. [CHAP. IX. This is also a very common case, the middle individual being many times repeated, and appearing as a series of fine FIG. 2900. FIG. 290^. striations parallel to the twin plane. Fig. 291 is another group, in which the ends i. and in. are twinned upon adjoining faces of the prism in the middle crystal n. Fig. 292 is the FIG. 291. FIG. 292. horizontal section of a similar triple group of Copper-glance, twinned upon faces of oo P. The angle of the prism in this species is so nearly = 120 (119 35') that the group is very similar in appearance to a regular hexagonal prism. Figs. 293, 294 are examples of cruciform twins produced by the penetration of two crystals of the combination oo P. oo ^oo.o P in Staurolite. In fig. 293 the twin plane is a face (o 3 2) of f ^oo, the vertical arms of the cross being in direct, and the transverse ones in the inverted position. CHAP. IX,] Rhombic Twin Crystals. 179 The latter are nearly, but not quite, horizontal, the angles be- tween the vertical axes of the two crystals being alternately FIG. 293. FIG. 294 91 36' and 88 24'. In fig. 294 the twin plane is the face (232) of f _/*!, the left-hand crystal being placed in direct and the right-hand one in reversed position. Here the axes of the two prisms cross at 58 46', and the re-entering angle between the brachypinakoid faces is 119 34', or nearly 120. These peculiarities are caused by the fundamental ratios a : b : c of this species being very nearly as \ : i : f ; the actual values are 0.48 : i : 0.67, which give a twin axis inclined to the vertical at nearly 45 in the first case, and 60 in the second. In the sphenoidal hemihedral forms of this system, the three pinakoids are possible twin planes ; but examples of such twinning are not common. The best known case oc- curs in Manganite where a combination containing f Pec or {36 5} as a sphenoid is twinned upon the brachypinakoid. Fig. 295 is a compound structure observed in the hemimorphic species, Electric Calamine; the twin plane is the basal pinakoid, whose normal is the vertical or hemi- morphic axis. As, however, the faces of 2 P 2 in these N 2 FIG. 295. i8o Systematic Mineralogy. [CHAP. IX. crystals only appear at the negative end of the vertical axis, it is necessary to consider the lower component as inverted about one of the lateral axes as well as about the twin axis, and the notation of the faces of the inverted form will differ according as this reversal takes place about a or b. In the first case the front transverse faces 3 P oo, GO P, 2 p2 will all be positive to the axis a, while the longitudinal or side faces, oo Poo and ^oo, on the same side will be positive above and negative below to the axis b ; and in the second the longi- tudinal faces on the same side will have similar signs, while the transverse ones will be negative to the axis a in the lower crystal. This difference is only geometrical, and there is no reason to consider either reading as preferable to the other. The same structure may also be explained by supposing the vertically reversed crystals as essentially penetrating the direct one, which, however, requires either the brachy- pinakoid or macropinakoid to be the twin face, an assump- tion which is not compatible with the exclusion of planes of symmetry from possible twin planes. Twin crystals of the oblique system. In this system any face is a possible twin plane except the clinopinakoid, or FIG. 296. plane of symmetry, and of the three crystal- lographic axes only the vertical is a possible twin axis. The type of most frequent occur- rence is that having the orthopinakoid as the twin plane, the twin axis being horizontal. When the same face is the plane of contact, the group resembles fig. 296, a hemitrope of the combination P. oo P. oo jPoo, common in Gypsum, the front half of the crystal being sup- posed to be the reversed one. The faces of the hemipyramids in the two crystals meet in re- entering angles above and parallel salient ones below, and their clinopinakoids in the same surface at either end. When the individuals are differently proportioned, and have their greatest length parallel to the inclined axis, they 21R Q4P 010 CHAP. IX.] Oblique Twin Crystals. iSi may form a complete cruciform penetration twin, the re- entering angles of the faces of the hemipyramid breaking the lines of the vertical edges in front and behind. These may be also considered as twinned by rotation upon the ver- tical axis, in which case the twin plane is horizontal and not a possible crystallographic face ; while on the former view the normal to the twin plane, being horizontal, is not a pos- sible crystallographic axis. The same type of twin crystal is very common in the FIG. 297. FIG. lio 2AST. species Orthoclase, two individuals of the combination oo P, oP, 2^00, 00 ^P 00 (fig. 297), being combined with partial penetration upon the clinopina- FIG. 299. koid in the groups, figs. 298, 299, each component preserving its individuality, except in the case of the prism faces, which, though apparently simple, are made up of parts of different crystals, joined along the diagonal dotted lines in the figures. This is known as the Carlsbad type of twin crystal, the groups being further distinguished as right- and left- handed, according as the reversed crystal is to the right (fig. 298) or left (fig. 299) of the direct one, a geometrical distinction which is only 182 Systematic Mineralogy. [CHAP. IX. recognisable so long as the penetration is incomplete. Fig. 298 is noted as having a horizontal twin axis, and fig. 299 a vertical one. Fig. 300 is a similar combination to fig. 296, the positive hemi-orthodome f being substituted for the more acute one FIG. 300. FIG. 301. \ \ This, when twinned by rotation upon a plane parallel to the base, as shown in the dotted line, gives fig. 301, where the faces of the prism meet in re-entering angles in front and projecting ones behind. The twin axis in this case is in- clined to the vertical at the angle 180 (90 + /3). This is sometimes called the Manebach type of twin in Felspar, Fig. 302 is a crystal of the same combination as the last, but of a nearly square section on the orthopinakoid ; this, FIG. 302. FIG. 303. when twinned on the clinodome 2 ^Poo, indicated by the dotted line, produces fig. 303, in which there are no re- entering angles. This is known as the Baveno type of twin in Felspar. CHAP. IX.] Triclinic Twin Crystals. 183 Twin crystals of the triclinic system. As there are no planes of symmetry in this system, any face is a possible twin plane. The observed cases are generally similar to those of the oblique system, with an additional one special to the system where the brachypinakoid is the twin plane. This is repre- sented in fig. 304, the right-hand half of the divided crystal being supposed to be rotated upon a normal to the brachy- pinakoid, which brings the opposite halves of the basal pinakoids together in a re-entering angle at the top, as indi- FIG. 304. FIG. 305. cated by the arrows, and in a corresponding salient one at the bottom ; and those of the hemi-macrodome in the same way are convex in front and concave behind. Supposing such a group to have only its lower faces developed, it would be impossible to distinguish it from a simple "crystal by con- siderations of form alone. If this structure is repeated with a third individual, as in fig. 305, the exterior components resemble the halves of crystals in their normal positions, divided by an intermediate parallel plate. This middle in- dividual may, however, be replaced by a large number of much thinner plates, in which case the re-entering angles of the basal pinakoids appear as fine striations parallel to the edge between oo ^Poo and o P upon the latter faces. These, known as polysynthetic striations, are especially characteristic of the Felspars crystallising in this system, there being no ap- parent limit to the number and fineness of such twin lamellae, which are, as a rule, easily recognised by the microscope, 1 84 Systematic Mineralogy. [CHAP. IX. even when not apparent to the naked eye. In many cases, however, they are perfectly visible without being magnified, as, for instance, in Labradorite. Fig. 306 is an example of complex twin grouping of triclinic crystals, the individuals i. n. and in. iv. being FIG. 306. twinned upon the brachypinakoid, as in the preceding example, and these are further compounded according to the Carlsbad type. This, though generally similar in appearance to fig. 305, differs from it by having similar faces arranged in pairs instead of alternating singly. The above are the principal types of twin crystals in the different systems. Other and more special cases will be noticed in treating of the minerals in detail. Irregularly developed crystals. In the artificial prepara- tion of crystals in the laboratory, it is a well-known practice to select some of the most regular individuals from those first deposited, and place them in such a manner in fresh portions of the solution that they may increase as equally as possible in all directions, for which purpose it is necessary to alter their positions from time to time. If, on the other hand, the growth upon any part is hindered, as, for example, upon the face in contact with the surface of the vessel, the deposition of the additional material will increase the faces remaining free to such an extent that their true character may not be readily seen. This condi- tion prevails to a great extent in natural crystals, and in fact one of the chief objects of determinative crystallography is the reduction of such distorted forms to their theoretical regularity. Only a few of the simpler cases can be given in this place. Fig. 296 is an octa- hedron with two faces (in) (III) prominently larger than the other six, in which the solid angles are no longer FIG. 307. CHAP. IX.] Distorted Crystals. i3 5 apparent, as the four faces similarly placed with respect to any principal axis do not meet in the same point, but in an edge parallel to the axis of a dodecahedral zone, or, in other words, the crystal is elongated in the zones [no], [oil], and [ i o i ], and shortened on [ 1 1 o], or shortened on the ternary axis (i 1 1). This is common in crystals of Nitrate of Barium deposited from solution upon a flat FIG. 308. FIG. 309. surface, the enlarged faces appearing as nearly regular hexa- gons ; and it is also readily obtained by cleavage from octa- hedra of Fluorspar. When the octahedron is elongated on a ternary axis the faces perpendicular to that axis may be completely obliterated, producing an acute rhombohedron, as in fig. 308. This is easily obtained by cleavage in Fluor- spar, and its octahedral character is as easily restored by cleaving off two tetrahedra in the directions shown in the dotted lines. Fig. 308, an octahedron distorted by elonga- tion on a binary axis, has a general re- semblance to a combination of two prismatic forms in the rhombic system. Fig. 311 is a rhombic dodecahedron elongated vertically, which converts the upright faces into a square prism, and the inclined ones being unchanged, the general effect is that of the combination of a prism and pyramid of diagonal position co P. P co in the tetragonal system. In the hexagonal system distorted crystals are also of fre- FIG. 310. 1 86 Systematic Mineralogy. [CHAP. IX. quent occurrence, producing great variation in shape. Some of the most familiar examples are afforded by the common combination R R oo R in Quartz, three of which are repre- sented in the following figures. In fig. 311 the four rhom- bohedral and two prismatic faces in the zone [2 7 1 2 ] are FIG. 311. FIG. 312. lengthened parallel to its axis ; in fig. 312 the crystal is elongated on the axes of the zones [121:0], [1212], and [1212], giving an apparent rhombic character to the com- FIG. 313. bination ; and in fig. 313 one face of one rhom- bohedron is prominently longer than the others, giving a kind of chisel-edged termination. Crystals of this kind generally with only one end developed are common among the brilliant groups of rock crystal found in the Western Alps of Dauphine and Piedmont. In the rhombic and oblique systems the commonest case of distortion is that of the un- equal development of the faces of the prism, one pair being broader than the other, giving a rhornboidal instead of a rhombic basal section. Imperfections in the faces of crystals. In addition to the irregularities arising from distortion and compound structure, crystals, when of large size, often appear with roughened, striated, or even partially hollow faces. The latter im- perfection is common in substances that crystallise easily, whether from solution, as salt and alum, from sublimed vapours as arsenious acid, or from molten masses, as lead, CHAP. IX.] Imperfections of Crystals. 187 silver, bismuth, and silicates produced in furnace slags. In all these cases crystals are often observed having their edges perfectly defined, while the faces themselves are hollowed out and reduced to very narrow surfaces adjacent to the edges, giving skeleton structures, in which the general ele- ments of the form are, however, usually recognisable without difficulty. In Fluorspar, octahedra with roughened faces are occasionally found, which are made up of minute cubes piled up like courses of masonry, the side of each successive course being diminished by the breadth of two cubes. The roughness of the face is, therefore, due to the step-shaped section of the pile. In the hexagonal system, instances of irregular single crystals, built up from smaller individuals of the same or different kinds, are very common in Calcite. In the north of England lead mines the obtuse rhombohedron known as Nailhead Spar, and the combination shown in fig. 144, are often aggregated in such a manner as to produce rough- faced rhombohedra and scalenohedra often of considerable size. Sometimes this step-shaped outline is apparent on some of the faces, while the others are comparatively smooth and regular. The same kind of structure may often be brought out in the most regularly developed crystals by the action of solvents, which produce the so-called corrosion figures, which show that in many cases the smallest crystals are fully as complex, or even more so, than those of larger size. The characteristic habit of particular crystals is often apparent in the most minute individuals ; for instance, the peculiar geniculated groups of Tinstone and Rutile (fig. 284), are developed when these oxides are crystallised from solution in melted borax or phosphate of soda before the blowpipe, even when the crystals must be magnified from 400 to 500 diameters to render them visible. The prismatic faces of Quartz crystals are very generally covered with horizontal striations, representing very minute portions of the faces of an acute rhombohedron. This is 1 88 Systematic Mineralogy. [CHAP. ix. generally called oscillatory combination, a tendency towards the formation of rhombohedral ends being supposed to have alternated with another towards prismatic elongation. In Beryl the prisms are striated vertically, as is also the case in Tourmaline, the continued repetition of the prisms of the first and second order producing nearly cylindrical forms. Another class of imperfection, where the faces of crystals are curved instead of plane surfaces, is characteristic of certain minerals, the most striking examples being afforded by Siderite, which occurs in rhombohedra having strongly curved faces ; Gypsum and Diamond : the latter, when in the form of the hexakisoctahedron, are often nearly spherical in shape. Habit of crystals. In describing minerals it is usual to speak of their crystals as affecting particular types, according to the character of the dominant or principal faces ; thus, in the cubical system, they may be cubic, octahedral, dode- cahedral, &c., as one or other of the principal forms prevail in the combination. In the other systems the types are pyramidal, sphenoidal, rhombohedral, or scalenohedral when the closed forms are most apparent, and prismatic when the development is mainly in the direction of the open forms. In the latter case several further distinctions are founded upon the relation of the height of the prism to the breadth of its base or closing pinakoid. The shorter forms, or those having their principal dimensions in the direction of the lateral axis, are said to be tabular, or, if very thin, platy or scaly : when the height is only a few times the breadth, they are short columnar, and as the relative length of the prismatic axis increases they become columnar, slender-prismatic, and acicular, or needle-shaped. In these terms the expression prismatic is not restricted to the forms assumed as vertical prisms, but is used with the general signification of any zone of prismatic planes, whether vertical or horizontal. The habit of considering any prominently defined axis as a prismatic one in describing the appearance of crystals is very CHAP. IX.] Irregular Aggregates. 189 general and convenient, but care must be taken not to con- found such * columnar ' forms with the more exactly deter- mined crystallographic prisms. Irregular grouping of crystals. Masses of crystals, when not arranged as symmetrically twinned forms, are spoken of as groups or crystalline aggregates. These are commonly found in hollow spaces or druses in the containing rock, attached at one end, with the faces terminating the opposite end, freely developed, the individuals of the group having a more or less radial arrangement diverging from the point of attachment. This, in general terms, may be considered as the most typical kind of grouping of well individualised crystals. When the aggregates are of a more compact kind, the indi- viduals are rarely recognisable with anything like their full number effaces, but appear, as a rule, as columnar or fibrous masses arranged in parallel or divergent forms. The latter, when in sufficient numbers, make up more or less spheroidal masses, which, according to the size of the spheroids, are spoken of as mamillary, remform, or kidney- shaped, and botryoidal, or grape-like masses or concretions. Other smooth spheroidal masses of substances, having no apparent definite structure, are generally called nodules. Parallel aggregates of a fibrous structure, such as those of Calcite and Gypsum, often form regular beds of a silky character on the face. These are known as Satin Spar ; the same structure is common in salt and alum, where the fibres are often bent or contorted. Aggregates resembling corals, mosses, and other organised forms, are common in Arra- gonite, the so-called flos-ferri, or flowers of iron, and native metals ; the latter are usually called dendritic forms, the same name being also applied to the plant-like stains of Brown Iron Ore and Peroxide of Manganese on rocks. Wire- like or filiform masses are very characteristic of native silver. Stalactites are irregularly shaped crystalline masses found in caverns hanging from the roof, and stalagmites are similar masses accumulated above the floor. These terms Systematic Mineralogy. [CHAP. X, are rather geological (as indicating methods of origin) than structural. When no indications of crystalline structure are apparent in a mineral aggregate it is said to be massive. Sometimes such masses are spoken of as amorphous. This, however, is an improper use of the term, which should only be applied to substances which fail to show crystalline structure when tested by optical and other methods, many perfectly well crystallised bodies often appearing structureless until so examined. CHAPTER X. MEASUREMENT AND REPRESENTATION OF CRYSTALS. THE angles between two faces in a crystal may be measured in two different ways, namely, directly from the inclination of a pair of jointed blades striding over the edge, and in- directly by determining the angle through which the crystal must be turned in order to obtain the reflected image of an object successively from both faces. The instrument used in the first method is the hand or contact goniometer, which has not been materially altered from the form in which it was originally made by Carangeot for Rome de Tlsle and Haiiy at the end of the last century, and is shown in fig. 314. It consists of a semicircular arc, divided into single or half degrees, according to size, having attached to it two steel blades, one of which, k m, is fixed, or rather has only a sliding movement in a straight line upon the pins c d, while the other has an angular as well as a sliding motion about c. The zero point of the graduation is on a line parallel to the direction of the fixed rule, so that when the latter is laid upon one of the faces containing the angle to be measured, and the movable one turned until it bears similarly upon the other face, the edge g i will indicate the value of the angle CHAP. X.] Contact Goniometer. 191 upon the divided arc. The object of the slots is to allow the length of the measuring arms to be varied to suit the size of the crystal. For the same purpose the arc is divided, and can be folded back upon a hinge at , the supporting k arm /being attached by a screw at the back, which can be taken out when greater freedom of manipulation is required, as, for instance, when working upon crystals in implanted groups. A more convenient arrangement is to have the arms separate from the measuring arc, in which case a com- plete circle may be advantageously used. Under the most favourable conditions angles can be measured by the contact goniometer to within half a degree, when the crystals are of a certain size. It is therefore best adapted for trial measurements upon large and imperfectly lustrous faces, or for use in making models. When greater precision is required, the more exact method depending upon reflection from the faces must be used. Fig. 315 represents Wollaston's reflecting goniometer in one of its simplest forms. It consists of a circular disc E E, with a divided rim reading to single minutes of arc by a vernier at R, movable about a horizontal axis passing through the bearing c at the top of the pillar B, with a milled head for turning it at G. This axis is hollow, and an inner one, moved 192 Systematic Mineralogy. [CHAP. X. by the milled head i, is connected with the arrangement carrying the crystal at K. This consists of two bent arms K M, movable about a pin L, and the carrier proper o, which has a rotatory as well as a sliding movement in the collar N, the crystal a being attached by a ball of wax to its outer end. The parts s T and u form a clamp upon the disc F for main- taining the circle in any particular position. The foot A is a block of wood ; in the larger instruments it is of metal, with levelling screws. In observing angles it is necessary to bring the edge exactly into the line of the axis of the instrument, which is done by the various movements of the carrier, which together form a kind of universal joint. Two signals are required, one of which, x, is seen by reflection, and the other, y, di- rectly. These may be any small, well-defined objects, such as a bright spot seen through a screen placed before a win- dow or a lighted lamp, and a window-bar, chalk-line or line CHAP. X.] Reflecting Goniometer. 193 of light, as far off as they may be conveniently seen. The edge is in adjustment when the image of a horizontal or vertical line reflected from either face coincides with that viewed directly from the same point, the eye being as close to the crystal as possible. The circle being clamped, the crystal is then turned by i until the reflected image of x is brought into coincidence with y, seen directly, when it is undamped, and the whole circle is turned by G until the same thing is seen from. the second face. The angle through which the circle is rotated will be the supplement of the dihedral angle required, if it was originally set to zero. As a rule, however, it is better not to start from this point, but to read the vernier after each adjustment, and take the differ- ence of the readings, repeating them for several different positions on the circle to eliminate the errors of eccentricity as much as possible. In all cases the observation must be often repeated to obtain results of any value. Numerous modifications and improvements of the re- flecting goniometer have been introduced for the purpose of facilitating the operation of centering and adjusting the crystal, and obtaining greater precision in observation. For the latter purpose a sighting telescope of low magnifying power is used, the observed object being the image of the cross wires in a second telescope, or collimator ; and for the former the carrier is made with a system of rectilinear and cylindrical, or sliding and rocking, movements analogous to those of the ' universal ' machine tools. The more exact modern instruments of this kind are usually constructed upon the pattern introduced by Mitscherlich in 1843 ; another construction, that of Babinet, in which the divided circle is horizontal, being also considerably used. The latter has the advantage of placing the crystal upright, and not as in the preceding form overhanging the carrier; but as it requires signals that are not in the same vertical plane, it is not so easily used when without a telescope. The large instruments made by Fuess, of Berlin, on this pattern are o 1 94 Systematic Mineralogy. [CHAP. X. remarkable for the extreme precision of their centring appa- ratus, which has two straight line movements at right angles to each other, and two cylindrical ones also at right angles, the former being used for centring or bringing the edge into view, and the latter for adjusting or making it vertical. This operation is facilitated by the addition of a lens of short focus to the objective end of the telescope, which for the time converts it into a microscope of low magnifying power, and allows the edge to be brought into coincidence with a vertical wire in the eyepiece. When the observations are made with the unaided eye or a single telescope, the signals observed must not be too near, or there may be an appreciable error of parallax. A convenient object is a small gas-flame about half-an-inch high, from 15 to 30 feet distant, the observations being made in a darkened room. When the instrument has two telescopes the error of parallax may be eliminated, but the loss of light is so considerable that crystals with very perfect faces are required to give distinct images of the signal. In all cases observations must be repeated several times, and, if possible, upon different crystals, the quality of the reflection obtained being specially noted. As a rule, the smallest crystals have the most brilliant faces, and therefore give the best results. A goniometer for measuring the angles of minute crystals with dull faces has recently been described by Hirschwald. It is of the Wollaston pattern, but the position of the faces is ascertained by bringing them into the focus of a com- pound microscope with an objective of very short focal distance, which, when adjusted, can be traversed horizontally by sliding movements, the adjustment of the face being complete when its whole surface is equally well denned in the field of the microscope as the latter is moved over it. The calculation of the crystallographic constants from the observed angles of crystals is the work of determinative crystallography, and the subject is beyond the scope of the CHAP. X.] Drawing Crystals. 195 present tieatise. The student is referred to the special works on the subject, more particularly to Miller's tract on Crystallography, and the works of Klein and Mallard. Methods of representing crystals. The larger number of figures in this volume, following the usual practice of works on Mineralogy, are drawn in so-called parallel perspective, which supposes the point of sight to be at an infinite distance from the object represented, so that all rays proceeding from the latter to the eye are parallel, or all lines and surfaces parallel to each other remain so in the drawing ; and in order to bring in a sufficient number of faces the plane of projec- tion is so chosen that all lines actually at right angles will be projected as acute or obtuse angles, and consequently all lines, except those parallel to the plane of projection, and all faces, will be shown in other than their true dimensions. This is done by adopting reduced lengths for the axes in the triaxial systems, which may be either diminished by the same amount, as in the so-called isometric projection, or two may be reduced in one proportion and the third in some other, as in the monodimetric projection ; or, finally, each axis may be reduced in a different proportion, giving the anisometric projection. Of these methods the first is not suited for representing crystals, as the distortion of the faces is too great, but either of the other two may be used, ac- coiding to circumstances, the anisometric projection, as a rule, giving the most natural figures, which, however, is accompanied with the disadvantage of an excessive fore- shortening of the basal section. The following table, by Weisbach, gives the elements of the projections most commonly used in this class of draw- ing : the notation refers to three equal rectangular axes taken in Weiss's order ; the first group of figures gives the proportional lengths assumed for the axes, the second the corresponding reduced lengths compared with their true lengths, as given in their orthographic projections, and the last the angles between the axes at the Origin o. The o 2 196 Systematic Mineralogy. [CHAP. X. second of the monodimetric and anisometric series respec- tively will be found to be most generally suited for repre- senting crystals. Projections Proportional lengths of axes a b c Reduced lengths of axes for real length = 1,000 a. be Angles between axes AOB AOC BOC Isometric . . Monod metric Anisometric . i i : i i 2:2 i 3: 3 1 4 : 4 5 9 : I0 6 17:18 8 31 : 32 10 49 : 50 0-817 : 0-817 ' 0*817 9-471 : 0-943 : o 943 0-324 : 0*973 : 0*973 0*246 : 0*985 : 0*985 0*493 : 0*887 : 0*985 0*333 : o'944 ' 0*998 0*250 : 0*968 : 0*999 0*200 : 0*980 : 1*000 120 i&X 1 34. 06' 157 Q , i74'46' 1 75- 52' 120 I 3 i.2 5 ; I34.'o6' 107. 49 96. 23' 94. 55 93- 58' 120 97. H' 93. 1 1' 9 i* 47 ' 95-n 90. 47' 90. 20 90. 10' In drawing the projection, the axis c is made vertical, and the other axes are then set out from the point o, at their proper angles, with a protractor. The proportional lengths are then laid off upon each side of the centre, and if the drawing is sufficiently large, these lengths may be sub- divided into equal parts. This gives the projection, or ' axial cross,' for three axes of the same length, or that of the unit form of the cubic system ; and therefore, by applying to the vertical axis the proper value of c as given in works on Descriptive Mineralogy, the unit axis of any tetragonal species may be found, and similarly, by altering a and c, those of a rhombic species corresponding to b = i. In the hexagonal system the projections of the lateral axes h and / are found from those of the cubic system by making the front axis of the latter a= \/y= 1732, and drawing lines to the extremities of the right and left axis b, which in this case corresponds to k. This gives the projec- tion of a right-angled triangle, or half the base of a rhombic prism, whose obtuse angles are 120; a line parallel to k&, bisecting a, truncates the acute angle or gives a third side of a regular hexagon, and lines drawn from these points of inter- section through the centre, o, will be the projections of the CHAP. X.] Projection of Oblique Axes. 197 axes required ; the vertical axis is then changed to the characteristic value of <:, as in the tetragonal system. In the oblique system the position of the clinodiagonal axis is found by laying off its projections upon the rectan- gular unit axes a and c that is, upon the former a sin. p, and upon the latter c cos. /3 these being the sides of a parallelogram whose diagonal corresponds to the unit-length of an axis having the characteristic inclination required, the proper lengths corresponding to the fundamental para- meters of the species being then substituted as in the rhombic system. In the triclinic system a similar principle is followed. Starting from three rectangular axes of the same length, and calling the angle between the pinakoids o i o and i o o, , O, GO O, 262. The plane of projection is the cube face (o o i ), and the common point of intersection of the faces the unit length of the vertical axis, is represented by the central point c. This position is the normal one for the octahedron, for those faces of the rhombic dodecahedron whose indices have the order o 1 1 and i o i, and those of the icositetrahedron whose order is 112. The first form, there- fore, will be represented by the projection of its horizontal edges, or the square whose diagonals are the unit lengths of CHAP. X.] Linear Projection. 2OI the lateral axes A and B, the second by the square described upon these lengths, and the third by the square whose diagonals are 2 A and 2 B. The remaining faces of the cube and rhombic dodecahedron, being parallel to the vertical axis, will, when made to pass through it, be represented by straight lines intersecting at 45, or the diagonals of the pre- ceding squares. The eight remaining faces of the icositetra- hedron having the order i 2 i and 211, intercept the vertical axis at 2 c ; and therefore, when brought back to c, the pro- jections of the first four will be the four lines enclosing a rhomb, whose diagonals are a and \b, and those of the second four the sides of a similar rhomb having the dia- gonals \ a and b. The zonal relations of the faces of the different forms entering into a combination are indicated in this projection by the crossing points of the lines, or the so-called zonal points. Thus (112) is the zonal point of a zone containing two adjacent faces of the rhombic dodeca- hedron, and the face of 2 Oz truncating their edge, the latter being common to two octants of the crystal; B gives the zone formed by two adjacent faces of the octahedron, the face of the rhombic dodecahedron truncating their edge and two faces of 2 O 2, and so on for many others, which may be de- duced from the figure. In the systems of lower symmetry the diagrams are less regular, and apparently less simple, but their construction is easily learnt when the leading principle is mastered. For the complete description of these the reader is referred to Quenstedt's works on Crystallography and Mineralogy. An ingenious application of this method in the construction of perspective figures of crystals is given in E. S. Dana's 'Text-book of Mineralogy,' p. 427. Spherical projection. This method, originally introduced by Neumann, and brought into general use in its present form by Miller, supposes the crystal to be placed within a sphere, both having a common centre, and lines normal to the faces to be drawn through the centre to the surface of the sphere on either side. These lines will obviously be the 2O2 Systematic Mineralogy. [CHAP. X. diameters of the sphere, and the points of intersection their poles ; and as the angle between the normals of two faces is equal to the supplement of their dihedral angle, the position of the poles will determine those of the faces by a system of points upon the spherical surface, which give rise to problems that can be solved by spherical trigonometry. If a sphere be supposed to be projected upon its equatorial plane, that being also parallel to the terminal pinakoid, or the equivalent face of the cube, the pole of that face will be the centre of the circle of projection, the poles of faces in the zone of the principal prism will lie in the circumference, their angular distance being given directly by the measured angles with the reflecting goniometer, and those of the iio 020 130 pyramid, or other inclined forms, will be at various intermediate points, all being so related that the poles of all the faces in a zone will lie upon the same great circle, which, from the property of the projection, will either be a straight line when it passes through the pole of the circle of projection, or an arc of a circle when in any other position. Fig. 317 is an example of the spherical projection of a crystal of Topaz, containing the CHAP. XL] Spherical Projection. 203 forms DO/*, 00/2, P, P, P, f/2, /oo, ooAo, o P, as shown in horizontal projection in fig. 318. In it the principal zones, or those that include the terminal pinakoid, are represented by diameters, which are projections of meridians of the sphere, whose direction is found by laying off at the centre the supplements to the interfacial angles of the prism from the pole of o i o ; the position of the pole of any intermediate face in these zone circles being found by taking in them a distance equal to the tangent of half the supplement ot the inclination of the face upon o P, the radius being considered as unity. Zones that are not perpendicular to the plane of projection, or do iot include o P, are represented by arcs of circles, these being the stereographic projections of great circles oblique to the equatorial plane. This system has the advantage of representing the faces of crystals in the most general manner namely, by points and therefore there is no limit to the number of them that can be included in a single figure. Some very remarkable examples will be found in Descloi- zeaux's ' Text-book of Mineralogy.' CHAPTER XL PHYSICAL PROPERTIES OF MINERALS I CLEAVAGE, HARDNESS, SPECIFIC GRAVITY, ETC. THE methods employed in the second great division of mineral ogical research, or the investigation of the struc- tural peculiarities of minerals, are essentially those of experi- mental mechanics and physics. All such investigations are based upon the assumed existence of physical molecules or indivisible particles of matter, all of the same kind and similarly arranged in the same substance. Without entering into the question of the actual nature of such molecules, or 2O4 Systematic Mineralogy. [CHAP. XI. atoms, it appears to be certain that a great part of the results of observations may be explained by assuming that the molecular centres occupy the points of a reticular system, 1 and that the molecules may be considered as con- centrated upon such points as attractive or repulsive centres of force. As a working hypothesis this involves the further supposition that if they are to be regarded as having definite size or dimensions in space, these dimensions must be very small as compared with the linear distance between any two. By the substitution of such centres of force for geo- metrical points in the reticular systems of the crystallo- grapher, we arrive at the idea of the physical as distinguished from the morphological crystal, in which the consideration of molecular arrangement precedes that of form, the latter being a consequence of such arrangement. These investi- gations are especially applicable to those minerals whose crystals appear in forms common to two or more systems, such as the combination of three pairs of rectangular planes, or the regular six-sided prism, the true character of which can only be determined by a knowledge of their physical characteristics. Without anticipating what will subsequently be treated in detail, it may be generally stated that the observations upon which conclusions as to molecular structure are founded are essentially those of elastic resistance to forces tending to 1 There are two conjugate forms of the relation between the molecular centres and the points of a reticulation. They may coincide, or the molecular centres may occupy the centres of gravity of the (tridimensional) mesh. In a strictly cubical system this merely shifts the position in space of the whole reticulation. But it is quite different in the well-known case of round shot in pile. The centres of these lie on the knots of a fourfold reticulation, dividing space into octahedra and tetrahedra in the numerical ratio, I : 2, parallel to the faces of a regular tetrahedron ; but their bounding planes form a network, each mesh of which is a right rhombic dodecahedron. Some writers use one, and some the other, hypothesis. CHAP. XI.] Cleavage. 20$ alter the equilibrium of the mass, including under this head the coarser mechanical agency necessary to produce fracture, as well as the more subtle evidence derived from the trans- mission of vibratory movements, as manifested in the action of light, heat, electricity, and magnetism. In some cases we find that the effects of these agents are similar in any direction, while in others there is decided dissimilarity in different directions, and these differences are intimately con- nected with crystalline symmetry. In addition to these there are two properties common to all solids alike, and therefore independent of structure namely, density and hardness, or specific cohesive power which are of value in Determinative Mineralogy, but it will be more convenient to consider first the structural property where relation to crys- talline form is most apparent. Cleavage. The resistance opposed by cohesion in homo- geneous amorphous substances to forces tending to displace or separate their particles is similar in any direction, so that there is no reason why they should yield more readily in one way than another when strained beyond their elastic limits by forces of any kind, and therefore, when so treated they will break up into fragments of essentially irregular form. It is, however, different in the case of crystals, which, in a large number of instances, when similarly treated, show- very decided tendencies to separate into fragments bounded by planes which are for the most part related in some simple manner to the unit form of the series proper to the substance ; and when the direction of such surfaces is known, a com- paratively slight cutting or wedging strain will be sufficient to produce a separation, while the resistance in other direc- tions may be considerably greater. This property is known as crystalline cleavage, and the surfaces of separation are called cleavage planes. It is directly related to crystalline structure, but has no relation to specific cohesive power or hardness as measured by the resistance to abrasion, the hardest known mineral, Diamond, being one of the most 2O6 Systematic Mineralogy. [CHAP XL commonly cleavable, as is also Gypsum, one of the softest ; while among those of intermediate hardness, Quartz, Garnet, and Pyrites, are scarcely cleavable, but break like masses of glass or other amorphous bodies. The readiest way of determining the cleavage of a crystal is to place the edge of a knife or small chisel upon a face parallel to that of some principal form, and strike a light blow with a hammer, when, if the direction is near that of a principal cleavage, a more or less flat-faced fragment will be removed. If, on the other hand, no cleavage is obtainable in the direction of the blow, the fractured surface will be uneven and irregular, or will show traces of step-shaped structure in the direction of the true cleavage plane. Thus Fluorspar, whose crystals are chiefly cubes, cannot be cleaved parallel to that form, but yields with the greatest ease in the direction of an octahedral face. Galena has an extremely perfect cleavage parallel to the faces of the cube ; and Zinc- blende to those of the rhombic dodecahedron. Some minerals, such as Mica and Gypsum, are very easily cleavable, and may with slight effort be divided by the finger-nail, or the point of a knife, or needle, into laminae of extreme thin- ness. In the case of Mica there seems to be no limit to the capacity for cleavage, as laminae may be obtained thinner than the edge of any cutting tool that can be brought to bear upon them. In some instances clesvages may be developed in im- perfectly cleavable crystals by strongly heating and sud- denly cooling them in water. Quartz crystals, when so treated, occasionally develop faces parallel to those of the unit rhombohedron ; and under ordinary circumstances they break with a fracture like that of glass. Easily cleavable minerals, such as Salt, Galena, Fluorspar, and Calcite, usually decrepitate, or fly to pieces, when suddenly heated, the frag- ments obtained being regular cleavage forms. In the cubic system, where all three axes are physically CHAP. XL] Cleavage. 207 equivalent, a cleavage parallel to any face of a form requires the existence of a similarly perfect one parallel to the remain- ing faces, and therefore the cleavage form will be a closed one, as are also the forms produced by pyramidal and rhom- bohedral cleavage in the tetragonal and hexagonal systems ; but the prismatic and basal cleavage in these systems, as well as all those in the remaining systems, can only give open forms, and therefore, to obtain regular fragments, a cleavage in two or more directions is required in the systems of lower symmetry. As, however, these are not always found, or, when present, are of very unequal value, the general cleavage tendency of crystals belonging to these systems is to produce parallel plates from the extreme development of a principal cleavage as compared with others. This is very strongly marked in the Mica group, whose crystals are cleavable without limit parallel to the base, but are exceed- ingly tough and strong in other directions. In describing minerals with two or more cleavages, it is necessary, there- fore, to indicate their quality as well as their direction. The terms used for this purpose are highly perfect \ as in Mica; very perfect, as in Fluorspar, Barytes, and Hornblende ; perfect, as in Augite and Chrysolite ', imperfect, as in Garnet and Quartz ; and very imperfect, when only traces of cleavage can be obtained. The following are the principal directions of cleavage observed in the different systems, with a few examples of each, from which it will be seen that only the simpler forms, or those with low indices, are possible as cleavage forms : i. Cubic system. Octahedral : Fluorspar, Sal-ammoniac, Diamond. Cubical : Common Salt, Galena. Rhombic dodecahedral : Zincblende. 208 Systematic Mineralogy. [CHAP. XL 2. Hexagonal system. Pyramidal of either order . . P, P2 Pyromorphite. Prismatic of either) ^ Pr> J Apatite, Red-zinc Ore, order . .) ' ( Cinnabar. Basal ... o P Beryl, Red-zinc Ore. Rhombohedral . R The Carbonates of the Calcite group, Quartz. 3. Tetragonal system. Pyramidal of either) p p f Scheelite, Copper Py- order . .) \ rites. Prismatic of either order . . ocP, coPoo Rutile, Tinstone. Basal . . . . oP Apophyllite. 4. Rhombic system. Prismatic . . GO P White-lead Ore, Na- trolite. Macrodomatic . Pec Barytes. Brachydomatic . .Aoo Barytes. Basal ... o P Topaz, Prehnite. Macropinakoidal . GO Poo Anhydrite. Brachypinakoidal . co Pcc Barytes, Antimony Glance. 5. Oblique system. Hemipyramidal . P Gypsum. Prismatic . . co P Hornblende, Angite. Clinodomatic . :Pao Azurite. Hemiorthodomatic Pcc Euclase. Basal ... oP Orthoclase, Epidote, Mica. Orthopinakoidal . GO Poo Epidote. Clinopinakoidal . ooPoo Orthoclase, Gypsum. CHAP. XL] Fracture, 209 6. Triclinic system. Hemiprismatic . oo P oo 'P Labradorite. Hemidomatic 'P oo Cryolite. Basal . . . oP Triclinic felspar group. Macropinakoidal . co^oo Cyanite. Brachypinakoidal . oo P Axinite. Besides the cleavage surfaces, other divisional planes indicating structure may be obtained in some minerals by special treatment. Thus, by filing away two opposite edges of a cube of Salt parallel to the face of a rhombic dodeca- hedron, and screwing it up carefully in a vice, an internal fracture parallel to a face of the latter form will be produced. A similar face parallel to J J? may also be readily pro- duced in Calcite. Another method consists in the use of a conical steel point such as a centre punch, which, when placed on the face of a crystal and struck, often develops a system of cracks parallel to some principal crystallographic directions. Fracture. The characteristic appearances of the sur- faces of minerals, broken in directions that are not cleavage planes, are described by the following terms : i. Conchoidal. This is the characteristic fracture of homo- geneous amorphous substances, the surfaces presenting an alternation of rounded ridges and hollows ; it is best seen in glass and imperfectly cleavable minerals, such as Quartz, Garnet, and Ice. According to the nature of the undula- tions of the surface, it is further characterised as flat- or deep-, coarsely or finely, conchoidaL 2. Smooth, when the surface, without being absolutely plane, presents no marked irregularities. 3. Splintery, when the surface is covered with partially separated splinters in irregular fibres, as in fibrous Hematite. 4. Hackly, when the surface is covered with ragged p 2io Systematic Mineralogy. [CHAP. XI. points and depressions. This is especially characteristic of native metals. In easily cleavable minerals it is, as a rule, difficult to develop any special fracture, but it may sometimes be done by striking a fragment a sharp blow with a blunt point, as that of a rounded hammer or pestle, when traces of charac- teristic fractures may occasionally be obtained, springing across from one cleavage surface to another. Hardness. By this term is meant the resistance of the surface of a mineral to abrasion when a pointed fragment of another substance is drawn rapidly across it without sufficient pressure to develop cleavage separation. When the latter is the harder substance it will scratch the former, but when it is the softer the point will be blunted without the surface passed over being affected. As there are very considerable differences between minerals in regard to this test, it is one of the most important of their physical constants; but as there is no means of expressing the results in absolute measure, recourse must be had to an indirect method, in which com- parative hardness is measured by a scale of typical minerals. This, known as Moh's scale of hardness, is as follows : 1. Talc. 6. Orthoclase. 2. Gypsum or Rock Salt. 7. Quartz. 3. Calcite. 8. Topaz. 4. Fluorspar. 9. Corundum. 5. Apatite. 10. Diamond. Breithaupt. while preserving these numbers, proposed to interpolate Mica as 2.5 and Scapolite as 5.5, but they have not been generally adopted. The softer numbers of the scale, Talc and Gypsum, may be scratched by the finger nail, and those up to 6 by a file or the point of a knife, while Quartz and the higher numbers are all harder than steel. In testing a mineral for hardness, it is applied successively to different numbers of the scale until one is found that can CHAP. XI.] Hardness. 2 1 1 be scratched by it. The positions are then reversed, and if it is found that the scale mineral will scratch that under ex- amination, both are said to be of the same hardness ; but if it is not scratched, its hardness is said to be intermediate between that of the particular number and the next harder one. Thus, the hardness of Barytes, which scratches but is not scratched by Calcite, and is easily scratched by Fluor- spar, is said to be 3.5, or between 3 and 4. Small differences in the hardness of substances may also be appreciated by drawing fragments of about the same size over a flat file, when the harder substance will give a sharper sound than the softer one ; and if they are both of lower hardness than quartz, the softer one will leave the largest amount of powder or streak on the file. The hardness of crystals that are easily cleavable often shows very decided differences in different directions. This subject has been investigated by Exner, who found that the minimum load upon a steel point necessary to produce abrasion when moving over a cube of rock salt parallel to the diagonal of the face was one-third greater than that required when the line of motion was parallel to an edge. Similarly the face of a rhombohedron of Calcite may be more easily scratched parallel to the principal cleavage than in any other direction. It is better, therefore, in making up a scale of hardness, to use irregularly broken fragments, when they can be obtained of sufficient purity, rather than perfect crystals, but when the latter are used the trial should be repeated with different faces. For cabinet or indoor use a complete scale made up of a few pieces of each number, with the possible exception of the diamond, should be arranged in a divided box, together with a flat file not too coarsely cut ; but for travelling use the numbers 3 to 8 will be found sufficient in most cases. The nail-trimming blade of a penknife is also extremely useful in trying the hardness of minerals. In testing cut gems, care is required, especially with those p 2 2 1 2 Systematic Mineralogy. [CHAP. XI. of doubtful authenticity, not to disfigure them, and it is therefore best in the first instance to apply the test of the file cautiously to the border of the specimen, where a scratch may be hidden by the setting when the stone is mounted. Tenacity. This term is rather loosely applied in mine- ralogy to the behaviour of minerals when subjected to the action of cutting or pulverising tools, the different degrees being distinguished as follows : 1. Brittle-. The substance breaks up into fragments without extending. By far the larger number of minerals are of this kind and no special examples need be given. 2. Ductile : The substance can be cut with a knife, but crushes to powder under a hammer. The property is seen in Copper Glance and Copper Pyrites, and in a higher de- gree in Silver Glance and Chloride of Silver; the latter cuts into shavings like horn, and can scarcely be powdered. 3. Malleable : The substance can be cut into shavings and beaten out under the hammer. The soft native metals, Copper, Silver, and Gold, are examples. 4. Flexible-. The substance when divided into thin plates can be bent, and remain so without breaking, as Talc, 5. Elastic-. A thin plate of the substance when bent springs back to its original form when the strain is removed. This happens with Mica plates that are not too thin. An- other remarkable example is the elastic Bitumen, or Elaterite of Derbyshire. Although not a common property, a few instances of extreme toughness are known in minerals, such as certain varieties of Serpentine, the massive Felspars, Jade, and Saussurite. These are, as a rule, uncrystalline substances, and their toughness is in no relation to their hardness. Malleable native Copper, especially when intimately mixed with siliceous vein stuff and some varieties of Hematite and Iron Pyrites, has the same property in a high degree. Probably the uncrystalline diamond or carbonado may be considered as combining toughness with extreme hardness. CHAP. XL] Density. 213 Density and Specific Gravity. The density of a substance is the mass of its unit volume expressed in units of weight. Specific gravity is the ratio of the density of a substance to that of another assumed as a standard, or of the weights of equal volumes of both. Water at the ordinary temperature of air, or at that of its maximum density, is usually adopted as the standard substance. In the metrical system the same number expresses the weight of a cubic centimetre in grammes, or both density and specific gravity. The term * density ' is generally used by French minera- logists, and ' specific gravity ' by those of England and Ger- many, in describing minerals. Rammelsberg, however, em- ploys the equivalent expression ' volume-weight.' Specific gravity is one of the most important factors in determinative mineralogy, as it is found to be constant or variable within small limits in different varieties of the same species, while the differences between different species is often very considerable, the observed range being from 0.75 0.90 in some liquid hydrocarbons, to 21 or 22 in the metals of the platinum group. It will, however, be seen in examining a classified list of the specific gravities of minerals, such as those published by Websky and the Bureau de Longitude, 1 that species of analogous composition and constitution are generally near together, and that upon this characteristic a certain rough grouping is possible, as in the following examples : 0-5 1.5. Most fossil resins, Petroleum and Bitumen. 1.02.0. Coal Lignite and carbonaceous minerals gene- rally, many hydrated Alkaline Sulphates and Borates, Nitre. 2.0 2.5. Sulphur, Graphite, Nitrate of Sodium, Salt, Gyp- sum, and most Zeolites. 2.5 2.75. Quartz, the Felspars, Talc, Serpentine, Calcite Emerald. 1 These, with other admirable tables of the physical constants of minerals, should be in the possession of every student. They are contained in the Annuaire for 1876 and later years. 214 Systematic Mineralogy. [CHAP. XI. 2.8 3.0. Aragonite, Magnesite, Dolomite, Tremolite, Wollastonite, Mica. 3.0 3.5. Apatite, Fluorspar, Epidote, the Pyroxene and Hornblende groups, Tourmaline, Olivine, Axinite, Diamond, Topaz, the Sulphides of Arsenic. 3.5 4.0. Spinel, Corundum, Siderite, Limonite, Strontia- nite, Celestine. 4.0 4.5. Rutile, Zircon, Barytes, Witherite, Zincblende, Copper Pyrites, Magnetic Pyrites. 4.5 5.5. Hematite, Magnetite, Iron Pyrites, the Ruby Silver Ores, and many compound sulphides not containing Lead. 5.6 6.6. Arsenic, Arsenical Pyrites, Oxides of Manga- nese, and many compound sulphides contain- ing Lead and Silver. 6.7 7.9, Antimony, Sulphide and Carbonate of Lead, Sulphide of Silver, Tinstone, and Pitchblende. 8.0 9.0. Tellurides of Gold and Silver, Sulphide of Mer- cury, and Copper. 9.0 u.o. Bismuth, Silver, and Palladium. 12.0 15.0. Mercury, and Amalgams. 15.0 20.0. Gold, Platinum, and several of its associated metals. Above 20. The native alloys of Osmium and Iridum. Metallic Iridum, when perfectly purified from the metals with which it is usually associated, is the densest substance known, having a specific gravity of 22.4. The determination of specific gravity is in principle very simple, the substance being first weighed in air, and then in water, the difference between the two weights gives the weight of an equivalent volume of water, and the quo- tient of the original weight by the difference will be the specific gravity. An exact determination is, however, a matter of considerable nicety, and involves the use of deli- cate balances, such as are only found in laboratories. The CHAP. XL] Specific Gravity. 215 general details of manipulation will be found in the larger treatises on practical chemistry. When the substance con- tains cavities, it is necessary to powder it before taking the specific gravity, and in such cases the determination is made by estimating the amount of water displaced by a known weight of the powder from a bottle that has been exactly filled with water and weighed at a standard temperature. As most minerals enclose more or less of hollow or air- filled spaces, it is not remarkable that they should, as a rule, be denser when powdered than in the solid state. In irregular aggregates the difference is often very considerable. Thus, the spongy substances known as float-quartz and pummice in masses are apparently lighter than water from the large amount of air enclosed, although when pulverised their true density is found to be between two and three times as much. In the determination of the specific gravity of mineral masses of large size and known weight, the method of gauging the volume of water displaced may be conveniently used. One of the best is that given by Mohr, which is sus- ceptible of considerable accuracy. The gauging vessel is a glass cylinder, which is filled with water to a standard point formed by a needle projecting from a slip of wood across the top, the exact level being attained when the front of the needle and its reflected image in the water coincide. The weighed substance is then carefully lowered into the cistern, when it displaces its own volume of water, with a correspond- ing rise of the surface level. The amount of displacement is measured by drawing the water into a graduated tube or burette until the original level is restored. A convenient size of graduated tube is the ordinary alkalimeter used in volumetric analysis containing 1,000 grains, and divided into 5 -grain spaces, or an equivalent one with metrical divisions. The level of the water may be adjusted with great nicety by a simple valve formed of a piece of glass rod inserted in the indiarubber delivery tube, the aperture of which can be 216 Systematic Mineralogy. [CHAP. XL varied by slight pressure of the ringer upon the tube. This method, which has the advantage of not requiring a correc- tion for temperature, is well adapted for taking the specific gravities of Coal. Limestone, and similar substances ranging from 2 to 3, which can be used in fragments of about half a pound weight. Jolly's spring balance, another contrivance for the ap- proximate determination of specific gravity, is recommended as being expeditious in use, fairly accurate, and dispensing entirely with the determination of absolute weight. It con- sists essentially of a pair of scale pans suspended one above the other ; the upper one is attached to the end of a coiled steel spring, and the lower one is immersed in a cistern of water standing on a bracket, whose position can be adjusted by a sliding movement worked by a rack and pinion. The face of the upright bar to which the arm carrying the spring is attached has a silvered mirror with a scale of equal parts engraved upon it fixed to it in front, and the upper scale pan carries a pointer. When in use the level of the water vessel is so adjusted that the lower scale pan may be freely immersed when a reading is taken by bringing the pointer into coincidence with its reflected image in the glass scale. The mineral is then placed in the upper scale, whereby the spring is distended to a certain point, which is determined by a second reading, which when deducted from the first measures the weight in air. By removing it to the immersed pan, the strain is diminished, and the pointer consequently rises to an amount determined by a third reading, which when subtracted from the second measures the loss of weight in water. It is of course essential to accuracy that the spring should deflect equally for equal weights applied, or it must not be strained to anywhere near its elastic limits, while at the same time it must be sufficiently free to work to move through a measurable distance by a moderate change of weight. The instrument, which is very highly spoken of by Von Kobell, is made at Munich at a cost of 24^. CHAP. XL] Sonstedfs Method. 217 The density of large masses of an approximately regular figure may be roughly determined by weighing them and calculating their cubic volume from their measured dimen- sions. The specific gravity is found by dividing the weight by the contents in cubic feet multiplied by 62.4 Ibs., or the weight of a cubic foot of water. The reverse operation of calculating the weight of a measured mass from the known specific gravities of its components is often useful on the large scale in estimating the probable yield of mineral de- posits, and the student may therefore be recommended to become familiar with it by practice. Specific gravity is in another class of approximate methods determined by immersion in a fluid of known density, when the substance, if heavier, will sink, but if lighter, it will float. This is specially useful in the discrimina- tion of cut stones ; the fluid used for this purpose, called after the inventor, Sonstedt's solution, is made by saturating a solution of iodide of potassium in water with iodide of mercury, which gives a liquid having a maximum density of 2.77, and which, when diluted with water, mixes without sensible change of volume, and therefore the specific gravity is proportional to the amount of the two fluids in the mix- ture. This is also useful as a means of separating mixed minerals for analysis, when they are so intimately associated as to be incapable of separation by hand ; as, for instance, the Felspars and lighter silicates in a rock may be roughly divided from the denser ferrous and magnesian ones by a solution of a specific gravity of about 2.75, when the first mineral will just float, while the latter will sink readily. The chief drawback to the use of this substance is in its ex- tremely poisonous character, and it can therefore be scarcely recommended except for laboratory use. An extension of the same principle adapted for the separation of the denser constituents of mineral sands has been proposed by Breon , who uses molten chloride of zinc or lead, or mixtures of both. These give fluids whose 218 Systematic Mineralogy. [CHAP. XII. density ranges from 3.0 to 5.0. As these salts can be melted in glass tubes, and are alike soluble in water, the separated substances can be obtained in the pure state with- out much trouble. They have been applied in the separa- tion of Tinstone, Rutile, Magnetite, and other heavy minerals from Quartz, and silicates in the microscopic investigation of sands. The same method is applied on the large scale in the separation of Gold from Galena and Iron Pyrites by a fluid of intermediate density, namely Mercury, in the Hungarian Gold-mill, although in this case the result is not quite so simple, the metal being to some extent, at any rate, dissolved in the separating fluid. CHAPTER XII. OPTICAL PROPERTIES OF MINERALS. UNDER this head are included the general and more apparent properties of colour, lustre, and translucency or opacity, which are common to all minerals alike, as well as others, requiring special methods of investigation applicable only to such minerals as are transparent. These include the deter- mination of the quality, and numerical values, of the re- fraction constants in particular directions, the so-called axes of optical elasticity, and the relation of these axes to the principal crystal! ographic lines, and the phenomena of pleochroism. Now, although the practical application of these methods is, as a rule, beyond the power of beginners, as they involve the use of exact and somewhat expensive apparatus and carefully prepared crystallographic material, the results obtained are in many cases of such great interest and importance, especially in the determination of crystallo- graphic characters from fragments of minerals in forms not otherwise definable, that a brief indication of their character CHAP, xii.] Wave Motion. 219 and the reasoning upon which they are founded may not be out of place here. The reader is referred to the treatise on ' Physical Optics,' by Mr. Glazebrook, for the more com- plete discussion of the subject. In order to understand how light may be used as an in- dicator of the structure of minerals, it will be necessary to consider as shortly as possible certain common optical terms, and first of all the nature of light itself. This, according to the undulatory theory of Huyghens, which forms the basis of all modern optics, is a consequence of the vibratory move- ment of molecules in an intangible fluid, or ether, which is assumed to fill the intermolecular space in all material substances solid, liquid, or gaseous alike ; and by its con- tinuity forms a medium for the transmission of such move- ments from their point of origin, the luminous source, to the optic nerve, where they become apparent as light, the cha- racter of the movements being essentially similar to those of waves traversing air, water, or any other substance. If a system of molecules in stable equili- FIG brium be supposed to be arranged in a % c d e line at regular distances apart, as in J /\~* * * * fig. 3 1 9, a straight line passing through ., them will include all their positions of repose. If, however, one of them be by some force dis- placed sideways, as for example a to a', the tendency of the attractive force of the adjacent particle b will be to bring it back to the original position, and it will after a time return, arriving at a with sufficient impetus to carry it on to a point at the same distance as a' on the other side of the line of equilibrium, and so on backwards and forwards, a periodical vibratory motion, analogous to that of a pendulum, being set up in a plane determined by the direction of the moving force. When the particle a is displaced it will exert a disturbing influence over the adjacent one , tending to draw it to itself as indicated by the diagonal arrow, which influence will be opposed by the attraction, of c acting along 22O Systematic Mineralogy. [CHAP. XII. the horizontal arrow. The movement of b will, there- fore, be in some intermediate direction, such as that of the vertical arrow, which it will swing parallel to a, and so on for any molecule in the train ; the commencement of the movements being progressive, each molecule starting when the preceding one has traversed a certain dis- FIG. 320. tance on its journey. If the particle a in fig. 320 be supposed to have just completed one vibration, having successively occupied all the positions indicated by the points o to ii in the order shown by the arrows, the mole- cule , which started when a was at i, will at the same moment be at n, moving downwards towards the centre line; c will be a little farther off at 10, but moving in the same way; d will be at the greatest distance above the line at 9 ; the points next following, e and f t will be travelling upwards, g will be passing through the central position, and so on to 0, which will be just about to com- mence its first movement. The points a to o therefore represent the positions of a contiguous series of vibrating molecules, supposing their movement to be stopped simul- taneously, and together they represent the successive posi- tions passed through by each one or the phases in its move- ments : the line joining them will be a wave line, having a crest at d, a hollow at /, and nodal points at a, g, and o, the horizontal line passing through them being the line of propagation of the wave. The total distance from a to o is called a wave-length ; ag and go are half, and ad, dg, g l % CHAI-. XIL] Wave Motion. 221 and lo quarter wavelengths, quantities that are usually represented by the symbols X, _, and -, Points in the 2 4 series which are half a wave-length or whose difference of phase = - are vibrating in opposite directions, as shown by the arrows in fig. 320. The surface containing the paths of the molecules is termed the plane of vibration, and the direction in which the wave movement progresses, the direction of propagation. In the simple case represented these are at right angles to each other, or the wave is a transverse rectilinear wave. The distance #3-09, termed the amplitude of vibration, determines the intensity or brilliancy of the light, and, as in the case of the pendulum, the time of vibration is constant, whether this distance be long or short, the movement of the particle when at its maximum speed being quicker in the former than in the latter case, but the total duration of one vibration is the same in either. The wave-length, on the other hand, being the distance travelled during the time of a single oscillation, will depend upon the period of the latter being longer or shorter, according as the motion is slower or faster. Upon these differences depends the colours of the light produced; that having the slowest period or greatest length of wave is called red, and that of the most rapid movement and shortest wave-length is violet, those of the colours orange, yellow, green and blue being of intermediate velocities and wave-lengths. When a luminous source pro- duces waves of only one length, the light is said to be homo- geneous or monochromatic, but when waves of different lengths are originated simultaneously the light is hetero- geneous, mixed, or white. The latter is the character of ordinary sunlight, monochromatic light being characteristic of the vapours of elementary bodies when intensely heated sodium producing yellow, thallium green, lithium red, &c. The light of candle, oil, or gas flames is also mixed, but 222 Systematic Mineralogy. [CHAP. XIL differs from sunlight in the relative proportions of the con- stituent colours, containing less blue and more yellow, some portion of the latter coloured light being of a kind never seen in sunlight. When a line of molecules vibrating in any direction is subjected to a new impulse acting in some other direction, its movements will be resolved according to the parallelo- gram of forces, and a new wave will be set up, compounded of both. In such cases the two waves are said to interfere, and the new one to be an interference wave. The effect produced by such interference may vary very considerably in character, according to the relation of the constituent waves. In the simplest case, where both are moving in the same direction, their planes of vibration being also the same, if one differs from the other by one or more complete wave- lengths, the amplitude of the new wave will be augmented, it being equal to the sum of those of the original ones. When the difference of phase is one half or any uneven number of half wave-lengths, the positions in the fresh wave will correspond to similar ones in the second, but their directions of motion will be dissimilar. If therefore the two waves be of different amplitudes, the phases of that pro- duced by their interference will correspond to those of the larger one, but its amplitude will only be equal to their difference that is, the intensity of the resultant wave will be considerably reduced. In the special case where both waves are of the same amplitude, that of the resultant = o, or the wave motion is entirely destroyed. For differences of phase other than the above, the interference waves produced are of different phases and amplitude. Thus, for waves cf similar amplitude and quarter wave-lengths apart, the aug- mented wave of interference has a difference of phase = J X in advance of the first, while for three-quarter wave-length distance it is retarded by J X. The wave motions originated by a luminous point are propagated uniformly in every direction from it as a centre, CHAP, xil.] Huygheris Wave Construction. 223 so that at any particular instant a point in any one wave will be in exactly the same condition of vibration as one in any other at the same distance from the centre, or, in other words, points of similar phase on the wave paths will all be equidistant from the centre, and a surface including the whole of them will be a sphere whose radius is the common distance. This surface is known as the wave surface or wave front, and the radii representing the wave paths as light rays ; any small portion of the surface and the rays determining it constitute a beam of light. As the points included in a spherical wave surface all commence their vibrations simultaneously, they will affect FIG. 321. the equilibrium of the particles beyond them in a similar manner, so that each may be considered as originating a particular wave. In fig. 321 K K' represent a portion of the surface of a wave originating at c, and P p' a similar portion at a later instant. The point B in the second will be reached by movements coming not only from A but from every other point of the surface K K'. To determine the effect of such 224 Systematic Mineralogy. [CHAP. XII. movements, suppose a series of circles LI/, p p', R R', &c., to be described upon the surface K K' like the parallels of latitude on a globe about CAB as a polar axis ; all points in the circumference of any one of these, such as p and P', will be equidistant from B, but at a different distance from those in any other circle. The waves proceeding from different circles will therefore have different distances to travel, and as their period and rate of progression are the same, they must arrive at B in different conditions of vibra- tion. If further we suppose for every one of the circles another to be constructed of exactly half a wave-length greater distance from B, the undulations proceeding from each of them will by their interference be extinguished, and there- fore have no effect at that point. Such circles can be de- scribed for every portion on the surface except A, and therefore the whole of the waves proceeding towards B will be extinguished except those on the line A B, and similarly only those on the prolongation of the line c P and c P' will reach/ and/' respectively. From this it follows that although every point on the wave surface is the centre of a new wave, the motion of the latter will only be apparent at the point where it touches the surface forming a common envelope to the whole of the waves of the same class. This is, in the case in question, the spherical surface pp'> or the wave front corresponding to the later period. Upon this principle is founded Huyghens' wave construction, whereby the wave front at any given moment may be found if the originating surface and the rate of propagation be given. This is done by describing upon points of origin, such as A p and P' semicircles whose radii are proportional to the velocity of wave transmission at those points, and drawing through the points / B/' the common enveloping surface, which is the second wave front as before. If the point of origin of the wave motion be situated at a great distance, the rays contained in a beam of light may be considered as sensibly parallel, and the included portion of CHAP. XII.] Wave Motion. 225 the wave front as a plane surface. This is termed a plane wave. Ordinary solar or daylight is of this character, the distance of the luminous centre, the sun, being so great that a beam of light must have a diameter of nearly a thousand miles for the radii limiting it to diverge by an angle of one second from parallelism, so that in most instances of natural illumination the beam may be considered as a cylinder made up of parallel rays. The power of transmitting wave movements uniformly in one or more directions supposes the medium possessing it to be homogeneous or uniform in molecular structure. If the structure is such that the rate of transmission is exactly the same in any direction, the medium is said to be isotropic. This represents the most complete and symmetrical kind of homogeneity. In the second kind of homogeneous media, known as anisotropic or heterotropic, the rate of progression of wave movements varies with their direction, but is constant for any similar direction. In heterogeneous or non-homogeneous media, on the other hand, there is no relation between direction and velocity of wave propagation, the latter being subject to variation even when the former is constant. The movements in such a medium can only be reduced to order by supposing them to be made up of portions of homo- geneous substances of different properties, and treating each one separately. The velocity of propagation of wave movements changes when they pass from one medium into another, the different velocities being related to each other according to a funda- mental proposition in mechanics, directly as the square roots of their coefficients of elasticity e, and inversely as the square roots of their densities ; or if c and c 1 represent the different velocities, their ratio will be expressed by - \/~d : v? 226 Systematic Mineralogy.' [CHAP. XII. When e is nearly of the same value in the two media the velocity will diminish as d increases, or generally the velo- city of propagation is less in the denser medium. In passing from one medium to another of different density, the entire wave movement is never completely transferred. If the second medium has the lesser density, the energy of a vibrating molecule at the limiting surface of the first will be more than sufficient to establish an equality of movement in a similar molecule in the second, and there- fore part of the original motion will be returned in the form of a backward wave, whose phase at any point will be exactly the same as that of the original wave would have been at a point similarly situated in front. On the other hand, if the second medium be the denser, not only will the original movement be entirely absorbed, but a return movement will be set up in the molecules of the first in the form of a wave of exactly opposite phase to the original one. In either case the change is accompanied with the same result a portion of the original movement only enters or is transmitted by the second medium, the remainder being diverted or reflected at the surface of con- tact. This proposition holds good in all cases. Whenever light passes from one medium to another, a portion is in- variably lost by reflection, and the intensity of the trans- mitted beam is diminished, the loss increasing with the difference in density of the two media, Where the difference is so great that none or very little of the light enters, the denser substance is said to be opaque ; but if all, or nearly all, is transmitted, it is transparent or translucent. No substance is known to be either completely opaque or per- fectly transparent, as the light reflected by the most charac- teristic examples of the former, properly polished metal surfaces, is always modified to some extent, showing that penetration has taken place, though only in a very small degree ; and those that can be reduced to a sufficiently thin laminae such as gold and silver, are actually found to be CHAP. XII.] Reflected Wave. 227 translucent, and the most limpid translucent substances in sufficiently thick masses sensibly dimmish the light passing through them. The light so lost is said to be absorbed. When the absorption affects waves of all lengths similarly, the result is only a diminution in the brilliancy of the light ; but selective or unequal absorption of particular kinds of waves alter the character of the transmitted light, giving rise to the phenomena of colour. The nature of the wave surface produced by reflection may be found by the construction given in fig. 322. If njf y/ 1 x r M \ a t. T S>\ \ \ \ / / / ,--: \>:\^ \ \ X - / / -' /'"'''' x O\. 4 \""V\-h-7""r'7'// ^A\ i //^ Mr > x > \ * x ^ MN in fig. 322 represent the surface of contact of two dis- similar isotropic media, and A the point of origin of wave- motion in the first, the arc B B' will be the section of the spherical wave-front at the moment when the ray Aa meets MN ; the more divergent rays Aa l to A# 4 on either side arriving at the same surface progressively at later periods. At each of these points a reflex wave-movement Q 2 228 Systematic Mineralogy. [CHAP. XII. originates, that from a , being the earliest, arrives at the point e at the same moment that those from a\ a 2 a 3 and# 4 reach e l e 2 e 3 and e respectively, and the outer rays A M and AN have just arrived at the surface MN. If, therefore, semicircles be described from a Q with the radius a e Q , from 0, with a i *u from a 2 with <2 2 e z> etc> > the curve forming their common tangent, the circular arc M e Q N, whose radius is C0 + a Q e , will be the wave-front required. If an arc be struck with the same radius from A passing through M N, it will represent the position that the original wave- front would have reached had the motion been continued in the first medium. Hence we see that the direct and reflected wave- surfaces are exactly similar in dimension, but reversed in position. The direction of reflection in the central ray A a Q is the same as that of its arrival or incidence i.e., normal to the reflecting surface ; but in the rays on either side these direc- tions differ, making longer angles with each other as the obliquity of incidence increases. If through each of the reflecting points, a l to # 4 , a line be drawn normal to M N, it will be symmetrical to the two rays at that point, or they will make equal angles with it and consequently with the reflecting surface. This is ex- pressed in the following pro- position : The angle of inci- dence is equal to the angle of reflection. The direction of rays pass- ing into the second medium is found by the construction shown in fig. 323. If AJ A 2 A 3 A 4 represent rays coming from a luminous source at a great distance, and therefore parallel, the surface represented by the line a l B normal to them will be the plane wave- front at CHAP. XII.] Refraction. 229 the moment of incidence of the ray A I a l with the surface limiting the two media M N. Supposing the density of the second medium to be such that the velocity of wave propa- gation in it is only one-half of that in the first, the semi- circles described about a l a% and a 3 , with radii corresponding to half the distances E I N, B 2 N, and B 3 N respectively will represent the positions of waves set up at these points at the moment when the most distant ray, A 4 , arrives at the point M, and, according to the principles previously laid down, the plane represented by the line NRj, which is tangent to all of them, will be the wave-front of the whole beam. As this is also a plane wave, the parallelism of the rays is not altered, but their direction, indicated by the lines normal to R! N passing through a\ a 2 and a 3 is changed, the individual rays being bent or refracted towards a normal to the plane of incidence passing through the same points. If, on the other hand, the first be the denser medium, the velocity of propagation will be doubled in the second, and the rays, as shown in fig. 324, will be bent away from the normals to the plane of incidence, or the refractive angles will be in- creased. The amount of the refraction is measured in fig. 325 for the rays A I o and o R t by the lines A! n l and RJ 3 , which are to each other the sines of their respective angles with the vertical line nop y or calling the larger angle * and the smaller one r* FIG. 324. sm. t sin. r n. This quantity, which is constant in an isotropic sub- stance for any angle of incidence, and corresponds to 230 Systematic Mineralogy. [CHAP. XII. the ratio of the respective velocities of wave transmission or , is termed the quotient ', exponent ', or index of refraction, v \ the last or the equivalent term, refractive index, being most generally used. The value of n varies for the greater number of transparent isotropic minerals between i -3 and 1 7, that of air being taken as unity, the maximum of 2*4 being attained in Diamond. A few anisotropic substances such as the Ruby, Silver Ores, and Cinnabar, afford examples of still higher refractive indices, which reach or even slightly exceed 3*0. A more oblique ray, such as A 2 o in fig. 325, whose angle of incidence is z', is refracted in the denser medium in the direction o R 2 , or has the larger angle of refraction r 1 , but the ratio of the quotient of their sines is not altered i.e., FIG. 325. sin. sin. i = n R3 within the first medium. sin. r 1 sin. r as before ; or the deviation of the refracted ray varies with the angle of incidence. The maximum obliquity of inci- dence is in the direction M o, or parallel to the dividing surface, in which it is clear there can be no refraction, as the path of the ray is entirely Also, if the ray meet the plane of incidence at right angles, or /= o, the deviation of the refracted ray r also = o, or there is no deviation, and the ray follows the same direction in both media without refraction. For any other direction sin. / is always greater than sin. r, as sin. r = -. If the first medium be denser than the second* the velocity of trans- CHAP. XII.] Total Reflection. 231 FIG. 326. mission will be greater in the latter than the former, the wave-front of the transmitted beam R! N, fig. 326, will make a longer angle with M N than that of the incident one M B I} the angles of refraction of the individual rays will be larger than their angles of incidence, and will increase with the deviation of the latter from the vertical, but more rapidly, as will be apparent by turning fig. 325 upside down and considering the positions of the refracted and incident rays to be transposed. In this case sin. r=m sin. /', and as sin. 90 = i when sin. / = , r = 90, or the refracted ray will coincide with the plane of separation. The construction for the special case n = 2 and r = 90 is seen in fig. 326, where the vertical line through N is the tangent plane or wave-front of the refracted beam, and M n the direc- tion of the refracted ray. The radius of the circle M K = 2 BJ N, and A! n = sin. /= 1 or /= 30. For this particular value of the refractive index, therefore, all rays whose inclination to the normal to the plane of incidence is thirty degrees or above are incapable of passing into the less dense medium, but are reflected at the surface of separation. This property is known as total reflection, and the minimum angle at which it takes place is the limiting angle of refraction, or the angle of total re- flection. When the denser medium is limited by parallel surfaces, the angle of incidence of the refracted ray at the second surface will be the same as that of its first refraction, or i' = r (fig. 327), and therefore its angle of refraction, r', on emergence into the first medium, will be the same as that of original incidence, t, or, in other words, the ray will resume K u 232 Systematic Mineralogy. [CHAP. xil. its original direction. A ray of light is not therefore altered in direction by passing through any transparent substance bounded by parallel surfaces. If, however, the surfaces are not parallel, the ray at emergence will be more or less changed from its original direction, supposing the in- clination of the surfaces not to be sufficient to produce total reflection. The amount of this deviation in direction may be calculated when the inclination of the limiting sur- faces and the refractive index of the medium are known, FIG. 327. FIG. 328. P" and conversely from the measured deviation produced by a refracting body or prism of known angle, the refractive index of the substance composing it may be found. The principle of this method is shown in fig. 328, where PP'P" is the section of a prism of the refractive angle n in a plane perpendicular to its refracting edge P. The ray A M incident at the angle / on P P' is refracted at the angle r towards M', where it becomes incident at the angle /' upon P P", is refracted at the angle r* 9 and follows the direction M'R CHAP. XII.] Minimum Deviation. 233 after emergence. If the line M' A' be drawn parallel to A M, the angle A' M R = I will measure the total deviation of the ray from its original course effected by the prism. Next, from the point M' draw the lines M' A' and M' T, the first parallel to the original direction of the ray A M, and the second to the normal n M, produce the common path of the first refracted and second incident ray M M' towards K, and continue the normals n M n' M' to their point of intersection at s. The angles formed at M' will have the following signification : n' M' K = i' = K M' T = r, A' M' T = n M A = i, n' M' R = /, and A' M' R = 3, or the deflection of the ray from its original course by the prism. But A' M' R = c A' M' K + K M' R = (T M' A' T M' K) + (RM'^' ;Z'M'K) or 3 = (/- r) + (r f - i') = i + r 1 - (r + i'). In the triangle PMM' a + (90 r) -f- (90 - f) = 180, consequently, a = r+ i' and c = /" + r 1 a. Fig. 328 is so constructed that t-=r' and r = i, for which case d = 2 / a = 2 r' a, and has a minimum value. If one angle is greater than the other, e.g. if / = r 1 + /3, or r 1 = / + /3, the deviation will be in the first case c) = 2 r' + /3 a, and in the second I = 2 i + /3 , either of which is obviously greater than 2 z* a. This latter, therefore, is the minimum angle of deviation, and is produced when the ray makes equal angles with the faces of the prism both at incidence and emergence. Further, as . a + a 3 = 2 t - a, t = ? ; and as 234 Systematic Mineralogy. [CHAP. XII. and sin." 2 or the refractive index of the substance, if isotropic, is equal to the sine of half the sum of the refractive angle of the prism and the angle of minimum deviation divided by the sine of half the angle of minimum deviation. This is the simplest and most direct method of determining refractive indices, as two faces of a crystal may be used as a prism if their angle is not too large ; from 40 to 70 are the most convenient angles when the refractive indices are moderately high (1-5 to 1-7). In all cases the determina- tion must be made for particular parts of the spectrum, either by using monochromatic light or by observing the deviation of the principal dark lines in the solar spectrum. For the former purpose the flame of a Bunsen's burner coloured red by a bead of sulphate of lithium ignited at the end of a platinum wire, yellow by chloride of sodium, or green by sulphate of thallium, is the most convenient kind of illumination. The angle a is measured with a reflecting goniometer and the deviation by the same instrument arranged to allow the prism and telescope to move inde- pendently. Another method of determining refractive indices origi- nally proposed by the Due de Chaulnes in 1767, and re- cently systematised and extended by Sorby and Stokes, 1 consists in measuring the displacement of the focal point of a lens or compound microscope adjusted to a distinct vision of an object in air when a parallel plate of known thickness of a denser substance is interposed between the object and the lens. The effect of this interposition is to bring the 1 Proc. Royal Sac. xxvi. p. 384. Journal of the Royal Micro- scopical Society, 1878. CHAP. XII.] Single Refraction. 235 apparent place of the image nearer to the lens, so that the latter has to draw back through a small distance, d, whose amount depends upon the thickness of the plate /, and its refractive index n, if isotropic, the relation of these quantities being expressed by the formula, n = r? This method has the advantage of being applicable to very thin and minute crystals or parallel plates, and is there- fore useful in the microscopic study of rocks and minerals, but the measuring apparatus, which may be either a vernier scale or micrometer screw, but preferably the latter, must be made with great accuracy, as the quantities to be measured are usually very small. In an isotropic substance the velocity of propagation of wave motion being alike in all directions, the refractive index will also be constant in any direction for light of any particular colour, being lowest for the red and highest for the blue end of the spectrum. Such substances are therefore said to be single-refracting or monorefringent. They include all transparent homogeneous gases and liquids, the latter with a few exceptions ; and among solids, both those that are amorphous and those crystallising in the cubic system, all other crystallised substances being aniso- tropic. The properties of a ray of ordinary light are similar in any radial direction about its line of transmission considered as an axis ; it appears to vibrate simultaneously in these directions. There is, however, good reason to suppose that this is not actually the case, and that the directions or azimuths of vibration change continuously, but so very rapidly that the changes are not perceptible to the eye. Upon this supposition the plane of vibration of a molecule, o, fig. 329, would be changed by a small angle at each pul- sation, or if the limit of its first vibration is #, on the line o N, that of the second will be somewhere to the right of N (tf the change is in that direction), that of the third, some- Systematic Mineralogy. [CHAP. XII. FIG. 329. what further on the same side, and so on, until the direction of vibration is on the line oe. The motion in any interme- diate plane, such as o c, may be considered as the resultant of two forces acting simultaneously, on towards N, and oe towards E ; and if the velocity be the same in any direction the amplitude of the vibration in oc will be exactly the same as those in the NS or EW planes, and consequently the intensity of the light will be unchanged, what- ever may be the azimuth of its plane of vibration. These rela- tions, however only hold good for isotropic media, as in ani- sotropic ones the velocity of wave propagation varies with the direc- tion, the differences being greatest between directions at right angles to each other. Supposing NS to be such a direction of maximum velocity, E w will be that of the corre- sponding minimum, and the molecule o will vibrate in either of these with its full proper intensity ; but when in the direction o c the intensity will not, as in the previous instance, be that due to the components on^oe, as these impulses having different velocities cannot arrive simultaneously at c, but each will act independently, and consequently the ray will be resolved into two, whose intensities will vary with their azimuths, one of them having its greatest intensity in the direction 0N, diminishing to on in the direction oc, and to nothing in o E, or in proportion to the cosines of the azi- muths, and the other having its maximum in o E and dimin- ishing to nothing in o N or o s, or as the sines of the azimuth ; but as these changes take place too rapidly to be followed by the eye, the resulting effect is the production of two rays, each of half the intensity of the original ray of ordinary light, and vibrating in one plane which is at right angles to the corresponding plane of the other ray. Such rays are said CHAP. XII.] Double Refraction. 237 FIG. 330. to be polarised in planes at right angles to their planes of vibration, and are no longer freely transmissible in any direction like those of ordinary light, and as a consequence of their differences in velocity they will have different refractive indices, or the substances producing them are doubly refracting or birefringent The general phenomena observed in a doubly refracting crystal are best seen in Calcite, in which the property is very strongly developed. 1 If abed (hg. 330) be the section of a cleavage rhombohe- dron so placed that a b and c d are the shorter diagonals of a pair of faces, a c and bd polar edges, and ad the principal axis, a ray of light, ;-, incident at n in a direc- tion normal to a b, will be divided into two parts ; one following the law of ordi- nary refraction will pass through unchanged towards o, while the other will be refracted, and assuming the original direction on emergence at / will travel in a direc - tion parallel to the first, but at some distance from it, to- wards e, so that by looking through the crystal in the direction no towards a brightly illuminated object on a dark ground, two images of it will be seen, one at o and the other at e. The first of these rays is called the ordinary and the second the extraordinary ray. If the direction of incidence be oblique, both rays will be refracted, but in different degrees : the ordinary ray following the law of sines and having a constant index, v and the hyperbolas are bordered with red within and blue without, is seen in the allied minerals Aragonite and White 1 The reader is recommended to colour these and the succeeding diagrams, when the differences will be more readily appreciated. CHAP. XII.] Rhombic Dispersion. 267 Lead Ore. In the latter, owing to the high specific refrac- tive and dispersive power, the phenomenon is very strikingly shown, the brushes appearing as broad red and blue stripes, without an intervening dark space. In a small number of crystallised substances belonging to the rhombic system, of which Brookite is the principal natural example, the axes of differ- FIG. 359. ent colours, though having the same median line, are not only widely dispersed, but lie in different planes, those for red. being in the plane indicated by $ one bar of the cross, while those for green are in the other bar at right angles to it. In white light these crystals give very peculiar interference figures, like fig. 359, the rings being replaced by a series of curves symmetrical to the cross upon a parti- coloured ground, the maximum of red being about the horizontal bar of the cross, and that of green near the vertical one. If, however, this is viewed in homogeneous red light, it is resolved into a series of lengthened oval rings, whose poles are at ;-, /, while with green light another series, more nearly circular in form, are seen, having their poles at r, r'. The same phenomena are perhaps more strikingly seen in the triple Tartrate of Potassium, Sodium, and Ammonium, known as Sel de Setgnette, which is not only more readily obtained than a crystal of Brookite, but, being colourless, shows the colours of the field more vividly. With a section of this salt the red and blue ring- systems may be seen superposed, when the middle part of the spec- trum is extinguished by a tolerably thick cobalt blue glass. In the oblique system, the optic axes corresponding to different colours have not necessarily a common median line, and therefore a new element is presented for conside- 268 Systematic Mineralogy. [CHAP.- XII. ration, namely, the dispersion of the median lines. This may take place in three different ways, each having a more or less characteristic effect upon the interference figures, and to these the names inclined, horizontal, and crossed dispersion have been applied by Descloizeaux, who first systematically investigated them. The first case, that of inclined dispersion, arises when the optic axes lie in the plane of symmetry dispersed at in- clinations which are usually small, their median lines making angles with each other varying from a few minutes to one or two degrees. Of the latter, therefore, only one, usually that for the middle of the spectrum, can coincide with the normal to the plate, and consequently those for the other colours will be unsymmetrically placed right and left of the centre. When such a crystal is placed with its axial plane parallel to that of one of the Nicols, there will be often seen a marked difference in the shape and size of the polar rings, one being nearly circular, while the other is a lengthened FIG. 360. FIG. 361. ellipse, as in fig. 360, the vertical bar of the cross is nearer to the latter than the former, and the colours are generally brighter about one pole than the other, the same order being observed when the plate is turned through 45, as in fig. 361 ; but the colours of the brushes in the latter position may be either opposed, one being blue inside and red outside, and the other blue outside and red inside, or vice versa, or similar, the phenomena being complicated by differences in the dis- CHAP. XII.] Oblique Dispersion. 269 persiori of the optic axes, as well as by that of their median lines. This kind of dispersion is best seen in Diopside, Gypsum and other minerals having large angles between their optic axes ; the contrast between the form of the polar rings is seen in the artificial salt, Platino- Cyanide of Barium. When no polar rings are apparent, as in the small angled Felspars, there is only a slight difference of brilliancy of the colours of the brushes, which is not clearly appreciable without practice in observing. When the plane of the optic axis is perpendicular to the plane of symmetry and parallel to the orthodiagonal, the latter may be either perpendicular to the first meridian line, or parallel to it. The former position corresponds to the second case, or that of horizontal dispersion, the axes corresponding to one colour only, being situated in a plane parallel to the orthodiagonal, while those for other colours are in planes making a slightly different angle with the clino- diagonal axis, but all their meridian lines lie in the plane of symmetry, so that the horizontal bars of the cross, in the in- terference figures for the extreme colours will not lie in the same line, but will be parallel to each other with more or less horizontal displacement. If therefore the plate is cut true for a mean colour, as yellow or green, the horizontal bar in the position of greatest obscuration will be bordered FIG. 362. FIG. 363. with red on one side and blue on the other, as in fig. 362, the corresponding colours of the brushes in the diagonal position being seen in fig. 363. 2/O Systematic Mineralogy. [CHAP. XII. In the third case, that of crossed dispersion, the median lines for all colours coincide in direction with the orthodia- gonal, but the planes of their optic axes cross at small angles. The colours of the rings and brushes are therefore disposed chequerwise, as in fig. 364, the maximum of red in the left FIG. 364. FIG. 365. polar field being above the horizontal bar and to the left, and that of blue, below and to the right ; while in the left one the positions are reversed, a similar contrasted arrange- ment is also apparent in the diagonal position, fig. 365. This kind of dispersion is well seen in Borax crystals, and to a lesser degree in Gay Lussite. In the triclinic system no direct relation is apparent between the crystalline form and the interference figures, two or more kinds of dispersion being sometimes seen in the same crystal. Determination of the sign of the double refraction. The positive or negative character of a doubly refracting mineral may be determined in most cases from a section showing its interference rings, by placing below the analyser a plate of another mineral whose sign is known, and observing the effect upon the figure. With uniaxial crystals the simplest method that can be used is to place above the section under examination a plate of some other uniaxial crystal, when, if both have the same sign, they will act together, producing the effect of an apparent thickening of the lower plate, and the interference rings will appear closer together ; but if they are of dissimilar signs, one will partially neutralise the effect of the other, and the rings of the first will be expanded as CHAP. XII.] Determination of Sign. 271 though its thickness had been reduced. This method is, however, but of very limited application. When a section of a uniaxial crystal, parallel or but slightly inclined to the optic axis, is examined in convergent polarised light between crossed Nicols, no coloured rings are observed except it be very thin, when a series of hyperbolic bands symmetrical to a central cross appear. If, therefore, a plate of sufficient thickness be placed so as to give a field of maximum brightness, in which position its planes of vibra- tion are at 45 to those of the Nicols, and a tapered wedge of Quartz, whose length is parallel to its optic axes, be passed below the analyser, first in the direction of one vibra- tion plane and then of the other, the hyperbolic bands will be seen in one or other position. The reason of this is, that Quartz being positive, its optic axis coincides with that of its minimum elasticity, and the particular direction in which it is without effect or appears to augment the thickness of the plate is the axis of like elasticity in the latter, while the other direction at right angles to the first is that of maximum elasticity in the plate, and opposed to that of the Quartz, which therefore acts as though the plate were thinned. If, therefore, the latter direction is that of the optic axis, the double refraction of the crystal is opposed to that of the Quartz, or is negative, while in the other case it is similar, or positive. The particular thickness required for this so-called compensating action varies with that of the plate, and there- fore a certain length of Quartz is required to give the neces^ sary range. As ordinarily made, the wedge is about if inche long and -j^ inch at the thickest end. The same method is applied with sections of biaxial crystals, showing the interference rings and brushes by in- serting the Quartz wedge, first in the direction of the line joining the optic axis, and then at right angles to it. If the rings appear to expand in the first position the crystal is positive ; but if they are unchanged or completely effaced, it is negative, and the expansion or production of the hyper- bolic bands will take place at right angles to the axial plane. 272 Systematic Mineralogy. [CHAP. XII. A third method which is specially applicable to uniaxial crystals and to biaxial ones in sections that are too thin to show the polar rings, depends upon the use of circularly polarised light, either in the polariser, or the analyser. If a plate of biaxial Mica sufficiently thin to produce a retarda- tion of phase of exactly a quarter wave length between the refracted rays be placed in the path of a plane polarised beam, with the plane of its optic axes that of 45 degrees to the polariser, the light will emerge in a condition of circular polarisation, and the dark line marking the plane of vibra- tion of the polariser will no longer be apparent. ! A section of a uniaxial crystal viewed under these conditions no longer presents the concentric rings and dark cross between cur^ea Nicols, but that shown in figs. 366-7, the rings being broken into four parts, which, when compared with their original forms, are expanded and contracted in alternate quadrants. The contracted parts having the greatest intensity of obscuration, those of the innermost ring will appear as two dark or coloured spots, and a line joining them will be either parallel or perpendicular to the axial plane of the Mica plate. If, therefore, the latter is always used in one posi- tion, with its axial line diagonal to the first and third quad- rants of the circle if the spots lie upon it, as in fig. 366, the crystal is negative ; but when they are in the second and fourth quadrants, or at right angles to it, as in fig. 366, the crystal is positive. The same thing is observed with biaxial 1 An explanation of this change of polarisation will be found in Spottiswoode's treatise on polarised light, p. 50. CHAP. XII.] Determination of Sign. 273 crystals of small angle, but in those of wide angles the rings are expanded above and below the horizontal line, so that if the field be supposed to be divided into four quadrants as before, a negative crystal will show a spot above the line in the first, and below it in the third quadrant (fig. 368) ; while in a positive one the spots will be below the line in the second, and above it in the fourth quadrant, as in fig. 369. If the light be circularly analysed as well as polarised, by the use of a second quarter undulation plate below the analyser, the interference figures will appear as complete rings without any dark cross, and will behave in the same way as those observed in a circularly polarising crystal, the rings expanding or contracting as the analyser is turned one way or the other. If the polariser and analyser be placed parallel with the axial plane of the plate under examination in the vertical line of the field, and the two quarter undula- tion plates with their axial planes crossed at right angles or at 45 to that of the plate, the latter will, if negative, behave as a right-handed crystal, or the rings will expand when the analyser is turned to the right ; but if positive, the rotation must be to the left to produce the same effect. This is one of the best methods for use with crystals whose interference rings are small and close together. Irregular polarisation. In some cases crystals belonging to the cubic system, when examined under polarised light, exhibit traces of double refraction not compatible with the assumption of uniform structure required by the system. The most notable examples are Alum, Boracite, and Senar- montite. These have been variously explained ; as for T 274 Systematic Mineralogy. [CHAP. XII. example, in Alum, by the assumption of lamellar structure due to the successive layers of the crystal not being in abso- lute contact, and therefore capable of polarising light in the same way as a bundle of glass plates, or by the existence of strains in the interior of the crystal, producing a structure analogous to that of unannealed glass, and in Boracite by the symmetrical inclusion of doubly refracting substances in very minute crystals. The polarising character of such mine- rals is not constant, differing in different parts of the same section, and being quite independent of any particular direc- tion of the crystal. Similar irregularities are often observed in uniaxial crystals, the interference rings and cross behaving like those of biaxial ones of small angle. Here again, how- ever, the disturbance is commonly local, and a part of the plate may often be found giving the proper figure, and in the disturbed figure the innermost ring is not a continuous curve as that of a truly biaxial crystal should be, but is made up of disconnected portions of circles of different radii. A general explanation of the anomalous optical behaviour of minerals applicable to all the known cases has been put forward at great length in an admirable memoir by Professor Mallard, 1 who supposes these phenomena to be indications of polysynthetic structure, simple crystals, either asymmetric or of systems of low symmetry, by many repetitions produc- ing groups which simulate simple forms of more complex symmetry. The presence of foreign substances in transparent minerals, as well as of twinned structure, is often very strik- ingly shown in parallel polarised light. Fig. 370 is an ex- ample of a section of an apparently homogeneous Quartz crystal perpendicular to the optic axis, in which all the light parts have one rotation and the shaded ones the opposite, thus proving it to be a really complex twin of many right- and left-handed individuals with irregular contact surfaces. 1 Annales des Mines > 3 ser. vol. x. p. 60. CHAP. XII.] Irregular Structure. 275 Fig. 371 is a section of a twinned group of Aragonite, in which the planes of the optic axes in the alternate indi- viduals are inclined to each other as shown by the lines and rings. If parts of the two adjacent bands be seen together FIG. 370. FIG. 371. in convergent light, both systems of interference rings, crossed in direction, will often be apparent at the same time, but by shifting the plate slightly either to right or left one of them will disappear. The same thing is seen in Carbonate of Lead and Strontianite, minerals whose crystals are analogous in structure. The subject of minute inclosures of foreign minerals, as well as that of the preparation of thin sections suitable for optical examination, has already been treated at length in the treatise on Rocks in this series, to which the student is referred. It may, however, be useful to remark that the sections required for showing interference rings are, as a rule, much thicker than those prepared for microscopic in- vestigation under parallel light. In the greater number of the cases the most useful thicknesses will lie between and ^ of an inch ; but the actual size of the fragment, apart from its thickness, is immaterial ; for example, the whole of the rings may be seen and the character of the double refrac- tion determined on a plate of Mica of two or three hun- dredths of an inch on the side. The preparation of sections for the polariscope is much facilitated by the existence of a perfect cleavage parallel to the optic axis in uniaxial, or to the first median line in bi- axial, crystals, as with these cleavage plates will usually suffice 276 Systematic Mineralogy. [CHAP. xn. without specially grinding or polishing. Among the former are the tetragonal variety of Sulphate of Nickel, which sepa- rates from a saturated solution at temperatures above 15 Cent. ; the native Phosphate of Copper and Uranium also tetragonal. The transparent crystals of Molybdate of Lead (Wulfenite) from Utah, which are tabular to the basal pinakoid, may be used without any preparation ; but as the specific refractive power is very high, only the thinnest will give a system of rings of any great width. Among biaxial sub- stances the best examples are the species of the Mica group, which are susceptible of almost unlimited cleavage, and are therefore well suited for illustrating the alteration of the rings with the thickness of the plate. Topaz, Sugar and Bichro- mate of Potassium have perfect cleavages normal to one of the optic axes, so that from them plates may be easily ob- tained showing the rings about one axis only. The following table shows the optical constants of the principal transparent minerals. It is for the most part com- piled from that given in the 'Annuaire du Bureau des Longitudes,' and the works of Descloizeaux and Groth. [Note to page 263.] The optical elements of a biaxial crystal are related in the following manner : Axes of elasticity a '. b \ c Refractive indices (min.) o : (mean) ft '. (max.) y Velocities L : 1:1 0/87 Coefficients of elasticity JL : JL : J_ The true angle of the optic axes with the median line is found, when the three refractive indices are known, by the formula Cos. V CHAP. XII.] Optical Constants. 277 OPTICAL CONSTANTS OF THE PRINCIPAL TRANSPARENT MINERALS. ISOTROPIC. Amorphous and Cubic Ray Refractive index Water Hydrophane, dry red Do. saturated with water ... Hyalite Opal, iridescent variety .... ,, Quartz, melted .... Fluorspar, green .... Alum . . . Sylvine yellow Analcime red Plate glass (mean) Agate, light coloured Rock Salt . . . . . . yellow Sal Ammoniac .... Boracite Spinal (rose colour) ..... red Arsenious Acid .... Garnet Almandine .... ,, Cinnamon Stone . Senarmontite Zincblende (yellow) .... Diamond (colourless) . (brown) .... Ruby Copper Ore , 336 387 439 437 446 457 433 458 482 487 530 537 543 642 667 712 748 772 741 2-073 2-341 2-414 2-487 2-849 ANISOTROPIC UNIAXIAL. Tetragonal positive Ray Indices Leucite Apophyllite . Scheelite Zircon . Phosgenite . Anatase red orange 1-508 I-53I7 1-918 1-92 2-114 2-554 1-509 I '5331 1-934 1-97 2-140 2-493 278 Systematic Mineralogy. [CHAP. XII. Tetragonal negative Ray Indices c Mellite yellow 1*52* I'^O Meionite .... > 1-560 1-595 Melinophane . . red 1-592 i-6n Idocrase .... yellow 1-717 1-719 Wulfenite . . . red 2-304 2-402 HEXAGONAL. Positive Ray Indices tt e Ice (mean index) yellow 309 Quartz . '544 1-553 Parisite red 5 6 9 1-670 Phenakite 654 1-670 Dioptase Greenockite green yellow 667 2-688 1-723 Cinnabar red 2-816 3-142 Negative Ray Indices Nitrate of Sodium Hedyphane . Pyromorphite Calcite . . yellow red yellow red green red ii yellow 1-336 1-463 I-465 1-486 I-503 1-537 1-576 1-578 1-626 2792 (a 586 467 474 659 657 612 542 577 584 641 767 3-084 3-088 Apatite. Dolomite Nepheline Pennine Emerald Tourmaline (green) Corundum (Ruby) . Ruby Silver Ore (dark) . Do. light (Proustite) . CHAP. XII.] Optical Constants. 279 ANISOTROPIC BIAXIAL. Indices Angle of optic axes Disper- Ray Min. Mean Max. Real 2V Apparent 2 E sion RHOMBIC-POSITIVE o / o / Thenardite . . red 470 Natrolite . ,, x *477 480 1-489 59*29 94*27 Struvite . '497 Harmotome . '516 Anhydrite . . .. Electric Calamine . yellow 1*614 576 1*617 1-614 1-636 46-09 43 '30 78-39 p > v Topaz (white) . ,, I'6l2 1*615 1*622 56*39 100-40 p > v Celestine 1*625 89-36 Barytes . ,, 1-636 i'637 1-648 63*12 p < v Olivine (Peridote) . red 1*661 1.678 1*697 87-46 p < v Zoisite _ 1*700 98*2 Diaspore . . . yellow 1*722 Chrysoberyl . ,, 1*747 1*748 1-757 42*50 p > v Staurolite . . . red I *753 p > V Anglesite i '874 i *88o 1-892 66-40 89*49 P < v Sulphur . . . yellow I-958 2-038 2*240 69*40 70 to 75 p < v -NEGATIVE Sulphate of Sodium . 1*440 Sulphate of Magne- sium 1*4325 1- 4554 1*4608 77*50 Sulphate of Zinc . 1*457 1*480 1*484 70*16 Nitre. . i '333 1*5046 1*5052 7*12 Aragonite i'53 1*682 1*686 17-50 30*14 p < v 'Andalusite. . . red 1-632 1*638 1*643 Autunite . . ,, - 1"S7 2 Cordierite . . . orange Carbonate of Lead . yellow OBLIQUE-POSITIVE 1-562 1*804 1*561 2*076 1*563 2*078 8*07 16-54 Ferrous Sulphate . 1*471 1*478 1*486 85*27 Gypsum 1*521 I'527 I'S^O 61*24 inclined Euclase . . ,, 1*652 1*671 87*59 inclined Anthophyllite . . red 1*636 81*05 p ~> v Diopside . . . yellow 1.673 1*679 1*703 58*59 111 '34 inclined Sphene . . . red 1*903 53 '30 p > v -NEGATIVE Borax . . . yellow Adularia S. Gotthard. i '447 1*519 1-469 I'524 1*472 1*526 69*43 59-23 121 "06 crossed lorizontal _ Eifel . . red Muscovite (ural) . yellow l*523q 1*5240 !'574 1*3*34 40*21 20*45 64*14 inclined p> V Tremolite . 1*622 87-31 inclined Aclinolite 1*629 80*04 inclined Epidote . . . red Malachite . . . 1*731 i*754 1-88 1*761 73*36 43 '54 89*18 inclinad inclined TRICLINIC-POSITIVE Sillimanite . 1-66 44 "o .NEGATIVE Sulphate of Copper . yellow Axmite . . . red i'5i6 1*672 i*539 1-678 1*^46 1*681 7i'38 96 158-13 p < v p < v Amblygonite (Monte- bras) . __ 1*592 __ Cyanite . . . 1*720 280 Systematic Mineralogy. [CHAP. XIII. CHAPTER XIII. OPTICAL PROPERTIES OF MINERALS Continued. Translucency. In systematic mineralogy, minerals are clas- sified as transparent, semi-transparent, translucent in different degrees, and opaque, according to their power of trans- mitting ordinary light through their mass ; these terms being used in the popular sense, without reference to the homo- geneity or colour of the substance. The test of transparency is the power of discerning an object through a parallel-sided plate or crystal of a certain thickness. Rock-crystal, Calcite, Gypsum, and Barytes, and among the ores of the heavy metals, Zincblende in its lighter-coloured varieties, are among the most transparent substances known. When the object is only imperfectly seen, the substance is semi-trans- parent ; when only a cloudy light like that seen through oiled paper or ground glass is transmitted, it is translucent ; when no light is transmitted, it is opaque. These terms are to a certain extent relative, particularly in the lower degrees, where the thickness of the substance must be considered, especially when it is dark coloured. Flint and Obsidian, for example, are said to be translucent at the edges, or in thin splinters, while in thicker masses they are apparently opaque. Ferric oxide and its hydrates are also fairly trans- lucent in minute microscopic crystals, but opaque when sufficiently large to be apparent without magnifying. Mag- netite, on the other hand, does not appear to be susceptible of transmitting light under any condition, and is therefore opaque, as are also the native metals, and most of the heavy metallic sulphides. Colour. When a transparent substance has the power of absorbing light-rays of different refrangibility unequally, it will, when viewed in ordinary light, appear to be of the colour of the light of greatest intensity transmitted. This CHAP. XIII. ] Colour. 281 may be a single colour or a mixture, its true nature being easily determined by examination with a simple spectro- scope. In describing minerals, however, only the apparent colour, as seen by the unaided eye, is taken into account ; such as are transparent and without selective absorption in white light being said to be colourless, while others are classi- fied according to a scale laid down by Werner, which has been adopted by mineralogists in all countries, and is one of the few instances in which a uniform terminology has been obtained. It is founded upon the use of familiar coloured objects as standards of reference, distinction being made between metallic and non- metallic colours as fol- lows : METALLIC COLOURS. Copper red. As in metallic Copper and Red Nickel Ore. Bronze red. Slightly tarnished Bronze and Magnetic Pyrites. Bronze yellow. Perfectly fresh Bronze, and newly frac- tured Magnetic Pyrites. Brass yellow. Freshly fractured Copper Pyrites. Golden yellow. Pure unalloyed Gold. Silver white and Tin white. These are used in a con- ventional sense for any brilliant opaque mineral without any strongly marked colour. Lead grey. Galena, Antimony Glance. Steel grey. Platinum, Fahlerz. Iron black. Magnetite, Graphite. NON-METALLIC COLOURS. There are eight of these namely, white, grey, black, blue, green, yellow, red, and brown, each being divided into numerous tints or varieties, in the following order, the purest or most characteristic tint being placed first. Whites. Snow-white, yellowish-white, reddish-white, greenish-white, bluish- or milk-white, greyish-white. 282 Systematic Mineralogy. [CHAP. XIII. Greys. Ash-grey, bluish-grey, greenish-grey, yellowish- grey, reddish-grey, smoke-grey, blackish-grey. Blacks. Velvet-black, greyish- black, brownish- or pitch- black, reddish-black, greenish- or raven-black, bluish-black. Blues. Prussian-blue, blackish-blue, azure, violet, lav- ender, smalt, indigo-blue, sky-blue. Greens. Emerald-green, grass -green, verdegris, celadon, mountain-green, leek-green, apple-green, pistachio-green, blackish-green, olive-green, asparagus -green, oil-green, sis- kin-green. Yellows. Lemon -yellow, sulphur, straw-yellow, wax- yellow, honey-yellow, ochre-yellow, wine-yellow, Isabella- yellow, orange-yellow. Reds. Carmine, aurora or fire-red, hyacinth-red, brick- red, scarlet, blood-red, flesh-red, cochineal-red, rose-red, crimson, peach-blossom red, colombine-red, cherry-red, brownish-red. Browns. Chestnut-brown, clove-brown, hair-brown, yellowish-brown, wood-brown or umber, liver-brown, black- ish-brown. According to intensity, colours are further qualified as light or dark, pale or deep. As a distinguishing character of minerals, colour is of very unequal value; being constant, or showing but slight variation from a single tint in particular species, such as the native metals, the crystallised salts of Copper, and most natural metallic sulphides ; while in the larger number of so-called non-metallic minerals a single species may, without any great variation of composition, be either colourless or show a considerable range of colours. In such cases, how- ever, a particular tint may often be taken as an indication of partial replacement of one metallic constituent by another, as, for example, the Silicates of Magnesium and Calcium, which, when pure, are colourless, pass through various shades of green to nearly black, in proportion as Magnesium is replaced in part by the analogous dyad metal, Iron. CHAP. XIII.] Colour. 283 When Manganese is substituted in the same way, as in certain Micas and other silicates, they become red or purple, and so on in many other cases. A mineral, ordinarily colourless, may also, if transparent, appear to be coloured, by reason of included foreign sub- stances. Familiar examples of this are afforded by the numerous varieties of Quartz : the purest, or Rock Crystal, being colourless and transparent, while Amethyst, Chryso- prase, Cairngorm, and Eisenkiesel, show different colours, including purple, green, brown, black, and red, due either to minute traces of metallic elements in combination, or to par- ticles of ferric oxide, carbon, or other opaque substances, visibly included. Such minerals are said to be allochro- matic, or adventitiously coloured, while those whose colour is uniform and due to their own proper absorption, are self- coloured, or ideochromatic. Strictly speaking, these latter are the only ones that can be properly said to be coloured minerals. Streak. Useful aid in the determination of minerals is in many cases afforded by comparing the colour of the mass with that of the streak or powder produced by rubbing on a file or upon a piece of unglazed porcelain. With trans- parent minerals this is generally of a lighter colour, and with opaque ones darker than that of the mass. For instance, Gold, Copper Pyrites, and Iron Pyrites are often- sensibly of the same brassy yellow tint, but the first gives a streak of its proper colour, while that of Iron Pyrites is black, and that of Copper Pyrites dark brown. Hydrated and Anhydrous Ferric Oxide, or, as they are commonly termed, brown and red Hematite, also differ sensibly in their streak, which is brown in the first and red in the second. Pleochroism. The selective absorptive power producing colour is, in those substances that are isotropic, constant in any direction, so that they will appear of the same colour whether in ordinary or polarised light, but in anisotropic 284 Systematic Mineralogy. [CHAP, xm. ones it may vary with the direction, so that a crystal may appear to be differently coloured by transmitted light, ac- cording to the direction in which it is viewed. This property, to which the general term of pleochroism is applied, depends upon the unequal absorptive capacity of the crystal for refracted rays vibrating in different planes in a manner analogous to the differences in optic elasticity. The most remarkable example is afforded by tourmaline, which absorbs the whole of the ordinary ray, whose plane of vibration is perpendicular to the optic axis, or the crystal is impermeable to light in the direction of that axis, while the extraordinary ray vibrating parallel to it passes freely, and in some cases almost without change of colour, in the direction of the lateral axes, the latter property being utilised as a method of obtaining a single plane polarised ray by double refrac- tion and absorption in the polariscope known as the Tour- maline tongs. In less extreme cases, the absorption parallel to the axis will only be for particular colours, and the light transmitted will show the residual colour, while in a perpen- dicular direction the light will be of another colour, usually complementary, or nearly so, to the first. As the difference will be greatest in directions corresponding to maximum differences in velocity, a uniaxial mineral may have two distinct axial colours, or be dichroic, and a biaxial one tri- chroic, a different tint being apparent in the latter in the direction of each of the three principal axes of optic elas- ticity, when the light transmitted parallel to either of them is examined separately. This may be done by a Nicol's prism placed with its shorter diagonal parallel to the plane of vibration of each ray successively. For instance, a rhombic crystal whose axes of form and optic elasticity coincide, when viewed in the direction of the vertical axis, or through the base, shows the colours of the rays trans- mitted parallel to a lateral axis, when the principal section of the Nicol's prism is parallel to that axis, and those proper to the vertical and a lateral axis may be seen in the same CHAP. XIII.] Dichroiscope. 285 way through either of the other pinakoids. Such differences of colour, though possible in all coloured doubly-refracting minerals, are not always apparent, the property being pos- sessed in very unequal degrees by different minerals, and in many of them it is extremely feeble. In such cases it may often be rendered apparent when heightened by contrast, the colours proper to the two axes in the same section being seen side by side at the same time, which can be done by using a rhomb of Calcite or double-image prism instead of a Nicol's prism. The most convenient arrangement of this kind is Haidinger's Dichroiscope, fig. 372. It consists of a long cleavage rhombohedron Fig 3?2 . of Iceland Spar, a, mounted in a tube, having a glass prism c b V of 1 8 refracting angle ce- mented to either end which parallelise the incident and emergent rays to the larger edges of the prism. A metal cap with a square hole, c, is fixed to one end, and a convex lens, d, is cemented to the glass prism at the opposite end, which brings the two images produced by double refraction in the Calcite exactly parallel to each other when viewed through the eye-piece cap o. In white light these images will be exactly alike except as regards intensity, but when a pleochroic mineral is placed in front of c they will be differently coloured, certain rays being extinguished in the ordinary beam, and their comple- mentaries in the extraordinary one. Some few minerals are so strongly pleochroic that the differences of tint are apparent to the unaided eye. lolite or Dichroite, for instance, appears to be dark sapphire-blue in certain directions, and pale smoky grey or brown in others. These, however, are not pure axial colours, but are the approximate complementaries to those rays that are most completely absorbed, and for exac determination the use of polarised light is necessary. Zircon, Diaspore, Hornblende, the darker coloured Micas, certain varieties of Chlorite, 286 Systematic Mineralogy. [CHAP. XIII. Andalusite, Axinite, and Epidote, are other examples of strongly pleochroic minerals. An interesting application of the principle of selective absorption consequent on direction is afforded by Dove's test of the optical character of the Mica groups, many species of which give apparently uniaxial interference figures when viewed in convergent polarised light, owing to the very small difference between their minimum and mean refractive indices. A polarising instrument is arranged for parallel light, and a plate of unannealed glass or calcite is viewed by a plate of mica used as an analyser. Supposing the latter to be truly uniaxial, its absorptive power will be equal for rays vibrating in any azimuth, and therefore no figure will be apparent, but if it be biaxial, however small the angle of its optic axis may be, every ray entering it will be divided into two, whose planes of vibration will be parallel to the axes of minimum and mean elasticity respectively, the one making the largest angle with the direction of the plane of the optic axes in the mica will be most completely ab- sorbed, and the interference figure corresponding to the other will be rendered visible, though usually in very faint colours. The success of the test depends on the power of absorp- tion, and therefore it is only suited to those micas that are somewhat coloured and transparent in moderately thick plates, amongst which, however, the apparently uniaxial varieties are not generally found. The same remark also applies to the use of Tourmaline plates as polarisers, the colourless varieties, being almost without absorptive power, are comparatively valueless as compared with the brown or green ones. Lustre. Light, when reflected from the faces of crystals or other surfaces, is partly returned in a regularly reflected beam, and partly irregularly reflected or dispersed, the joint effect of reflection and dispersion being to produce upon the surface the peculiar appearance known as lustre, glance, or CHAP. XIIL] Optical Lustre. 287 brilliancy. In the definition of lustre, which is often very useful in the proper determination of minerals, two points are considered, namely, its quality or kind, which is specific and depends upon the refractive energy of the substance, and its intensity or degree, which varies in the same sub- stance with the character of the reflecting surface. The kinds of lustre, commencing with the highest, are : 1. Metallic lustre. This is the peculiar and brilliant appearance seen upon a perfectly polished metal surface. It is essentially characteristic of the native metals and heavy metallic sulphides, and of the few dark-coloured transparent ones, such as the Ruby Silver Ores and Cinnabar having refractive indices above 2*5. 2. Adamantine lustre. The typical example is the Dia- mond, but it is also characteristic of transparent minerals, whose refractive indices are from r8 to 2-5, which include the natural Sulphide, Carbonate, Chloride, Tungstate, and Molybdate of Lead. The heavy lead glass, known as flint glass or crystal, has also an adamantine lustre when polished. 3. Vitreous lustre, or that of a glass not containing Lead, is characteristic of most of the transparent crystals known as gems, whose refractive indices are below i '8, Quartz or rock crystal being a most familiar example. Some of the minerals of this class, when dark-coloured or imperfectly transparent, show a resinous lustre, as that of boiled pine resin or colo- phoninm. 4. Fatty or greasy lustre resembles that of a freshly oiled reflecting surface, and is characteristic of slightly transparent minerals, such as Serpentine, Nepheline, and Sulphur. 5. Nacreous lustre, or that of the Mother-of-Pearl shell, is a common characteristic of minerals having very perfect cleavages, and is best seen upon cleavage surfaces, such as those of Gypsum and Stilbite. 6. Silky lustre is essentially characteristic of imperfectly translucent and fibrous aggregates of crystals. The fibrous 288 Systematic Mineralogy. [CHAP. xin. variety of Gypsum known as Satin Spar, and the native Alums, are among the best examples. There being no absolute standard of classification in regard to lustre, intermediate terms are often used in de- scribing that of minerals which do not exactly correspond to a particular kind in the judgment of the observer. Thus anthracite is said to be semi-metallic in lustre, certain varieties of Carbonate of Lead metallic-adamantine, &c. The definitions based upon the refractive power of the sub- stances given are those of Professor W. H. Miller. It is not uncommon to find different faces of the same crystal different in lustre, and such differences are often of considerable value in fixing the position of the forms. Thus in Quartz, of the two rhombohedra making up the unit hexagonal pyramid, that considered as the positive one is generally brighter than the other ; and the so-called rhombic faces, those of the acute pyramid of the second order 2^2, are so much more brilliant than those of the associated forms that they may be often detected by the naked eye, even when extremely small. As regards degree or intensity of lustre, minerals are said to be splendent, shining, glistening, or glimmering, as the proportion of dispersed to reflected light increases ; when no distinct reflection is obtained, the substance is said to be dull. The terms are, however, even looser than those de- fining the quality of the lustre. Iridescence. The appearance of a colour, either singly or in variegated bands and patches on their surfaces, when viewed under oblique incident light, is a well-marked cha- racter of many minerals, some of the most striking examples being furnished by the massive cleavable Felspar of Labra- dor, which very generally shows patches of deep blue alter- nating with green upon the brachydiagonal cleavage planes, and in rarer instances a much more extended range of colour, including rose-red and orange-yellow. Hypersthene also shows a bronze red tint upon the same surfaces. These CHAP. XIII.] Iridescence. 289 appearances have been attributed by different investigators either to structures proper to the mineral, such as small pores or cavities regularly arranged, or to the interference phenomena produced by very minute crystals of Horn- blende or other minerals, interspersed in the same regular manner. In other cases, minerals which are transparent and colourless, sparkle with a golden light by the reflection from interspersed opaque crystals, producing the so-called avanturine structure seen in Felspars and Quartz, the in- cluded substances being in the first case scales of Hematite or Goethite, and in the last flakes of a golden-coloured Mica. The large Calcite crystals from Lake Superior are sometimes coloured in the same way by small crystals of native Copper disseminated through them. The brilliant iris of the Opal is attributed to the presence of minute faces crossing the mass in different directions, and in one variety known as Hydrophane, the opalescence dis- appears when the substance is soaked in water. In minerals with very perfect cleavage coloured rings are frequently seen at different points in the interior. These are the ordinary colours of thin plates produced by minute films of air included between cleavage surfaces. They may be seen in almost any large clear specimen of Mica, Gypsum, or Calcite. Surface iridescence or peacock colour is also due to the formation of very thin layers of one substance upon the surface of another, and therefore marks the commencement of alterations in the second. It is usually spoken of as iridescent tarnish, and is seen in many sulphides, such as Antimony Glance, Copper Pyrites, and Purple Copper Ore. The Hematite of Elba is also remarkable for the brilliant rainbow colouring often seen upon the crystals. Asterism. Certain varieties of Corundum known as star sapphires, when ground to a spherical form and polished, show a pale blue six-rayed star when a strong light is re- flected from the polished surface. This is due to repeated u 290 Systematic Mineralogy. [CHAP. XIII. parallel twinning producing a finely lamellar structure, the laminae of which act in the same way as the lines in a diffrac- tion plate. The biaxial Mica of South Burgess, Canada, shows a similar but more sharply denned star when held in front of a candle flame or other luminous point. This is at- tributed to the inclusion of minute crystals of uniaxial Mica arranged along lines crossing at 60. The same thing may be seen in thin sections of Labradorite, Aragonite, Proustite, Brookite, and generally in transparent minerals that either contain minute crystals of other substances symmetrically enclosed, or whose crystals show repeated parallel twinning. Thin plates of Aragonite and Strontianite, when held at a distance of about six or eight feet from a candle flame, often show an extended series of diffraction spectra on either side of the central image. Fluorescence, or the property of rendering visible rays of higher refrangibility than are ordinarily apparent in white light, is not a very common property of natural minerals, but it is well marked in certain varieties of Fluorspar, es- pecially the large transparent crystals from Weardale, which transmit an emerald or grass-green light, but reflect a deep sapphire-blue. More striking examples are furnished by the liquid hydrocarbons known as Petroleum, which are colourless, or transmit various tints of yellow to brown, but by reflection show the light blue rays above the violet of the ordinary spectrum. Phosphorescence, or the power of emitting light in a dark place is characteristic of a certain small number of minerals, and may be variously developed by exposure to sunlight, friction, heating, or an electric discharge. The first method is completely effective with Diamond and some varieties of Zincblende, and less perfectly so with Strontianite and other earthy carbonates. The second method is applicable to Quartz, two pieces of rock crystal or any other variety of this mineral emitting a pale yellow light when rubbed to- gether in the dark. This is still more markedly shown by CHAP. XIV.] Phosphorescence. 291 loaf sugar. Heating is, however, the most generally effective method, as many minerals which are rendered phospho- rescent when their temperature is raised are not so affected by sunlight. Fluorspar, when heated to 200 or 300 degrees Cent., becomes strongly luminous, the light being usually blue ; but with the green variety known as Chlorophane it is of a brilliant emerald green. Phosphorite in the same way emits a yellow light. Topaz, Diamond, Calcite, and some silicates also phosphoresce when heated, but to a less degree than the typical examples, Fluor and Phosphorite. Phospho- rescent minerals generally lose that property when strongly heated, but it may be more or less restored by subjecting them to a series of discharges from an electrical machine. Crookes has shown that, when exposed to electric cur- rents of high tension in an extremely rarified atmosphere, Ruby and Sapphire phosphoresce with intense red and blue, and Diamond with a vivid green light. The best examples of phosphorescence are afforded, however, by the sulphides of the earthy metals Barium, Strontium, and Calcium which, though not natural minerals, are prepared by heating the native sulphates of these metals with carbon. Sulphide of Barium is the so-called Bologna phosphorus, and was the first substance in which the pro- perty of phosphorescence by sunlight was discovered. These substances are applied in the production of clock faces, which emit sufficient light to show the time in the dark. CHAPTER XIV. THERMAL AND ELECTRICAL PROPERTIES OF MINERALS. Thermal relations of minerals. A crystallised substance placed in the path of a pencil of rays emitted from a heated body affects the latter in the same way as it would a beam of light that is, it may either transmit them freely without 292 Systematic Mineralogy, [CHAP, xiv, sensible absorption, or it may absorb them entirely or in part. The former case, of which Rock Salt is the most complete example, corresponds to heat transparency, or diathermancy, and the latter, exemplified by Alum, to heat opacity, or athermancy. Sulphur and Fluorspar are also diathermanous, but less perfectly so than Rock Salt, while Tourmaline, Gypsum, and Amber are nearly as opaque to heat as Alum. Heat rays are also subject to the same laws of reflection, refraction, and polarisation, as those of light, being refracted singly by Rock Salt or other isotropic substances of sufficient heat-transparency, and doubly by those crystallising in the anisotropic systems. A parallel beam of heat- rays from a Rock Salt lens falling upon two Mica plates is more completely transmitted by the latter when their planes of polarisation are parallel than in any other position, the amount being reduced to a minimum when they are crossed, showing an extinction exactly analo- gous to that of luminous rays under similar conditions. The thermal conductivity of amorphous or cubic minerals is ip like manner similar in any direction, while in those crystallising in the other systems it varies with the crystallo- graphic symmetry, being more perfect in some directions than others, or, in other words, such minerals show axes of conductivity analogous to those of form and optic elasticity. The principal investigations upon this subject are those of De Senarmont, who determined the conductivity in sec- tions of crystals cut in known directions by coating one surface with wax, 1 and inserting into a hollow in the centre the point of a platinum wire heated by a lamp at some distance. When the heat received is transmitted equally in all directions, the wax wi}l be melted in a circular patch around the wire ; but if the rate is unequal, it will be an ellipse whose axes will correspond to those of maximum 1 A modification of the apparatus in which the heating is effected by an electric current is shown in Tyndall's Heat, a Mode of Motion, p. 202, 4th edition. CHAP. XIV.] Dilatation by Heat. 293 and minimum rates of transmission in the particular section under investigation. In Quartz the conductivity is higher in the direction of the optic axis than at right angles to it, while in Calcite it is less, the two minerals in regard to this property being positive and negative in the same way as they are optically. In the rhombic system the three crystal- lography axes are also axes of dissimilar thermal conduc- tivity, while in the oblique and triclinic systems the latter, like the optic axes, are not directly related to the axes of form. The dilatation of crystallised substances by heat is also, in all but those of cubic symmetry, attended by change of form, as the rate of expansion may be much greater in one principal crystallographic direction than another. This subject has been elaborately investigated by Fizeau, by a method of extraordinary delicacy, in which a plane surface of the crystal under investigation is covered by a very slightly concave glass plate, the curvature being so large as to allow the formation of Newton's rings when the surface is illumi- nated by a monochromatic yellow light The crystal is supported upon a tripod of platinum, and heated in an air- bath, when if the expansion be uniform the surface will approach the glass regularly, and the rings change their positions symmetrically, but if it be unequal the surface will be distorted, and the distance from the glass will change more in some directions than others, so that the position of the rings will also be distorted, and by careful observation of these the alteration of form due to very small changes of temperature may be determined. The general conclusions derived from these experiments are as follows : Cubic crystals have three axes of dilatation correspond- ing to those of form, the coefficient of expansion being similar for all, and consequently for any direction, in the crystal Uniaxial crystals have a principal axis of dilatation 294 Systematic Mineralogy. [CHAP. XIV. corresponding to a particular coefficient of expansion, which may be either greater or less than that in directions perpen- dicular to it. In extreme instances the expansion may be positive in one direction and negative that is, the substance may contract at right angles to it ; but, in any case, the arithmetical mean of the values obtained along three axes at right angles 1 to each other, will correspond to that observed along a line inclined at 54 40' to them. The relation of these lines is that of the trigonal interaxes, or the normals to the octahedral faces to the three principal axes of the cube. In the rhombic system, the axes of dilatation, like those of optic elasticity, correspond to those of form ; in the oblique system, one of the former is parallel to the axis of symmetry, the orthodiagonal, but the others make different angles, not only with the other crystallographic axes, but with those of optic elasticity and thermal conductivity. The relation of these different directions for Orthoclase is shown in fig. 373, taken from Fizeau's memoir, which re- presents the distribution of two of each of the three kinds of axes ; E 2 and E 3 being two of the axes of optic elasticity, the first and second median lines of the ol optic axes ; D 2 and D 3 two of the three ' a axes of dilatation, c 2 and C 3 two of the axes of conductivity, and A A the clino- diagonal, and c~c the vertical axis. In Gypsum the direction of the two axes of dilatation and the optic median lines very nearly coincide. In the triclinic system, no determinations of the axes of dilatation have been made, but from the analogy of those of optic elasticity they probably have no simple relation to the axes of form. As a consequence of their unequal linear dilatation in 1 These are in the tetragonal system three crystallographic axes, and in the hexagonal, the principal axis, one lateral axis, and the inter- axis at right angles to the latter. FIG. 373- CHAP. XIV.] Dilatation by Heat. 295 different directions, the geometrical characters of anisotropic crystals are subject to change with alterations of temperature. For example, Calcite elongates in the direction of the prin- cipal axis, and contracts at right angles to it, so that the angle of the polar edges of the rhombohedron becomes more acute by heating. The amount of this alteration was found by Mitscherlich to be 8' for a range of 90, the observed angle at 10 being 105 4', while at 100 it was only 104 56'. Aragonite and Gypsum are examples of minerals in other systems whose rates of linear expansion in different directions are sufficiently different to show varia- tions in the angles when heated. In the greater number of instances, however, the alterations are too small to be directly measurable, but the principle is important, as es- tablishing the difference between substances crystallising in different systems, even when they may have forms of exactly similar geometrical characters. For example, the coinci- dence between a cube and a rhombohedron of 90 is only true for one particular temperature, as the latter will become either more acute or more obtuse when heated or cooled, according to the molecular arrangement of its substance, while a cube is rectangular at all temperatures. The parameters of the unit or fundamental form of a series in like manner are altered by heating, but as the same change extends to all the forms of the series, these rela- tions are not changed thereby. For instance, if the para- meters of the form (i i i) be changed by heat from a : b : c to a 1 : b' : d, those of the form (221) will change at the same temperature from a : b : 2 c to a' . b' : 2 c', but the symbols will not be altered, as the new length of the vertical axis in the second will be twice the new lengths of the others. This is expressed in the statement that the principle of the rationality of the axes is independent of temperature. The effect of heating substances is generally to increase their volume and diminish their density, and this is accom- panied by a change in optical properties, more particularly 296 Systematic Mineralogy. [CHAP. XIV. in the refractive power, which is, as a rule, reduced, and in biaxial crystals, where the principal indices alter unequally, the change often affects the position of the optic axes. These changes are most apparent in the earthy and alkaline sulphates. In Barytes, Celestine, and Sulphate of Potassium, the inclination of the optic axis is increased, but in Felspar and Gypsum it is diminished, by heating. In the latter mineral the alteration is very considerable, even when the tempera- ture is but slightly changed ; the apparent angle of the optic axes which lie in the plane of symmetry is about 90 for red light at the ordinary temperature of the air, but at 1 1 6 it is o, or the substance is apparently uniaxial. At higher temperatures the axes diverge again, but in a plane nearly at right angles to the original one. On cooling, the same changes take place in the reverse order. In some varieties of Felspar the change in angle of the optic axes is to some extent permanent that is, it does not return exactly to the original value if the crystal has been exposed to a red heat. Electrical properties of minerals. All minerals become electric by friction, but the positive or negative character of the electricity developed varies according to circumstances. For testing, an electroscope is used, consisting of a light metallic needle with a knob at either end, and suspended in the centre by a thread of raw silk, or balanced upon a steel point by an agate cap. This, when rendered positively or negatively electric by a rod of glass or sealing-wax, is either attracted or repelled by an excited mineral, according as the latter is charged with opposite or similar electricity. If the mineral under trial happens to be a conductor of electricity, it will be necessary to insulate it in order to obtain an effect upon the electroscope. Several minerals may be rendered electric by pressure, Calcite possessing this property in the highest degree ; a cleavage fragment of Iceland Spar, when slightly pressed between the fingers^ becoming positively electric. Topaz, Aragonite, Fluorspar, and Quartz are similarly affected, but in a less degree. CHAP. XIV.] Magnetic Properties. 297 Minerals that become electric by change of temperature are said to be thermo- or pyro-electric. Axinite, Tourma- line, Calamine, Topaz, and Boracite are among those that show this property best. When the crystals show opposite electricities at different points, they are said to be polar, and the points at which the changes of electricity are observed are poles. These cannot, however, be distinguished as positive and negative, as both kinds of electricity are de- veloped at either pole alternately, one during heating, and the opposite one in cooling. It is customary, therefore, to call the points which are rendered positively electric by heating or negative by cooling, * analogue poles,' and those becoming negative by heat or positive by cold, ' antilogue poles.' The positions of these vary in different crystals. In the prominently hemimorphic species, Tourmaline and Electric Calamine, the property of thermo-electricity is most characteristically developed, and are the poles at opposite ends of the principal axis. Boracite has eight poles corre- sponding to the solid angles of the cube, while in Quartz the electric poles are situated at the ends of the three lateral axes. Magnetic Properties of Minerals. Nearly all minerals containing iron are magnetic, but in very unequal degrees. Practically, only native Iron, Magnetite, and Pyrrhotine, or Magnetic Pyrites, are sufficiently magnetic to affect a com- pass needle upon a pivot, but with a more delicate astatic instrument many other minerals, including ferrous sulphates and silicates, show slight traces of magnetism. Native magnetic oxide of iron, or Lodestone, is always strongly magnetic, and is often found in masses showing distinct polarity, but natural magnets capable of supporting con- siderable weights are rare. Masses of Magnetite of a regu- lar figure, not naturally polar, may, however, be rendered so by touch, in the same way as a steel bar is magnetised. The allied minerals, Chromic Iron and Franklinite, are generally magnetic, but it is not certainly known whether this is a 298 Systematic Mineralogy. [CHAP. XV. special property or caused by finely interspersed Magnetite. Spathic Iron Ore, or Ferrous Carbonate, becomes strongly magnetic when heated to redness from the formation of magnetic oxide, as do most of the Sulphides of Copper and Iron when fused in an oxidising atmosphere. For common purposes of testing, a magnetic needle with a brass or agate centre, to be used upon the same pivot as the electroscope, will generally be sufficient. When in use, it should be covered with a glass, to protect it from the disturbing action of currents of air. CHAPTER XV. CHEMICAL PROPERTIES OF MINERALS. IN the determination of the nature of a mineral a knowledge of the kind of matter constituting it that is, of the elemen- tary substances it may contain, and the relative proportion of such substances to each other when more than one are present is of primary importance. The technical methods whereby such knowledge is attained are known as qualita- tive and quantitative chemical analysis, and it will be assumed that the reader is familiar with the details of such methods, as laid down in the treatise on chemical analysis by Professor Thorpe in this series, or in some other standard work on the subject. A certain knowledge of qualitative analysis is of great use in the identification of minerals of obscure habit, and for this purpose a course of simple testing by methods not requiring the resources of a complete analytical laboratory has been developed by mineralogical chemists, the results obtained by such tests being given in systematic descrip- tions of minerals under the head of chemical characteristics, or some equivalent term. CHAP. XV.] Blowpipe Apparatus. 299 The methods employed in testing are divisible into two groups, the first involving the application of heat to minerals, either alone or in the presence of certain reagents, being known as analysis by the dry way, and the second, where liquid solvents or reagents are used, as the wet way. In the former, or dry way, the mineral under examination is sub- jected to the flame of a lamp, urged by a blast of air, which is usually produced by the mouth blowpipe, although the blast of a bellows or the flame of a Bunsen burner may also be used. The latter arrangements are, however, only to be found in laboratories, while the blowpipe is for the mine- ralogist essentially a portable instrument, and may, with the necessary apparatus and fluxes, be packed into a case of small volume, forming a portable laboratory, ready for use with very small preparation, and as such is invaluable to the travelling mineralogist. The most convenient form of blowpipe is that of Gahn, as modified by Plattner. This consists of a brass or German silver tube, from 8 to 9 inches long, diminishing in diameter from one-third of an inch above to one-sixth or one-seventh below. A trumpet-shaped mouthpiece is fitted to the larger end; the smaller one fits air-tight in a cylindrical chamber about half an inch in diameter, from which a jet- pipe about an inch long projects at right angles ; this is also- slightly coned, from one-sixth to about one-tenth of an inch diameter, and is terminated by a conical nozzle of platinum perforated with a hole about ^ inch diameter. For some purposes, a second nozzle, with a larger aperture, is de- sirable. All the parts are ground together so as to fit air-tight without screws. The trumpet-shaped mouthpiece is more convenient in use than the cylindrical form, from the support given to the lips when the blowing is carried on con- tinuously for some time. Of the other forms of blowpipe, that known as Dr. Black's is the best, when made of an appro- priate size, the great fault of most of the cheaper instruments being the disproportionately small diameter of their tubes. Systematic Mineralogy. [CHAP. xv. The flame used with the blowpipe may be either that of a candle, a lamp, or gas jet, the best being that of an oil lamp with a flat wick, the top of the wick being cut with a forward slope of about 20. When in use', the jet of the blowpipe is held parallel to the slope of the wick, and a current of air is forced into the flame by the action of the muscles of the cheeks, breathing being kept up through the nostrils in a manner which, though difficult of description, is easily learned with a little practice. When the jet is placed so that the air enters the flame at the higher side of the wick, a large proportion of the unconsumed gases of the dark interior part is carried forward, producing a pointed flame of a certain brilliancy, which is of a neutral or non-oxidis- ing character, and is called the reducing flame. If, however, the point is laid above the middle of the wick, so that the air is brought into contact with the dark part of the flame, total combustion of the gases is produced, and the resulting flame is small, slightly luminous having the characteristic blue colour of burning carbonic oxide, and much hotter than the reducing flame. This is called the oxidising flame, as any substance heated in it is subjected to the full effect of the unconsumed oxygen of the blast or of the adjacent atmosphere. The maximum oxidising effect is obtained immediately before the point of the flame. The power of producing a clean flame, or one that is entirely reducing or oxidising, is one of the first essentials to success in blowpipe work, and should therefore be practised by the learner, with the tests recommended by Plattner. These are borax beads, saturated with molybdic acid and oxide of manganese re- spectively ; the former becomes colourless when melted for some time in an oxidising flame, but turns black in a reduc- ing flame ; while the latter is of a dark violet colour in the oxidising flame, which is entirely discharged by the reducing flame. To obtain these effects perfectly by the use of the blowpipe alone, considerable nicety of manipulation is re- quired. It is also essential that the metallic oxides used CHAP. XV.] Blowpipe Apparatus. 301 should be pure, especially that of manganese, which must be free from iron, otherwise the action of the reducing flame will be obscured by the bead taking a more or less green tint. The apparatus required (in addition to the blowpipe and lamp) is as follows : A pair of forceps with platinum points, closing by a spring. Pieces of platinum wire about three inches long, bent into a loop, about one-eighth of an inch diameter, at one end. A small platinum spoon, and a piece of platinum foil. A jeweller's hammer, small bright steel anvil, and an agate mortar. Pieces of glass tube, about a quarter of an inch bore, in lengths of three inches, open at both ends, and a few shorter pieces closed at one end. A few watch glasses ; a small bar magnet, which may have a chisel point at one end. Charcoal for supports : the best are pieces of straight grained pine charcoal, about three inches long, and three-quarters of an inch to one inch square ; these are, however, with difficulty obtainable, and should therefore be used carefully. For most purposes, artificially moulded blocks, formed of charcoal powder cemented with starch and subsequently carbonised, are sufficient ; they are made in various sizes, and may be purchased of dealers in chemical apparatus. Hard-wood charcoal, made from brushwood, is to be avoided, as it usually decrepitates when heated, besides leaving a large amount of ash. The use of a plate of aluminium as a support for minerals giving coloured sublimates has been recommended by Colonel Ross. The most essential fluxes, or reagents, are : Borax, calcined, but not fused into a glass ; phosphorus salt (ammonic sodic phosphate) ; dried carbonate of soda, which must be free from sulphate ; nitre ; bisulphate of potassium ; nitrate of cobalt ; fluorspar : cupric oxide ; and test papers, both of litmus and turmeric. In addition to these, a few liquid reagents, such as hydrochloric and nitric acids, and ammonia, are useful, although it is better to dis- 302 Systematic Mineralogy. [CHAP. XV. pense with them as much as possible, as they cannot be easily carried when travelling. The course of operations followed in a systematic ex- amination of minerals by the dry way is the following : i. Heating in tube closed at one end. A fragment of the mineral (generally called the assay) is placed at the bottom of a tube closed at one end, and heated first over the flame of a lamp, and subsequently in the blowpipe flame. Hydrated oxides and salts by this means give off water, which con- denses in visible drops on the cooler surface of the tube. Nitrates, and the higher oxides of manganese give off oxygen, which can be recognised by the ignition of a glowing splin- ter of wood. A few minerals, such as Sulphur, Arsenic, An- timony, Mercury, and their oxides and sulphides, sublime without residue ; but are redeposited in the case of anti- mony and arsenic in the form of black metallic mirrors, a short distance above the assay, while the sulphides of the latter element are recognisable by the red or yellow colours of their sublimates. The higher sulphides, such as iron pyrites, give a sublimate of sulphur ; arsenical iron pyrites gives both sulphur and arsenic, recognisable by the red and yellow sublimates, corresponding to the minerals Realgar and Orpiment. These, however, are products of the de- composition, and do not exist as such in the mineral. Ara- gonite, Calcite, Magnesite, and Dolomite give off carbonic acid, with the formation of caustic lime and magnesia, when strongly heated, while Siderite leaves a residue of magnetic oxide of iron. The carbonates of zinc, copper, and lead are also decomposed, producing oxides of the metals. The sulphates of alumina and ferric oxide give off sulphuric and sulphurous acids, the acid character of the vapours being recognised by a slip of test paper inserted in the mouth of the tube. 2. Heating in the open tube. The fragment of mineral under examination is placed about half an inch above the lower end of the tube, which is held in a slightly inclined CHAP. XV.] Fusibility. 303 position, and the blowpipe flame is directed upon it, so that the assay is heated in a full current of air, when sulphides give off sulphurous acid, which is easily recognised by its characteristic odour. Arsenides and selenides give the peculiar odours characteristic of Arsenic and of Selenium, and sublimates of their oxides, which deposit at a greater or less distance from the assay, according to their degrees of volatility. Antimony gives a similar sublimate of antimonious acid. Many sulphides which do not give an indication of sulphur in the closed tube are decomposed with the formation of sulphurous acid m the open tube. 3. Fusibility. For this test a fine splinter of the mineral, held in the platinum forceps, is exposed at the point of maximum heat in the oxidising flame. This supposes it to be very refractory ; but in some instances exposure to the flame without blowing is sufficient to effect fusion. The range of the melting points of minerals being very great, even excluding those that are fluid at ordinary tempera- tures water, mercury, &c. it has been proposed by Von Kobell to express fusibility by a scale of typical minerals, analogous to that employed in describing hardness. This scale comprises the six following numbers ; but, owing to the indefinite nature of the gradations between the different numbers, it is not much used : (1) Antimony. Glance. Fuses readily in the flame of a candle. (2) Stilbite. Fuses without the help of the blowpipe. (3) Almandine Garnet. A large or thick fragment can be melted before the blowpipe. (4) Actinolite Hornblende of Zillerthal. Thin splinter melts in extreme point of the oxidising flame. (5) Felspar (Adularia of S. Gothard). Similar to No. 4; but fuses with greater difficulty. (6) Bronzite of Kupferberg. Thin splinters can only be rounded on the edges. When the edges of a splinter cannot be rounded or 304 Systematic Mineralogy [CHAP. XV. softened in the hottest part of the flame, the mineral is said to be infusible. This statement is of course only true in relation to the means employed : many substances that are infusible by the mouth blowpipe can be easily melted by the oxyhydrogen blowpipe or other powerful sources of heat. The manner of fusion, as well as the character of the fused mass, is a point of considerable importance, and must be carefully observed, some minerals fusing quietly to a glass or enamel, while others intumesce or swell up, from loss of water or other volatile components, and become scoriaceous or slaggy masses ; others, again, give a mass which crystallises on cooling. It is important in trying fusibility to direct the flame upon a fine point of the fragment under examination, so as to induce fusion as quickly as possible, otherwise an oxidis- ing action, interfering with the result, may be set up. Thus the double sulphides of copper and iron are fusible, but if heated for some time in air below their melting point, they lose sulphur, and give a residue of oxide of copper and magnetic oxide of iron, which is practically infusible. In the same way, silicates containing ferrous oxide, although fusible, may by calcination be partly resolved into silica and ferric oxide, both of which are infusible. When trying the fusibility of easily reducible metallic minerals, care must be taken that the points of the platinum forceps in contact with the assay are not strongly heated, as an alloy of the platinum with the more fusible metal may result. It is safer in such cases to try the fusibility upon charcoal. Flame colour tests. These may often be observed simul- taneously with the trial of fusibility, the splinter of mineral used for the latter purpose in some instances giving a charac- teristic colour to the flame when first exposed to the point of the oxidising flame. It is more satisfactory, however, to make a special trial. Strontium, Lithium, and Calcium minerals CHAP. XV.] Flame Reactions. 35 give a red colour, which is best seen when the mineral, after being previously heated to redness in the reducing flame, is moistened with hydrochloric acid, and exposed in the outer blue envelope of the flame, without blowing. This is especially the case with the sulphates of barium and strontium, which may be partially reduced to sulphides in the reducing flame upon charcoal ; the sulphides being converted to chlorides, by moistening with hydrochloric acid, give compounds which are eminently volatile at the high temperature of the blowpipe flame. Lithium gives a brilliant crimson ; calcium a yellowish red ; and strontium a purplish red tint. When the two latter substances are together, the colour due to lime is seen first, and then that of strontia. When the red flame is examined by a dark blue glass screen, the light due to lime and lithia is almost entirely extinguished, while that of strontia is but little altered. Sodium compounds, even in very minute quantity, colour the flame a deep yellow, which completely effaces the light due to other volatile bodies ; the yellow colour may, however, be cut off by the blue glass. Minerals containing barium give a feeble yellowish green colour to the flame when very strongly heated, especially when moistened with hydrochloric acid. Copper gives a bright emerald green, except in the state of chloride, when the flame is blue with a purple border. Phosphates, when moistened with sulphuric acid and exposed to the outer part of the flame, show a momentary coloration of a pale, bluish green ; and borates, when similarly heated, give a brighter green colour. Potassium gives a very characteristic reddish violet colour to the flame, which is completely hidden by even a very small proportion of sodium. If, however, the yellow light of the latter is cut off by the blue glass, which allows the violet rays to pass freely, a small quantity of potash may be de- tected even in the presence of a relatively large amount of soda. x 306 Systematic Mineralogy. [CHAP. XV. In order to determine the presence of alkalies in silicates which are not decomposable by acids, they should be pre- viously heated with fluoride of ammonium in the platinum spoon to decompose them, when the bulk of the silica is volatilised, and the residue can be examined in the flame with the blue glass. Chloride of calcium may be used for the same purpose, as the lime coloration is but slightly transmitted through the glass. The alkaline metals may be determined with more cer- tainty by the spectroscope, 1 the characteristic bright lines being, for their oxides Sodium, a bright yellow line. Calcium, one green and one red line. Lithium, a red line, which is at a greater distance from the soda (D line) than that of lime. This is cut off by the blue glass. Potassium, a dull red line beyond that of Lithia ; this passes through blue glass. Strontium, an orange line near the D line, several red lines, and a blue line. Barium, a group of several green lines close to each other. The rare alkaline metals, Caesium and Rubidium, are also readily detected by their spectra ; while with the ordi- nary tests they may be confounded with Potassium and Lithium. For the detection of boracic acid in silicates, such as Axinite and Tourmaline, the mineral is heated on a platinum wire, with a mixture of Fluorspar and Bisulphate of Potash, when the characteristic green line is produced. Heating on charcoal. For this purpose a rectangular prism, cut from a piece of straight-grained charcoal, having the rings of growth perpendicular to the ends and parallel to the length on two faces, is preferable to a moulded charcoal 1 Browning's small direct-vision spectroscope with photographed micrometer, mounted on an upright pillar, is a convenient form of instrument for this purpose. CHAP. XV.] Heating on Charcoal. 307 block. A slight depression being made by scraping the surface with the point of a knife near one end, the assay fragment is placed in it, unless the mineral decrepitates by heat, when it must be previously powdered, and the flame is directed downwards upon this point, the charcoal being held with a slight upward inclination, with its length parallel to the direction of the flame. The points to be observed are fusibility or infusibility, the production of phosphorous or arsenical odours in the same manner as in the open tube, change of colour, or the production of an alkaline mass. The latter reaction is characteristic of Calcite and Aragonite, which, when strongly heated, leave an infusible residue of caustic lime, giving an alkaline reaction with moistened turmeric paper. Many sulphides and other compounds containing iron are converted into infusible masses, which show magnetic properties. The principal special reaction upon charcoal, however, is the production of coloured in- crustations of the oxides of volatile metals, which are pro- duced from their combinations, either with or without an actual metallic residue. These incrustations cover the sur- face of the charcoal, commencing at a distance of an inch or less from the assay, according to the volatility of the metal. The deposit produced by Zinc minerals is yellow when hot and turns white in cooling ; that of Cadmium of a brownish yellow usually more or less irised, neither giving a globule of metal. Antimony and Bismuth compounds yield brittle metallic globules, the former with a thick white incrusta- tion edged with blue, and the latter one of lemon yellow colour. Lead minerals are easily reduced, giving a malleable globule of metal, and a yellow incrustation ; but when notably argentiferous, the incrustation is more or less crimson at the inner edge. Minerals containing Tin, Copper, or Silver as principal constituents are reduced to the metallic state without producing an incrustation on the charcoal. Tests with soda upon charcoal. The whole of the reac- tions depending on the reduction of metallic minerals and the X 2 308 Systematic Mineralogy. [CHAP. XV. formation of coloured sublimates on charcoal may be facili- tated by adding a small quantity of carbonate of soda to the substance under examination, which forms a slag with the infusible constituents ; and with minerals containing Tin a small quantity of cyanide of potassium should be used to facilitate the reduction and to protect the reduced metal from oxidation. The most convenient way of applying this flux is to mix the assay in a finely powdered state into a paste with about its own volume of dried carbonate of soda upon the palm of the hand by adding a few drops of water. The mixture is made and removed by a small iron spatula or the blade of a knife, and the paste is spread over the charcoal, care being taken to heat it up gradually, so as to dry it before exposing it to the full strength of the flame. When sufficiently large, the metallic globules produced are tested as to their brittle or malleable character by the ham- mer and anvil ; but when, as generally happens, they are small and interspersed through a mass of slag, the assay with the adjacent portion of the charcoal must be removed and pulverised in the agate mortar. The charcoal is then re- moved by carefully washing the contents of the mortar in .a gentle current of water, leaving the reduced metal, which, if malleable, will be found in flattened scales or spangles. Other special uses of soda are in the decomposition of infusible silicates, the detection of Manganese, and of Sulphur in insoluble sulphates, or the lower sulphides that do not give a sulphur sublimate when heated. When Quartz or an infusible silicate, powdered and mixed with carbonate of soda, is heated on charcoal, the mixture effer- vesces from the escape of carbonic acid, and a fused alkaline silicate is formed, which is soluble in water, and may be decomposed by mineral acids with the production of silica in the gelatinous or soluble form, which may be rendered insoluble by heating to redness. This is the ordinary method adopted for rendering these minerals soluble for analysis in tne laboratory, and the method may be sometimes CHAP. XV.] Tests on Charcoal. 309 adopted with advantage on the small scale by the blowpipe. Minerals containing manganese, when fused with soda in a full oxidising flame, give a green enamel-like mass of manga- nate of sodium. This test may be made upon charcoal, but generally some nitre is added to the mixture, and the fusion is effected on platinum foil. The test for sulphur consists in fusing minerals such as the sulphates of barium, stron- tium, or calcium with soda upon charcoal in the reducing flame until the bulk of the melted flux is absorbed by the coal, the slaggy matter remaining, and then the adjacent portions of the charcoal are cut out and placed upon a bright coin or plate of silver, with the addition of a few drops of water, in which the alkaline sulphide formed dis- solves, producing a dark brown or black stain of sulphide of silver upon the metal. This is a very simple and delicate test ; care must, however, be taken that the soda used is free from sulphates, which point must be previously determined by testing it upon the metal alone. Fritting tests on charcoal Some infusible metallic oxides which under ordinary circumstances are colourless, become coloured when moistened with a solution of nitrate of cobalt and strongly heated upon charcoal. The colours are blue with alumina, flesh red or pink with magnesia, and green with oxide of zinc. The first and last of these are in fact the ordinary Cobalt blues and greens used as water colours. It is essential for this test that the substance tried should be infusible, or otherwise a cobalt blue glass or enamel will be formed. Tests with vitrifiable fluxes. The salts used for this purpose are Borax (di-sodic borate) and microcosmic salt or salt of phosphorus (ammonio-sodic phosphate), which melt into a clear glass at a red heat. The active agents are in either case the combined equivalent of boracic and phos- phoric acids respectively, borax being an acid salt, and phosphorus salt, though a neutral or bibasic, phosphate, loses its ammonia when heated, becoming a bibasic sodic 3IO Systematic Mineralogy. [CHAP. XV. phosphate with one atom of phosphoric acid free. The acids themselves may therefore be, and are sometimes, used instead of their sodium salts, but they are inconvenient from their excessively hygroscopic properties. The sodium salts of boracic and phosphoric acids have, when melted, the power of dissolving up nearly all metallic oxides, producing glasses which are characteristically coloured by even very minute quantities of such oxides as are of strong colouring power. The test with borax is one of the most useful in the whole series of blowpipe operations, and is performed by melting in the loop of a platinum wire a bead of borax, which should be perfectly colourless both hot and cold. A small quantity of the mineral to be tried, pre- ferably in a fine powder, is then added, the bead is remelted in one or other of the blowpipe flames until the substance is completely dissolved, when the bead is allowed to cool, and the colour due to the particular oxide and flame employed will be recognised. The following are the most characteristic reactions obtained in this way with compounds of different metals and borax : Iron : In reducing flame dark (bottle) green ; in oxidi- sing flame, yellow while the bead is hot, but becomes nearly colourless when cooled. Manganese : Colourless in reducing flame, deep violet or amethyst in oxidising flame. Chromium : Grass green in both oxidising and reducing flame. Uranium : Green in reducing, yellow in oxidising flame. Cobalt : Deep blue in both flames. This colouration is produced by an exceedingly minute quantity of oxide of cobalt, and is probably the most delicate of all the tests for this metal. Nickel : Reddish to brown hot, yellowish to dark red cold in oxidising flame, which is rendered blue by an ad- dition of nitre. In the reducing flame the colour disappears, the bead becoming grey with finely divided metallic nickeL CHAP. XV.} Tests with Borax. 3 1 1 These reactions refer to chemically pure preparations in mi- nerals ; they are generally obscured by the presence of cobalt. Copper-. Green while hot, turning blue on cooling in oxidising, and sealing-wax red in reducing flame. There is a great difference between the colouring power of the two oxides of this metal, the bead, which is perfectly transparent in the oxidising flame, being coloured with cupric oxide, becomes opaque from the formation of cuprous oxide or suboxide of copper in the reducing flame. This result is most easily obtained by heating the green bead on charcoal or touching it when melted with a piece of tinfoil so as to detach a minute globule of tin, which has an ener- getic reductive action upon the cupric oxide. In this way a small trace of copper may be recognised in the presence of iron, in spite of the strong colouring power of the latter metal in the reducing flame. The colours obtained with beads of salt of phosphorus are generally similar to those of borax, but there are some characteristic differences, especially in the reducing flame. Thus Iron gives a yellow or reddish tint instead of the dark green obtained with borax ; Vanadium gives a yellowish brown in the oxidising and green in the reducing flame, both being green with borax ; Uranium gives green in the oxidising flame. It is often of more importance for determinative purposes to know the colours given by minerals containing more than one element of strong colouring power than that of the components taken separately in a pure state, as in such cases very characteristic reactions are given by the combi- nation. This is specially the case with the iron compounds of certain metallic oxides ; thus, Tungsten in the form of tungstic acid, gives, with salt of phosphorus, a colourless or yellow bead in the oxidising, and a blue one, which is green while hot, in the reducing flame ; but if Iron is present, as in Wolfram (natural tungstate of iron), the latter colour is changed to a brownish red. 312 Systematic Mineralogy. [CHAP. XV. Another example is afforded by titanic acid, which, when pure as in Rutile, Anatase and Brookite, gives, with salt of phosphorus, a bead colourless or cloudy in the oxidising, and red, passing into violet when cold, in the reducing flame. The latter colour is, however, changed to brownish red when iron is present, as in titaniferous iron ores, and the violet colour can only be brought out by the addition of tin or zinc to the bead. The presence of titanic acid in iron ores may be rendered apparent by a method devised by Gustav Rose, which is of great interest. If the dark-coloured clear bead when completely saturated in the reducing flame be trans- ferred to the oxidising flame, it loses colour, but becomes opaque from the separation of titanic acid TiO 2 , which is much less soluble in melted phosphate of soda than the lower oxide Ti 2 O 3 . By flattening the cloudy bead between the forceps while still hot it may be rendered translucent, and when examined by the microscope is seen to contain minute crystals, having the characteristic square-based form of Anatase. These are perhaps more readily seen if a drop of water is added to the bead on the glass slide, when the vitrified sodium phosphate dissolves, releasing the crystals, which remain suspended in water, and are more visible. Quartz and many silicates, when heated in sodic phos- phate, do not dissolve, but leave an opaque mass usually called a silicious skeleton. This, according to Gustav Rose, is actually crystallised silica of the variety of low specific gravity known as Tridymite, a mineral found only in rocks of volcanic origin. The subject of the microscopic character of crystals formed by other oxides has recently received considerable attention from Sorby and other investigators, whose researches should be consulted by the student. Tests by the wet way. The use of the methods of quali- tative analysis by the wet way, in the determination of minerals, is necessarily restricted to those that can be applied CHAP. XV.] Tests by the Wet Way. 313 without requiring any great amount of apparatus or labora- tory appliances, and, as a rule, should only be used when the blowpipe tests are insufficient. They are therefore of most value in the examination of silicates and other insoluble compounds of the earthy and alkaline metals which are not readily recognisable by the dry way. The most useful reagents are hydrochloric, nitric, and sulphuric acids, ammonia, sulphide of ammonium, * caustic potash or soda, * phosphate of soda, * nitrate or chloride of barium, * oxalate of ammonia, * nitrate of silver, * molybdate of ammonia. These, with the exception of those marked *, are liquids, and must be kept in stoppered bottles, while the others may be kept either as solutions or in the dry state, the former being most convenient when they are often used, but when they are only required occasionally it is better to make the test solutions when wanted. In every case distilled water, or such as is free from sulphates and chlorides, is to be used. The principal pieces of apparatus, in addition to those already mentioned as requisite for use with the blowpipe, are a few porcelain capsules, the largest about 2 inches across, test tubes, a few small beakers and funnels, paper niters up to 2\ inches diameter, or sheets of filter paper, a wire filter stand, and, if possible, a small platinum crucible. The course of examination followed should be similar to that adopted in systematic analysis with substances of un- known composition, and in all cases the mineral should be finely powdered, the test for solubility in water being first applied. Among the minerals readily soluble are the dif- ferent alkaline Chlorides and Sulphates, the Alums, the Sulphates of Magnesium, Zinc, Copper, Iron, Nickel, &c. ; less so are Sulphate of Calcium (Gypsum), and Arsenious acid. Hydrochloric acid is next used, first without and then with the aid of heat, and subsequently the same acid in a more concentrated form when necessary. Minerals Systematic Mineralogy. [CHAP. XV. may be completely, partially, or not at all soluble by this treatment. The solution is attended with effervescence in the case of carbonates and sulphides, carbonic acid and sulphuretted hydrogen being respectively evolved. The latter may be recognised by its action upon a slip of test paper made with acetate of lead, which is blackened when exposed to the current of gas at the mouth of the test tube or beaker used for making the solution. The different carbonates belonging to the Calcite-Aragonite group vary considerably in their solubility in acids. Thus, calcic car- bonate in both forms effervesces readily with very weak hydrochloric or even acetic acid, while those of magnesium and iron, as well as the more complex varieties containing two or more of these metals, are but slowly acted upon even by strong acid in the cold. Compounds containing the higher oxides of manganese dissolve in warm hydrochlo- ric acid with evolution of chlorine. Silicates belonging to the Zeolite group are decomposed with a separation of silica in the gelatinous form, while others, such as Labra- dorite, leave a pulverulent or granular residue of silica, the metallic bases passing into solution as chlorides. In doubtful cases the effect of the acid may be established by filtering the insoluble residue, and testing the clear solution with ammonia and phosphate of soda ; if these reagents produce no precipitates, the mineral has not been acted upon. Nitric acid is used for the decomposition and solution of the higher sulphides and arsenides, and of native metals, most of which are attacked with the evolution of red fumes of peroxide of nitrogen, the production of nitrates of the metals, and, in the case of sulphides of sulphates, with a separation of sulphur. Nitrates being, however, inconvenient for most analytical determinations, it is customary to convert them into chlorides by adding hydrochloric acid, and evaporating to dryness previously to commencing the systematic exami- nation of the solution. CHAP. XV.] Examination of Silicates. 315 Gold, Platinum, and the allied metals, being insoluble in either nitric or hydrochloric acid alone, are brought into the soluble form as chlorides by the use of aqua regia, a mixture of nitric and hydrochloric acids, or any similar combination producing free chlorine. Sulphuric acid is principally useful in decomposing fluorides, producing sul- phates and hydrofluoric acid. Experiments of this kind must be performed in a leaden or platinum vessel, the finely powdered minerals being heated with strong sulphuric acid: the mouth of the crucible is covered with a glass plate protected by an etching ground of wax, upon which a design is marked by removing the wax with a steel point. Wherever the glass is laid bare, it is corroded by hydrofluoric acid vapour, reproducing the lines drawn upon the glass plate. Liquid, or rather a strong watery solution of hydrofluoric acid, is also an exceedingly convenient reagent for the decomposition of silicates where it is desired to obtain the metallic bases, especially those of the alkalies, the whole of the silica being removed as hydrofluosilicic acid gas ; but it cannot be used safely except in laboratories provided with good ventilation, owing to the corrosive action of its vapour on all glass and most metallic objects. The usual method of decomposing silicates insoluble in acids by fusion with carbonate of sodium in a platinum crucible over a spirit lamp or Bunsen gas burner, as adopted in quantitative analyses may be imitated on a small scale before the blowpipe, the fusion being effected upon charcoal instead of on platinum. In such cases a small quantity of borax should be added to the alkaline flux to prevent the latter being absorbed by the pores of the charcoal, and to form a well-defined bead of the fused mass that can be removed from the charcoal without loss. The bead so obtained is dissolved in dilute hydrochloric acid, the solu- tion evaporated to dryness, and the residue moderately heated, which renders the silica insoluble, while Lime, Magnesia, Alumina, and other bases may be removed by 316 Systematic Mineralogy. [CHAP. XV. digestion with dilute hydrochloric acid or soluble chlorides. The following are among the most characteristic reactions obtained by the wet way with the more abundant metallic oxides and mineral acids, and, as tests that can be used without requiring much refinement of apparatus or manipu- lation, are useful to the mineralogist. Alumina. This base, when contained in the hydro- chloric acid solution of a silicate after the silica has been separated by evaporation and heating, gives a white gelati- nous precipitate with ammonia, which is soluble in solution of caustic potash. When both Iron and Alumina are present, as is the case with many silicates, potash must be used to precipitate the former as ferric hydrate, and after filtration the Alumina is thrown down by adding carbonate of ammonia to the solution, the excess of potash having been previously neutralised by hydrochloric acid. The precipitate, when dried and ignited, is tested with cobalt solution in the manner described on p. 309. Iron. The solution of any iron mineral, when brought to the condition of a ferric salt by heating with a few drops of nitric acid, gives a dark blue precipitate with a drop of solution of yellow prussiate of potash. The ammonia pre- cipitate is rusty brown with ferric, and green with ferrous salts, the latter passing into the former when exposed to the air. Lime. In concentrated solutions a white precipitate of sulphate is produced by sulphuric acid, which is, however, sensibly soluble in water, so that this reaction cannot be used with weak solutions ; in such cases a precipitate may be produced by the addition of alcohol. Oxalate of am- monia is the most sensitive test, giving a white precipitate in very dilute solutions, after other bases separable by ammonia have been removed. Magnesia. This base may be recognised in the ammo- niacal solution remaining after the removal of Alumina, Iron, Lime, and other bases by ammonia and oxalate of CHAP. XV.] Tests for Magnesia and Baryta. 317 ammonia, by adding solution of phosphate of soda, which produces a crystalline precipitate of ammonio-magnesium phosphate. The formation of the precipitate is facilitated by well stirring the solution with a glass rod, and allowing it to stand for some hours. In the systematic examination of minerals containing Silica, Alumina and Iron Oxides, Lime and Magnesia, when the tests are applied in the order given, the whole number of bases may be verified in the same solution. It is essen- tial when Magnesia is present to have a considerable quan- tity of chloride of ammonia in the liquid, in order to keep it dissolved until the other bases are separated, otherwise it may be precipitated as hydrate of magnesia by ammonia. Baryta. This base is recognised by the insolubility of its sulphate in water, the smallest trace of it in a solution producing, with a drop of sulphuric acid or of solution of a soluble sulphate, a cloudy precipitate, which, however, may require some considerable time to form when the quantity is very small. Strontia behaves similarly, but the sulphate being more soluble takes a longer time to precipi- tate. When a solution containing both of these bases in hydrochloric acid is evaporated to dryness, ignited, and digested with alcohol, chloride of strontium dissolves, leaving a residue of chloride of barium, and the properties of both can be established by their flame reactions. Most of the so-called heavy metals are remarkable for giving dark-coloured precipitates when treated with sulphu- retted hydrogen or an alkaline sulphide, and they are divisible into groups according to the condition of the solution necessary to ensure precipitation. Thus, Gold, Silver, Lead, Copper, Bismuth, Arsenic, and Antimony can be separated as sulphides from an acid solution, while with Iron, Nickel, Cobalt, Manganese, and Zinc, the solution must be alkaline, as the sulphides of these metals are decomposed by acids. Aluminium and Chromium are also members of the latter group, but the precipitates formed with sulphide of ammo- 3 1 8 Systematic Mineralogy. [CHAP. XV. mum are not sulphides, but hydrates of alumina and chromic oxide. Whenever sulphuretted hydrogen is used, the solu- tion of the metallic bases must not contain much free nitric acid, as in that case no precipitation of sulphides will be effected until the oxidising action of the acid upon the sulphuretted hydrogen, which is attended with a separation of sulphur, is exhausted. The colours of the precipitates produced by sulphuretted hydrogen with the more abundant metals are as follows : Copper . . Brownish black. Lead . . . Bluish-black. Antimony . . Orange red. Arsenic . . Bright yellow. Aluminium . . White gelatinous hydrate of alumina. Chromium . . Green hydrate of chromic oxide. Zinc . . . White. Manganese . . Flesh colour. Nickel \ -PT , ~ , , > . . Black. Cobalt J The steps to be taken for the separation of the different metals existing in a mixture of precipitated sulphides will be found in the treatise on chemical analysis previously men- tioned, and it will not be necessary to go further in detail upon this point here, as for most purposes required by the mineralogist the metals contained in such mixtures may be more readily determined by the dry way with the blow- pipe. Silver, when in solution, may be detected, even when in very minute quantity, by means of hydrochloric acid or any chloride, which produce a white curdy precipitate of chloride of silver soluble in ammonia, and becoming grey or violet, and ultimately black when exposed to sunlight. The chloride is readily reduced to the metallic state when mixed with carbonate of soda and heated on charcoal. Lead. Solutions of this metal in nitric acid give a white CHAP. XV.] Tests for Silver, Gold, Copper, &c. 319 precipitate of sulphate of lead with sulphuric acid, and a crystalline deposit of chloride of lead with hydrochloric acid, insoluble in the cold, but dissolving readily in boiling water. The presence of silver in lead can be best deter- mined by the method of cupellation on bone-ash, a special modification of the process for use with the blowpipe having been contrived by Plattner. In this way very minute silver beads having the characteristic lustre and colour of the metal may be obtained, but it is a useful precaution to verify their properties by dissolving them in nitric acid and testing them with salt. If the silver should contain gold, it will remain as a black, insoluble speck in the acid solution. Gold may generally be best recognised by the dry way, but when in a moderately concentrated solution, it is pre- cipitated as a brown metallic powder by sulphurous and oxalic acid or ferrous sulphate. In a weak solution, chlo- ride of tin produces a purple or ruby-red coloration, or when sufficiently concentrated, a substance known as purple of Cassius, whose colour is due to finely-divided metallic gold. Copper. Solutions containing this metal as chloride or sulphate, when slightly acid, are decomposed by metallic iron with a deposition of copper. Thus a knife-blade is coppered in a few minutes when plunged into such a solu- tion. Ammonia gives a deep blue colour to the solution when the proportion of copper is very small, and yellow prussiate of potash is a still more delicate test, producing a brown precipitate, or, when only the minutest trace of copper is present, a reddish-brown tint to the liquid. Sulphuric acid. The test for this acid is the inverse form of that for baryta, a solution of nitrate or chloride of barium producing a white precipitate of sulphate insoluble in acids. Hydrochloric acid is detected by means of nitrate of silver solution, which produces chloride of silver, as de- scribed under the head of Silver. 320 Systematic Mineralogy. [CHAP. XV. Phosphoric add. When in minute quantity, this may be best detected by adding molybdate of ammonia to the solution, which must be acidified with nitric acid and heated for a short time, when a yellow precipitate contain- ing about 6 per cent, of phosphoric acid is produced, and subsides slowly. When the proportion is larger, it is separated as ammonio-magnesian phosphate, in the same way as described for magnesia, except that some soluble magnesian salt, as sulphate or chloride of magnesium, is added instead of phosphate of soda. This test may be performed in the presence of ferric oxide and alumina if a small quantity of citric acid is present, as these oxides are not precipitated by ammonia when organic matter is present. Quantitative analysis. Although the methods of quali- tative testing are sufficient in the greater number of instances for the identification of well-established minerals, there is a large remainder even of these which cannot be so identified, a knowledge of their actual composition being required in addition ; and the same holds good with those of doubtful or unknown constitution, especially when the latter are from new localities. For those, therefore, who may desire to extend the field of mmeralogical knowledge, a practical acquaintance with the methods of quantitative analysis as applied to minerals, or, at any rate, to the more abundant and simply constituted species, is essential, but the larger number of students, who may require only a good sight knowledge of known minerals, may be content to accept the results furnished by analytical chemists without further inquiry. In this case, however, a knowledge of the method of calculating the results of analyses, or the deduction of formulae from percentage quantities, will often be found of great use. Constitution of Minerals. Among minerals are included not only elementary substances, but combinations of two or more elements forming the classes of compounds known as CHAP. XV.] Chemical Composition. 321 oxides, sulphides, acids, bases, haloid salts, oxysalts, sulpho salts, double salts, anhydrides, and hydrates. Such combinations, being fixed, have a definite constitu- tion. Thus, quartz invariably contains 7 parts of silicon, and 4 of oxygen ; calcite, 10 of calcium, 3 of carbon, and 12 of oxygen ; fluorspar, 20 of calcium, and 19 of fluorine ; and so on. The proportion of the different components being constant in those forms that are normally constituted, de- viations from the normal types can be shown to be caused either by foreign substances existing as mechanical im- purities, or by partial substitution of one or more of the component elements by others of analogous properties, according to the laws of isomorphism. In the former case, the proportion between the essential constituents, notwith- standing their absolute diminution in quantity, is unchanged ; while in the latter, the change, though less simple, consists in the substitution of one element for another in the propor- tions of their chemical equivalents. Thus, a specimen of calcite containing 10 per cent, of silica, clay, or other con- stituents insoluble in hydrochloric acid, cannot be con- sidered as differing essentially from the normal composition, if the remaining 90 per cent, is so constituted as to make up the proportion 10 : 3 : 12, or 9 : 27 : io'8, which are the ratios of calcium, carbon, and oxygen in calcite when in a pure state. Variation in composition, due to the second of the above causes, or the partial substitution of analogous elements, is commonly observed in the same mineral, as almost all specimens of calcite show a deviation from the typical composition by containing small quantities of the elements magnesium, manganese, iron, or zinc. These, however, are held to be in partial substitution of the normal amount of calcium, and the analyses, when interpreted according to the theory of equivalent proportions, are found to be in accordance with the normal constitution. This theory supposes every element to have a combining 322 Systematic Mineralogy. [CHAP. XV. value, or quantity peculiar to itself, and that its compounds with other elements are formed either in the ratio of that quantity, or of one or more simple multiples of it. Thus, supposing A and B to be two elements, they may form compounds : zA+B, A + B, 2A-\-$B, A + 2B, &c. It is further supposed that analogous compounds may be made with another element C : 2 A + C, A + C, 2 A -f 3 C y A + 2C, in which the values of A remain unaltered ; and in like manner, other elements, D, F, &c., may be substituted for A, giving a series of compounds with B and C, in which the latter are unchanged. The special quantities of the elements so substituted, or the combinations, are said to be equivalents ; and if the weight of any one be known or assumed, the others may be referred to it, whereby a series of constant numbers expressing the combining proportions of the different elements is obtained. These are called chemi- cal equivalents. In forming such a series it is necessary to fix upon some one element as a basis, and this choice is necessarily an arbitrary one. For this purpose the elements oxygen and hydrogen have been chosen : the former as being the most abundant element in nature, and the latter as being the lightest In the oxygen series of equivalents, introduced by Berzelius, oxygen was taken at i oo, with the result of giving inconveniently large values to most of the elements, especially to such metals as silver, gold, antimony, &c. In spite of this drawback it was for a long time current in France and Germany, and is used in the greater number of works upon mineral chemistry of the first half of the present century, a period which has been more prolific in discovery in this branch of science than those immediately preceding or following. In England it has been customary to use Dalton's scale, upon the basis of which hydrogen is assumed as unity, being the lightest of all the elements, from which, assuming its combination with oxygen in water to be in the proportion of single equivalents by weight, the equivalent of CHAP. XV.] Atomic Weight. 323 the latter is found to be 8, that of water HO, or protoxide of hydrogen, 9, and so on. Latterly, however, a modifica- tion of Dalton's original hypothesis, founded upon the weight of equal volumes of the elements when in the gaseous form, has come into general use among chemists, and the older schemes, founded upon considerations of weight alone, have been practically abandoned. This is founded upon the proposition known as Avogadro's law: namely that equal volumes of all gases contain an equal number of molecules. By the term molecule is meant a quantity of an element, or compound of elements, capable of independent existence, but so small as to be incapable of further division. The molecule of a compound is a complex of still smaller portions of the component elements, which are known as atoms, an atom being denned as the smallest combining proportion of an element. A molecule must therefore contain at least two atoms which, in the case of an element, are both of the same kind, but in that of a compound are of dissimilar kinds (or those of the constituent elements). The atomic weight of an element is the weight of a volume of its vapour expressed in terms of the weight of a similar volume of hydrogen ; the unit weight adopted being that of one cubic centimetre of hydrogen under the normal pressure of 760 mm. of mercury at o Centigrade. The molecular volume of an element or compound is that corresponding to two volumes of hydrogen ; the molecular weight of an element is therefore usually double its atomic weight. When elements or compounds can be obtained in the state of gases, their molecular weights may be determined by direct experiment, otherwise a vapour density must be assumed upon considerations founded upon analogies drawn from known compounds supposed to be similarly constituted,. Such determinations must necessarily be doubtful. The atomic weight of an element is determined from the analysis of some one of its best denned and most stable Y 2 324 Systematic Mineralogy. [CHAP. XV, compounds, the molecular constitution of the latter being assumed. The researches of Dulong and Petit have shown that the atomic weights of elements are to each other in- versely as their specific heats. The following table contains the atomic weights of the elements as far as they have been accurately determined. The symbol prefixed to each is held to signify an atomic unit, when used in combination, in the construction of molecular formulae. TABLE OF THE ATOMIC WEIGHTS OF ELEMENTS. Name Symbol Class Atomic weight Aluminium . , Al. II. IV. VI, 27-3 Antimony (Stibium) Arsenic . Sb. As. III. V. I. III. V. 122 75 Barium . Ba. II. IV. 137 Beryllium (Glucinum) Bismuth . Be. Bi. II. V. 9-33 208 Boron . B. III. II Bromine Br. I. III. V. VII. 80 Cadmium Cd. II. 112 Calcium . Ca. II. IV. 40 Carbon . C. II. IV. 12 Cerium . Ce. IV. 92 Coesium . Cs. I. 133 Chlorine Cl. I. III. V. VII. 35'5 Chromium Cr. II. IV. VI. 52 Cobalt . Co. II. IV. 59 Copper (Cuprum) Didymium Erbium . Cu. Di. Er. II. II. II. 63'4 96 112-6 Fluorine . Fl. I. 19 Gallium . Ga. VI. 69*8 Gold (Aurum) Au. I. III. 196 Hydrogen Indium . H, In. I. III. 1137 Iridium . Ir. II. IV. VI. 198 Iron (Ferrum) Fe, II. IV. VI. 56 Iodine . I. I. III. V. VII. 127 .Lanthanum La. II. 93 Lead (Plumbum) Pb. II. IV. 207 Lithium . Li. I- 7 Magnesium Mg. II. 24 CHAP. XV.] Table of Elements. 325 Name Symbol Class Atomic weight Manganese . Mn. II. IV. VI. 55 Mercury (Hydrargyrum) Hg. II. 200 Molybdenum . Mb. II. IV. VI. 92 Nickel . Ni. II. IV. 59 Nitrogen N. I. III. V. 14 Niobium Kb. V. 94 Oxygen . Osmium . O. Os. II. II. IV. VI. 16 108 Palladium Pd. 11. IV. 106 Phosphorus . P. I. III. V. 31 Platinum Pt. II. IV. 198 Potassium (Kalium) K. I. III. V. 39 Rhodium Rd. II. IV. VI. 104 Rubidium Rb. I. Ruthenium Ru. II. IV. VI. 104 Sulphur . S. II. IV. VI. 32 Selenium Se, II. IV. VI. 79 Silver (Argentum) Ag. I. III. 1 08 Silicon (Silicium) Si. IV. 28 Sodium (Natrium) Na. I. III. 23 Strontium Sr. II. IV. 88 Tantalum . Ta. V. lS2 Tellurium Te. II. IV. VI. 128 Thallium TL I. III. 204 Thorium . Th, IV. 234 Tin (Stannum) Sn. II. IV. 118 Titanium TL 11. IV. 248 Tungsten Wolfram Tu.1 Wj IV. VI. 184 Uranium u. II. IV. VL 240 Vanadium V. IIL V. 51-4 Yttrium . Y. II. 617 Zinc Zn. II. 65 Zirconium Zr. IV. 90 [Elements marked I. are monads ; n. dyads ; III. triads ; IV. tetrads ; v. pentads ; vi. hexads ; and vn. heptads.] The elements are classified according to their atomicities, or combining power, as measured by the number of hydro- gen atoms with which they combine to form definite com- 326 Systematic Mineralogy. . [CHAP. XV. pounds. Thus hydrogen unites in the following manner, with Chlorine, i equivalent to form hydrochloric acid HC1. Oxygen, 2 water H2<3. Nitrogen, 3 ammonia H3N. Carbon, 4 marsh gas H4C. in which instances the quantities i, 2, 3, 4 express the combining power, or quantivalence, of the respective ele- ments, which are said to be monad (i.), dyad (IT.), triad (HI.), or tetrad (iv.), according to the special number of hydrogen atoms attached to them. Besides these, there are higher ratios of quantivalence, known as pentads (v.), hexads (vi.), and heptads (VIL). In the table the different elements are classified by these numbers, and it will be seen that one element may have several distinguishing atomicities, or that it may form several different types of compounds. Thus, sulphur is dyad in sulphuretted hydrogen, H 2 S ; tetrad in sulphurous anhydride, O 2 S (oxygen being dyad) ; hexad in sulphuric anhydride, O 3 S. The hexad forms of chromium, aluminium, manganese, iron, nickel, and cobalt are typified in the compounds known as sesquioxides, or those containing two equivalents of metal to three of oxygen, or i : i^. In such cases the double equivalent of the metal is usually considered as a unit, and is represented by a barred symbol, whose atomic weight is double that of the ordinary atom. Thus Al repre- sents A1 2 ; e. Fe 2 ; Al O 3 , A1 2 O 3 ; e C1 3 , Fe 2 C1 3 , &c. This arrangement is specially convenient in representing the composition of minerals where the same metal occurs in two states of combination, as is often the case. Construction of chemical formula. The number of atoms of the different elements entering into the constitution of a mineral, when its composition has been determined by analysis, is found by dividing the percentage proportion of each element by its atomic weight, and, subsequently, CHAP. XV.] Chemical Formula. 327 dividing the quotients so obtained by the smallest among them, which gives a series of numbers standing in a simple relation to each other, which, when reduced to whole num- bers, give the number of atoms required. For example, Baryte, or Heavy Spar, gives the following analysis : Barium 58-80 . . 137 . . ^^ = 0-429 = i Sulphur 1373 . . 32 . . Mfp =0-428=1 Oxygen 27-47 . . 16 . . 2 -P =1-717 = 4 lOO'OO The numbers in the last column being in the ratio i : i 14, the formula required is BaSO 4 . When three or more elements are contained in a mineral the formula obtained by writing down the number of atoms of the constituents side by side is known as an elementary or empirical formula. This is merely the simplest nu- merical expression obtainable, not expressing any opinion on the probable arrangement of the components. In most cases, however, it is necessary to obtain some idea of the grouping of the constituents, for which purpose a knowledge of the principles of chemical classification is requisite ; and the formulae constructed by these means are known as rational or constitutional formulae. The range of compounds occurring in minerals is, however, comparatively small It will only be necessary here to consider the principal types of composition known as acids, bases and salts. According to modern views, an acid is a compound of hydrogen with an electronegative element or combination of elements known as a compound radical. Hydrochloric acid HC1, hydrobromic acid HBr, and hydrofluoric acid HF1 are examples of the first kind, or acids with simple radicals. The acids of compound radicals may contain either oxygen or sulphur in the radical, but they are otherwise analogous in constitution ; the former are called oxygen acids, and the latter sulphur acids. They 328 Systematic Mineralogy. [CHAP. XV, are further distinguished according to the number of atoms of hydrogen, as monohydric with one, dihydric with two, or trihydric with three atoms. The constitutional formulae of acids are constructed on the hypothesis of containing one, two, or three atoms of oxygen or sulphur, one half of which is united with an equal number of atoms of hydrogen, and the other half with an acid radical, which may be either mono- di- or tri-valent, according to the character of the third element. Thus, nitric acid has the. following formula : Elementary, Constitutional. HN0 3 H-0-(N0 2 )' Sulphuric acid . . H 2 SO 4 H 2 =O 2 =(SO 2 )" Phosphoric acid . H 3 PO 4 H 3 3O 3 =(PO)'" Sulpho-carbonic acid . H,CS 3 H 2 =S=(CS)" where the accents represent the equivalency of the radical. When the molecule of an acid is broken up by the re- moval of the whole amount of hydrogen, and the corre- sponding quantity of oxygen required to form water, H 2 O y or one atom of the latter to two of the former, or in the case of a sulphur acid of sulphur to form sulphuretted hydrogen, H 2 S, a compound is obtained known as the anhydride of the acid, which is, in fact, an oxide of the radical. In the case of mono- and tri-hydric acids two molecules are re- quired in order to express the results in whole numbers of atoms. Thus : Nitric acid (2 mol.) or 2 (HNO 3 ) less I eq. H 2 O giving \ O = N 2 O 5 Sulphuric acid (i mol.) H 2 S0 4 i H 2 O (SO 2 ) O = SO, Phosphoric acid (2 mol.) 2 (H 3 P0 4 ) 3 H.O (pg) O, - P 2 O 5 Carbonic acid (i mol.) H 2 CO 3 I H.O (CO) O = CO 2 Sulpho-carbonic acid (i mol.) H,CS 3 I H 2 S (CS) S = CS, CHAP. XV. ] A cids and Bases. 329 The compounds in the last column are therefore called anhydrides of the corresponding acids, or generally but im- properly, anhydrides, without further qualification. In the older works on chemistry they are called anhydrous acids, the acids denned as above being considered as hydrated acids. A base is defined to be a combination of hydrogen with a compound radical of an electro-positive character, con- sisting of an electro-positive element or metal united with oxygen or sulphur, the former being called an oxygen, and the latter a sulphur, base. The constitution of bases is represented similarly to that of acids, the constituent atoms of oxygen being con- sidered as combined to the extent of one-half with an equal number of hydrogen atoms, and the other half with an equivalent atom of the metal, producing, as in the case of the acids, monohydric and polyhydric bases, or hydroxides, or with sulphur as hydrosulphides, thus : i i H-O-K is Hydropotassic oxide, or of the type (HO) R ii ii H 2 = O 2 = Ba,, Hydrobaric oxide (HO) 2 R in in H 3 = O 3 = Bi,, Hydrobismuthic oxide (HO) 3 R The Roman figures represent the equivalency of the metallic element, and in the type formulae in the last column R stands for any metal of corresponding equivalency. And of the analogous sulphur compounds : i i H S K is Hydropotassic sulphide, or of the type (HS) R H 2 =S 2 = Ba,, Hydrobaric sulphide (HS) 2 R" in in H 3 = S 3 = Bi Hydrobismuthic sulphide (HS) 3 R Basic anhydrides are produced in the same manner as the acids by the removal of the hydrogen, with sufficient oxygen or sulphur to form water or sulphuretted hydrogen 33O Systematic Mineralogy. [CHAP. XV. from the hydrometallic oxides or sulphides respectively. Thus : 2 HOK H.,O = K 2 O known as potassic oxide or potash. H 2 O 2 Ba H 2 O = BaO ,, baric oxide or baryta. u f \ TJ; i TT r TJ; n / bismuth oxide, or strictly di- 2H 3 3 Bi- 3 H 2 Bi 2 3 | bismuthlc tr ; oxide> And from the analogous sulphur compounds 2 HSK - H 2 S = K 2 S or dipotassic sulphide. H 2 S 2 Ba - H 2 S = BaS baric sulphide. 2 H 3 S 3 Bi - 3H 2 S = Bi 2 S 3 dibismuthic trisulphide. The last of these is the mineral known as Bismuth Glance. The anhydrides of bases are therefore the oxides and sulphides of their respective metals. A salt is considered to be a combination formed by the action of equivalent quantities of an acid and a base upon each other when the whole of the hydrogen and one of the atoms of oxygen are removed, as water. Thus : Nitric acid . . H-O-(NO 2 )) (K-O-(NO 2 ) or K NO, potassic nitrate, and f = ( potassic n Hydropotassic oxide . H-O K } (H 2 O water. Sulphuric acid . . H 2 = O 2 = (SO 2 ) \ ( Ba = O 2 = (SO,) or BaSO 4 \ = \ baric sulphate, and Hydrobaric oxide . H 2 = O 2 = Ba j (2 H 2 O water Acids and bases containing like amounts of hydrogen combine in equal molecules, but when they are unlike the molecular relation of acid to base in a salt is dissimilar. Salts formed by acids of a simple radical, such as hydro- chloric acid, HC1, and hydrofluoric acid, HF1, are called Ha- loid Salts, and those with acids of compound radicals Oxy-salts and Sulpho- salts. A neutral salt is that resulting from equi- valent quantities of acid and base ; it is also called a normal salt. An acid salt is a combination of a molecule of normal salt with one or more molecules of acid ; while a basic salt is similarly a normal salt combined with one or more mole- cules of base. Bisulphate of potassium < f j 2 gQ 4 f * s an CHAP. XV.] Constitution of Salts. 331 example of the former, and Malachite or basic carbonate of Copper j H U Q?6 I" of the latter class ' Acid and basic salts may in some cases be free from hydrogen, that is, they may consist of a normal salt com- bined with the anhydrides of the acid or base respectively. Of this character are acid bisulphate of Potassium K2 ?S 4 \ bU 3 J which is obtained by heating the salt w! 4 1 , basic chro- W 2 bO 4 J mate of Lead 2 pb Q 4 j , and the oxychlorides of Lead pbO 2 1 and 2 p b( j I forming the rare minerals Matlockite and Mendipite. Double salts are compounds of two different salts, which may be either similar or dissimilar in class or constitution. r KCI Thus, Carnallite < AT pi is a compound of two similar haloid salts, the chlorides of Potassium and Magnesium : Cryolite < AJ THA of the two haloids of dissimilar consti- tution; Blodite < jyj- a QQ 4 of two similar sulphates ; Potash K SO 1 Alum A1 c 2 4 > of two dissimilar oxysalts (sulphates) and AU03U 12 J Chlorapatite ^ i a ^> 2 ^ ? of an oxysalt and a haloid, phos- 3Ca 3 r 2 ^8 J phate and chloride of calcium. The acid sulphate of po- tassium TT 2 gQ \ niight also be regarded as a double sul- phate of potassium and hydrogen, except for the special signification attached to the term acid. Many minerals, especially alkaline sulphates and other easily soluble salts, give off water with more or less readiness when the crystals are exposed to the air, in some cases without, but more readily by, heat. Such water is usually regarded as not essential to the constitution, or as water of 332 Systematic Mineralogy. [CHAP. XV. crystallisation, when it is given off at the boiling point of water or a little above it, and when the salt so dehydrated takes the same amount again when dissolved and recrystal- lised. The water so combined is often distinguished by the symbol Aq. Thus, Potash Alum, containing 24 equiva- lents of water, is represented by A1 known as Alstonite, has the same symmetry as its constituents Witherite and Aragonite ; but Baryto-Calcite, which is of similar constitution, belongs to the oblique system, proving the type RCO 3 to be actually trimorphous, although no carbonate of a single base is known to crystallise in the latter system. Similar cases are presented in the following series, which establish isomor- phous relations between the dyad sulphates and carbonates and the combination of both : Rhombohedrai. Rhombic. Oblique. RCO 3 Calcite CaCO 3 Aragonite CaCO 3 Baryto-Calcite RSO 4 Dreelite An S lesIte PbSO Glauberite Lanark CHAP. XVI.] Polysymmetry. 339 Substances of the above kind that are both isomorphous and heteromorphous are said to be isodimorphous or isotri- morphouS) according to the number of different crystalline systems in which they occur. Polysymmetry. One of the most important series of mine- rals, known as the Hornblende-Augite group, is represented by the general formula R"SiC>3, where R n =Ca, Mg, Fe, or Mn. The members of this group are not isomorphous in the sense of having the same crystallographic symmetry, as some of them occur in the rhombic, others in the oblique, and others in the triclinic system ; but a general similarity in the geo- metrical elements of the four is observed. Thus, Diopside or Augite, of the type M^g-^ 3 ( cr y sta ^i s i n S in the oblique system, has the fundamental parameters a : b : c i '094 : i : 0-591, and/3=74 ; while those of Tremolite or Hornblende 3MgSiO 3 } are ^ :b:c = '544 : i : 0-294, and /3= 75 15'. The parameters of the axes a and c in Augite are therefore approximately double those of Hornblende, while the angle ft is nearly the same in both species. A second group, ^^eSiO 3 i ' re P resente d by the species Bronzite, Hypersthine and Enstatite, is rhombic, with the parameters a : b : c 1*031 : i : 1*177. These may be compared with those of Augite, if the latter be referred to a system of axes that are rectangular or nearly so, which is done by considering the face (ooi) as (101), which gives the parameters a : b : c = 1-052 : i : 0-296, and /3=894o'. If the same face be further noted as (104), the following close approximation between these new oblique parameters and the rhombic ones becomes apparent : Bronzite . 1-031 Augite . 1-052 c ft 1-177. 90 1-182. 89 40', Z 2 340 Systematic Mineralogy. [CHAP. XVI. Similar approximations between the elements of sub- stances of analogous or identical composition, but crystal- lising in different systems, are observed in the rhombic and hexagonal varieties of Sulphate of Potassium and in Albite and Orthoclase. They are included by Rammelsberg under the general head of isomorphism, but the special term polysymmetry has been applied to them by Scacchi. When a mineral contains both dyad and hexad bases, it may, by the progressive substitution of one metal for another of the same class, vary considerably both in composition and physical characters without change of form. One of the best examples is afforded by Garnet, which occurs in many varieties, differing considerably both as regards colour and density, but all crystallising in the cubic system the rhombic dodecahedron being the dominant form ; the ob- served range of the four principal bases being as follows : Alumina . (AiO 3 ) o to 22 per cent Ferric Oxide . (FeO 3 ) o to 30 Lime . . CaO o to 37 Magnesia . MgO o to 22 When, however, the proportions of the bases in a lime- alumina and a lime-iron garnet respectively are reduced to the above values, it is found that in the first case Ca : At= 3 : i, and Ai : Si=i : 3, and in the second Ca : Fe=3 : i andFe : Si=i : 3 ; while in both Ca : Si=i : i. Whence it appears that the two compounds are of the analogous com- position Ca 3 AiSi 3 Oi 2 and Ca 3 FeSi 3 O l2 , and that those containing both alumina and ferric oxide are isomorphous mixtures of both types in varying proportions. In addition to these, other varieties are known contain- ing the following silicates Mg 3 AlSi 3 12 Mg 3 FeSi 3 12 Fe 3 AJSi 3 12 Fe 3 FeSi 3 12 Mn 3 AiSi 3 O 12 Mn 3 FeSi 3 O ia CHAP. XVI.] Isomorphism. 341 either independently or in combination with the calcium silicates given above. The term Garnet, therefore, is not special to any one of these compounds in particular, but distinguishes a group of isomorphous silicates, which, however much they may differ qualitatively, have the above ratio, 3 : i : 3, for their dyad and hexad metals and silicon respectively common to all, or may be represented by the generalised formula, II VI R 3 RrSi 3 O 12 , which covers every possible variety of compo- sition indicated by the above special types. The isomorphism of compounds, not containing the same number of elementary atoms, supposes the substi- tution of the elements to take place in the proportion of their equivalence, two atoms of a monad replacing one of a dyad element, &c. This is seen in the Diaspore group, where H 2 in Gothite and Manganite represents Be in Chryso- beryl, and Fe, Mn, or Zn, in the analogous dimorphous species, Magnetite and Franklinite of the spinel series. Another example is afforded by Oxygen and Fluorine, O replacing F1 2 , or R 2 O=RF1, and RO=RF1 2 . This is seen in Topaz 5, c-pi 5 f? which is rhombic and isomorphic with Andalusite A4SiO 5 . The isomorphism of analogous compounds of monad I n vi (R), dyad (R), and hexad (R) elements is apparent in the Augite group of silicates which, in addition to the varieties already mentioned as represented by the constitution RSiO 3 , contains others both in the augite and hornblende series, in whose composition sodium, aluminium, and ferric silicates form part, in addition to the dyad metals Ca, Mg, Fe, &c. Of these, the following, Babingtonite ^|f^ j , Achmite Na 2 Si0 3 ^ Na ? Si0 3 -) FeSiO 3 > , and Aegirite 2RSiO 3 >, appear in the augite 2^Si 3 O 9 J eSi 3 O 9 J 342 Systematic Mineralogy. [CHAP. xvi. Na 2 SiO 3 form, while Arfwedsonite RSiO 3 \ assumes that of horn- blende. These formulae suppose Na 2 SiO 3 and RSiO 3 to be equivalent molecules, three of which correspond to one of ferric silicate FeSi 3 O 9 . In one instance, in the augite group, the R elements are completely absent: this is ,~D ciiO "I ** in Spodumene 6 AJO- <~\ f > where R is replaced by 2R 4zTlol 3 wg J I =Li, Na, and e by Ai. Here 2A1 2 is equivalent to 3R. niv vi The isomorphism of RRO 3 and R 2 O 3 is illustrated by the case of Titanic iron ore, a term applied to several minerals of varying composition, containing Iron, Titanium, and Oxygen, and usually some Magnesium, but which, ac- cording to Rammelsberg, can be represented by the general expression w -FeTiO 3 \ ? a n having the crystalline form of /zre 2 L^ 3 j Hematite, or Fe 2 O 3 . This view is not universally adopted, as another hypothesis supposes them to contain Ti 2 O 3 , the blue oxide of titanium. The aluminous varieties of augite and hornblende may be similarly represented by \ , Q 3 > . 2 3 J The most remarkable examples of isomorphism combined with dissimilarity of constitution are afforded by the alkaline nitrates ; potash-nitre, or saltpetre, KNO 3 , being crystallo- graphically almost identical with Aragonite, while nitrate of sodium, NaNO 3 , is equally close in form to Calcite. There are two cases of isomorphism of minerals not of analogous constitution among the class of silicates. These are Spodumene and Petalite, and Anorthite and Albite. The former are both oblique and closely allied in form, but completely dissimilar in composition Spodumene, being R 6 A4 4 Si 15 O 45 , or a bisilicate ; and Petalite R r} Al 4 Si 30 O 75 , or a quadrisilicate. CHAP. XVI.] Growth of Crystals. 343 Similarly, in the second case, both minerals being triclinic and isomorphous members of the lime-soda felspar group Anorthite, CaAlSi 2 O 8 , is a monosilicate ' and Albite, Na 2 AlSi 6 O 16 , a trisilicate. When a crystal of a salt is placed in a solution of some other similar salt of an isomorphous metal brought to the crystallising point, it will increase in size by the addition of layers of the new salt, which will be symmetrically disposed about the planes of the nucleus, exactly in the same manner as would have happened had the growth been contained in the original solution. The form will therefore be preserved, but the crystal will obviously be only a mixture of hetero- geneous substances, and its composite nature will be appa- rent if there is any marked difference in physical characters between the different constituents. One of the best examples of this kind of structure is furnished by the double sulphates I VI I of the alum series R 2 R 2 S 4 O 16 24Aq, where R 2 maybe either VI ammonium, potassium, or some other monad metal, and R 2 either Chromium, Iron, or Aluminium. Two of these, the ammonia-aluminium and potassium-aluminium salts are colourless, while the chromium and iron salts are strongly coloured, the former being dark green and the latter violet, so that crystals formed from the solutions of two or more of them present strongly contrasted alternating bands of colour upon a cross section ; or, if one of the colourless salts is from the outer layer, they may appear as transparent octahedra with coloured centres. Crystals of this kind, although illus- trating the phenomena of isomorphism in a graphic manner, are obviously only mechanical mixtures whose heterogeneous character is plainly visible, and cannot therefore be repre- sented as compounds of isomorphous bases in the same sense as those of dolomite, pearl spar, and other minerals are, where the combination extends to the individual crystal- line molecules. They are, however, of considerable import- 344 Systematic Mineralogy. [CHAP. XVI, ance as illustrations of facts which occur in nature tolerably frequently. Thus crystals of Vanadinite, 3PbV 2 O 8 PbCl 2 , from Russia, are occasionally found to contain a nucleus of the isomorphous species Pyromorphite, 3PbP 2 O 8 PbCl 2 , which is of the same hexagonal form. Crystals of Tour- maline when transparent are also commonly observed to be banded in different colours which correspond to differ- n ences of composition in the R bases. In the felspar group, apparently homogeneous crystals are often made of alter- nations of the isomorphous minerals Albite and Orthoclase, and these being colourless, it is often difficult to distinguish one from the other, the use of optical tests being necessary in such cases. The same thing probably occurs in many other minerals, and is the cause of the discrepancies between the theoretical composition as required by the formula, and the results obtained by analysis. In this respect minerals differ essentially from crystallised salts artificially prepared, which may, by particular manipulation, be obtained in a state of almost absolute purity, while the former almost invariably contain some matters foreign to their essential constituents. Another interesting case of crystals made up of alter- nating layers of isomorphous compounds of different com- position is occasionally seen in the arsenides of Nickel and Cobalt (NiAs 2 , CoAs 2 ). These are both cubical, and found in large lead-grey crystals apparently perfectly uniform in composition, but, when exposed to damp air, become oxi- dised with the formation of basic arseniates of the respective metals that of cobalt being pink and that of nickel pale green, so that the crystals when broken across often weather in layers which are alternately coated with pink and green incrustations, according as one or other metal predominates in the particular layer. It is probable that the universal presence of gold in minute quantities in such minerals as galena, PbS, and iron CHAP. XVII.] Alteration of Minerals. 345 pyrites, FeS 2 , may be due to a mechanical isomorphous intermixture of this kind, as all these species are cubical in form, and there is no reason to suppose that the gold is in chemical combination as it may often be extracted by the process of solution in mercury known as amalgamation. CHAPTER XVII. ASSOCIATION AND DISTRIBUTION OF MINERALS. MINERALS, when exposed to the action of air, water, car- bonic acid, and other agents of a similar kind tending to produce alteration in chemical composition, show very unequal degrees of stability ; some species, such as gold, diamond, and graphite, the different forms of carbon, quartz and tin ore being practically unalterable, as they are neither susceptible of change by oxidation, nor reduction in air at the ordinary temperature, and almost, if not quite, insoluble in meteoric or ordinary spring waters ; while, on the other hand, soluble and hydrated salts, especially those containing the alkaline metals, and the dyad forms, iron, manganese, and calcium, are, in a greater or less degree, liable to change either in form or composition under ordinary atmospheric vicissi- tudes. The following are some of the most genera 1 cases : i. Alteration by loss of water. This is called efflores- cence, and is characteristic of minerals containing water of crystallisation which may in dry air be given off either entirely or in part. Crystals of Laumonite, j 9f^.A ] +4Aq, lose I A1 2 51 3 U 9 J water by exposure and fall to pieces, although the change may be very gradually effected. Crystals of gypsum, CaSO 4 2Aq, which are perfectly transparent when fresh, are often found in dry countries to become opaque either wholly or in part from a partial dehy- dration when exposed to the air. 346 Systematic Mineralogy. [CHAP. XVII. 2. Solution and absorption of water. Minerals whose crystals lose their form by absorption of atmospheric mois- ture, and are ultimately converted into solutions are said to be deliquescent ; this property is common to many of the soluble alkaline salts, such as nitrate of soda, common salt, sal ammoniac, &c. 3. C fiange by oxidation of one or more constituents. This is a very common occurrence in ferrous or manganous com- pounds, and is generally known as rusting. It is most rapidly developed in the soluble salts of these metals. Thus, n Ferrous sulphate (FeSC^yAq) is of a well-defined consti- tution and form, but the crystals can only be preserved in absolutely dry air, or in the vapour of a hydrocarbon, as under ordinary conditions of exposure they become dull and rusted through the production of ferric salts, a very large series of which are known in nature, and are produced by progressive oxidation, the ultimate product of such alter- ation being a Ferric hydrate (H 6 Fe 4 O 9 ) and acid ferric sulphate. In the same way, carbonates containing the same base, such as Siderite, Pearl spar, Dolomite, become inva- riably rusted externally when exposed to the air, even when the proportion of iron present is but small. Manganous compounds which, when fresh, are of a delicate rose red, are even more susceptible, as they turn brown by exposure to sunlight, and ultimately become brown or black by the formation of manganic-oxide MnO 2 ,the brown oxide Mn 3 O 4 , or their hydrates. For this reason, specimens of minerals such as Rhodonite MnSiO 3 and Diallogite MnSO 4 are gene- rally kept in the dark, or the cases containing them in museums are screened from direct light. Most metallic sulphides and arsenides are similarly liable to change by oxidation in damp air, with the formation of oxysalts of their constituents. Thus, Galena PbS gives rise to Anglesite PbSO 4 , Zinc blende ZnS to Zinc vitriol ZnSO 4 7Aq, Cobalt speiss CoAs 2 to the hydrated arseniate CHAP. XVII.] Action of Air and Carbonic Acid. 347 Co 3 As 2 O 8 . +8Aq, known as Cobalt bloom, and Nickel speiss NiAs 2 to the corresponding Nickel salt or nickel bloom. Bisulphide of iron FeS 2 , which is among the commonest of minerals, forming the dimorphous species Pyrites (cubical), and Marcasite (rhombic), besides being found in various isomorphous mixtures with other metallic sulphides and arsenides, yields, by the simultaneous oxidation of both con- stituents, Ferrous sulphate or Green vitriol FeSO 4 , and Sulphuric acid, the former salt going through the changes previously noticed, the ultimate product being Ferric hydrate and basic ferric sulphate ; but when aluminium or sodium compounds are within reach, the iron salts may be com- pletely destroyed with the formation of the sulphates of these metals, Glauberite, Gypsum, and Alum. This group of reactions is one of the most important in the whole range of natural chemistry, as it is concerned in the production of the quantity of soluble sulphates present in most terrestrial waters, and which, in the case of mineral springs, often amounts to a considerable percentage. This change, which is generally known as vitriolescence, goes on more rapidly in the rhombic and granular varieties of pyrites than in the crystallised cubes ; and when specimens of such minerals are kept in cabinets they often develop the unpleasant pro- perty of * eating up their labels,' that is, the labels are rotted and destroyed by the sulphuric acid formed, the change being accompanied with the formation of capillary crystals of ferrous sulphate. Sulphuric acid is also formed by the oxidation of sul- phurous acid in the steam jets of volcanoes or fumaroles. In such localities it is common to find the felspathic com- ponent of the rocks reduced to a mass of clay variegated with parti-coloured patches, which are essentially alum and ferric sulphates. 4. Change by action of carbonic acid. All natural waters, whether terrestrial or atmospheric, hold more or less car- bonic acid in solution, and, as such, are a cause of alteration, 348 Systematic Mineralogy. [CHAP. XVII. which, though less energetic than sulphuric acid in any special case, is, as a whole, more important from the univer- sality of its action. The effects produced are of two kinds : ist, the solution of carbonate of calcium, magnesium, and analogous metals, and of their phosphates and fluorides, is promoted, as these salts are not sensibly soluble in pure water, but dissolve more or less readily in water saturated with carbonic acid ; and 2nd, the double silicates containing alkaline metals and aluminium, typified by the felspar group, are attacked and decomposed with the separation of the alkaline silicate, which dissolves and is ultimately decom- posed with the production of alkaline carbonates and soluble silica, while the insoluble aluminium silicate becomes hy- drated, forming the mineral Kaolin or China clay. This change, known as kaolinisation, affects many minerals besides felspars, and is probably concerned in the production of most clay deposits. Silicate of lime CaSiO 3 is also decom- posed by carbonic acid water, as are also ferrous silicates, but less readily than the former. Waters containing alkaline carbonates, especially when concentrated as in hot springs, also have a marked solvent effect upon silica, which is after- wards deposited as opal, hyalite, or chalcedony, or even quartz. The sulphides of the heavy metals, lead, zinc, or copper, are slowly converted into carbonates under the action of atmospheric waters containing carbonic acid, and are therefore commonly found to have been so changed in mineral veins at or near the surface. 5. Change by reducing agents. These are, to some ex- tent, inverse reactions to those involving oxidation. Ferrous sulphate, when kept in contact with decomposing organic matter, whether animal or vegetable, is converted into sulphide, crystals of iron pyrites having been produced in this way both naturally and artificially. Hydrated ferric oxide is in like manner reduced by decaying vegetable substances and carbonic acid, producing ferrous carbonate. Water containing alkaline sulphates may, in contact with CHAP, xvil.] Pseudomorphism. 349 organic matter, produce sulphides of copper, lead, &c., by direct action upon those metals, as is shown by Daubree to have taken place in the basins of certain thermal alkaline springs at Plombieres, where copper pyrites and antimonial grey copper ore have been produced from Roman coins im- bedded in the mud of the spring. 6. Alteration by chlorides. The mineral Atacamite, or oxychloride of copper, yj p U Q 2 >, is a tolerably common product of the alteration of copper pyrites in the mining districts of Chili, Peru, and South Australia. It is readily produced when sulphuretted copper ores are exposed to the joint action of air and sea- water, and is probably due to a similar action in the mines in question which are situated in hot, dry countries, with little or no rainfall, and where, in consequence, the alkaline chlorides in the rocks have not been completely washed out. Probably most of the chloride of silver found in mineral veins is to be attributed to the same cause as a product of the alteration of sulphide of silver. Evidence of alteration. Pseudomorphism. The trans- formation of a mineral by any of the methods previously described may be more or less evident, according to the nature of the change and the extent to which it has pro- gressed. The earliest stages of alteration are marked prin- cipally by change of colour or lustre in the faces of the crystals, which become superficially altered while preserving their original structure and composition within ; while, on the other hand, the alteration may be so great, and the de- velopment of new minerals so completely effected, that the derivation of the latter can only be inferred by generalisation upon evidence obtained in other cases. Such evidence may in some cases be obtained by direct experiment ; but in the larger number of instances it is furnished by what are known as pseudomorphs, i.e. minerals that appear in crystalline forms not compatible with their chemical constitution, and 3150 Systematic Mineralogy. [CHAP. XVII. must therefore have been altered by a partial or complete modification of their constituent elements, while retaining the forms proper to their original composition. The study of this most interesting branch of mineralogy has been systematised by Haidinger, Blum, Volger, and other ob- servers, and the observed cases have been classified under the following heads : 1. Pseudomorphism by substitution. This implies a gra- dual replacement of the original substance by another, by means of simultaneous solution and deposition, without involving chemical action. The pseudomorphs of Quartz, or other varieties of Silica, after Calcite, Fluor Spar, Barytes, and similar minerals, are of this kind : the siliceous matter having been deposited from solution, part passu, as the ori- ginal crystal was attacked and dissolved ; the general order observed in such cases being that the replacing substance is less soluble than that forming the original crystal. The fossilisation of the remains of plants and animals by Silica and other minerals is also to be referred to this kind of action. 2. Pseudomorphism by incrustation. In this case one mineral having been deposited upon another, and the older one subsequently removed, evidence of former existence of the latter is supplied by the hollow impression of its crystals retained in the second or incrusting species. As instances of this kind may be mentioned : Quartz upon Fluor Spar, Iron Pyrites upon Barytes, Quartz upon Barytes, Chlorite upon Dolomite, and Siderite upon Barytes. In all these instances, the free or outer surface of the incrusting mineral is generally developed according to its own form, while the under side of the layer forms an empty mould of the crystal of the mineral upon which it was originally deposited, but which has been since removed. It often happens, however, that the hollow space so produced is subsequently filled either with the incrusting substance or some third mineral, with the production of a substitution-pseudomorph of a more complex kind than those of the previous case. CHAP. XVII.] Paramorphism. 351 3. Pseudomorphism by alteration. Under this general head are included the following particular cases : (a) By loss or diminution of constituents. (fr) By gain or increase of one or more constituents. (c) By interchange or substitution of one or more con- stituents. The first of these cases is exemplified by pseudomorphs of Anhydrite after Gypsum, where the change is a simple dehydration ; Calcite after Gaylussite j ^aCO^ } ' where the carbonate of sodium is removed ; and native copper after Cuprite (Cu 2 O), where there is reduction or removal of oxygen. The following are examples of the second case : Gypsum after Anhydrite, involving the addition of water ; Malachite after Cuprite (Cu 2 O), by addition of oxygen, carbonic acid, and water ; and Anglesite (PbSO 4 ) after Galena (PbS), by simultaneous oxidation of both lead and sulphur. The third and last case is wider in scope than either of the preceding, and includes the larger number of 'observed pseudomorphs, some of the most prominent being the iollowing : Limonite (H 6 Fe 4 O 9 ) after Iron Pyrites (FeS 2 ), and Siderite (FeCO 3 ), by loss of sulphur and addition of water in both instances ; White Lead ore (PbCO 3 ) after Galena (PbS), by loss of sulphur and gain of carbonic acid ; and Kaolin (Al 2 Si 2 O 7 2Aq) after Felspar (R 2 Al 2 Si 6 O lfi ), by the loss of an alkaline silicate and the addition of water. The term paramorphism is applied to a particular class of pseudomorphism where a mineral occurs in a form proper to its composition, but having the structure proper to a dimorphous mineral of the same composition. Ex- amples of this are furnished by the change of oblique, prismatic crystals of sulphur into an aggregate of rhombic crystals without change of exterior form, aragonite with 352 Systematic Mineralogy. [CHAP. XVII. calcite structure, and augite with hornblende structure in Uralite. The derivative character of pseudomorphs is evident from the absence of the structural peculiarities, such as cleavage, lustre, &c., proper to the forms imitated, and as a rule they are dull, made up of amorphous material, very generally hydrated, such as Kaolin and Steatite, dull or waxy on a fractured surface, and usually spongy or hollow, especially when produced by total replacement; but in some cases of pseudomorphism by alteration, the change may be effected without alteration of volume, so that the pseudomorph may be as compact as the original crystal. This is especially well seen in the common case of Limonite pseudomorphs after cubes of Iron Pyrites, where the latter, or original, substance is completely transformed into a com- pact mass of the former, while preserving all the external characters of the crystals, even to the twin striations upon the faces. Here the proportion of the unaltered constituent Iron (467 per cent.) in the molecule of Pyrites, is to that in the molecule of Limonite (60 per cent.) as i to 1-3, while their specific gravities are in the inverse ratio of 1-4 to i, or 5 'o for Iron Pyrites and 3-6 for Limonite. In the parallel case of pseudomorphism of Limonite after Siderite, the change is attended with diminution of volume, as both are of nearly the same density, 3-5, but Siderite contains only 45 per cent, of iron against 60 per cent, in Limonite, so that the volume of the pseudo- morphs can only be about |ths of that of the original crystal. Actually, however, the volume is considerably less, as Siderite invariably contains more or less of the isomorphous bases Lime, Magnesia, and Manganous Oxide, which are either removed in solution as carbonates or separate as Pyrolusite, or other manganese ores, by oxidation. These differences do, however, actually correspond to structural differences in the resulting minerals on the large scale, as brown Iron Ores (Limonite) produced by the alteration of CHAP. XVII. ] Origin of Minerals. 353 masses of Pyrites are usually dense or compact in structure, while those produced similarly from Spathic Carbonates (Siderite) are, as a rule, spongy or cellular. In many cases the alteration of minerals is attended with very considerable increase of volume, as, for example, in the conversion of metallic Copper into Malachite, or Iron into Limonite. In the latter instance the increase of volume is from eight- to tenfold, so that the change may be attended with considerable mechanical action when effected in a con- fined space. Numerous examples of this action may be seen in old wrought-iron work, which has been long exposed to the weather, where rusting has gone on between surfaces originally in contact, but which have been thrust apart by the rust formed between them. An analogous action may be assumed as taking place in the change of complex silicates into alkaline carbonates, silica and clay, owing to the greatly increased volume of these products as compared with that of the original mineral. Origin of mimrals. Questions as to the probable origin and method of formation of minerals are among the most interesting in the whole field of mineralogy, but the material available for their solution is comparatively small. In many instances the evidence of pseudomorphs is sufficient to show a secondary origin, o r transformation from a pre-existing combination. The researches of Daubree upon deposits formed in the conduits of the thermal springs supplying mineral baths in France and Algiers, which have been in use since the Roman conquest of Gaul, have shown conclu- sively that minerals of the class of Zeolites may be readily produced by the action of slightly alkaline heated waters upon the rocks they traverse when continued for a period of many centuries; and similarly, the condition of the bronze objects found in the remains of Assyrian and other ancient cities, by their conversion into Ruby Copper ore and Malachite afford not only a proof of the essentially secondary character of these minerals, if it be needed, but also, in A A 3 54 Systematic Mineralogy. [CHAP. xvn. some degree, a measure of the time required for their for- mation. With minerals of a more complex character, especi- ally among Silicates, direct evidence can rarely be obtained, and in such cases, therefore, the undesigned production of compounds similar in form and composition to natural minerals in slags and other furnace products, or what are usually known as artificial minerals, is of special significance. The following are amongst the minerals that have been most frequently observed in furnace products definitely crys- tallised: LimeAugite (CaSiO 3 ) in the slags of several blast fur- naces smelting iron ore. Manganese Augite (MnSiO 3 ), or approximating to that composition, which resembles the natural mineral Babingtonite, but is not exactly similar. This is found in slags produced in the Bessemer process of steel- making. Potash Felspar (KAiSi 3 O 12 ), in minute crystals in the walls of furnaces smelting the copper-schist of Mansfeld, exactly similar to the natural mineral, but of rare occur- rence. Humboldtilite, in modified square prisms, very much larger than those of the natural mineral which occurs in the lavas of Vesuvius: these are commonly seen in the older blast-furnace slags of South Staffordshire. Iron Chry- solite, or olivine (Fe 2 SiO 4 ), corresponding in composition to the doubtful mineral species Fayalite. This occurs very commonly in the slags of puddling furnaces, and also at times in those produced in smelting lead or copper ores, or generally where slags that are essentially ferrous silicates are formed. The crystals have the form of the isomorphous mineral chrysolite or olivine (Mg 2 SiO 4 ), but the latter is not found in furnace products, for the obvious reason that slags consisting mainly of magnesian silicates are not susceptible of formation under the ordinary conditions of working, owing to the refractory character of such compounds, and the use of magnesia in fluxes is therefore carefully avoided. Mag- netite (Fe 3 O 4 ) is common in octahedral crystals in the slags CHAP. XVII.] A rtificial Minerals. 355 produced in the later stages of the puddling process, when the amount of iron taken up is in excess of that required to form a definite silicate with the silica present. It is also produced when steam is passed at a red heat over ferrous sulphide, which has probably been the mode of formation of the brilliant, artificial crystals found occa- sionally in the deposits formed in furnaces smelting pyritic silver ores at Freiberg. Galena, PbS, in brilliant, cubical crystals and columnar aggregates, is tolerably common in deposits apparently the re- sult of sublimation in the throats of blast furnaces smelting lead ore, and where the ore contains zinc, Blende (sulphide of zinc) and oxide of zinc (the latter not occurring as a natural mineral) may be deposited in a similar manner. When metallic sulphides are roasted by burning in heaps in the open air, numerous minerals are formed by partial oxidation of the more volatile constituents, especially sulphur and arsenic, and deposit in the cooler portions of the heap in a manner analogous to that observed in solfataras and other volcanic emanations. Among these are Sulphur^ Arsenwitsaa'd(\s 2 O 3 \ Realgar (hs\ and Orpiment (As 2 S 3 ). This association of realgar crystals with sulphur is common at the solfatara of Naples. Sal Ammoniac (NH 4 Cl) is occasionally deposited in the same manner upon waste heaps over burning coal slack. Specular hematite (^O z ) in minute, brilliant crystals, is occasionally found on surfaces of salt- glazed pottery (such as drain-pipes, &c.) ; these are a conse- quence of the action of steam upon ferric chloride (Fe 2 Cl 6 ), which results in the production of ferric oxide and hydro- chloric acid. In this process, common salt is thrown into a kiln when the clay goods are brought up to a bright red heat, and in the presence of water vapour is decomposed with the formation of a glaze (silicate of soda) upon the heated surface of the ware, hydrochloric acid and some ferric chloride from the iron contained in the clay being volatilised. This reaction explains the formation of the very brilliant crystals of specular A A 2 356 Systematic Mineralogy. [CHAP. XVI I. iron ore found upon the lavas of Vesuvius, Ascension Island, and other volcanic centres, hydrochloric acid and ferric chloride being commonly found in the steam emitted from fumaroles during and after periods of eruption. Graphite in the form of crystalline scales, and occasion- ally in masses of considerable size, and closely resembling the natural mineral, is a common product of blast furnaces smelting iron ores, but the method of its formation, namely, separation from solution in molten cast iron, cannot be con- sidered as analogous to any process likely to produce it in nature. Water at high temperatures, when it is made to act under conditions by which the formation of vapour is prevented, or when under considerable pressure, has a powerful solvent action upon many substances which are not affected by it under ordinary temperatures and pressures, and may in such a case give rise to considerable danger to substances sub- mitted to its action. The principal experiments upon this point are due to Daubree, who found that hard glass, a homogeneous silicate of lime and soda, may be converted into crystallised quartz and pyroxene at a comparatively low temperature in this way. It is probable that an action of this kind may be concerned in the production of crystalline rocks containing quartz, with orthoclase and other silicates, by the slow rearrangement of masses originally homogeneous, when solidified by the process which is generally known as devitrification, or the transformation of a glassy substance into an opaque mass containing crystals. Another agent of considerable importance in the produc- tion of minerals is probably boracic acid, which, though practically fixed when exposed alone to a very high tempera- ture, is sensibly volatile even at the boiling point of water in an atmosphere of steam. The method of occurrence of minerals containing borax, especially tourmaline, in veins in granite and other rocks, seems to indicate a formation by a process analogous to sublimation; but more direct evidence CHAR XVII.] Artificial Minerals. 357 is afforded by the presence of boracic acid in the condensed steam issuing from volcanic vents and furnaces in Tuscany and other places, and which are the principal source of sup- ply of this mineral for commercial purposes. By the solvent action of boracic acid, or borax, at very high temperatures, many refractory or ordinarily infusible substances may be made to combine and crystallise from fusion. In this way, Ebelmen produced such minerals as Spinel (MgAi0 4 ) and Chrysoberyl (BeAiO 4 ) from mixtures of magnesia and alu- mina, and glucina and alumina respectively, boracic acid being used as a solvent. The colours of the natural minerals were imitated by the addition of oxide of chromium for red, iron for black, and cobalt for blue spinel. Alumina was also converted into crystals of corundum and ruby by heat- ing with borax. The analogous method of producing crys- tallised titanic acid by the use of salt of phosphorus, due to Gustav Rose, has already been noticed at page 312. Hydro- fluoric acid and fluoride of silicon have also been used to induce combination between silica and metallic oxides. In this way Staurolite has been formed by passing hydrofluoric acid through alternating layers of silica and alumina and the analogous silicate Topaz, which contains fluoride of alu- minium, by the action of fluoride of silicon upon alumina. For further information upon this most interesting class of subjects the reader is referred to the various memoirs published by Ebelmen, Daubree, Senarmont, and others. A compendious notice of these will be found in Percy's ' Swiney Lectures on Geology,' published in the ' Chemical News,' vol. xxiv., and Daubree, 'Etudes de Geologic Syn- thetique,' Paris, 1879. Speaking generally, two great and contrasted groups of causes may be said to be concerned in the production and modification of terrestrial minerals. These are, i. The pro- duction by heat within the crust of the earth of homo- geneous silicates of the alkaline and other light metals, analogous to glass. Silicates either appear at the surface Systematic Mineralogy. [CHAP. xvn. by eruption, or remain, as deep-seated molten masses, to undergo more or less complete devitrification and differentia- tion into aggregates of quartz, felspars, and other silicates, phosphate of calcium, magnetic iron ore, metallic sul- phides ; 2. The conversion of these aggregates by the action of air and carbonic acid, atmosphere and thermal waters, into the various classes of hydrated silicates and metallic oxides, alkaline and earthy carbonates, metallic sulphates, &c. These actions are essentially compensatory, the tendency of the second being towards the production of quartz, soluble silica, clay, brown iron ore, alkaline carbonates, and carbonate of lime; the latter, being removed in solu- tion, can only be returned to the general circulation by being brought within the range of the earth's internal heat. That causes of this kind have been in action from a very early period of the earth's existence as a solid body is evident from the identity of the minerals found in the oldest rocks with those recurring in other places in similar rocks, which can be shown to have been formed at very different geo- logical periods. Daubree, from his researches on the synthesis of me- teorites, has suggested the probability of an earlier period of universal scorification, when heavy silicates, such as olivine and masses of the heavier, metals, may have been formed within the crust of the earth and depressed below the range of our immediate observation, but of whose existence evidence is furnished by their presence in me- teorites and volcanic masses. This view supposes the formation of basic silicates by the action of heat alone to have preceded that of quartz, and the silicates of the felspar group, where the intervention of water is assumed to be essential. Association and grouping of minerals. This subject is intimately connected with the preceding, as it is only by a study of the relative positions of dissimilar minerals in the same aggregate or mass that their order of succession or CHAP. XVII.] Par agenesis. 359 relative ages can be made out. . The branch of mineralogy devoted to this class of investigation, and which stands in very close relation with geology, is usually known as Para- genesis, a term first applied by Breithaupt in 1849, in a work upon the associations of minerals observed in the veins worked in different mining districts. The further develop- ment of the same class of observation consequent upon the application of the microscope to mineralogical investigation, and more particularly to the constitution of rock masses, has given rise to another special branch known as Petrology, which forms the subject of a companion volume in this series. The nature of these associations can best be considered in the description of individual minerals, but some of the more prominent groups may be mentioned here. Quartz occurs in association with almost every other mineral, but is more commonly found together with ortho- clase, felspar, and other so-called acid silicates, than with those containing less silica. It is also very frequently found with mica, tourmaline, rutile, tin-stone, topaz, and the richer silver ores. Together with the amorphous varieties of silica (agate, chalcedony, &c.), it accompanies hydrated silicates of the zeolitic group in basalt and vesicular lavas, where it is obviously of secondary origin, as is also the case when it occurs in mineral veins traversing limestone strata. Labradorite (soda-lime felspar) is almost invariably found with pyroxene, hypersthene, and titaniferous iron ore, form- ing the rocks known as norite, basalt, &c. Of the different minerals containing iron, Magnetite is commonly associated with all rocks containing ferrous and magnesium silicates, and less so with quartz or micaceous schists, where hematite and titaniferous iron ores are more generally found. Magnetite occurring in large masses worked for iron ores, usually contains iron pyrites, chlorite, garnet, hornblende, and apatite in small quantities. He- matite deposits in stratified rocks as a rule contain as associates quartz, barytes, fluorspar, calcite. and aragonite. 360 Systematic Mineralogy. [CHAP. XVII. Spathic iron ore is generally associated with sulphides, such as iron pyrites, copper pyrites, galena, &c., but very un- equally in different localities, and also with the various iso- morphous carbonates of calcium, zinc, and manganese, and the products of the alteration of the latter such as pyro- lusite (MnO 2 ). Iron Pyrites is the most abundant of all the metallic sul- phides, and is widely diffused through rocks and mineral deposits of all kinds. In small quantities it is found in clays and other rocks impermeable to water, and in coal, and other carbonaceous deposits, where it is protected against oxida- tion. When in large masses, it is commonly associated with copper pyrites, arsenical pyrites, and the various sul- phides and arsenides of nickel and cobalt, gold and silver. It also forms a general constituent of mineral veins con- taining the ores of tin, copper, and lead. Tinstone and its associates constitute a very special group of minerals, which, though restricted to a small number of areas, are often very abundantly developed in particular localities within these areas. In this group are included Tinstone (Stannic oxide), tourmaline, topaz, wolfram, scheelite, iron pyrites, mispickel, copper pyrites, chalcedony, fluorspar, hematite, pitchblende, and occasion- ally bismuth ores. Galena, or sulphide of lead, the principal ore of that metal, forms part of numerous well-marked groups of minerals, which as a rule are characteristic of particular districts. Among its more usual associates are the pro- ducts of its own alteration, sulphate, carbonate, and phos- phate of lead, and zincblende (ZnS), calamine (carbonate and silicate of zinc), iron and copper pyrites, and the 4 waste 'or earthy minerals, calcite, fluorspar, aragonite, dolomite, and barytes, when in slate and limestone districts ; in addition to which quartz, and occasionally zeolites, are found when the veins are in siliceous rocks, such as granite, gneiss, &c. When in company with antimonial minerals, CHAP. XVII.] Association of Minerals. 361 gold and silver ores usually enter into the group, as in the Hartz and Hungary. Nickel and cobalt ores, when found in quantity, i.e. as rich arsenides, and not merely as mixtures with iron pyrites, severally accompany native arsenic and various arsenides, the products of their oxidation (pharmacolite, cobalt bloom, nickel bloom), and the different ores of silver and bismuth. Copper ores are very commonly found in connection with minerals containing magnesia, such as hornblende, chlorite, serpentine, and dolomite, and also with quartz. Copper pyrites, the most abundant ore of this metal, has two principal lines of association, the first being with iron pyrites in more or less intimate mixture, forming the so- called coppery pyrites, and the second with copper-glance (Cu 2 S) and erubescite (FeCu 3 S 3 ), forming a series richer in copper. Native copper, and the various oxides and oxy- salts, carbonates, sulphates, and phosphates of this metal, are common products of the alteration of the sulphuretted minerals ; but in the district of Lake Superior the metal occurs exceptionally and in enormous quantities over a very large area practically without other copper ores, and in as sociation with quartz, calcite, and zeolitic minerals. Rock Salt (NaCl), though often found in very large masses in a perfectly pure state, is generally associated with salts of calcium and the alkaline metals, especially gypsum and anhydrite. Where the deposits have been isolated in such a manner that the more soluble salts contained in the original salt water have been preserved, a numerous series of double sulphates and chlorides of potassium, magnesium, &c., are developed. Among these are Kainite. Carnallite, Kaluszite, Boracite, Polyhalite, &c., besides bromide of magnesium. This is only seen on a very large scale at Stassfurt in Prussia, and Kalusz in Gallicia. As regards the frequency or scarcity of the occurrence of minerals, it may be useful to remember that a substance may be rare in two different waysbeing either widely dis- 362 Systematic Mineralogy. [CHAP. XVII. tributed, but in minute or even invisible quantities in other minerals, or restricted to a few localities, where, however, it may be found isolated in quantity. Sulphide of cadmium may be taken as an instance of the first kind, having only been found in a single locality as an independent mineral (Greenockite), and in a few minute examples; but as an isomorphous associate with the corresponding sulphide of zinc, it is present in the larger number of samples of zinc- blende, so that some tons of the metal are made annually by fractional distillation of the first deposits of zinc oxide obtained in the zinc works, which are usually found to be cadmiferous. Other examples are afforded by the rare alka- line metals ccesium and rubidium, common in certain mineral waters, but almost unknown in individual minerals ; also by thallium, gallium, and similar elements existing in spectroscopic traces in common minerals such as pyrites, zincblende, &c. Cryolite (6NaFl+AiF16) is an example of the second kind of rarity, it being almost entirely restricted to one spot -on the coast of Greenland, where, however, it is found in such masses that it can be utilised as a commercial source of soda and alum. INDEX- ACI A CICULAR crystals, 188 -rl _ Acids, 327 Airy's spirals, 256 Albite, forms of, 162 twin forms of, 183 Allanite, form of, 155 Allochromatic minerals, 283 Alteration of minerals, 345 Alum group, 337 Alumina, tests for, 309, 316 Analogue poles, 297 Angle of optic axes, 263 Anglesite, form of, 140-141 Anharmonic property of zones, 34 ratios of planes, 32 Anisometric projection, 195 Anisotropic media, 225 Anorthic system, 156 Antilogue poles, 297 Antimonic oxide, -dimorphous, 334 Apatite, pyramidal hemihedron of, 104 Apatite group of minerals, 336 Aragonite, twin forms of, 177 Association of minerals, 358 Asterism, 289 Asymmetric system, 156 Atacamite, production of, 349 Athermancy, 292 Atomic weight, 323 Atom?, 323 Augite group, 339 Augite, as a furnace-product, 354 Avanturine, 289 Avpgadro's law, 323 Axinite, forms of, 162 Axis, optic, 238 angle of, 263 Azurite, form of, 155 BABINET'S goniometer, 193 Babingtonite, forms of, 162 Baryta, test for, 317 Barytes, form of, 139, 140 Bases, 329 CLE Baveno type of twin crystal, 182 Biaxial crystals, 257 principal sections of, 244 wave-surface, 241 Binary system, 145 Bisectrices of optic axes, 242 Biowpipe, 299 Borax, form of, 155 as a blowpipe flux, 309 Botryoidal aggregates, 189 Brachydiagonal axis, 129, 156 quarter pyramids, 158 Brachy dome, 134 pinakoid, 134, 160 Bravais-Miller hexagonal notation, 75 Breithaupt on Paragenesis, 359 Breon's method of separating minerals, 217 Brezina's plate, 259 Brittleness, 212 Brookite, 138 Brushite, form of, 154 /^ALCIFERRITE, 338 \~s Calcite, combinations of, 96, 97 99, loo irregular aggregates of, 187 twin crystals of, 172, 173, 174 Caledonite, forms of, 154 Carbon, 334 Carbonate of lime, dimorphous, 334 Carlsbad type of twin crystal, 181 Chemical constitution, 320 properties of minerals, 298 Chrysolite, as a furnace-product, 354 Circular polarisation, 254 Cleavage, 205 qualities of, 207 Cleavages, principal : Cubic system. 207 Hexagonal system, 208 Tetragonal system, 208 Rhombic system, 208 Oblique system, 208 Triclmic system, 209 364 Index. CLI Clinoclase, forms of, 154 Clinodiagonal axis, 147 Clinodpmes, 150 Clinopinakoid, 151 Clinorhombic system, 145 Colour, 280 Columnar crystals, 188 Combinations of cubic system, 63 Constants, optical, 277 Contact goniometer, 191 twins, 168 Conversion of notation, 20 Copper, tests for, 319 glance, twin forms of, 178 ores, occurrence of, 361 pyrites, forms of, 125 Corundum group, 336 Crookes on Phosphoresence, 291 Crossed dispersion, 269 Cryolite, rarity of, 362 Crystal, etymology of, 7 Cube, 37, 48 combinations of, 64, 65, 67, 68, 69, 70, 7 2 > 73 hemihedral forms of, 58 twin crystals, 169 Cubic system, symmetry of, 37 general form of, 39 Curved faces of crystals, 188 DAUB REE on origin of minerals, .353. 356 Deltoid-dodecahedron, 57 Density, 213 Deviation, minimum, 232 Diamond, curved crystals of, 188 Diaspore group, 337 Diathermancy, 292 Dichroiscope, 285 Dichroism, 284 Dihexagonal prism, 82 pyramid, combinations of, 78 Dimorphism, 333 Disperson of optic axes, 265 of the median lines, 267 Ditetragunal prism, 116 pyramid, 113 Ditrigonal prism, 107 Double refraction, 237 determination of sign, 270 Doubly-oblique system, 156 Ductility, 212 Dyakisdodecahedron, 53 combinations of, 71, 72 EBELMEN'S artificial minerals, 357 Eingliedrig, 156 Elasticity, 212 Electric calamine, forms of, 165 twin forms of, 179 Electrical properties of minerals, 296 HEX Equivalents, chemical, 321 Essential qualities of minerals, 2 Exner on hardness in different directions, T7ACE determined by two zones, 31 JT Faces, geometrical relations of, 24 Felspar as a furnace-product, 354 Fibrous aggregates, 189 First median line, 242 Flame reactions, 304 Flexibility, 212 Flos ferri, 189 Fluorspar, irregular forms of, 187 twin crystals of, 169 Fluorescence, 290 Formulae, chemical, 326 Fracture, 209 Fuess's goniometer, 193 Furnace products, 354 Fusibility, 303 GALENA, as a furnace-product, 355 association of, 360 Garnet group, 340 Gold, tests for, 319 Goniometer, 191 Graphite, artificial, 356 Greenockite, forms of, 165 rarity of, 362 Gypsum, alteration of, 346 curved crystals of, 188 twin forms of, 180 HABIT of crystals, 188 Haidinger's dichroiscope, 285 Hardness, 210 Hausmannite, twin crystals of, 175 Hematite, association of, 360 Hemi-brachydome, 160 Hemihedral cubic combinations, 70 forms, 13 cubic forms, 49 cubic diagrams, 59 Hemimacrodomes, 159 Hemimorphism, 163 Hemiorthodomes, 150 Hemitrope crystals, 168 Heteromorphtsm, 333 Heterotropic media, 225 Hexagonal axes, projection of, 196 basal pinakoid, 84 hemihedral forms, 88 holohedral combinations, 85 diagrams, 84 Weiss 's notation, 74 prism, 83 of second order, 83 prism, twin crystals of, 173, 174 pyramid, So Index. 365 HEX Hexagonal pyramid of second order, 81 pyramid of third order, 103 scalenohedra, 92 symmetry, 73 tetartohedra, 104 trapezohedral hemihedra, 91 twin crystals, 172 Hexakisoctahedron, 39 Hexakisoctahedron, combinations of, 69, 70 Hexakistetrahedron, 55 Hirschwald's goniometer, 184 Holohedral cubic diagram, 49 forms, 13 tetragonal combinations, 118 tetragonal diagram, 117 Homeomorphism, 336 Horizontal dispersion, 269 Hornblende, forms of, 155 Hydrochloric acid, test for, 319 TCOSITETRAHEDRON, 41 -I combinations of, 65, 66, 67 Ideochromatism, 283 Ilmenite, crystalline form of, no Imperfections of crystals, 186 Inclined dispersion, 268 Inclined hemihedra, cubic, 55 Inclosures in minerals, 275 Index of refraction, 230 determination of, 234 Interference figures, 260 waves, 222 Iridescence, 288 Iron, reactions of, 316 Irregular groups of crystals, 189 polarisation, 273 Isochromatic curves, 253 Isometric projection, 195 Isomorphism, 335 I-.otropic media, 225 J OLY'S spring balance, 216 T ABRADORITE, association of, 359 J-* twin striation in, 183 Laumonite, alteration of, 345 Lead, tests for, 318 Levy's notation, 23 Lime, tests for, 316 Limonite, formation of, 352 Linear projection, 200 Lustre, 286 TV /TACLED crystals, 168 1VJL Macrodiagonal axis, 129, 156 quarter pyramid, 158 Macrodome, 133 Macro pinakoid, 134, 160 PAR Magnesia, tests for, 309, 316 Magnetism, 297 Magnetite, association of, 359 Mallard on polysynthetic structure, 274 Malleability, 212 Mammillary aggregates, 189 Manebach type of twin crystal, 182 Median lines of optic axes, 242 Mica, asterism of, 290 Miller's notation, 18 rhombohedral notation, 74 Minimum deviation, 233 Mitscherlich's goniometer, 193 Mitscherlich on isomorphism, 335 Mohr's method of gauging, 215 Mohs' scale of hardness, 210 Molecules, 323 physical, 203 Molybdates, forms of, 127 Monoclinic system, 145 Monodimetric projection, 195 Monosymmetric hemihedrism, 144 Monosymmetry, 152 ATAUMANN'S notation, 21 1\ Negative biaxial crystals, 243 Nicol's prism, 247 Nitrate of barium, distorted forms of, 185 Nodules, 189 Notation of crystals, 16 OBLIQUE axes, projection of, 197 nemipyramids, 147 prism, 149 rhombic pyramid, 147 rhomboidal pyramid, 157 system, 145 twin crystals, 180 Octahedron, 44 combinations of, 62, 63, 64, 65, 66, 68, 69, 71 distorted forms of, 184 twin crystals, 169 Olivine, forms of, 142 Optic axis, 238 angle of, 263 Optical classification, 245 constants, 277 properties of minerals, 218 Origin of minerals, 353 Orthoclase, twin forms of, 181 Orthodiagonal axis, 147 Orthohexagonal notation, 74 Orthopinakoid, 151 pARAGENESIS, 359 L Parallel grouping of crystals, 165 Parallel hemihedra, cubic, 52 Parameters of hexagonal pyramid, deter- mination of, 79 rhombohedron, determination of, 101 366 Index. PAR Parameters of tetragonal pyramid, deter- mination of, 114 Penetration twins, 168 Pentagonal dodecahedron, 54 combinations, 70 tetartohedral, 61 twin crystals of, 170 Phosphorescence, 290 Phosphoric acid, test for, 320 Physiography, 3 Plagihedra cubic, 51 Plane of composition, 167 of contact, 167 Pleochroism, 283 Polariscope, 249 Polysymmetry, 339 Polysynthetic crystals, 274 Positive biaxial crystals, 243 Projections, elements of, 196 perspective, 195 Pseudomorphism, 349 Pyramidal hemihedrism, hexagonal, 103 Pyrites, alteration of, 352 twin crystals of, 170 Pyroelectricity, 297 ^VUANTIVALENCE, 326 \ f Quartz, association of, 359 distorted crystals of, 185 double rotation of, 256 right and left-handed crystals of, 255 tetartohedron of, 108 twin cry.vtals of, 175 Quenstedt's projection, 200 P ATIONALITY, principle of, 8 Xv Reflected waves, 227 Reflecting goniometer, 191 Refraction, 228 double, 237 Regular solids possible as crystals, 9 Reniform aggregates, 189 Reticular point systems, 14 Rhombic combinations, 136 dodecahedron, 47 combinations of, 64, 66, 67, 68, 70, 72 distorted forms of, 185 hemihedral forms of, 58 - twins of, 169 forms, diagram of, 134 hemihedral forms, 143 prism, 131 pyramid, 129 determination of parameters of, 131 sphenoids, 143 symmetry, 128 twin crystals, 177 Rhombohedra, normal or hemihedral, 94 of second order, no of third order, 109 of longer polar edges of, scalenohedra, 98 TAU Rhombohedra of middle edges of scaleno- hedra, 97 of shorter polar edges of scalenohe- dra, 98 limits of, 96 tetartohedral, 109 twin crystals of, 172, 173, 174 Rhombohedral combinations, 95, 96, 97, 99, ioo_ _ ot positive and negative, 95 notation, Miller's, in Naumann's, 101 Right and left handed rotation, 255 Rock-salt, association of, 361 Rocks, definition of, 5 Rose, G., on titanic acid, 312 on tridymite, 312 on carbonate of lime, 335 Rotatory power of quartz, 254 of cinnabar, 255 Rutile, twin crystals of, 176 C ALTS, 330 O Satin spar, 189 Scale of hardness, zio< Scalenohedra, limits of, 97 twin crystals of, 173, 174 Schraufs' hexagonal notation, 74 Senarmont's experiments, 292 Siderite, curved crystals of, i8& Silica, allotropic, 334 Silver, tests for, 318 Sodalite, twin crj stals of, 169 Sonstadt's solution, 217 Sorby's method of finding index of re> fraction, 234 Species, definition of, 3 Specific gravity, 213 Sphenoidal combinations, 125 twin crystals, 176 Spherical projection, 201 Spinel, twin crystals of, 171 - group, 337 Stalactites, 189 Stalagmites, 189 Star sapphire, 289 Staurolite twin forms, 178 Stauroscope, 238 Streak, 283 Striations on quart?, crystals, 187 Struvite, forms of, 165 Sulphate of magnesium, 144 Sulphur, allotropic, 333 forms of, 136, 137 Sulphuric acid, test for, 319 Symmetry, axes of, 10 of crystals, 8 crystals classified by, n limitation of possible, HTABULAR crystals, 188 J_ Tautozonality, 29 condition of, 30 Index. 367 TEN Tenacity, -21-2 Tetartohedral forms, 13 Tetartohedron, cubic, 60 Tetarto-pyramids, 157 Tetragonal, basal pinakoid, 117 hemihedrism, 120 prisms of first order, 116 pyramid, 114 pyramidal hemihedrism, 126 pyramids of second order, 115 pyramids of third order, 126 pyramids, limits of, 116 pyramids, twin crystals of, 175 scalenohedra, 123 sphenoids, 1 24 symmetry, 112 trapezohedra, 121, 127 twin crystals, 175 Tetrahedron, 57 combinations, 72, 73 twin crystals of, 169-170 Tetrakishexahedron, 45 - combinations of, 67, 68, 69 Thermal relations of minerals, 291 Thermoelectricity, 297 Tinstone, association of, 360 twin crystals of, 176 Titanic acid, trimorphous, 334 Topaz, 137 ' Total reflection, 231 Toughness, 212 Tourmaline, forms of, 164 Translucency, 280 Trapezohedral, tetartohedrism hexago- nal, 105 Triakisoctahedron, 43 combinations of, 66, 67 Triakistetrahedron &6 ZWE Triclinic axes, projection of, 198 hemiprisms, 159 quarter pyramids, 157 system, 156 twin crystals, 183 Trigonal pyramids, 106 Tungstates, forms of, 127 Twin axis, 167 grouping, 166 plane, 167 T TNI AXIAL crystals, 238 \J behaviour in parallel polarised light, 250 in convergent polarised light, 251 VON KO BELL'S scale of fusibility, 303 stauroscope, 258 WAVE motion, 219 Weiss's notation, 17 White lead ore, twin forms of, 177 Witherite, forms of, 143 Wollaston's goniometer, 191 '7EOLITES, formation of, 353 iLe Zinc, test for, 309 Zone axis, 29 plane, 30 symbol, determination of, Zones, 29 Zwei- und eingliedrig, 145 30 PRINTED BY FOTTis\vooDE AND co., NEW-STREET SQUARE LONDON RETURN EARTH SCIENCES LIBRARY TO *> 642-299; LOAN PERIOD 1 1 MONTH 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Books needed for class reserve are subject to immediate recall DUE AS STAMPED BELOW FORM NO. 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