Received SYSTEM OF CRYSTALLOGRAPHY, WITH ITS APPLICATION MINERALOGY JOHN JOSEPH GRIFFIN, TRANSLATOR OP "ROSE'S MANUAL OF ANALYTICAL CHEMISTRY." AUTHOR OF "CHEMICAL RECREATIONS." GLASGOW: PUBLISHED BY RICHARD GRIFFIN AND COMPANY. AND THOMAS TEGG, LONDON. MDCCCXLI. CONTENTS. PREFACE, PAGE vii PART I. PRINCIPLES OF CRYSTALLOGRAPHY. SECTION I. OF THE AXES OF CRYSTALS, , PAGE 1 Axes, 1. Poles, 9. Normals, 10. SECTION II. OF THE PLANES OF CRYSTALS, 3 Parts of a Crystal: planes, edges, and solid angles, 15. Geometrical Defini- tions, 16. Notation of Planes, 19. The Equator, 20. The Meridians, 20. Polaric Positions, 21. The Planes P, 22. The Planes M, 23. The Planes T, 24. Combinations of P,M, and T, 25. The Planes MT, 26. Definitions relating to Plane Trigonometry, 37. Problems respecting the Axes of rhombic sections, 50. Value of angles produced by replacement, 59. Notation adapted to ex- press the relative magnitude of Planes in combinations, 69. Polaric Positions round the Equator, 70. Geometrical Relations of the angles of an Equator, 79. Control over the correctness of Measurements and Calculations, 84. The term Prism defined, 86. Forms of the Equators of Prisms, 87. The planes PM, 88. The planes PT, 101. Pentagonal Dodecahedron, 108. Tetrakis- hexahedron, 111. Oblique Rhombic Prisms, 117. Zones, 122. The planes PMT, 124. Regular Octahedron, 127. Isosceles Octahedrons, P X MT, 128. Icositessarahedron, 3P-MT, 131. Triakisoctahedron, 3P+MT, 147. Scalene Octahedrons, 162. Six varieties of the Scalene Octahedron, 168. Hemi- hexakisoctahedron, 3P_MT+, 177. Method of denoting the polaric positions of planes on the combinations of Scalene Octahedrons, 188. Hexakisocta- hedron, 6P_ MT+, 194. A group of all the varieties of Octahedrons or Pyra- mids that can possibly occur, 198. Forms of the Equators of Pyramids, 199. Synopsis of Planes belonging to Prisms and Pyramids, 200. Abridged Sym- bols for Complex Octahedrons, 200. SECTION III. OF PRISMS AND PYRAMIDS, AND THEIR COMBINATIONS WITH ONE ANOTHER, 69 SECTION IV. OF THE CLASSIFICATION OF CRYSTALS, ...'... 71 System of six classes, with five orders in each class, 213. Directions for putting a Crystal into a proper position for examination and description, 214. j v CONTENTS. SECTION V. OF THE POSSIBLE LIMIT TO THE VARIETY OF PLANES THAT CAN OCCUR ^ UPON CRYSTALS, AGE ' Properties of the seven fundamental Forms of Crystals, 215. No other Forms than these seven can possibly occur upon Crystals, 225. Limit recognised, 226. SECTION VI. OF CRYSTALLOGRAPHY NOTATION, 77 Principle of the present Notation, 227. Twin Crystals, 229. Examples of Notation, 230. SECTION VII. OF CLEAVAGE AND PRIMITIVE FORMS, 80 Nature of Cleavage, 231. How indicated, 232. Primitive Forms only hypo- thetical, 233. Hauy's Primitive Forms, 234. The doctrine of Primitive Forms not practically useful, 234. Secondary Forms, 235. Both Primitive and Secondary Forms are merely examples of the seven Fundamental Forms or their combinations, 236. SECTION VIII. OF FORMS AND COMBINATIONS, . ... '*.'. ' 85 Forms, 237. Combinations, 239. Methods of indicating the General Aspect of Combinations, 243. Examples of Comparative Notation, 243. Intelligible Notation incompatible with the assumption of Primitive Forms, 245. SECTION IX. THE FIVE ZONES, 91 SECTION X. THE LAW OF SYMMETRY, 93 Classification and Nomenclature of Symmetrical and Unsymmetrical Forms, 253. Homohedral Forms, 258. Table of all the possible kinds of Homohedral Forms, 262. Uniaxial, Biaxial, and Triaxial Forms, 262. Hemihedral Forms, 263. Tetrahedron, 265. Theory of the Hemioctahedrons, 266. The four kinds of Hemihedral Forms with inclined planes, 267. Right and Left, or Direct and Inverse, Hemihedral Forms, 269. Theory of Right and Left Combinations, 270. Hemihedral Forms with parallel planes, 271. Hemioctahedrons of the Oblique Prisms, 272. Hemihedral Forms of each particular Zone, 276. Forms commonly but erroneously called Hemihedral, :'81. Tetartohedral Forms, 284. Discrimination of Homohedral, Hemihedral, and Tetartohedral Forms, 285. SECTION XI. A THEORY OF CRYSTALLISATION, 106 Eidogens, 288. Their Origin and Properties, 289. Illustrative Examples, 294. SECTION XII. THE USE OF SPHERICAL TRIGONOMETRY IN CRYSTALLOGRAPHY, . J 19 Table of Trigonometrical Formulae for calculating the relations between the sides and angles of Triangles, 298. Definitions of Algebraic terms, 299. Of Solid Triangles, 300. Description and Application of the Formulae, 308. Logarithms of Numbers, 319. Calculation of Right angled Plane Triangles, 322. Table of Indices, 323. The axes of Forms belonging to a given Zone are multiples of one another for the same Mineral, 326. Construction of Symbols, 327. Calculation of Oblique angled Solid Triangles, 328. Calcula- tion by means of Quadrantal Solid Triangles, 331. Table of Square Roots, 333. General directions for the Analysis of Crystallographic Combinations, 334. Abridgement of Formulae, 335. SECTION XIII. AN INQUIRY INTO THE VARIETY OF FORMS AND COMBINATIONS WHICH OCCUR UPON THE CRYSTALS OF MINERALS, . . ,149 Rose's System of Crystallography, 337. Classification of Crystals according to six systems of Axes of Crystallisation, 338. Discrimination of the Crystals of these six systems, 340. CONTENTS. V 1. TJie OctaJiedral System of Crystallisation, 341. Explanation of Unipolar, Bipolar, and Tripolar Normals, 342, 349. Octa- hedron, 344. Cube, 350. Rhombic Dodecahedron, 354. Icositessarahe- dron, 369. Triakisoctahedron, 385. Tetrakishexahedron, 395. Hexakisocta- hedron, 408. Tetrahedron, 432. Hemiicositessarahedron, 437. Hemitria- kisoctahedron, 447. Hemihexakisoctahedron with inclined faces, 450. Pen- tagonal Dodecahedron, 453. Hemihexakisoctahedron with parallel faces, 462. The Aspect of complex crystals of the Octahedral system of Crystallisation, useful as a means of discriminating their component Forms, 470. 2. The Pyramidal System of Crystallisation, 472. Quadratic Octahedrons, 473. Horizontal Planes, 481. Quadratic Prisms, 482. Dioctahedrons, 483. Eight-sided Prisms, 485. Hemihedral Forms, 487. Zones, 489. Mathematical Investigations, 490. 3. The Rhombohedral System of Crystallisation, 517. Six-sided Pyramids, 519. Horizontal planes, 529. Six-sided Prisms, 531. Twelve-sided Pyramids, 536. Twelve-sided Prisms, 537. Rhombohedrons, 538. Scalenohedrons, 561. Aspect of Complex Crystals belonging to the Rhombohedral System, with the symbols of Forms that replace the edges and angles of predominant Combinations, 569. 4. The Prismatic System of Crystallisation, 570. Table of Characteristic Combinations belonging to the Prismatic System of Crystallisation, 572. The Rhombic Octahedron, 573. Indices of the Rhombic Prisms, M X T. P X M. P X T, 576. Analysis of Combinations of the Prismatic Sys- tem, 577. Aspect of Complex Crystals belonging to the Prismatic System, 579. 5. The Oblique Prismatic System of Crystallisation, 580. Axes, Forms, and Combinations of this System, 582. North Combinations and East Combinations discriminated, 582. Examples of North Combinations, 585. Examples of East Combinations, 586. Classification, 587. Mathemati- cal Analysis of the Combinations of this System, 588. Miscellaneous Remarks on Calculations peculiar to the Forms of the Oblique Prismatic System, 598. 6. The Doubly Oblique Prismatic System of Crystallisation, 600. Mathematical Analysis of the Combinations of this System, 601. SECTION XIV. MB. BROOKE'S POPULAR SYSTEM OF CRYSTALLOGRAPHY, . PAGE 322 Resolution of Mr. Brooke's Primary Forms and their Modifications, or Secon- dary Forms, into the seven Fundamental Crystallographic Forms, P, M, T, MT. PM, PT, PMT, 602. Evidence of the mischief that flows from assuming the existence of primary forms, 602. SECTION XV. ON THE UTMOST POSSIBLE ABRIDGMENT OF EXACT CRYSTALLOGRA- PHIC NOTATION 329 Qualifications of good Notation, 603. Modes of Abridgment, 603. Table \ of Abridged Notation, 604. Objectionable Notation, 605. SECTION XVI. TABLE OF SINES AND TANGENTS, .... . 335 IV CONTENTS. SECTION V. OF THE POSSIBLE LIMIT TO THE VARIETY OF PLANES THAT CAN OCCUR UPON CRYSTALS, PAGE 74 Properties of the seven fundamental Forms of Crystals, 215. No other Forms than these seven can possibly occur upon Crystals, 225. Limit recognised, 226. SECTION VI. OF CRYSTALLOGRAPHIC NOTATION, 77 Principle of the present Notation, 227. Twin Crystals, 229. Examples of Notation, 230. SECTION VII. OF CLEAVAGE AND PRIMITIVE FORMS, 80 Nature of Cleavage, 231. How indicated, 232. Primitive Forms only hypo- thetical, 233. Hauy's Primitive Forms, 234. The doctrine of Primitive Forms not practically useful, 234. Secondary Forms, 235. Both Primitive and Secondary Forms are merely examples of the seven Fundamental Forms or their combinations, 236. SECTION VIII. OF FORMS AND COMBINATIONS, ....... 85 Forms, 237. Combinations, 239. Methods of indicating the General Aspect of Combinations, 243. Examples of Comparative Notation, 243. Intelligible Notation incompatible with the assumption of Primitive Forms, 245. SECTION IX. THE FIVE ZONES, 91 SECTION X. THE LAW OF SYMMETRY, 93 Classification and Nomenclature of Symmetrical and Unsymmetrical Forms, 253. Homohedral Forms, 258. Table of all the possible kinds of Homohedral Forms, 262. Uniaxial, Biaxial, and Triaxial Forms, 262. Hemihedral Forms, 263. Tetrahedron, 265. Theory of the Hemioctahedrons, 266. The four kinds of Hemihedral Forms with inclined planes, 267. Right and Left, or Direct and Inverse, Hemihedral Forms, 269. Theory of Right and Left Combinations, 270. Hemihedral Forms with parallel planes, 271. Hemioctahedrons of the Oblique Prisms, 272. Hemihedral Forms of each particular Zone, 276. Forms commonly but erroneously called Hemihedral, *J81. Tetartohedral Forms, 284. Discrimination of Homohedral, Hemihedral, and Tetartohedral Forms, 285. SECTION XI. A THEORY OF CRYSTALLISATION, 106 Eidogens, 288. Their Origin and Properties, 289. Illustrative Examples, 294. SECTION XII. THE USE OF SPHERICAL TRIGONOMETRY IN CRYSTALLOGRAPHY, . 1 19 Table of Trigonometrical Formulae for calculating the relations between the sides and angles of Triangles, 298. Definitions of Algebraic terms, 299. Of Solid Triangles, 300. Description and Application of the Formulae, 308. Logarithms of Numbers, 319. Calculation of Right angled Plane Triangles, 322. Table of Indices, 323. The axes of Forms belonging to a given Zone are multiples of one another for the same Mineral, 326. Construction of Symbols, 327. Calculation of Oblique angled Solid Triangles, 328. Calcula- tion by means of Quadrantal Solid Triangles, 331. Table of Square Roots, 333. General directions for the Analysis of Crystallographic Combinations, 334. Abridgement of Formulae, 335. SECTION XIII. AN INQUIRY INTO THE VARIETY OF FORMS AND COMBINATIONS WHICH OCCUR UPON THE CRYSTALS OF MINERALS, 149 Rose's System of Crystallography, 337. Classification of Crystals according to six systems of Axes of Crystallisation, 338. Discrimination of the Crystals of these six systems, 340. CONTENTS. V 1. Tlte Octahedral System of Crystallisation, 341. Explanation of Unipolar, Bipolar, and Tripolar Normals, 342, 349. Octa- hedron, 344. Cube, 350. Rhombic Dodecahedron, 354. Icositessarahe- dron, 369. Triakisoctahedron, 385. Tetrakishexahedron, 395. Hexakisocta- hedron, 408. Tetrahedron, 432. Hemiicositessarahedron, 4S7. Hemitria- kisoctahedron, 447. Hemihexakisoctahedron with inclined faces, 450. Pen- tagonal Dodecahedron, 453. Hemihexakisoctahedron with parallel faces, 462. The Aspect of complex crystals of the Octahedral system of Crystallisation, useful as a means of discriminating their component Forms, 470. 2. The Pyramidal System of Crystallisation, 472. Quadratic Octahedrons, 473. Horizontal Planes, 481. Quadratic Prisms, 482. Dioctahedrons, 483. Eight-sided Prisms, 485. Hemihedral Forms, 487. Zones, 489. Mathematical Investigations, 490. 3. The Rhombohedral System of Crystallisation, 517. Six-sided Pyramids, 519. Horizontal planes, 529. Six-sided Prisms, 531. Twelve-sided Pyramids, 536. Twelve-sided Prisms, 537. Rhombohedrons, 538. Scalenohedrons, 561. Aspect of Complex Crystals belonging to the Rhombohedral System, with the symbols of Forms that replace the edges and angles of predominant Combinations, 569. 4. The Prismatic System of Crystallisation, 570. Table of Characteristic Combinations belonging to the Prismatic System of Crystallisation, 572. The Rhombic Octahedron, 573. Indices of the Rhombic Prisms, M X T. P X M. P X T, 576. Analysis of Combinations of the Prismatic Sys- tem, 577. Aspect of Complex Crystals belonging to the Prismatic System, 579. 5. The Oblique Prismatic System of Crystallisation, 580. Axes, Forms, and Combinations of this System, 582. North Combinations and East Combinations discriminated, 582. Examples of North Combinations, 585. Examples of East Combinations, 586. Classification, 587. Mathemati- cal Analysis of the Combinations of this System, 588. Miscellaneous Remarks on Calculations peculiar to the Forms of the Oblique Prismatic System, 598. 6. The Doubly Oblique Prismatic System of Crystallisation, 600. Mathematical Analysis of the Combinations of this System, 601. SECTION XIV. MR. BROOKE'S POPULAR SYSTEM OF CRYSTALLOGRAPHY, . PAGE 322 Resolution of Mr. Brooke's Primary Forms and their Modifications, or Secon- dary Forms, into the seven Fundamental Crystallographic Forms, P, M, T, MT. PM, PT, PMT, 602. Evidence of the mischief that flows from assuming the existence of primary forms, 602. SECTION XV. ON THE UTMOST POSSIBLE ABRIDGMENT OP EXACT CRYSTALLOGRA- PHIC NOTATION 329 Qualifications of good Notation, 603. Modes of Abridgment, 603. Table of Abridged Notation, 604. Objectionable Notation, 605. SECTION XVI. TABLE OF SINES AND TANGENTS, . 335 VI CONTENTS. PART II. APPLICATION OF CRYSTALLOGRAPHY TO MINERALOGY. SECTION I. ROSE'S TABULAR ARRANGEMENT OF MINERALS, ACCORDING TO Six SYS- TEMS OF CRYSTALLISATION PAGE 1 SECTION II. A CATALOGUE OF CRYSTALLIZED MINERALS, SHOWING THE COMBINA- TIONS THAT OCCUR IN NATURE, . . . ... . . .14 List of Authorities quoted, '".',. 14,32 Class 1. Octahedral System of Crystallisation, . . . .-.. . 15 2. Pyramidal System of Crystallisation, . . . . . . . 32 3. Rhombohedral System of Crystallisation, . . .... .43 4. Prismatic System of Crystallisation, . . . . ... 61 5. Oblique Prismatic System of Crystallisation, 77 6. Doubly Oblique Prismatic System of Crystallisation, . . . . 91 SECTION III. A SYSTEMATIC ARRANGEMENT OF THE CRYSTALS FOUND IN THE MINERAL KINGDOM, WITH A LIST OF THE MINERALS COMMON TO EACH CRYSTAL. 95 Crystallographic Classification, 95 Explanation of the Mineralogical Characters employed to discriminate the Minerals that Crystallize in the same Form, 95 Class 1. Complete Prisms, 97 - 2. Complete Pyramids, 100 3. Complete Prisms combined with Incomplete Pyramids, . . . 104 4. Incomplete Prisms combined with Complete Pyramids, . . .110 5. Incomplete Prisms combined with Incomplete Pyramids, . . .114 6. Incomplete Pyramids, 122 SECTION IV. A DESCRIPTIVE CATALOGUE OF THE MODELS OF CRYSTALS EMPLOYED TO ILLUSTRATE THIS SYSTEM OF CRYSTALLOGRAPHY. 123 INDEX TO MINERALS DESCRIBED. . 139 PREFACE. THERE are many systems of crystallography in print, but none in general use. The different systems hitherto published have failed to satisfy the wants of the public. They are either too difficult to learn, or when learnt, too troublesome for service. Hence crystallography is little studied, either by chemists or mineralogists ; and the consequence is, that, for want of a language in which observations can be recorded, the study of crystallisation also is shunned. How trifling is the annual addition made to our knowledge of the crystalline forms of minerals, and how vague and inaccurate are the descriptions which chemists give us of the forms of nearly all the crystallised products of the laboratory ! Yet no one questions the importance of studying the productions of crystallisation, or the necessity of employing crystallography to record the facts which the examination of crystals brings to light. Indeed, since the discovery of " Isomorphism," everybody has become desirous of knowing how to describe the crystals that fall in his way ; the mineralogist, the chemical analyst, and the manufacturing chemist, are alike solicitous to record their observations ; but all have been so disheartened by the failure of former attempts to learn crystallography, that when they now recur to this science, it is only to demand how it can be learnt EASILY. The present publication is an attempt to answer this favourite ques- tion ; to show the way, not merely how to describe a crystal, but how to do it easily. Whether the attempt is successful or not, every reader will determine according to his peculiar standard of easiness. Persons accustomed to the investigations peculiar to abstract science, and those who have a distaste for calculations, will not agree on what constitutes an easy system of crystallography, and therefore will not coincide in their estimate of the value of this production. I trust, however, that in forming his opinion, the reader will not omit to take into consid- eration, how difficult a thing it is to give a popular character to any abstract science, and how especially difficult when that science is one which has to do with so vast a multitude of complex and troublesome details as those which constitute crystallography. To describe these de- yiii PREFACE. tails briefly and intelligibly, to bring them under general laws, and to show in plain Janguage the practical use of these laws in investigating the forms of minerals, is a task of such extreme difficulty, that success, however desirable, is of problematical attainment. In expressing my opinion that a popular system of crystallography was a desideratum, I by no means intend to undervalue the existing works on this science, When I state that they are not in use because they are too difficult to learn, or too troublesome for service, I allude in these expressions, less to the qualities of the publications, or the qualifications of their authors, than to the attainments of the major part of the stu- dents by whom crystallography ought to be learned. The students who enter our schools of chemistry, mineralogy, metallurgy, and mining, possess almost universally but a slender stock of mathematical know- ledge ; while the existing books on crystallography, or at least all that have any pretensions to science or system, are remarkable for contain- ing investigations which require very high mathematical attainments. " Crystallography," says Whewell, " is essentially a mathematical subject. The striking mixture of simplicity and complexity which here, as in other parts of nature,- but yet more here than in any other part of nature, offers itself to our notice, depends upon the combination of the primary forms belonging to the above systems [of crystallisation] with the geometrical and numerical laws by which other forms are de- rived from these. To trace the properties of such derived forms, and of their combinations, necessarily requires some considerable portion of mathematical calculation, which may, however, be of several kinds. Spherical trigonometry, solid geometry, and analytical geometry of three dimensions, may, any of them, be made to answer the purposes of the crystallographer. Haiiy and Mohs, proceeding in the manner which, of the three, implied the least extended acquaintance with mathe- matics, employed in most instances particular constructions and calcu- lations founded on solid geometry ; and though they thus want the con- ciseness, beauty, and generality of other methods, they are perhaps, in consequence of this, intelligible to a wider circle of students" Re- port on the Recent Progress and Present State of Mineralogy, addressed to the British Association, 1832. In illustration of these crystallographic calculations, which imply "the least extended acquaintance with mathematics," I beg to lay before the reader a few extracts from Haiiy's " Traite de Cristallogra- phie," Paris, 1822. 2 tome 8vo, pp. 1340. Tome I. page 452. ,Znxy2y ^ x+nxy+y^ \ nxy-\-x nxy x+y) A / / V ( PREFACE. Tome I. page 452. 2nxy2y - V / t :: A/ / V V c? -f % onxy + oy oa?/ nxy-x + y Tome I I. page 112. ^Otef&xfitf-. 13p"+4tf Tome II. page 113. <&= 4 X ffP Vox II. page 114. 75 du: di \: 4 ' 13 L_ +4 I 2 ^ r 9 15 , 36^ '13 V 4 By way of contrast to the foregoing specimens of calculations, assumed to be, from their facility, " intelligible to a wide circle of stu- dents," I take leave to quote, on the following page, a few scientific cal- culations, from one of the most original and popular writers among the crystallographers of Great Britain, I mean Professor WHEWELL, from whose Memoir 011 " A General Method of Calculating the Angles made by any Planes of Crystals, and the Laws according to which they are formed" these equations are extracted. See Philosophical Transactions of the Royal Society of London, for the year 1825. b PREFACE. 2 02 o o PQ S \ 5M ^5 ^^ > PREFACE. XI This "considerable portion of mathematical calculation," and these illustrations, show the discordant principle which keeps the existing works on crystallography out of general use. The doctrines they develop are based on a higher philosophy than comes within the reach of the ordinary student of crystallography. The books are written in an unknown tongue and cannot be read. Reflecting on these matters, it occurred to me, that the mathematical difficulties of crystallography were not inherent in the science, but re- sulted from the manner in which it was commonly treated, and that it was only necessary to adopt a different method of treating it, to be enabled to remove many of these difficulties, and to present the science in a much simpler and more attractive form than had been hitherto ac- complished. It were unnecessary to do this if the mathematical attain- ments of students of crystallography could be easily and extensively in- creased ; but since experience shows this to be impossible, the next best proceeding is to simplify and remodel the science so as to adapt it to the circle of knowledge possessed by the popular student. What mat- ters it, if we lose a little of the elegance and precision, by avoiding the splendid difficulties, of abstruse mathematics ? Is it not better that our science should be even imperfectly mastered by the multitude, than be entirely restricted to the service of a few accomplished mathe- maticians? Led by this idea, I began to examine the mutual relations of the forms of crystals, and the methods by which their geometrical properties could be investigated by a student possessing a minimum of mathematical knowledge ; and having done this, I contrived a series of symbols, by which the results of these investigations could be easily, intelligibly, and accurately recorded. The principle of notation upon which these symbols are founded, occurred to me at an early period of my inquiry into this subject ^ and though it is a principle which, from its fertility in symbols, proved susceptible "of extensive application in the science, it is, at the same time, extremely simple in its nature. I cannot perhaps introduce the few remarks which I have to make in this Preface, better than by giving a preliminary notice of the principle of the proposed notation, and a sketch of the crystallographic machinery by the employment of which the general laws of the science are subse- quently developed. We begin, then, to lay the foundation of crystallography by assuming the existence of what is termed a system of axes. These axes are the three lines which indicate the length, the breadth, and the thickness of a crystal. They are respectively denoted by the signs p a m a and t a . These axes necessarily cross one another in the centre of the crystal at right angles, and the length of each of them depends upon the length, the breadth, or the thickness of the crystal. To promote per- spicuity, p a is assumed to be the principal or perpendicular axis, m a to be the middle or minor axis, or that which passes from the front to the xii PREFACE. back of a crystal, and t a to be the transverse axis, or that which passes from the left to the right side. When a crystal is held tip in front of an observer, its longest diameter is put in the place of p a , and its shortest in the place of m a . To indicate differences in the length of the three axes, indices are put under the small letter a , thus : p+ m t a , which indices intimate that p a is longer than t a and m a shorter than t a . These differences are more specifically shown by figures, as p a m a t a , or more indefinitely by letters, as p| m a t a . In the next place, we consider the relation of the external planes of the crystal to these three axes. A careful investigation shows that the planes of all crystals are reducible to seven different kinds. Of these, three kinds are characterised by the property of cutting one axis, three other kinds by that of cutting two axes, and one kind by that of cut- ting three axes. These peculiarities give us the power to indicate the seven different kinds of planes by merely naming the axes which they cut. The notation thus afforded is short, simple, and precise. For P signifies 2 planes that cut axis p a . M * 2 do. m a . T 2 do. t a . MT 4 do. axes m a and t a . PM 4 do. p a and m a . PT 4 do. p a and t a . PMT 8 do. p a m a and t a . No crystal can possibly present planes that differ from all the above seven kinds. This remarkable fact can be easily demonstrated. The planes marked P are horizontal and form the top and bottom of a crys- tal. Those marked M are vertical and form the front and back of a crystal. Those marked T are vertical and form the left and right sides of a crystal. These planes have fixed positions, from which they can- not swerve without instantly losing their identity. If the plane P, for example, were to incline ever so little from its absolutely horizontal posi- tion, it must assume such a position that, if sufficiently extended, it would cut either one or two of the horizontal axes, and thereby become equal to plane PM, or PT, or PMT, according to its particular inclina- tion towards m a or t a or both. In the same manner it can be shown that if the plane M changes its situation, it must become MT, PM, or PMT, and that if the plane T changes its situation, it must become MT, PT, or PMT. From these and other considerations which are stated at length in the fifth section of the following work, it can be proved that the planes which are denoted by the seven symbols P, M, T, MT, PM, PT, PMT, are all the kinds that can possibly occur upon crystals. These seven symbols, therefore, constitute the whole alphabet of crystallography, but brief as this alphabet is, the powers of the cha- racters are such as to enable it distinctly to name the innumerable crystals with which nature and art present us. PREFACE. Xlll I proceed to notice the few contrivances, by the adoption of which we are enabled to bestow upon these seven symbols the degree of descrip- tive energy proper to qualify them for this extensive duty. The planes differ in respect to the number that is proper to each hind, which number is 2, 4, or 8. All the planes of each kind constitute a " Form," and according as the planes cut one, two, or three axes, the forms are termed uniaxial, biaxial, or triaxial. The uniaxial forms, P, M, T, contain each two planes to the set ; the biaxial forms, MT, PM, PT, contain each four planes to the set ; and the triaxial form, PMT, contains eight planes to the set. On certain minerals these forms occur with all their planes, on others with only half their proper number, and on a third kind with only a fourth part of their number. These differ- ences are denoted by marking the half-form with the prefix |, and the quarter-form with the prefix \ ; as MT, PMT. Although the planes or Forms of crystals are limited to these seven kinds, there are innumerable varieties of each kind. Thus, of the biaxial form MT, there is a multitude of varieties, which differ from one another in the relation which the length of axis m a bears to that of t a , that is to say, in the relative distances from the centre of the crystal at which the two axes, m a and t a , arc cut by the set of four planes which constitute the form MT. The same may be said of the biaxial forms PM and PT, and of the triaxial form PMT; the differences in the last of which relate to two particulars, namely, the comparative length of p a to t a , and of m 3 to t a . In consequence of this variation in the quality of each of the seven kinds of forms, it is necessary to provide a discriminating index for the symbol of each variety. This discriminating index is a vulgar fraction placed between the letters which compose the symbol of the form, and of which fraction the upper figure shows the length of the axis named by the left hand letter, and the lower figure the length of the axis named by the right hand figure. Thus, Mf T denotes a variety of the form MT, in which axis m a bears to axis t a , the ratio of 2 to 3. The cross section of this form is a rhombus, whose two diagonals have the relation of 2 to 3. This method of notation is extremely simple ; it is capable of universal application to rhombic forms ; and it has a very important mathematical use, since the index of every form indicates the angle at which the external planes of the forms incline upon one an- other. This inclination is, in all cases, twice the angle whose cotangent is the fraction contained in the symbol. Thus, f = .6667 is the cotan- gent of 56 18' f and the obtuse external angle of the form Mf T is 112 37'. In like manner, the obtuse external angles of the forms Pf M and PIT are all 112 37', and their acute external angles are 180 112 37' = 67 23.' Hence, all the planes which surround a crystal in three direc- tions, at right angles to one another, can be denoted by some combina- tion of the symbols of the first six Forms, P, M, T, M X T, P X M, P X T; XVI PREFACE. tallometrical method, not to prisms, but to pyramids. But the separation of the planes of crystals into horizontal, vertical, and oblique, leads also to the necessity of establishing a fixed point of view for crystals, in order that terms so strictly relative as horizontal, vertical, and oblique, may acquire a definite or positive acceptation. I propose, therefore, that the point of view of a crystal shall be in the prolongation of the minor axis m a ; that the crystallographer, when examining a crystal, shall always be assumed to look towards the south ; and that a crystal, while under examination, shall be considered to be polarised. Then, the uppermost part of the crystal will be the zenith pole = Z, the lower- most part, the nadir pole = N, the part directly facing the spectator, the north pole = n, the part opposite, the south pole = s, the side opposite the spectator's right hand, the west pole = w, the side opposite, the east pole = e. The acceptation of these arbitrary terms, enables us to describe particular positions or points on a crystal, with a most conve- nient degree of accuracy. This advantage is not limited to the above six poles, for the compound terms Zn, Zs, Ze, Zw, Nn, Ns, Ne, Nw, and nw, ne, se, sw, enable us to refer, with equal perspicuity, to twelve polar positions, each intermediate between two of the six primary poles, and the terms Znw, Zne, Zsw, Zse, and Nnw, Nne, Nsw, Nse, enable us farther to refer to eight other polar positions, each respectively equidistant from three of the six primary poles. It is in many cases convenient to refer to lines which pass from the centre of the crystal, and terminate in one or other of the above-named twenty-six poles. These lines are termed Normals. The normals which terminate in poles denoted by one, two, or three letters, are termed respec- tively unipolar, bipolar or tripolar normals. By means of these poles and normals we can refer with great accuracy to twenty-six different points on the surface of a crystal, and to the angles which the normals make with one another at the centre of the crystal, and which they also make with other lines that connect their poles. Thus, the inclination of a / Unipolar Normal = 54 44' Tripolar Normal to any adjacent< Bipolar Normal =35 16' I Tripolar Normal = 70 32' Great use is made of this principle in investigating the forms of the crystals which belong to that class whose length, breadth, and thickness are alike. For other practical purposes, I assume every crystal to be capable of division by sections, as follows : By a horizontal section passing through the poles n, e, w, s, and parallel to the axes m a and t a . I call this sec- tion the Equator. It separates the Zenith half of the crystal from the Nadir half. There are four other sections, all vertical, and which are called Meridians. The two first of these are of most importance. The north meridian passes through the poles n, Z, s, N, and parallel to the axes p a and m a , and separates the east from the west half of the PREFACE. XV11 crystal. The east meridian passes through the poles e, Z, w, N, and paral- lel to the axes p a and t a , and separates the north from the south half of the crystal. The north-east meridian passes through the poles Z, ne, N, sw. The north-west meridian through the poles Z, nw, N, se. The two first and two last meridians are situated at right angles to one another, but the common intersection of the four meridians is an angle of 45, and all the meridians make an angle of 90 with the equator. By means of any two of these sections, situated at right angles to one another, a crystal is divided into equal quarters, and there are six possible combinations of this kind. By means of the four meridians a crystal is divided into long octants or eighths. By means of two meridians and the equator, the crys- tal is divided into octants of another kind. By means of the five sections together, a crystal is divided into sixteenths. If we want to distinguish one of these quarters, eighths, or sixteenths, from all the rest that can be produced, we can do it easily by naming its polar situation. Thus, the sixteenth ZnV, is that which is on the zenith side of the equator, the north side of the east meridian, the west side of the north meridian, and the north side of the north-west meridian. The outer angles of this sixteenth are at the poles Z, n, and nw, and the edges formed by the intersection of its meridians and equator, have the positions of the Z, n, and nw normals. The plane angles of the internal faces, and the interfacial angles of the internal edges of this sixteenth can be readily determined from these data. The surface planes of a crystal, which are tangents to the above five sections, constitute collectively the Zones of those sections. Thus the planes M, T, MT, produce the prismatic or equatorial zone ; the planes P, M, PM, the north zone; the planes P, T, PT, the east zone; and the planes PMT, the north-east and north-west zone. Each of these zones is a belt of planes surrounding the edge of the corresponding section, and the axis of the zone is at right angles to the given section. The form PMT produces two zones, whose axes cross at a right angle. Such are the foundations of the System of Crystallography which is communicated in the following work, respecting the execution of which I will now take the liberty to make a few explanatory observations. The First Part is devoted to an account of the " Principles of Crystal- lography." This may be considered partly in the light of a Popular Introduction to this science, and partly as a series of original essays on the leading topics of Crystallographic research. The work was at first intended to contain only a brief account of the new system of notation, but being sent to the press as it was written, bit by bit, it gradually came to embrace a much wider range of subjects than the author ori- ginally proposed, until it finally comprehends a tolerably complete sketch of modern Crystallography. In consequence of the composite character of the work, it presents a XV111 PREFACE. greater amount of elementary matter, especially on mathematical topics, than it is usual to intermix with original investigations ; while on the other hand, it contains much more argument and discussion than is perhaps adapted to an elementary treatise. Hence the critical reader may notice a want of unity in the several parts of the work, and the occurrence of some apparently needless repetitions. But it is hoped that faults of manner of this kind will not be found very detri- mental, and will be considered to be compensated by the substantial merits of the associated information. The first six sections of the work relate to topics already spoken of in this Preface, namely, to the Axes, Poles, Normals, Planes, and Sec- tions of Crystals ; to Prisms and Pyramids and their combinations ; to the Classification of Crystals; to the possible limit to the variety of Planes that can occur upon Crystals ; to the demonstration of the suffi- ciency of three rectangular axes for all useful references in Crystallo- graphy; and to the description of crystallographic Notation. This scientific machinery may appear cumbrous, but it is useful. The equa- tor and meridians are as indispensable to the crystallographer as to the geographer, while axes and poles are no less necessary in the analysis of crystals, than are pistils and stamens in the discrimination of plants. The seventh section relates to Cleavage, and to the doctrine of Primitive Forms ; of these I need only say, that I have proposed an easy and exact method of denoting Cleavage by symbols, and argued for the total relinquishment of the doctrine of primitive forms, as one highly injurious to the science. Section eighth contains explanations relative to Forms and Combinations. Section ninth treats of the five Zones. Section tenth of the law of Symmetry, and of the distinction between Homohedral, Hemihedral, and Tetartohedral Forms. Section eleventh contains a new theory of crystallisation, founded on an opinion that electricity is the mainspring of crystallisation. The reader will perceive that the first six sections are entirely con- fined to descriptions arid explanations, and that all theoretical and general views, all inferences drawn from foregone statements of facts, are confined to the five last sections. By this separation and arrange- ment of particular and general truths, I have endeavoured to lead the popular student by the least difficult path to a knowledge of the science. I considered it, however, to be unnecessary to enter into much detail on theoretical matters, and I have therefore devoted but forty pages to the whole five last sections. But the speculative reader will not, I trust, hold these sections to be deficient in interest because they are limited in extent. He will find them to contain a variety of new views on some of the most curious topics of this branch of natural science ; and although a thorough investigation of the objects discussed in them was unneces- sary for the mere illustration of " the art of describing crystals," yet their farther pursuit, in reference to the physical theory of crystallisa- PREFACE. XIX tion, would neither be without interest to the student, nor advantage to science. Section twelfth contains a popular account of the use of Spherical Trigonometry in Crystallography. The object of this section is to pre- pare and explain a collection of trigonometrical formulse to be used empirically in crystallographic calculations, by persons who possess too little mathematical knowledge to prepare the formulae, and yet suffi- cient to use them in the investigations proper to Crystallography. The method here followed of presenting the mathematical part of the sub- ject in a state ready for the technical use of the crystallographer, appears to me to remove many of the difficulties which have heretofore impeded the popular study of this science. The thirteenth section contains an inquiry into the variety of forms and combinations which occur upon the crystals of minerals. There are three objects proposed in this section. One is, to show the agree- ment between the natural crystals of minerals, and the mathematical forms and combinations described in the preceding twelve sections. Another object is, to give a partial explanation of the system of Crys- tallography proposed by Weiss, and modified by Rose, Naumann, Mohs, Miller, and others, and to prove that the new notation is adapted to describe all the forms and combinations belonging to that system, and represented to be all that occur in nature. The third object is, to show the methods by which the mathematical calculations requisite for these several purposes, can be made empirically by means of the prepared Formulse and general Rules communicated in the twelfth section. This last object is one of great importance to the student, and the solution of the problem that it presents, has received my best attention. It appeared to me at the beginning of my inquiry, that if the mathe- matical calculations could not be rendered short and easy, it would be needless to entertain any hope of ever bringing Crystallography among the number of the popular sciences. I set myself therefore to try what could be done towards simplifying and organising the calculations that were indispensably requisite to discriminate and identify the co-existing forms of complex combinations. This research produced a body of Crystallographic Analyses, which is, I believe, the most comprehensive and systematic that has ever been published. All the forms and funda- mental combinations of each of Rose's six systems of Crystallisation, namely, the Octahedral, the Pyramidal, the Rhombohedral, the Pris- matic, the Oblique Prismatic, and the Doubly Oblique Prismatic, have been fully investigated, and their relations are pointed out, on the one hand to crystallised minerals, and on the other hand, to the symbols employed to denote them. One peculiarity of the mathematical pro- cesses given in this part of the work is, that they are never arbitrary, but are always founded upon the simple principle of finding the relation of certain unknown to certain known sides arid angles of the solid triangles XX PREFACE. produced by the ideal analysis of the crystals subjected to examination. By a systematic adherence to this principle, the calculations are brought into a regular train, the reason of each of them is distinctly shown, and the method of operating is, by frequent repetition of the same thing, made habitual and easy. It will be noticed, that many of the equations given in this section afford results that are not rigidly but only approximately accurate. This arises in some cases from the circumstance that angles are reckoned only to minutes, not to seconds. In other cases, from the adoption of logarithmic quantities having but four instead of seven figures after the point. The errors thus produced are of very little consequence. I have used short numbers, because the wearisome cal- culations introduced by the affectation of extreme accuracy, interrupt the train of reasoning, and disgust the reader entirely, instead of com- municating to him a more exact amount of information. Independent of any merit which this section may possess on the score of its mathematical simplifications, it presents a variety of new methods of crystallographic research, of which some may probably prove interesting to the more accomplished crystallographer, for whom the popular parts of the work can have no attraction. Dispensing, as I have done, with all notation, except that which refers to a single system of three rectangular axes; dispensing with primitive forms, with primary forms, with fundamental forms; with oblique prisms, and with doubly oblique prisms ; assuming new views in relation to the difference be- tween homohedral and hemihedral forms ; subdividing the forms of the " oblique prismatic system" on grounds entirely new; and making other important changes in different departments of the science ; it became necessary to make many new arrangements, and introduce a little new phraseology, to compensate for the loss of so many of the creations, and so much of the machinery, of former crystallographers. Whether the suggested alterations are amendments on the science, or merely exam- ples of retrograde steps, time and experience will determine. The fourteenth section of the work contains an account of Mr. Brooke's popular method of describing crystals by means of primary forms and tables of modifications. This section is intended to show that the doctrine of primary or primitive forms is useless and mischiev- ous, being equally unadapted for popular and for scientific nomenclature. The fifteenth section contains an inquiry into the means by which crystallographic notation may be made to attain the utmost limit of brevity. I have endeavoured to establish certain principles which ought always to govern the contrivance of notation, and to show how far extreme brevity of expression is practicable, and how far it is desirable. The sixteenth section contains a table of sines and tangents adapted for working the equations described in the preceding sections ; which, although too brief to replace the ordinary Tables of Logarithms, will PREFACE. XXI often prove a useful substitute, especially from its containing the INDICES of the Symbols of all commonly occurring varieties of the seven crystallo- graphic Forms. The Second Part of the work contains an account of the application of Crystallography to the investigation of the forms of crystallized Minerals. The first section presents a tabular arrangement of minerals accord- ing to the six systems of crystallisation, described in the thirteenth sec- tion of Part I. The catalogue of minerals is translated from the one published in GUSTAV ROSE'S Elemente der Krystallographie. It shows the crystallographic system, and, approximately, the chemical composi- tion, of every particular Mineral or Isomorphous group of minerals. The second section contains a catalogue of all the crystals or natural combinations peculiar to the minerals named in Rose's catalogue. The minerals are arranged in six classes, on the principles described in sec- tion thirteenth. Each class opens with an account of its peculiar axes, forms, and fundamental combinations. Under each mineral, its diffe- rent natural crystals, or secondary forms, are described in symbols, each in a different line, with an abridged reference to several well known mineralogical works, which contain figures or descriptions of the crys- tals thus particularized. The compilation of this catalogue cost me a great deal of labour, yet, after all, it is but an imperfect production, and gives only a provisional and approximate view of what I wished it to represent. When the calculations, upon which these symbols are founded, were made, I was not acquainted with the methods of investigation which are now printed in the thirteenth section. Indeed, the analytical formulae of that sec- tion were contrived during, and subsequently to, the writing of this catalogue of minerals. They are the fruits of my attempt to reduce into precise notation and regular order, the vague and often discordant statements of the mineralogists whose works I have quoted in the text. Had I possessed this body of mathematical aid when I began my task, it would have been better performed; but even the best provision as to method, would not supply that want of material which is found by an author who attempts to study such a science as Crystallography in such a town as Glasgow, where there is no public collection of crystallised minerals to refer to, and where no public library contains the literature of a science so " dry" as Crystallography. The crystallographers of Berlin, and Freiberg, and Paris, would need to study in this city a little while, to be able to feel the weight of this difficulty. But for the kind- ness of Dr. THOMAS THOMSON, to whom I am indebted for the loan of several scarce books, my catalogue of minerals would have been even more inaccurate than it appears at present. But although the table is not to be depended upon as absolutely XX11 PREFACE. correct, it nevertheless presents a great amount of information respect- ing the natural crystals of minerals, and it affords a convenient method of comparing the recorded crystallographic combinations with those which may be presented by the crystals of any mineralogical collection to which the crystallographer may happen to have access. The use of such a table is to show the mineralogist what varieties of each mineral have been observed and described, and so lead him to the discovery of new and unrecorded combinations. It is scarcely necessary to add, that the thirteenth section of the first part of this work is at once an introduction to, and a commentary on, the second section of the second part. The former contains in- structions how to do, what the latter presents done; imperfectly done, indeed, but still presenting the basis of a work which will be highly use- ful when rendered complete. The third section of this part contains a systematic arrangement of natural crystals, with a list of the minerals which are common to each crystal. This section is a counterpart to the last. One presents an account of all the crystals proper to each mineral. The other pre- sents an account of all the minerals which are common to each crystal. The object of the former is to aid in completing the natural history of each mineral. The object of the latter is to direct the mineralogist how to discover the name of a mineral from an examination of its form. The classification of crystals adopted in this section, is necessarily different from the classification of minerals followed in the last section ; the principles of classification being those which are described at page xv. The classes, orders, genera, and groups in which the minerals are here arranged, are sharply defined and strongly contrasted, and it appears to me, that the classification in question, with the subordinate contrivances which accompany it, are well qualified to constitute a use- ful GUIDE TO THE DISCRIMINATION OF MINERALS, the Contrivance of which is the principal object of this section. I shall endeavour to show the practical use to the Mineralogist, of the analytical method presented in this section, by contrasting it with MOHS'S instructions for the use of his celebrated " Characteristic" I quote from Haidinger 's Translation of Mohs's Treatise on Mineralogy, vol. i, pp. 383387. " It will be useful to give a short explanation of the process used in the determination of minerals. " If a mineral is to be determined, first its Form, if this be regular, must be ascer- tained, at least as far as to know the system to which it belongs. Then Hardness and Spe- cific Gravity must be tried with proper accuracy, and expressed in numbers. It is sufficient however, to know the latter to one or two decimals. The specific character requires these data ; they are also of use in the characters of the classes, orders, and genera. After this examination, the Characteristic may be applied, and it will at the same time point out what other characters are still wanting; so that a mere inspection of the mineral, or a very easy experiment, as for instance, to try the streak upon a file, or still better, upon a plate of porcelain biscuit, will very often be sufficient. The given individual is now car- ried through the subordinate characters of the classes, orders, genera, and specie?, one PREFACE. XX111 after the other, comparing its properties with the characteristic marks contained in the characters of these systematic unities. From their agreement with some, and their differ- ence from other characters, we infer, that the individual belongs to one of the classes, to one of the orders, to one of the genera, and to one of the species. Having advanced in this manner to the character of the species, it will in some instances he necessary, and in all cases advisable, for the sake of certainty, to have recourse to the dimensions of the forms. This is particularly necessary, if the genus to which the mineral belongs, contain several species having forms of the same system, as is the case in the genus Augite-spar. The common goniometer in most cases will suffice for determining the dimensions of the forms, the differences in the angles being in general so great, that they cannot easily be missed, even by the application of this instrument. If the differences be small, and their distinc- tion require on that account a higher degree of accuracy, it will be necessary to recur to the reflective goniometer. " It will seldom be necessary to read over the whole of any character of a class, order, genus, or species, excepting those which comprise the individual ; one term that does not agree sufficing for its exclusion. Thus even the characters of the orders, though the long- est, will not be found troublesome. " The application of the Characteristic has been facilitated in a great measure byseparating the absolute characteristic marks from the conditioned ones. It becomes still more easy and expeditious, by taking particular notice of some characters, which might be termed prominent. Such are a metallic appearance ; a high degree of specific gravity, particularly if the appearance be not metallic ; and a high degree of hardness. The observation of these will immediately decide whether an individual can belong to any particular class, order, genus, or species. It is understood, that if it be not thereby excluded, the other characters must next be examined, till either an excluding one be found, or if not, the individual may be considered as belonging to that class, order, &c., with which it has been compared and found to agree. " An individual, which has been carried through the characters of the classes, orders, genera, and species, and whose systematic denomination has thus been found, is said to have been determined. " In illustration of this, let us take the following example. Let the form of the mineral which is to be determined, be a combination of a scalene eight-sided pyramid, of an isosceles four- sided pyramid, and of a rectangular four-sided prism; the cleavage parallel to the faces of two rectangular four-sided prisms, in diagonal position to each other ; form and cleavage therefore pyramidal, or belonging to tJie pyramidal system. Let Hardness be 6.5; Specific Gravity 6.9. " In this case, both hardness and specific gravity are prominent characters, and exclude the individual at once from the first and third, but not from the second class ; with the characters of this class, its other properties also perfectly agree. Hence the individual belongs to the second class. " Comparing the properties of the individual with the characters of the orders in the second class ; hardness and specific gravity will be found too great for the order Haloide ; hardness too great for the orders Baryte and Kerate ; both of them too great for the order Mala- chite and Mica ; and specific gravity too great for the order Spar and Gem. But in the character of the order Ore, both hardness and specific gravity fall between the fixed limits, and cannot exclude the individual from this order. The other parts of this character are now to be taken into consideration. If the appearance of the individual be metallic, its colour must be black, otherwise it cannot belong to the order Ore. But the appearance is not metallic ; therefore the colour of the individual is quite indifferent ; that is, this condi- tional characteristic mark does not affect the individual, and consequently cannot decide. Since the appearance is not metallic, the individual must exhibit adamantine or imperfect metallic lustre. The first will be found, particularly in the fracture. The following cha- racteristic marks refer to minerals of a red, yellow, brown, or black streak ; and as the individual gives none of these, its streak being uncoloured, these characteristic marks do not come into consideration. The next mark requires, that if hardness be = 4.5 and less, the streak should be yellow, red, or black ; but hardness is = 6.5, therefore the colour of the streak indifferent. If hardness be = 6.5 and more, and streak uncoloured ; then specific gravity must be = 6.5 and more. Now this condition takes place ; hardness is = 6.5, streak is uncoloured. But also the conditioned character takes place, specific gravity being = 6.9, which is greater than 6.5. " In regard to the individual, which is to be determined, all the characteristic marks con- XXIV PREFACE. stituting the Character of the order Ore, may be divided into two parts. The first part contains those which refer to the individual ; the second those which do not ; the last evi- dently cannot be decisive. But with the first, all the properties of the individual concur. These properties agree consequently with the whole character of the order, as far as it is applicable to the individual, and determine it to belong to the order Ore, or, in shorter terms, to be an Ore. " It will be advisable to beginners, who do not yet possess a sufficient practice in the use of the Characteristic, also to compare the characters of the remaining orders, which will enable them to find out any error they might have committed in the comparison of the individual with the characters of the preceding orders. In the present case, the non- metallic appearance excludes the individual from the orders Metal, Pyrites and Glance; hardness from the order Blende; and both hardness and specific gravity from the order Sulphur. This fully confirms the above determination, and we must now return to the order Ore for comparing the properties of the individual with the generic characters which the order contains. " Considering again hardness and specific gravity as prominent, the individual will be immediately excluded from the genera Titanium-ore, Zinc-ore, and Copper-ore, but not from the genus Tin-ore. The forms of the pyramidal system, and the uncoloured streak, show that it belongs to this genus. If we compare the individual with the remaining generic characters, we find that it is excluded from the genus Scheelium-ore by its too great hardness, and too little specific gravity; from the genera Tantalum- ore, Uranium- ore, Cerium-ore, Chrome-ore, Iron-ore, and Manganese-ore, by hardness and specific gra- vity, both of them being too great; as also by its uncoloured streak, which only agrees with that genus from which the individual differs most by its hardness and specific gra- vity. From all this we infer that the individual cannot belong to any other than to the fourth genus, and that we are therefore entitled to give it the name of Tin-ore. " This genus contains but one species. The conclusion that the individual must belong to this species, might nevertheless be erroneous. There could exist a second species of this genus. Hence we must accurately consider the dimensions of the forms. If these coincide with the angles given in the character, the highest degree of certainty, that the individual belongs to or is pyramidal Tin-ore, will be obtained." The determination of a Mineral, after the method described in Sec- tion III. Part II. of this work, requires the following particulars: 1. Crystallographic Forms of the combination, described in symbols. 2. Axes of the combination. 3. Lustre or transparency. 4. Hardness. 5. Streak. 6. Specific gravity. 7. Cleavage. These particulars are given by MOHS, as follows: 1. CRYSTALLOGRAPHIC FORMS: a) A Scalene eight-sided pyramid. This is a dioctahedron of the pyramidal system = p x ni y t z , p x m z t y . b) An isosceles four-sided pyramid. This may be P X MT, or P S M, P X T. c) A rectangular four-sided prism (meaning a square prism). This may be either M, T, or MT. Hence the combination is one of the four following: M,T.P X MT, p x m y t z , p x m z t r M,T.P X M, P X T, p x m v t,, Pxin z t y . MT.P X M, P X T, p x m y t z , p x m,t y . MT.P X MT, p*m y t z , p x m,t y . A sight of the crystal would enable us to discard two of these sym- bols, which we cannot do from Mons's description of the combination, because he does not tell us whether or not the four-sided pyramid and PREFACE. XXV four-sided prism belong to the same or to different zones. We have also to assume, since he does not condescend upon that point, that the planes P do not form part of the rectangular (quadratic) prism. 2. Axes = p x m a t a . 3. Lustre = non-metallic ; adamantine. 4. Hardness = 6.5. 5. Streak = uncoloured. 6. Specific gravity = 6.9. 7. Cleavage = m, t, mt. This combination is immediately referred to Class 4, page 110, be- cause the forms M,;T, or MT, without P, constitute an Incomplete Prism, and because the forms P X MT or P X M, P X T, either with or without p x m y t 2 , p x m 2 t y , constitute a Complete Pyramid. The combination is referred to Order 1, because the equator of M, T, or MT, is square. It is referred to Genus 2, because the axes are p x m a t a . It is referred to Group a, because the prism M, T, or MT, has four vertical planes. See Part II. page 111. The group thus referred to, contains fifteen Minerals, among which the twelfth in order agrees in hardness with our specimen. It also agrees in lustre, as tl signifies non metallic and translucent. It dis- agrees in streak, as br signifies brown, and not uncoloured; but in point of fact, MOHS, in another place, states the streak of the given Mineral to be uncoloured to pale brown. Finally, it agrees in specific gravity, as 7 is very close to 6.9 ; and there is no other Mineral in the same group which has any thing near the same hardness, without differing very greatly in the specific gravity. The Mineral which thus agrees in its characters with the given speci- men is Oxide of Tin. By means of the Index, we find the place of Oxide of Tin in the second section, page 34, and examine its cleavage, which is m,t,mt. P1M,P|T, which partly agrees with MOHS'S state- ment, although it contains something more. We then look down the list of the natural crystals of Oxide of Tin, to find if any of them agree with the given combination, and among the varieties marked 4, I, which mean Class 4, Order 1, from which Order we have just been re- ferred, we find two crystals which agree very nearly with the given characters. These are MT. P1MT, p 3 m 2 t, p 3 mt2. MT. PIM, PIT, p3m 2 t, p 3 mt 2 . In the first of these combinations, the prism and quadratic pyramid belong to the same zone ; in the second, they belong to different zones. Our determination of the specimen can go no farther, because Moris has neglected to declare in his data, which of these two varieties he meant to refer to. This single investigation shows that the method of analysis which is d XVI PREFACE. here proposed, is preferable to that of MOHS'S, not only on the score of simplicity and facility, but also in point of precision. The fourth and last section of the work contains a description of the Models of Crystals employed to illustrate, not merely the general prin- ciples of this science, but its practical, mathematical, and mineralogical details. To the descriptions given in this section, I need only add, that the possession of models of this kind is indispensable for the compre- hension of the science, by every one who does not study it accord- ing to the strict rules of mathematics. It is, perhaps, possible for an accomplished geometer to learn the principles of Crystallography with- out seeing either crystals or models ; but the popular student need not attempt to learn this science without the aid to be derived from models of crystals, which afford that constant tangible correction of his errone- ous ideas, which is indispensable to his progress and success. I have purposely avoided, in this work, the discussion of two subjects of considerable interest to the crystallographer. One of these is Isomor- phism ; the other is the Optical Properties of Minerals. In Rose's catalogue of Minerals, the species are arranged in Isomor- phous groups, and as 1 have followed this arrangement, the reader has the opportunity of comparing with one another, many groups of natural crystals of the so-called Isomorphous minerals. The section also in which I have brought together the different minerals which crystallise in the same form, presents other Isomorphous groups which are deserving of notice, while at the same time it affords groups of dimorphous, isodimor- phous, and plesiomorphous crystals, which illustrate other relations of the forms of, minerals in an interesting and striking manner. Unfortunately, these catalogues are neither sufficiently correct nor sufficiently exten- sive, to warrant the drawing of conclusions exact enough either to con- firm or overturn the Law of Isomorphism, such as it was announced by Mitscherlich : " The same number of atoms combined in the same way, produces the same crystalline form, and the same crystalline form is independent of the chemical nature of the atoms, and is determined only by their number and relative position." If a hundred examples can be drawn from the following tables to confirm this law, we can also point out a hundred other examples that are repugnant to it. It would be unwise to reject or neglect a principle which is so important as Iso- morphism, if it prove to be true, but it is unsafe to adopt as a law of nature, a rule which seems liable to so many exceptions. I am aware that Isomorphism is considered, by a large body of philosophers, to be firmly established, but I doubt whether they have allowed due weight to the evidence which is arrayed against it. The hope that it might be true, has probably induced many to exercise a biased judgment on its merits. It is much to be desired that the salts and minerals, commonly assumed to be isomorphous, should be thoroughly investigated, and the PREFACE. XXV11 acknowledgment of their absolute isomorphism be conceded only upon satisfactory crystallographic evidence. The optical characters of crystals are passed over with a very slight notice, in consequence of my inability to afford the time that would be required to investigate the subject so carefully as to be enabled to write a popular account of it. Not that the mere description of the principal facts known respecting double refraction and the polarization of light would be attended with much difficulty ; for in fact, the Notation and Crystal- lographic machinery employed in this work are adapted for readily con- veying information on such points as those. But the consideration of the optical properties of minerals, in relation to the theory of crystalli- sation, which I have partially developed in the eleventh section of this work, would lead to a field of speculation so wide, that I cannot at pre- sent venture to enter upon it. I therefore leave this interesting subject for future investigation. I once intended to add to this Treatise a third part, on the applica- tion of Crystallography to Chemistry. But when I came to examine the descriptions which chemical writers give of the forms of the crys- tallised products of the laboratory, I found them to be universally so vague, that I could not attempt to translate them into symbols. The crystallographer who would write a catalogue of crystallised chemical products, has a long task before him, for he must make as well as mea- sure the substances that it would be necessary to describe. And not only must he make the crystals, but he must do so with a close examina- tion of the circumstances of the crystallisation of each of them. He must accurately determine their chemical composition, and that also of the mother liquor from which they are taken. He should notice the effects produced by changes of temperature, by light, air, and mechani- cal obstructions ; by chemical, electrical, and magnetical action, and so on. This is a task not likely to be soon undertaken ; for although the result would be highly useful to science, the labour would be miserably unprofitable to the philosopher. The task, however, must be done, and well done, before we can derive substantial advantages from the doctrine of Isomorphism. GLASGOW, November 2d, 1840. PART I. PRINCIPLES OF CRYSTALLOGRAPHY. PART I. PRINCIPLES OF CRYSTALLOGRAPHY. SECTION I.-OF THE AXES OF CRYSTALS. 1. CRYSTALLOGRAPHY is the art of describing crystals. 2. A crystal is a homogeneous inorganic solid body, which possesses length, breadth, and thickness, and is bounded by plane faces. 3. The measures of the length, breadth, and thickness of a crystal are three imaginary lines which pass through its centre, cross one another there at right angles, and terminate at its surface. 4. These imaginary lines are called Axes. 5. The longest axis is the principal or perpendicular axis. Its sym- bol is p a . Its position is vertical. 6. The next longest axis is the transverse axis. Its symbol is t". Its position is horizontal, and it passes from left to right. 7. The shortest axis is the minor axis. Its symbol is m a . Its position is horizontal, and it passes from the front to the back of the crystal. 8. The point of view of a crystal is in the prolongation of the axes m a . Hence the observer of a crystal has to hold it before him with the longest axis in a perpendicular position, and the broadest surface exposed to his eye. Variations from this rule will be explained hereafter. 9. The crystallographer is farther assumed to be always looking towards the SOUTH when examining a crystal. The front of the crystal, or the face that is turned towards him, is then exposed to the north; the parallel face, or that which is farthest from him, is exposed to the south; the face opposite to his left hand, to the east, and the parallel face, opposite to his right hand, to the west. The upper portion of the crystal is the Zenith portion, and the lower portion is the Nadir portion. The horizontal plane which separates these two portions, is the Equator. The symbols for these terms of position are as follow: North, n. South, s. East, e. I Zenith, Z. West, w. Nadir, N. Z (for Zenith) and N (for Nadir), but none of the other four letters, are written in capitals, in order that N = Nadir may never be mistaken for B 2 PRINCIPLES OF CRYSTALLOGRAPHY. n = north. The terminal points of the axis p a are in Z and N. Those of the axis m a are in n and s. Those of the axis t a are in e and w. p| is the zenith pole of the axis p a . PN is the nadir pole of the axis p a . m* is the north pole of the axis m a . m a is the south pole of the axis m a . t a is the east pole of the axis t a . t^is the west pole of the axis t a . 10. Any point on the surface of a crystal may be a. pole, and is named in reference to its proximity to any two or three of the above six cardinal points ; as nw = north west, Ze = Zenith east (midway between these two poles), &c. A. direct line between a pole and the centre of the crystal is a NORMAL, every variety of which is named after the pole to which it is perpendicular. Thus a line from the centre of the crystal to the pole Znw is the Znw normal. This arbitrary attribution of polarity to the axis of crystals, is in- tended to facilitate reference to the different faces, edges, and angles of crystals. 11. The axes of crystals vary in length, as the crystals that they belong to vary in length, in breadth, or in thickness. The symbols p a m a t a employed without addition, denote the axes to which they relate to be of equal length. Crystals whose axes are all equal, are called equiaxed crystals. When long axes are to be denoted, the symbols are written p+ m.|_ t_|_; and when short axes are to be denoted, the symbols are written pi ml tl. The sign plus (+) employed in this manner, signifies more than unity; the sign minus (-) signifies less than unity. 12. When an exact value is to be given to the symbols, the signs + and - are replaced by figures that indicate the precise lengths of the respective axes. Thus, when the principal, minor, and transverse axes have the relation of 3, 1, and 2, the symbol is written p m^ t|. 13. When the length of an axis is unknown, or when it is variable, that is to say, liable to be more or less than unity, its symbol is subscribed x . When the three axes, p a , m a , t a , are all of different, but of unknown lengths, they are written p| m* t*. 14. ILLUSTRATIONS BY MEANS OF THE MODELS OF CRYSTALS. [The reader will observe, that each of the Models of Crystals is marked with the letters P, M, T, and it is proper in this place, once for all, to inform him, that these letters have the following significations: P signifies the zenith pole of the axis p a . M signifies the north pole of the axis m a . T signifies the west pole of the axis t a . In examining a model, it may be held in the left hand, or placed upon a support, with the face M on a level with, and exactly opposite to, the observer's eye, 8. The several posi- tions of top and bottom, front and back, left and right, are then easily discriminated. On a few of the models that exemplify complicated forms, the Nadir, south, and east poles are marked, as well as the Zenith, north and west. See Models 95, 69, 117. This is done to prevent any misconception that might take place respecting the direction of any of the axes of such forms. The Zenith pole of any model is readily found by holding the solid in such a manner that the letters M and T stand the right way uppermost: p| will then be at the top of the model.] PRINCIPLES OP CRYSTALLOGRAPHY. 3 Model 1. The cube. The axes are p a m a t a . Model 2. The short quadratic prism. The axes are pi m a tV Model 3. The long quadratic prism. The axes are p+ m a t a . Model 6. The rhombic prism. The axes are pi m a 1$.. Model 12. The obtuse quadratic octahedron. The axes are pi m a t\ Model 13. The acute quadratic octahedron. The axes are p+ m a t a . Model 15. The regular octahedron. The axes are p a m a t a . Model 21. The rhombic octahedron. The axes are p+ ml t a . The absence of any sign under the letter a signifies that the axis to which the letter a belongs is to be considered as unity == 1. SECTION II. OF THE PLANES OF CRYSTALS. 15. Crystals are bounded by planes. Where two planes meet they form an edge. Where more than two planes meet, they form a solid angle. 16. The PLANES of crystals are flat faces bounded by straight lines. As respects their form, they are of three kinds, trilateral, quadrilateral, and multilateral. TRILATERAL FIGURES, or Triangles, are bounded by three straight lines, and have three angles. Considered in reference to their sides, they are of three kinds : a, Equilateral, or equal-sided, when the figure has three equal sides. See the planes of Models 15 and 117. b, Isosceles, or equal-legged, when the figure has two sides equal. See the planes of Models 12 and 13. c, Scalene, or unequal-legged, when the figure has three unequal sides. See the planes of Models 116 and 21. Triangles are also of three kinds, when described according to the nature of their angles: d, Right-angled, when the triangle has one right angle. A square divided into two halves by a diagonal line, produces two right-angled triangles. e, Obtuse-angled, when the triangle has one obtuse angle. See the small triangular planes on Model 40. These are obtuse isosceles triangles, as are also the planes of Model 119.^ Acute-angled, when the triangle has three acute angles. See the triangular planes on Model 73. g, The three interior angles of every triangle are equal to two right angles. h, If two angles of a triangle be given, the third is equal to the difference between their sum and two right angles, or 180. i, Each of the angles in an equilateral triangle is ^ of two right angles, or of one right angle, and therefore contains 60. -j, In every right-angled triangle, the sum of the two acute angles is equal to one right angle, and therefore con- tains 90. k, In every isosceles right-angled triangle, each of the acute angles is equal to half a right angle, and therefore contains 45. I, QUADRILATERAL FIGURES are those contained by four straight lines quadrangles or four-angled figures. All their angles together are equal to four right angles, or 360. Quadrilateral figures are of six kinds, according to the nature of their sides: m, a square, has all its sides equal, and all its angles right angles. See the planes of Model 1. , a rectangle or oblong, has all its angles right angles, and its opposite sides, but not all its sides, equal. See the planes M and T of Model 2, and the six vertical planes of Model 7. o, a rhombus or lozenge, has all its sides equal, but its angles are not right angles. See the planes P of Models 6, 84, 87. But the four angles of a rhombus are together equal to four right angles, or 360; that is to say, one of its acute and one of its obtuse angles are together equal to 180, the acute angle being exactly as much less than 90, as the obtuse angle is greater. p, a rhomboid, has its opposite sides equal to one another, but all its sides are not equal, and its angles are not right angles. See the planes P of Model 11. q, a trapezium, is a four-sided figure, whose opposite sides are not parallel. See the planes of Model 22. r, a trapezoid, has two sides parallel to each other, and two not so. See the eight similar planes on Models 76 and 115. 4 PRINCIPLES OF CRYSTALLOGRAPHY. s, MULTILATERAL FIGURES are those contained by more than four straight lines also called Polygons, or many-angled figures. A polygon is regular when all its sides are equal, and irregular when its sides are unequal. A 5-sided figure is a pentagon, a 6-sided, a hexagon, a 7-sided, a heptagon, an 8-sided, an octagon, a 10-sided, a decagon. The crys- tallographic polygons have not the same regularity as the geometrical polygons. Thus the pentagonal dodecahedron, Model 91, and the isosahedron, Model 92, both differ in the form of their faces from the geometrical solids which have the same names. t, All the angles of a multilateral figure are together equal to twice as many right angles as it has sides, minus four right angles or 360. That is to say, if you count the sides of a polygon, multiply the number by 180, and deduct 360 from the product, the result is equal to that obtained by adding together all the angles of the polygon. See 79. , Wherever two of the lines which bound a plane meet, they form a plane angle. Of any two such angles, that one is the greater which has the greater opening or divergence, whatever may be the comparative lengths of the lines by which the angles are formed. If a circle be described from the vertex of any angle, the arc intercepted between the legs of the angle is called the measure of the angle, and the number of degrees which the arc contains, is said to be the number of degrees in the angle. A circle is commonly divided into 360 degrees, marked 360. A right angle contains 90, for 2 straight lines dividing a circle into four parts, produce at the centre 4 right angles, and 3 ^ = 90. Each degree of the circle is subdivided into 60 minutes, marked 60', and each minute into 60 seconds, marked 60". An angle of 43 degrees, 15 minutes, and 25 seconds, is therefore marked 43 15' 25". An angle greater than a right angle is called an obtuse angle. See the plane angles of the plane P of Model 7. An angle less than a right angle is called an acute angle. See the plane angles of Model 117. 17. The EDGES of different crystals have different degrees of sharpness, which differences are estimated by measurement with the goniometer, and are expressed in degrees, minutes, and seconds. An edge which is formed by two planes that meet at a right angle, is denoted by 90. A sharper edge is denoted by any degree from 89 59' to 0. A blunter edge by any degree from 90 1' to 180. I do not describe the goniometer, because it is figured and described in almost every elementary work on Mineralogy. (See PHILLIPS'S Intro- duction) page xxxi.) It consists of a jointed pair of straight edges which can be applied to a crystal where two planes meet to form an edge, and afterwards to a semicircular scale, whereby the divergence of the planes is measured, and the value of the angle determined. Those who do not possess a goniometer, may measure the angles of the Models of crystals by means of a flat rule and a semicircular brass sector, such as commonly form part of a case of drawing instruments ; but small natural crystals cannot be measured in that way, nor is it indeed either a con- venient or accurate mode of measuring. 18. The SOLID ANGLES of crystals are named according to the number of planes which meet together to produce them, as, three-faced angles, six-faced angles, eight-faced angles, and so forth. They constitute the points or corners of crystals. Discrimination of different kinds of Planes. 19. ALL THE PLANES THAT BOUND A CRYSTAL, CUT ONE, TWO, OR THREE OF ITS AXES. The Denominations of the Planes are the initial letters of the names of the Axes which they cut. Thus PRINCIPLES OF CRYSTALLOGRAPHY. 5 P is the denomination of the planes that cut the axis p a . M is the denomination of the planes that cut the axis m a . T is the denomination of the planes that cut the axis t a . MT is the denomination of the planes that cut both m a and t a . PM is the denomination of the planes that cut both p a and m a . PT is the denomination of the planes that cut both p a and t a . PMT is the denomination of the planes that cut p a m a and t a . These planes occur in sets, which differ in their number as follows : Of P, M, T, there are 2 to each set. Of MT, PM, PT, there are 4 to each set. Of PMT, there are 8 to the set. Sometimes the planes occur in portions of sets, and such a portion is generally the half or the quarter of a complete set. All the varieties are subject to occur in half sets, but only the set PMT is commonly liable to be quartered. The symbol which denotes a half set, is the vulgar fraction J. and a quarter set, the vulgar fraction J, prefixed to the sym- bol or denomination which denotes the whole set of planes as, J PM, 1 PMT, &c. The dividing power of the fraction extends only over the symbol which stands betwixt itself and the next following comma, unless several symbols which are all to be divided by the same fraction, are placed within parentheses, as -J (PM, PMT), in which case the fraction divides the whole of the enclosed symbols. 20. The following diagram is intended to show the relative positions of these sets of planes in reference to the system of three axes. The principal axis is denoted therein by p% the transverse axis by t a , and the minor axis by m a . The crystallographer is assumed to be in front of the diagram, and his eye to be situated in the prolongation of the axis m a . The classification of planes into vertical, horizontal, and inclined, which follows hereafter, is made in reference to this assumption. There are several terms used in relation to Crystallographic Sections, which may be conveniently explained here. A Crystallographic Section is the outline of a plane produced by the imaginary cutting of a crystal through the centre into two equal and similar halves. Such sections can be made in a multiplicity of directions, but there are only five which are of practical importance in the present system of crystallography. Of these five sections, one is horizontal and four are vertical, as follow : The Equator, or horizontal section, which divides the zenith from the nadir portion of the crystal, see 9, is shown in the diagram by the lines 61293. It bisects the poles m a m* t| t. The North Meridian, or first vertical section, is shown by the lines M M 2 M 3 Mj. It bisects the poles p| p*< m a m^. The East Meridian, or second vertical section, is shown by the lines T T 2 T 3 T,. It bisects the poles p| p t a t a . The North-cast Meridian, or third vertical section, is shown by the lines 1658910. It bisects the poles p| p^, and cuts the equator at the ne (north-east) and sw (south-west). PRINCIPLES OF CRYSTALLOGRAPHY. The North-west Meridian, or fourth vertical section, is shown by the lines 23471211. It bisects the poles pz p$j, and cuts the equator at the nw and se. The positions of these sections are fixed and invariable, and have no dependence upon the form, or the number of planes, belonging to any single crystal. T 3 21. The application of the terms of position, quoted in 9, will be easily understood in reference to the planes, edges, and angles of the imaginary crystal which is represented in the above diagram. The point c being the centre of the crystal, and the plane 6 1293 being the equator, all the lines that proceed from point c, or from any other part of the equator, upwards, must strike planes, edges, or angles that are con- tained in the zenith portion of the crystal ; all the lines that proceed from the same points downwards, must strike planes, edges, and angles that belong to the nadir portion ; and all the lines that proceed from c horizontally, affect neither zenith nor nadir, but are directed towards the north, east, south, or west poles, or to some situation between those car- dinal points. Consequently, the line c p| points to the Zenith, c p^ to the Nadir, c m* to the north, c m* to the south, c t* to the east, c t to the west. And any line pointing from c diagonally as respects the axes, PRINCIPLES OF CRYSTALLOGRAPHY. 7 must point to Zn, Ze, Zs, Zw, or to Nn, Ne, Ns, Nw, or to some inter- mediate situation, as Znw or Nse. In examining these relations, the reader is requested to recollect the explanation of the polaric positions given in 9. Fixed Positions shown in the Diagram. Planes: Positions. Edges: Position 1 2 8 7 Z - 10 11 Ns 5 4 10 11 N 1 7 Ze 1 5 11 7 e 5 11 Ne 2 4 10 8 w 2 8 Zw 1 2 4 5 n 4 10 Nw 7 8 10 11 s Solid Angles : T 6 3 T 1 Zn 1 Zne M 3 9 M 1 Zw 2 Znw M 3 T 1 Znw 7, Zse M M 2 T 3 T 1 nw 8 Zsw M M 2 T 2 T ne 4 Nnw T 1 T 3 M 3 M 1 sw 5 Nne M 1 ' M 3 T 2 T se 10 Nsw Edges: 11 Nse 1 5 ne Lines : 2 4 nw All lines that are drawn upon 8 10 sw a plane have the same posi- 7 11 se tion as that plane; thus, 1 2 Zn T T 1 M M 1 1 82 7 4 5 Nn are all Z 7 8 Zs ILLUSTRATIONS. A. Place Model 1, the cube, in a position for ex- amination, and observe the situations of its planes, edges, and solid angles: ( 14), The uppermost plane is Z, the lowermost, N. The front plane is n, the back, s. The plane opposite to your left hand is e. That opposite to your right hand is w. The two front vertical edges are ne and nw. The two back vertical edges are se and sw. The front upper edge is Zn, the lower, Nn. The back upper edge is Zs, the lower, Ns. The upper side edges are Ze and Zw. The lower side edges are Ne and Nw. The two upper front solid angles are Znw and Zne. The two upper back solid angles are Zsw and Zse. The two lower front solid angles are Nnw and Nne. The two lower back solid angles are Nsw and Nse. B. Place Model 63 in position before you. What is to be noticed re- specting this form, the rhombic dodecahedron, is, that its 12 planes have PRINCIPLES OF CRYSTALLOGRAPHY. exactly the same polaric positions as the 12 edges of the cube, Model 1 ; that 6 of its solid angles, being those that are formed by the meeting of four planes in each, have the positions of the 6 faces of the cube ; and that 8 of its solid angles, being those that are formed by the meeting of three planes in each, have the positions of the 8 solid angles of the cube. C. Place Model 15 in position. Observe that its 8 planes have the polaric positions of the 8 solid angles of Model 1, and of the 8 three- faced angles of Model 63 ; that its 6 solid angles have the positions of the 6 planes of Model 1, and of the 6 four-faced angles of Model 63 ; and that its 12 edges have the positions of the 12 edges of Model 1, and of the 12 planes of Model 63. Model 15 is called the octahedron. D. Model 32. The 6 octangular planes of this model have the polaric situations of the 6 square planes of Model 1 . The 1 2 rectangular planes have the positions of the 12 rhombic planes of Model 63. The 8 hex- angular planes have the positions of the 8 triangular planes of Model 15. E. Model 33. The planes of this model are the same in number and in polaric position, as those of Model 32, but they differ in their relative sizes, and, as a consequence, in their forms. F. Model 34. The planes of this form are also the same in number and polaric position as those of Model 32, but they differ in relative size and in form from the planes of both the preceding Models. G. It follows from the foregoing observations, that each of the three Models 32, 33, and 34 contains all the planes of the three Models 1, 63, and 15 ; but that they all differ among themselves in the relative sizes of the planes ; so that in Model 32, the cube predominates, and the dodecahedron and octa- hedron are subordinate : In Model 33, the octahedron predominates, and the cube and the dodecahedron are subordinate : In Model 34, the dodecahedron predominates, and the cube and the octahedron are subordinate. H. But although the planes peculiar to the three simple crystals alter their size and shape when they form part of the three complex crystals, they never change their polaric positions. The angles across their edges are, therefore, always the same. Any one plane of the cube inclines upon any other plane of that form, at an angle of 90. Any one plane of the rhombic dodecahedron makes upon any other plane an angle of 120. Any one plane of the regular octahedron makes upon any other plane an angle of 109 28'. It matters not whether you measure these angles by applying the goniometer to Models 1, 63, or 15, or to Models 32, 33, 34; the results are the same". Polaric positions are fixed and invariable, and the planes of crystals are said to belong to one form or to another, according as they are found to occupy one or other polaric position. PRINCIPLES OF CRYSTALLOGRAPHY, Particular Description of each set of Planes. 22. Of the Planes P There are two of them, and their positions are shown by the lines 1287 and 5 4 10 11 in the diagram in 20. One of these planes forms the top and the other the bottom of the crystal. They are HORIZONTAL and parallel to one another. They cut the axis p% and are parallel to the axes m a and t a . The symbol P signifies two planes, which are all that belong to this set) or complement, of planes. The symbols which denote these two planes separately are PZ for the one that cuts the pole p|, and PN for that which cuts the pole p^. 23. Of the Planes M> There are two of them, and their positions are shown by the lines 1 23456 and 7 8 9 10 11 12 in the diagram 20. One forms the front and the other the back of the crystal. They are VERTICAL and parallel to one another. They cut the axis m a , and are parallel to the axes p a and t a . The symbol M signifies two planes. The front plane by itself is denoted by Mn, the back plane by Ms. The former cuts the pole m^ the latter the pole mt 24. Of the Planes T. There are two of them, and their positions are shown by the lines 2 3 4 10 9 8 and 1 6511 127. One forms the left and the other the right side of the crystal. They are VERTICAL and parallel to one another. They cut the axis t a , and are parallel to the axes p a and m a . The symbol T signifies two planes. Separately, the plane 2 3 4 10 9 8 is marked Tw, and the plane 165 11 12 7 is marked Te. The former cuts the pole t; the latter cuts the pole t a . 25. When the symbols P, M, T, denote the planes of an equiaxed crystal ( 1 0), the symbols are written without addition. But when one of the axes of the crystal which the symbols relate to, is longer or shorter than the other two axes, then the signs + or -, or a number, as the case may be, is subscribed below that symbol which is the representa- tive of the planes that cut the longer or shorter axis ( 11, 12). Thus: P,M,T, intimate that the Planes P, M, T cut the Axes p a m a t a at equal distances from the centre of the crystal. P_,M,T intimate that the Axis p a is cut by the Planes P nearer to the centre of the crystal than the Axes m a and t a are cut by the Planes M and T. P +J M_,T intimate that the Axes p a m a t a are cut by the Planes P, M, T unequally, p a being cut at the greatest, and m a at the least distance from the centre of the crystal. P 3 ,M!,T 2 intimate the same general fact as P + ,M_,T, but give a more precise account of the three different dimensions of the axes. M JO PRINCIPLES OF CRYSTALLOGRAPHY. The lengths of the Axes are found by measuring the lengths of those Planes of the crystal to which the axes are parallel. Thus, taking Model 1 for an example, the length of the axis p a is equal to the height of the planes M or T, the length of the axis m a is equal to the width of the planes P or T, and the length of the axis t a is equal to the width of the planes P or M. EXAMPLES OP THE FORMS P, M, T. Model 1. The Cube. P,M,T. The annexed figure, and Model 1, both re- present the cube. This form has six planes, namely, the set P, the set M, and the set T. The axes are p a m a t a . The planes are conse- quently square, for the prolongation of any one axis would cause four of the planes to be rectangles. There are necessarily twelve edges and eight corners. The equator is a square with the angles situated ne, se, sw, nw. The north meridian is a square with the angles Zn, Zs, Nn, Ns. The east meridian is a square with the angles Ze, Zw, Ne, Nw. The north-east meridian is a rectangle with the angles Zne, Zsw, Nne, Nsw. The north-west meridian is a rectangle with the angles Znw, Zse, Nnw, Nse. The symbol to express this form is P,M,T. Model 2. Short Quadratic Prism. PJ,M,T. This form, like the cube, has six planes, namely, the set P, the set M, and the set T. But its axes are p a m| t|, in which it differs from the cube, which is equiaxed. The planes P are squares, like the same planes of the cube, because, with the planes M and T, and with m a and t a equal, the planes P cannot be any thing else than squares. The planes M and T are rectangles, which form is the necessary consequence of the shortness of the axis p a in relation to the axes m a and t a . The equator is a square, and all the meridians are rectangles, the angles of which have the same positions as the angles of the equator and meridians of the cube. The symbol to express this form is P_,M,T ; or Pi,M,T. Model 3. Long Quadratic Prism. P 1M ,M,T. The difference between this form and the two preceding, results solely from the comparative length of the axis p a . The planes are the same, but the axes are pi m| t\. The equator is a square. The meridians are all rectangles. The angles are the same as those of Models 1 and 2. The symbol to express this form is P + ,M,T; or P 1J4: ,M,T. Model 5. Rectangular Prism. P + ,M_,T. The rectangular prism agrees with the cube in having 6 planes, in PRINCIPLES OF CRYSTALLOGRAPHY. 11 three pairs, each pair cutting the two ends of one axis at right angles. It differs from the cube in having the axes m a and t a unequal, be the length of the axis p a what it may. It differs from the quadratic prisms in the same particular. The planes of Model 5 are all rectangles. The equator, and the four meridians, are rectangles, the angles of which have the same positions as the angles of the equator and meridians of the cube and the quadratic prisms. The axes of the model measure pf 5 m^ t^. The symbol is P + ,M_,T; or P I5 ,M 12 ,T 13 . Model 80. P.P + M_T. Example of a combination which has the planes P without M and T. Model 70. M.P + M_T. Example of a combination which has the planes M without P and T. Model 111. T,MT + .PT + . Example of a combination which has the planes T without P and M. Model 61. M,T.PM + ,PT + . Example of a combination which has the planes M and T without P. A great many other combinations of this kind may be seen upon glancing over the set of models, but the above are sufficient to illustrate the preceding illustration of the symbols P, M, T, and to show that the planes which they represent never change their polaric positions when they appear upon a solid, either separately or together, in combination with various other planes. Consequently, if you want to know whether a crystal possesses the planes P, M, T, you have only to assume one of its planes to be = PZ, and then to hold that plane in the position of PZ and look for the other 5 planes of P, M, T, at the poles N, n, e, s, w. 26. Of the Planes MT. There are four of them, and their positions are shown by the lines T M M 2 T 2 , M T 1 T 3 M 2 , T 1 M 1 M 3 T 3 , M 1 T T 2 M 3 . Every single plane cuts the two axes m a and t a and is parallel to the axis p a . They constitute together the four sides of a VERTICAL PRISM, whose edges bisect the axes m a and t a , and whose axis coincides with the axis p a . The four planes are parallel two and two, and all of them are equivalent to one another. The symbol MT signifies the whole four planes, which in this case constitute the complement, whereas in all the foregoing cases, two planes constitute a complement. The positions and symbols of the individual planes of this set are as follow : That between the poles m* and t is MTnw. That between the poles m* and tf is MTne. That between the poles m s a and t is MTsw. That between the poles m a and t* is MTse. 12 PRINCIPLES OF CRYSTALLOGRAPHY. 27. The half of this set of planes is denoted by JMT, which signifies two vertical planes whose side edges bisect the two axes m a and t a . 28. When the symbol MT denotes the planes of a form whose axes m a and t a are equal, the symbol is used without addition. The alter- nating angles on m a and t a of the vertical prism which is formed by the intersection of the four planes MT, are in that case all right angles (90), and the form of the equator of the crystal is a square, with its angles due n, e, s, w. 29. But when the axes m a and t a are unequal, the symbol of the planes requires subscription with the sign + below the representative of the longer axis. Thus : MT_j_ denotes the vertical planes of a prism, the equator of which is a rhombus, having its longer diagonal parallel with the axis t a , and its shorter diagonal parallel with the axis m a . In this case, the angles formed by the intersection of the four vertical planes are not right angles, but alternately obtuse at m a and m a , and acute at tj and t; the one angle being as much wore than 90 as the other angle is less than 90, so that in every instance, the angle at m a added to the angle at t or the angle at m a added to the angle at t a , is equal to 180. In like manner, the symbol M + T denotes the vertical planes of a rhombic prism, having its obtuse angles on the axis t a and its acute angles on the axis m a . In this case, as in that before mentioned, the angles of two different prismatic edges taken together are equal to 180. And this is to be generally understood of the angles formed by the inter- secting planes of rhombic prisms, so that if one angle of such a prism be known, the discovery of the other angle is made by subtracting the sum of the known angle from 180. See 16, o. When the subscribed signs + and - do not denote the relative lengths of axes with sufficient precision, it is necessary to follow the method of numerical notation which has been described in paragraph 12. 30. It sometimes happens that we find upon the same crystal the planes MT_j_, MT_|_, where one + signifies a greater quantity than the other +. In this case, it is best to replace + by a number in both symbols. But if it is impossible to find the lengths of the axes, so as to be able to de- scribe them numerically, the next best method is to double the symbol IJI for the longer axis. Thus, MT+, MTl|l. By this means the general relations of the axes, as respects length, are indicated as well as they can be without numbers. A similar expedient may be adopted to indicate an extremely short in comparison with a short axis, in cases where the exact comparative lengths of the two axes are unknown. Thus, in M_T, M~T, the sign Z signifies very short) or shorter than the measure indicated by the sign -. See 73. EXAMPLES OF THE FORMS MT,MT + ,M + T. 31. Model 63. Rhombic Dodecahedron. MT.PM,PT. PRINCIPLES OF CRYSTALLOGRAPHY. 13 The four vertical planes of this form, that is to say, those that are directed to the polaric points ne, nw, se, sw, are the planes MT. See 21, Illustration B, or the annexed figure, in which one of the planes of this set is marked with M and T at the n and w poles. The equator of this form, as observed in 28, is a square. There are no meridians, because the prism MT runs to infinity upon the axis p a , and we are not now, in the explanation of this or any other variety of the form MT X , regarding the nature of the terminations of the prism. 32. Model 4. Eight-sided Prism. Quadratic prism with the lateral edges replaced. P + ,M,T,MT. This model has all the planes of the long quadratic prism, Model 3, with the addition of the four planes that constitute the set MT. The planes MT are distinguished on the model from the planes M and T by being narrower than those planes. They are all at right angles to the planes P. The axes of this crystal are pJjL m a t a . The equator is an octagon, having the sides M, T, exposed to n, e, s, w, and the sides MT exposed to ne, nw, se, sw. The symbol which expresses this form is P + ,M,T,MT. 33. Model 32. The Cube combined with the rhombic dodecahedron and the regular octahe- dron, as represented in the marginal figure. Model 33. The octahedron combined with the cube and the rhombic dodecahedron. Model 34. The rhombic dodecahedron combined with the cube and the octahedron. See Illustrations 21. The planes MT are marked upon Model 32, and they can be easily discriminated from all different planes upon the other two models, if their polaric positions be attended to. 34. In each of these three forms, the equator is an octagon, as is the equator of Model 4. Whenever, indeed, a prism contains the planes MT in addition to the planes M and T, and has no other vertical planes, the equator is always an octagon, whatever may happen to be the form of its meridians. In other words, the two four-sided prisms which pro- duce the planes M, T, and MT, occurring upon the same crystal, cut off. each other's edges and produce a single vertical prism of eight sides. 35. Model 6. Rhombic Prism. P_,MT + ; or P T %,MT|. The set of planes MT_j_ constitutes the four vertical planes of this model. To ascertain the numerical value of the sign +, the axis m a can be measured from n to s, and the axis t a from e to w. By this means m a 14 PRINCIPLES OF CRYSTALLOGRAPHY. is found to bear to t a the relation of 10 to 13, so that the exact symbol for this form is Mi T 13 , or MT^J. The equator of this form is a rhombus, with its obtuse angles at the poles n and s, and its acute angles at the poles e and w. 36. Model 7. Regular Six-sided Prism. P X ,T_,MT + ; or P 04 ,T 0866 , MT, 732 . The planes of this model are the set MT + , in combination with the sets P and T. The axes of the model are p m.|_ t% but the horizontal axes of the set of planes MT+ are not the same as the horizontal axes of the model. The length of the axis m a is the same, both of the form MT_|_ and of Model 7, and it is ascertained by measurement from n to s. The length of the axis t a of the model is also ascertained by measurement from e to w. But the axis t a of the form MT_j_ is the longer diagonal of the rhombus that would have been produced had the crystal or combination been, like Model 6, destitute of the planes T. The occurrence of the latter planes has served to cut off the acute edges of the rhombus and to convert it into a regular hexagon. It is therefore impossible to ascertain the length of the axis t.|- of the symbol MT + by lineal measurement of the distance between the two extreme points, and an evil is produced for which we have no remedy but trigonometrical calculation. We can, indeed, by measuring the angle at m*, easily determine the value of the angle removed from t, (See 16 0,) but having done that, we have still to determine the length of the axis or line from t to t", and this is what we can only do properly by means of trigonometry. PRINCIPLES OF CRYSTALLOGRAPHY. 15 Trigonometrical Determination of the Comparative Lengths of the Axes of Rhombuses. 37- I shall begin by explaining several general terms that are em- ployed in trigonometry, and then show how the information furnished in that explanation is to be applied to the solution of problems in crystal- lography. 38. A straight line drawn from the centre to the circumference of a circle is called its radius; as oQ, oc, oq, oC in the diagram in page 14. These lines are all equal to one another. 39. A straight line drawn through the centre of a circle and terminated both ways by the circumference, its called is diameter ; as the diameters c(7and Qq. 40. A quadrant is one of the four parts into which a circle is divided by two diameters intersecting each other at right angles. The circle being divided into 360 degrees, the quadrant contains 90. The circle in th diagram is divided by the diameters Qq and Cc into the four quadrants Qoc, coq, qoC, CoQ. 41. An arc is part of a circle separated from the rest of it by the two legs of an angle, or otherwise. The following arcs are shown in the diagram : Arcs of 30. Arcs of 60. Arcs of 90. Arcs of 120. Q* ec Qc ew wq cw cq wC we qW qC Ce CE EQ CQ Qqis 180. 42. The difference between an arc and a quadrant, is called the com- plement of that arc. The difference between an angle and a right angle is called the complement of that angle. Thus, the arc Qe is the comple- ment of the arc ec, and conversely ec is the complement of the arc eQ. Again, the angle woq is the complement of the angle woe, and con- versely, woe is the complement of woq. 43. Two arcs whose sum is a semicircle, or two angles which are to- gether equal to two right angles, are called supplements of each other. Thus, the arc Qe is the supplement of the arc eq, and the angle Qow is the supplement of the angle woq. 44. The straight line that joins the extremities of an arc is called its chord. Thus, the line ew is the chord of the arc ecw. The line Qoq is the chord of the arc (the semicircle) Qcq ; and the line EW is the chord of the arc ECW. 45. A straight line drawn from one extremity of an arc, perpendicular to the diameter that passes through the other extremity, is called the sine of that arc, or the sine of the angle which the arc measures ; and the part of the diameter that is intercepted between the sine and the arc is called the versed sine of the arc or the angle. Hence, the sine of an arc is half the chord of its double. Thus, the line wa is the sine of the arc ivq, or 16 PRINCIPLES OF CRYSTALLOGRAPHY. of the angle 0oq, and the line q, intercepted between the sine wa and the arc wq, is the versed^ sine of the same angle. Again, the line WC is the sine of the arc WC, or of the angle Wo (7, and the line CC, inter- cepted between the sine WC and the arc WC, is its versed sine. Lastly, the line EW is the chord of the arc ECW, which is the double of the arc CW, and the portion CW is the half of that chord. 46. The cosine of an arc is the sine of its complement. Thus, wa is the sine of the arc wq, and we is its cosine, or the sine of its complement we. Conversely, we is the sine of the angle cow, and wa is its cosine, being the sine of its complement woq. And in the under quadrant, WC is the sine of the arc WC, and Wa is its cosine ; or Wa is the sine of the arc Wq, and WC is its cosine. 47. When one of the legs of an angle is prolonged beyond one of the extremities of an arc, and is cut there by a straight line that touches the other extremity of the arc, and is at the same time perpendicular to the other leg of the angle, the straight line so intercepted between the two legs of the angle, and beyond the arc, [is called the tangent of that angle or that arc. The shorter of the two legs of the angle is called the radius of the arc, and the longer leg is called the secant. The form pro- duced by the three straight lines in contact is a right-angled triangle, and the right angle of the triangle is situated where the tangent touches the radius and the extremity of the arc. Thus, ocw is a right-angled triangle ; and c is its right angle ; we is the arc ; oww the secant ; and cw the tangent. The triangle coe has the same properties as cow, and the tangent is ce. Again, oCW is another right-angled triangle ; C is the right angle ; CW is the arc ; oC the radius, oWthe secant, and CW the tangent. The triangle vCE is similar to oCW, and the tangent is CE. The tangent of an arc of 45 (the half of a right angle) is equal to the radius of the arc = 1. The tangent of an arc of more than 45 is greater than the radius. The tangent of an arc of less than 45 is less than the radius. 48. The coversed sine, cotangent, and cosecant of an arc, are respec- tively the versed s\ne, tangent, and secant of its complement as, the cotangent of an arc of 60 is the tangent of an arc of 30, and so on. 49. In a book entitled " MATHEMATICAL TABLES, by Charles Hut- ton/' and in many other works relating to Logarithms, there is to be found a table entitled a " Table of Natural Sines, Versed Sines, Tan- gents, Secants, Cosines, Coversed Sines, Cotangents, and Cosecants," in which the comparative lengths of the lines indicated by these terms is expressed in numbers, for every degree and every minute of a quadrant ; and the table is so arranged, that if the value of the angle measured by any arc be given, the length of its sine, cosine, tangent, secant, &c., can be seen on reference to the page of the table where that degree is printed ; or, if any sine and its cosine, or any tangent and its radius, be given, the value of the corresponding arc and angle can also be deter- mined therefrom. In a subsequent section of this work, I shall give a short table of the same kind, sufficient to show its nature and use. PRINCIPLES OF CRYSTALLOGRAPHY. 17 PROBLEMS. 50. The rhombus e s w o, depicted in the diagram, is the equator of a crystal, and the angle e o w is found, by measurement with the goniometer, to be 120. Required, the length of the axes os and ew. The angle of 120 is equal to the angle of an edge of Model 7 at m*. 51. We see at once that the rhombus is divided by its two axes into 4 equal and similar right-angled triangles, and we take it for granted that any thing that is true of one of these triangles is true of all the others. We begin, therefore, by taking the triangle cow, which measures at o the half of 120, or 60. In this triangle, co is the radius, which is to be always taken as unity =r 1 ; cw is the tangent ; and ow is the secant. We turn to the Table of Natural Sines, &c, and at 60 we find the following numbers; (Radius being =1) : Angle Radius Tangent Secant 60 1 1.7320508 2.0000000. Consequently, the line co is to the line cw, as 1 is to 1.7320508, and the axis so bears to the axis ew the same relation. If therefore the axis so be equal to m a and the axis ew to t, and if the lines es, sw, wo, oe be the four planes of the set MT + , the exact numerical substitute for the sign + is 1.7320508, and the symbol for this variety of the rhombic prism is MT,. 732 . We see at the same time that the breadth of the planes MT,. 73 2 from o to w is equal to the length of the axis m a from o 18 PRINCIPLES OF CRYSTALLOGRAPHY. to s, because sow is an equilateral triangle, which has three sides of equal length. 52. Let us again examine the triangle cow. If the angle at o is = 60 and cw is the arc that measures that angle, cw is its sine, and its cosine, or the sine of its complement, is wa, which is evidently equal to co. If now, we turn to the Table of Sines, &c., and seek there for the sine and cosine of an angle of 60, instead of seeking for its tangent and secant, we find them to be as follows : Sine. Cosine. 8660254 5000000 which X 2 = 1.7320508 1.0000000 Now co bears the same relation to cw that co does to cw, or so to ew: this result is therefore the same as the former, and gives for the rhombus ecwo, the axes m* t a . 732 . 53. The rhombus oEn W is the equator of a crystal, and the angle Eo W is found by measurement with the goniometer to be 60. Required, the length of the axes on and EW. ' 54. We proceed as in 5 1 , to ascertain the comparative lengths of the three sides of the right-angled triangle oCW, in which the angle CoWis 6 j =r 30. The line oCis the radius = 1, CWthe tangent, and oWthe secant. Referring to the Table, we find the following num- bers : Angle Tangent Secant 30 .5773503 1.1547005. Consequently, the line CW'is to the line Co, and the axis EW is to the axis on as .5773503 is to 1.0000000. Now .5773503: 1.0 1 1 1.0: 1.7320508 which result is the converse of that obtained in 51. Hence, if o?i be = m_|. and EW be equal to t a , the symbol for the set of planes that have the equator oEnW is Mi 732 T. We see also in this case, as in the former, that the breadth of the planes M+T is equal to the shorter axis, which here is t a ; for, as 2.0 is the double of 1.0, so is 1.1547005 the double of 0.5773503. 55. Let us again examine the triangle oCW, or rather let us begin with the given angle of 30, and refer it to the arc CW and the angle CoW. In this case, CW is the sine of that angle, and Wa is its cosine, which is evidently equal to the line Co. Referring to the Table of Sines, we find Angle Sine Cosine 30 5000000 8660254 which X 2= 1.0000000 1.7320508 This is the same result that we arrived at in 52, save that the numbers apply to different axes. The reason of this is rendered obvious by a PRINCIPLES OF CRYSTALLOGRAPHY. 19 slight examination of the two equators drawn upon the diagram. In the upper equator, the axis t a is long, and the axis m a is short, whereas in the lower equator, the axis t a is short, and the axis m a is long. 56. The inference to be drawn from the foregoing examinations is, that when the equator of a crystal is a rhombus, which has its shorter diagonal on the axis m a and its longer diagonal on the axis t a , the length of the axis t a is to that of the axis m a as the tangent of the arc that measures half the obtuse angle of the rhombus is to unity. Longer Axis Shorter Axis Tangent of 60 Radius = 1 t a : m a :: 1.7320508 : 1.0 57. There is a rhombic equator, whose axes m a and t a have the relation of 10 to 13. Required, the measure of the obtuse angle at m a , that is to say, the measure of the angle of incidence of the plane MT + ne upon the plane MT_^_nw. 58. 10 to 13 is the same ratio as 1 to 1.3. 10 or 1 is the radius of the arc that measures the required angle, and 13 or 1.3 is its tangent. We seek in the Table of Sines, &c., and in the column of tangents, for 1.3, which number is easily found, as all the numbers run in regular order. The nearest number to 1.3 that can be found is 1.3000904 which is the tangent of an arc of 52 26'. This is of course the half only of the obtuse angle at m a which consequently measures twice 52 26' or 104 52'. It will be found by measurement with the goniometer that this is very nearly the value of the obtuse angle at m a of the rhombic prism repre- sented by Model 6, and described in 35. 59. The acute angles of a rhombic equator are cut off by two straight lines which separate portions that are isosceles triangles. Reguired, the value of the angles formed by the straight lines with the residual portions of the sides of the rhombus. The angles of the rhombus are given at 120 and 60. This case is represented by the rhombus eswo in the last diagram : ez and xw?, are the two straight lines ; eze and xww are the two triangles cut off; ezs, sxw, xwo, oez, are the four new angles whose value is required. The problem is solved in the same manner as the geometrical proposi- tion, that " the angles made by one line meeting another, between its ex- tremities, are together equal to two right angles." Let the line wx be supposed to meet the line sw, then the angles wxw and wxs are together equal to two right angles = 180. Now the angle esw is given at 120, consequently the angle cxw is 120, and the angle w?xw, which is the half of cxw, is 60. Further, the angle wwx is given at 60, the angle cwx, which is the half of wwx, is 30, wherefore the angle wxw must be 60. Hence, the angle wxs is 180 60 = 120. But if the angle wxs is 120, so also are the angles xwo, oez, and ezs. Therefore all the new angles are 120, and the resulting figure is a regular hexagon, similar to the equator of Model 7. 20 PRINCIPLES OF CRYSTALLOGRAPHY. 60. The same problem as 59 w given, but is now to be considered in reference to the separation of the obtuse angles of the rhombus. Diagram and measurements of the rhombus as before ; but the straight lines now in question, are those marked zx and ew ; the triangles cut off are zsx and eow ; and the new angles produced are ezx, zxw, wwe, and ivee. Principle of solution, as already explained. Let the line zx be supposed to meet the line sw, then the angles zxs and zxw are together equal to two right angles, or 180. Now the angle xww is given at 60 and the angle xwc is consequently = 30, and as xwc is equal to sxz, the angle wxz is equal to two right angles, minus the angle sxz ; or to 180 30 = 150. But the angle zxw is equal to the angles vrwe, wee, ezx, and consequently each of the four new angles is = 150. 61. It will be observed that the value of the new angles produced by the replacement of the angles of a rhombus by straight lines is not altered by the approximation of the straight lines either to the middle point of the rhombus or the vertex of the replaced angles. It will also be observed, that each new angle that is produced when a straight line cuts off the angle of a rhombus, is equal to a right angle added to half the value of the angle that is cut off"; and the reason is very obvious. The angle sxw, for example, has been found to be = 120, and it con- tains the right angle zxw, which is 90, and the angle sxz, which is 30, or the half of the acute angle xww, which is 60. Again, the angle zxw has been found to be =150, and it contains the right angle zxw, which is 90, and the angle wxw, which is~60, or equal to the angle csx, which is the half of the obtuse angle of 120, zsx. 62. From this digression into trigonometry, I return to the considera- tion of the forms that are produced by the complement of planes MT X . 63. Model 8. Six-sided Prism. P X ,T_,MT + , or P .65,T 071 ,MTi. 732 . This model contains the planes MT + in combination with P and T. The angle at in* is 120, consequently the axes of the planes are mf, t a .732 The planes T are parallel to the axis m a , and consequently the angles formed by T upon MTi. 732 ought to be 120, which the goniometer proves them to be. The axis t a of the model is found by measurement across the plane P, to be 0.7 of the length of the axis m a , and the axis p a is found to be 0.65 of the axis m a . Of course, the angles formed by T and MT + upon P, are all angles of 90. The difference between the forms represented in Models 7 and 8 de- pends entirely upon the dissimilar lengths of the axis t a . 64. Model 9. Six-sided Prism ; a made, double, or twin crystal. P X ,T + ,MT + , or P M ,T!,, MT 2 . 12 . Each prismatic half of this model contains the planes MT+, in- com- bination with the planes P and T. The angle at m* is 129 30'. The half of this angle is 64 45', the tangent of which is 2.12. By lineal measurement the axes of the model are found to be p a . 2 m a t a . 2 . PRINCIPLES OF CRYSTALLOGRAPHY. 21 The angles formed by the planes MT 2 .i 2 upon T, are found by adding 90 to half the acute angle of MT 2 . 12 . Now, as the obtuse angle is 129 30/ the acute angle must be 180 129 30'. = 50 30'. The half of this is 25 15' which added to 90 gives 115 15', and this is in fact the angles that are formed by T upon MT + , as may be determined by the goniometer. The method of denoting the made I shall describe hereafter. 65. The angle formed by each of the planes MT, upon M or T, Model 4, can be readily calculated upon the principle explained in 5961. The angle formed by M upon T is 90, and that formed by MT upon MT is the same. Therefore, in the combination, the angle of 90 between M and T is cut off by a straight line, and $g + 90 = 135, which will be found, by measurement with the goniometer, to be the value of the angle of every vertical edge of Model 4. 66. Model 10. Twelve-sided Prism. P_, M, T, MT + , M + T. Besides the two terminal planes of this model, which are the set P, there are upon it the sets M,T, MT+, and M + T. Of these twelve ver- tical planes, six are large and six small. One of the latter, the north plane, is marked M, and one of the large planes, the west plane, is marked T. Parallel to these may be found the corresponding planes Ms and Te. Upon putting the crystal into position for examination, it will be seen that the large planes are similar in number and position to the six vertical planes of Model 7, and that the small planes are equal to planes which may be supposed to have cut off the six vertical edges of Model 7. These edges we have found to measure 120, and according to 61, if they were cut off by a plane equally, the resulting edges should be 1 f + 90 = 150. Now upon measuring the angle formed by a large upon a small side plane of Model 10, it is found to be 150; so that this form would seem to consist of two hexagonal or six-sided prisms cutting one another, which is really the case, for the model con- tains, first, the set of planes MT L732 , which gives an angle of 120 at m*. This angle may be found by measuring the incidence upon one another of the two large planes on each side of the north plane, applying the goniometer horizontally across the plane M. The situations of the four planes of the set MT,. 732 are nV, n 2 e, s 2 w, s 2 e ; and their acute edges are cut by the two large planes T . 866 , situated e and w. Next, there is a rhombic prism of precisely the same relative dimensions but having m a long, which produces the four smaller side planes that are situated nw 2 , ne 2 , sw 2 , se 2 . The symbol for these four planes is M^T. The acute edges of this set of planes are cut by the two small planes M 0>866 , situated n and s. The proof that all this is correct, lies in the fact that all the vertical edges of Model 10 have angles of 150, which could not be the case if M + T was different in the relative lengths of its two axes from MT+. 67. But, it may be asked, how happens it that the planes of the form 22 PRINCIPLES OF CRYSTALLOGRAPHY. M, M + T are small, and those of the form T, MT + , large ? The answer is, that the entire form T,MT + , although it is most visible upon the combination, is in faet of smaller dimensions than the form M,M + T. Its planes therefore approach nearer to the centre of the com- pound, and leave less room for the planes of the other form. It often happens, indeed, in the combination of SIMILAR forms, that they are not EQUAL, and whenever this is the case, the largest form that enters into the combination, is that of which least is to be seen upon the solid. The planes that approach nearest to the central point of a com- pound form necessarily cut off the planes, or part of planes, that project to a distance from the centre. Thus, it will be seen that the planes MT on Model 4 are narrower than the planes M and T. Now the equators of MT and M, T, are both square, but though similar, they are not equal, for if the diameters of MT and M, T, be measured across the plane P, that of MT will be found to be longer than that of M, T. The planes MT are consequently farther from the centre of the combination than the planes M, T, and on that account they are smaller. 68. Models 32, 33, 34, which have been several times referred to, all contain the same planes, and in the same positions, and they are denoted by the same symbols. Yet the three forms are quite different from one another, and the difference arises entirely from this circumstance, that the planes upon each form though similar to those upon the others, are not equal to them. Model 32 consists of a small cube in combination with a large dodecahedron and a large octahedron ; Model 33 of a small octahedron in combination with a large dodecahedron and a large cube ; Model 34 of a small dodecahedron in combination with a large cube and a large octahedron. In all these cases, the form which is most visible is the smallest of all the forms that belong to the combination. 69. It is evident from these considerations, that it is of great import- ance to us to have a method of distinctly and conveniently describing the variations in the relative size of planes which arise from the causes that I have just investigated. The method which it has occurred to me is best adapted for this purpose, is to write the symbols that express the large planes, by which I mean those that are most visible, in capital letters, as P, M, T, MT. PM, PT, PMT ; and the symbols that express the small planes, by which I mean those that are least visible, in small letters, as p, m, t, mt. pm, pt, pmt. Upon this plan Model 4, is P+,M,T, mt. Model 10, is P x , m.^, T >866 , MTj. 732 , m,. 732 t. Model 32, is P,M,T, mt. pm, pt, pmt. Model 33, is p, m, t, mt. pm, pt, PMT. Model 34, is p, m, t, MT. PM,PT, pmt. It sometimes happens that the planes upon a crystal are of three kinds as respects their comparative size : namely, 1> very large ; 2, very small ; 3, intermediate. I may instance Model 35, in which the square planes are very large, the triangular planes very small, and all the others inter- PRINCIPLES OF CRYSTALLOGRAPHY. 23 mediate. The first kind may be expressed in print by capitals, P, M, T ; the second kind by small letters, p, m, t ; the third kind by small capi- tals, P, M, T. In manuscript, these varieties are distinguished by writing the first in capitals, the second in small letters, and the third in small letters with two lines scored below them. EXAMPLE. Model 35. P,M,T, MT. PM, PT, PMT, ipmt. 70. Polaric Positions round the Equator. In the description of Model 10, I have used the signs n 2 w, nw 2 , and some others, not yet ex- plained, to indicate certain polaric positions. It will be proper to explain these terms fully, in order to prevent ambiguity. Suppose the equator to be a circle, and suppose radii proceeding from the centre of the circle, (see o in the diagram in 37, and the equivalent point c in the diagram in 20,) in all directions towards the circumfer- ence. The extremities of these lines indicate polaric positions, and the planes which cross them occupy the particular positions that the lines point out. Let the circle CQcq, in 37, be this equator, and let oC be a radius pointing northward. Then C on the line ECW will be the polaric position n, Q the position e, c on the line ecw the position s, and q the position w. A radius striking the arc (7q, midway between C and q would point out the position nw ; exactly opposite would be se, and at right angles to these would be ne and sw. A radius striking the arc Cq midway between the positions n and nw, would point out the position that I have marked n 2 w, which means nearer to n than to w. The cor- responding position, midway between the positions nw and w is marked nw 2 , which means nearer to w than to n. EXAMPLES. 71. A crystal that has the planes M,T, MT, MT+, M+T, exhibits one plane of each set in the north-west quarter of the crystal, arranged in the following order: n n 2 w nw nw 2 w M, MT+, MT, M + T, T 72. A crystal that has the planes M, T, MT, MT+, MT+-, M+T, M+T, exhibits them in the following order: n n 3 w n 2 w nw nw 2 nw 3 w M, MT+-, MT+, MT, M+T, M+T, T Any other quadrant of the equator of the combination would exhibit the planes in a corresponding arrangement, as: e ne 3 ne 2 ne n 2 e n 3 e n T, M+T, M+T, MT, MT+, MT+, M 24 PRINCIPLES OF CRYSTALLOGRAPHY. 73. The twelve planes which form a zone round the equator of model 10, have the following positions: e ne 2 n 2 e n n 2 w nw 2 w Northern : T, e m + t, se 2 MT+, s 2 e m s MT+ S 2 W m + t, sw 2 T w Southern : T, m + t, MT+ m MT+ m + t, T In Phillips' s Mineralogy, page 174, there is a figure of a crystal of Fluorspar, which has 32 vertical planes, the arrangement of which is as follows, taking the nw quadrant as an example : n n 4 w n 3 w n 2 w nw nw 2 nw 3 nw 4 w M, MT^_, MTijl, MT+, MT, M+T, MjT, M+^T, T The signs _j_, IJI, ^ ? in all these symbols indicate the great, greater, and greatest distance of the poles of the axes referred to, while the figures 2, 3, 4, indicate the great, greater, and greatest proximity of the poles to whose symbol they are added. The figures which indicate the lengths of axes are consequently in an inverse ratio to those that indicate the proximity of the planes to particular poles. The greater the figure that multiplies the length of any axis, the further are the residual por- tions of the planes that appear upon a combination, removed from the poles of that axis. 74. It will be recollected that the axis t a of the form MT + is longer than the axis t a of the form MT, and that the axis t a of the form MTj is still longer than the axis t a of the form MT + : the axes m a being taken at unity in each case. On the other hand, it is shown in the above series of positions, that when these sets of planes have cut one another in a combination, the planes of MTf appear nearer to the planes M than do the planes of MT + , and that the planes of MT + appear nearer than the planes of MT. Hence the polaric positions of individual planes indi- cate the comparative lengths of the axes of the rhombic prisms to which they belong, so that when we see upon a crystal a plane situated betwixt two others, which we know to be M and MT, we may be sure that the symbol of the plane in question is MT + , since all the planes of the form M + T, must of necessity appear upon a crystal, if they appear at all, between MT and T, and cannot by any possibility occur between M and MT. The reason of this has been explained in 67, where it is shown that when the planes of rhombic prisms are cut by other forms, it is generally the portions that are attached to the poles of the shorter axes of the rhombic prisms that remain upon the combination. The portions attached to the longer axes are cut off and lost. 75. Model 11. Right Ehomboidal Prism, P,M_, JM+T. This model contains the planes P and M, in combination with the planes ^MT, that is to say the half of the set MT. See 27. Upon applying the goniometer to MT and M across the nw edge, we find the angle to be 120. Again applying it to the sw edge, we find it to be 60. PRINCIPLES OP CRYSTALLOGRAPHY. 25 The two opposite angles in ne and se agree with these. The equator is a rhomboid. Deducting 90 from 120, we have 30 as the value of the angle formed by a line passing from the south to the north pole, and another passing from the north to the west pole. This angle is exactly that formed by the rhombus of 120 and 60 as described in 53 55: for 30 X 2 = 60, = angle of the acute edge at m*, and 180 60 = 120, = angle of the obtuse edge at t. Hence the precise symbol for the planes of model 1 1 is P,M, -JMi 732 T nw, se. The polaric positions are added to show which two planes of the set M_|_T are present. The relation which this form bears to the large rhombic prism M 1732 T of which it may be supposed to be a segment, is shown in the diagram in p. 14, by the lines EZsrW, which lines represent the form of the equator of model 11. Another method of explaining the form would be that of referring it to the rhomboid exwo, which forms the half of the rhombus eswo, depicted in the diagram on page 14. Considered in this light, and having regard to its dimensions, Model 1 1 would be a made consisting of 2J or of 10 rhombic prisms of 120 and 60, combined so as to form a single crystal; and the symbol for it would be P X ,MT I732 X 10. 76. Designation of the Planes and Axes of Crystals by separate Symbols : In many cases it is difficult to add to the symbols that denote the planes of crystals, the signs that denote the lengths of their axes, without endangering the perspicuity of the symbols that designate the planes. It is better therefore when crystals require a complex symbol, to denote the length of their axes by a separate symbol. Thus in the case of model 11, the best method of denoting the length of its axes appears to me to be this: P,M,lM li732 T nw, se.p/ 2 mf 3 1 These measurements are taken along the edges or across the planes of the crystal, and serve to give a general idea of its principal dimensions, independently of the information communicated by the other symbols in respect to the number and disposition of the planes of the crystal. See Section 1., 114. 77. Model 91- Pentagonal Dodecahedron, MT 2 .PM 2 ,P 2 T. When model 91 is put into position, it will be found to possess four vertical planes which form part of a rhombic prism that has obtuse angles of 126 52' at the poles m* and m^. Now the half of 126 52' is 63 26', and the tangent of this arc is 1.999859- Hence the formula for this set of 4 planes is MT^^a,, or briefly MT 2 . The planes PM 2 , P 2 T, which are also possessed by the form exhibited in model 91, will be explained in subsequent sections. I am treating here only of the vertical planes of crystals. 26 PRINCIPLES OP CRYSTALLOGRAPHY. 78. Model 68. Tetrakishexahedron, MT 2 ,M 2 T.PM 2 ,P 2 M,PT 2 ,P 2 T. Place this model in position, and observe that it has eight vertical planes, divided from one another by 4 vertical edges at ne, nw, se and sw, and meeting in 4 angles at n, e, s, and w. The two northern and two southern planes, make up the four sides of a rhombic prism = MT_j_. The two eastern and two western planes make up the sides of a rhombic prism = M+T. These two prisms intersect one another, and therefore mutually cut off their acute angles. Upon measuring the obtuse angle at m, it is found to be, or it ought to be, 126 52', but some of the models are incorrect, and measure only about 120. The half of 126 52' is 63 26', the tangent of which arc is 1.999? as I have shown above. This gives the formula MT 2 for the southern and northern planes. The obtuse angle at t also measures 126 52', which gives the formula M 2 T for the eastern and western planes. The planes that are not vertical will be de- scribed hereafter The horizontal axes of this model are m a t 3 . 79- Geometrical relations of the angles of an Equator, or other section of a crystal. The equator of Model 68 is an octagon or figure of 8 sides, and it may serve to afford us an illustration of the use to be derived from the geometrical proposition contained in 16, t The mineral kingdom presents us with several varieties of the tetrakishexa- hedron, which differ among themselves in the relative lengths of the axes of the rhombic prisms whence their planes are derived, as MT|, MT 2 , MTJ, &c. In many works on mineralogy, these different forms are described by merely giving the angles which their planes form upon one another; such as, for example, the angle formed by the plane MT 2 n 2 w upon the adjoining plane M 2 T nw 2 , measured by the goniometer across the nw vertical edge. I will suppose this angle to be given at 143 8' and that it is required to learn from that datum what is the value of the sign + in the symbol MT + , M + T. 80. The equator has eight angles and 8 sides. According to the rule given at 16, t, I multiply the 8 sides by 180, which gives 1440, and then I deduct 360, which leaves 1080. This should be equal to the sum of all the eight angles added together. But one of the angles is given at 143 8', and there are four similar angles at nw, ne, sw, and se. Now 143 8' X 4 is 572 32', which, deducted from 1080, gives 507 28' for the value of the four intermediate angles at n, e, s, and w. Dividing 507 28' by 4, I have 126 52' as the value of a single obtuse angle, and this, divided by 2 is 63 26', which, as I have shown above, is the angle whose tangent gives 2 for the equivalent of the sign + in the formula MT + ,M + T, producing the symbol MT 2 , M 2 T. 81. Several other useful problems in crystallography can be solved in a similar manner. Thus, suppose we had the form MT + ,M + T, and knew the measurement of the obtuse angle at m* to be 126 52', and that we were required to deduce from that measurement alone the value of the 8 angles of the equator of the form. Of course 4 of the angles, namely, those at n, e, s, w, are 126 52', but the other 4 are unknown. PRINCIPLES OF CRYSTALLOGRAPHY. 27 To find them, I multiply 126 52' by 4, which gives 507 28'. This sum, deducted from 1080, leaves 572 32', which is the aggregate value of the 4 other angles. I divide again by 4, which gives 143 8' for the value of the single intermediate angle : consequently the equator has alternate angles of 126 52' and 143 8'. 82. We learn from the foregoing calculations, that 1080 is the constant aggregate value of the eight angles of an equator that has eight sides, being 8 X 180 360 = 1080. From a similar calculation, we learn that 720 is the constant aggregate value of the six angles of an equator that has six sides, 6 X 180 360 = 720; and that 360 is the con- stant aggregate value of the four angles of an equator that has four sides, 4 x 180 360 = 360. Again, if we divide 1080, the constant aggregate value of the 8 angles of an octagon, by 4, we have a product of 270. This is the aggregate value of 2 of the angles. Now it very often happens, that 4< of the angles of an octagon are equal to one another, and the 4 others are also equal to one another, but different from the first four ; an example of this is given in 80. When this is the case, one angle of each kind added together is = 270, so that, if we know the value of either of these angles, that of the other is ascertained by deducting the known angle from 270. Finally, the six angles of a hexagon, which are together always = 720, very frequently consist of 4 angles of one kind and two of another kind. Then of course 2 of the first and 1 of the second kind of angles are together equal to 360. If you know the sum of the odd angle, that of the two others will be equal to 360 minus the odd angle. If you know the value of one of the pair of angles, you have but to double its sum and deduct the product from 360 to find the value of the odd angle. The knowledge of these relations frequently enables one to understand descriptions of crystals so imper- fectly given as to be otherwise unintelligible. 83. Problem. The vertical planes of Model 47> which represents a combination of the cube with the pentagonal dodecahedron, the cube being subordinate, are m, t, MT 2 . Required, the value of the angles formed by m upon MT 2 , and by MT 2 upon t. The obtuse angle at m* of the form MT 2 , is found by seeking in the Table of Tangents for the arc of the tangent 2.000000. That arc is 63 26', and the double of it is equal to the angle at m*, which is 126 52'. Then ~^-' + 90 = 153 26'. This is the value of the angle formed by the plane m upon the plane MT 2 . When the obtuse angle of a rhombus at m* is 126 52', the acute angle at t* is 180 126 52' = 53 8'. And ~ + 90= 116 34'. This is the value of the angle formed by the plane t upon the plane MT 2 . Upon measuring the angles of Model 47 with the goniometer, you will find that these results are correct. 84. Control over the correctness of this reckoning* The planes m, t, MT 2 give eight sides, and therefore eight angles, to the equator; and 28 PRINCIPLES OF CRYSTALLOGRAPHY. these angles must be in sets of four and four alike ; namely, 4 angles formed by m upon MT 2 , and 4 formed by t upon MT 2 . Now : 4 x 153 26' = 613 44' \ 4 x 11 6 34' = 466 16' J = which is the constant aggregate value of the angles of an octagon. 82. 85. This method of examining the geometrical relations of the angles of the equator, or of any other section of a crystal, affords the means of controlling the correctness of the mechanical measurements of the external angles of crystals. It is a principle of control similar to that which the chemist has over the results of an analysis in the knowledge which he possesses of the value of equivalent proportions. There are many pub- lished measurements of the angles of crystals, which stand greatly in need of the corrections that could be made by using this power of control. 86. Definition of the word PRISM It will be evident to the examiner of the foregoing data, that the symbols M, T, MT, M + T, MT + , serve to denote every vertical plane that can possibly occur around the equator of any crystal ; for all planes whose positions are n, e, s, w, are denoted by M or T, and all that have intermediate positions, as ne, nw, se, sw, n 2 e, ne 2 , &c., are denoted by some variety of MT X . Consequently, if we arbitrarily assume these vertical planes to be, in conjunction with the horizontal planes P, the COMPONENTS OF PRISMS, we have in P, M, T, MT, MT_|_, M + T, a formula which comprehends every variety of prism which can possibly occur in nature or in art that is to say, of prism considered independently of inclined or pyramidal terminations. I pro- pose therefore to restrict the meaning of the word prism to forms which contain these planes and no others. 87. Forms of the Equators of Prisms. For a practical purpose that will be hereafter explained, we may here take a view of the different forms that can be assumed by the equators of prisms under every possi- ble change of circumstance I believe that these forms may all be reduced to five principal varieties, as follow : Axes. Planes. Equators. M,T = a square. MT = a square. M,T,MT = a square combined with a square. M,T + = a rectangle. M + ,T = a rectangle. MT + '== a rhombus. M + T = a rhombus. MT + ,M + T = a rhombus with a rhombus. M,M + T = a square combined with a rhombus. T,MT + = a square combined with a rhombus. M,M + T = a rectangle combined with a rhombus. T,MT + = a rectangle combined with a rhombus. PRINCIPLES OF CRYSTALLOGRAPHY. 29 88. Of the Planes PM. There are four of them. The first plane is shown by the lines TpS^SmSGin the diagram in 20. The others are not shown on the diagram. The second plane passes from 6 m a 3 to T 2 p T 3 ; the third from T 2 p T 3 to 9 m a 12; and the fourth from 9 m a 12 to T p| T 1 . The four planes have the following polaric posi- tions : Zn, Zs, Nn, Ns. Every one of these planes cuts the two axes p a and m a , and is parallel to the axis t a . They are all INCLINED. They are equivalent to one another and are parallel two and two. They form together a HORIZONTAL PRISM, whose axis coincides with the axis t% and whose edges are directed towards Z, n, N, s. The symbol for the complement of four planes is PM. The symbol for half the complement is 1 PM. The planes are individually denoted by PM Zn, PM Zs, PM Nn, PM Ns. The north- meridian forms the cross section of the horizontal prism PM, and coincides with the plane p| m* p m*. The form of this north meridian depends upon the relative lengths of the axes p a and m a . , If these axis are equal, the north meridian is a square. b, If the axis p a is longer than the axis m a , the north meridian is a rhombus with the acute angles upon the axis p a . c, If the axis m a is longer than the axis p a , the north meridian is a rhombus with the acute angles upon the axis m a . d, In the first case, the symbol for the planes is PM. In the second case, it is P_j_M. In the third case, it is PM_|_. The instructions given in 50 61 for determining the relative lengths of the axes m a and t a situated in a rhombic equator, apply equally to the determination of the lengths of the axes p a and m a situated in a rhombic meridian. EXAMPLES OF THE PLANES PM, PM + , P + M. 89- Model 63. Rhombic Dodecahedron, MT.PM,PT. See 21 illustration B, and 31. The Zn plane of the form PM is marked on this model with P at the Z pole and with M at the n pole. The whole four planes of PM meet at their acute points, and form a zone round the crystal in the direction Z,n, N,s. The north meridian, which is a square with its angles in Z, n, s, N, cuts across this set of planes. The planes PM incline upon the planes MT at an angle of 120. 90. Model 12. Obtuse Quadratic Octahedron, PM + ,PT + . The Zn plane of this model is marked with P at the Z pole, and with M at the n pole. The angle across the edge at the n pole is found, by measurement with the goniometer, to be 83 38', and the angle formed by the plane Zn upon the plane Zs, measured over the apex, is found to be 96 22'. Consequently, the axes are p a m_|_, and the form of the north meridian is a rhombus with its acute angles on m a . To learn the comparative lengths of the two axes, we divide 96 22' by 2, which gives 48 1 1', and seek in the Logarithmic Tables for the tangent of this angle, 30 PRINCIPLES OF CRYSTALLOGRAPHY. which is 1.1177846. This number is the length of the axis m a when p a is taken as unity. Now 1.1177846 is to 1, as 19 is to 17. We may therefore describe PM + as P 17 M 19 , or as PM|f , or as PM M18 . I think that the last method of notation is the best, because it gives the readiest reference to the Table of Tangents, &c., whenever we wish to learn the value of the edges of a crystal from the length of its axes. If the lengths of the axes of this model were reversed, so as to make p a the longer axis, the symbol for the form would be P,. 118 M, and the north meridian would be a rhombus with its acute angles at Z and N. 91. Model 13. Acute Quadratic Octahedron, P + M,P + T. The Zn plane of this model is marked with P at the Z pole and with M at the n pole. The angle formed by the plane Zn upon the plane Nn across the equator atn, is 137 10'; and that formed by Zn upon Zs over the apex Z is 42 50' ; as will be found upon applying the goniometer to the model. The north meridian is therefore a rhombus, with its acute angles at Z and N. The half of the obtuse angle 137 10' is 68 35', the tangent of which is 2.5495160; so that the axes have the relation of p 5 a m|, and the symbol for the set of planes is P 5 .iM 2 or P 2%55 M. If the obtuse angle at m*, instead of being 137 10', had been 136 24', then the axes would have had the simple relation of pi ml ; because the half of 136 24' is 68 12', the tangent of which is 2.5001784. The symbol for the planes would then have been P 2 . 5 M or P|M. The differ- ence between 137 10' and 136 24' is one that cannot be properly dis- criminated by measurement upon the model; but 137 10' is the measure- ment derived from the crystals of a mineral called Anatase, the general form of which is represented by the model. 92. Model 32. ^ Cube combined with the r h O mbic dodecahedron, Mode 33. V d ^ octahedron _ p,M,T,MT.PM,PT,PMT. Model 34. ) In these three models the planes PM, are distinguished by their equal inclination upon the planes P and M, with both of which they make angles of 135 or ^ -f- 90. On model 32 the four planes are inclined rectangles; on model 33, they are also inclined rectangles; and on model 34 they are octagons. On all the 3 models, the positions are Zn, Zs, Nn, Ns. 93. Model 14. Quadratic Octahedron in combination with an Acute Quadratic Octahedron, the former subordinate, pm, P + M, pt, P + T. The plane marked P at the pole Z on model 14 is pm Z 2 n. The plane marked M at the pole n is P + M Zn 2 . The angle formed by the plane pmZ 2 n upon the plane pmZ*s, measured over the apex, is 90. This is not the correct angle of the mineral which this model was intended to represent, but it has been made so by accident. 90 is the angle which denotes a square north meridian, so PRINCIPLES OF CRYSTALLOGRAPHY. 31 that the planes pm belong to an equiaxed form, arid the symbol is simply pm. The angle formed by the plane P + M Zn 3 upon the plane P + M Nn 8 across the edge marked M, is 136 47'. The half of this angle is 68 23^', the tangent of which is 2.5246392. Hence the symbol for the planes is P 2 .52 5 M, or briefly but less exactly Pf M. The symbol for the model is therefore pm, Pf M, pt, Pf T. 94. The form of the north meridian of the combination represented by model 14 is an octagon. We will examine the value of its angles: 2 X 90 at Z and N =180 00' 2 x 136 47' at n and s = 273 34' All the angles known are therefore = 453 34' But the eight angles of the meridian are together equal to 1080 00'. See 82 Deduct 453 34' = Value of the four known angles. Leaves 626 26' = Value of the four unknown angles. And the fourth part of 626 26' is 156 36J. This is the value of each of the four angles formed by the incidence of the planes pm upon the planes P 2 .5 25 M, as will be found by applying the goniometer to the model. 95. Model 91. Pentagonal Dodecahedron. MT 2 .PM 2 ,P 2 T. See 77. The letter P marked upon this model shows the edge formed by the meeting of the planes PM 2 Zn and PM 2 Zs. Upon applying the goni- ometer to these two planes, across this edge, the value of the angle of their incidence is found to be 126 52'. The half of this is 63 26', the tangent of which is 1.999589. Hence the formula for the set of planes PM + , two of which appear at the top of the model, and two at the bottom, is PM 2 . 96. Model 47. The Cube combined with the Pentagonal Dodecahedron, the former subordinate. p,m,t,MT 2 .PM 2 ,P 2 T. See 83. The plane pZ is marked P upon model 47. The plane m n is marked M. The plane t w is marked T. The planes MT 2 are the vertical planes situated between the planes m and t. The planes PM 2 are the front and back inclined planes situated between the planes p and m. The follow- ing is the proof. The angle at Z of the planes PM 2 Zn and PM 2 Zs, is 126 52'. See 95. Now as the plane P cuts these two planes equally, it should make angles of 153 26' with both of them; for 1^' + 90 = 153 26' and upon applying the goniometer to the model, this will be found to be the correct angle. See 77, 95. 97. Model 68. Tetrakishexahedron. MT 2 ,M 2 T.PM 2 ,P 2 M,PT,,P 2 T. See 78. 32 PRINCIPLES OF CRYSTALLOGRAPHY. The planes PM 2 ,P 2 M, form a zone of 8 planes round the edge of the north meridian. The planes PM 2 occupy the positions Z 2 n, Z 2 s, N 2 n, N 2 s. The planes P 2 M occupy the positions Zn 9 , Zs 2 , Nn 2 , Ns 2 . The angles are given at 78 to 82, and all that is said there respecting the means of investigating the properties of the equator of this form, applies equally to the investigation of its north meridian. 98. Model 79. Oblique Rectangular Prism. M,T + . 1PM+ Zn,Ns. Px m a t|, The position which this crystal is made to assume, seems, like the posi- tions of Models 2, 6, 7, 10, and others, to militate against the general rule, 8, that a crystal is to be held with its longest axis in a perpen- dicular position. It is to be remembered, however, that the length of a prism, or, what comes to the same thing, the length of its axis p a , is an extremely variable quantity, so that crystals whose side planes are invariable as respects the relation of the axes m a and t a , often differ greatly in their length. The plane marked M on Model 79 is the plane Mn. The plane marked T is the plane T w. The plane marked P is the plane PM + Zn. The angle which this plane forms with M, across the edge Zn, is 113 8'. Deducting 90 for the value of the right angle of the prismatic plane Mn, we have 23 8' for the value of half the acute angle of the rhombus PM + . The sine of this angle is 3928722 and the cosine is 9195931. Its cotangent is 2.3406928. The radius is 1. The axis p a of the form PM bears therefore to the axis m a the ratio of 1 to 2.34, and the symbol of the set of planes is PM 2 3t . But only two planes of the set are present, and these have the positions Zn and Ns. The three axes of the model are approximative^ equal to p 2 ml tg. Hence the exact symbol for the model is M,T + , J PM 2 . 3t Zn,Zs. p 2 ms tg. The north meridian of Model 79 has the same geometrical properties as the equator of Model 11. See 75. 99 Model 83. Obtuse Ehombohedron. MT + . J PM + Zn,Ns. The angle formed by the plane MT_j_ nw upon the plane MT + ne, measured across the edge marked M, is about 105 5'. Some of the models are not quite exact in the angle. The half of 105 5' is 52 32J', and the tangent of the latter is 1.3051896. Hence the planes MT+ are MT 13 . See 35. The angle formed by the plane PM+ Zn upon the north vertical edge is about 102. Deducting 90 we have 12 for half the acute angle of the rhombus PM + . The sine of 12 is 2078117, its cosine 9281476, its radius 1.0, its cotangent 4.7046301. Hence the symbol for PM_|_ is PM 4>7 . But there are only two of the set of planes PM + present on the combina- tion, which two planes have the positions Zn,Ns. In the foregoing calculations I have taken first the axis m a and after- wards the axis p a for unity ; but it would be better to consider the axis m a as unity in both cases, since that axis is common to both the forms PRINCIPLES OF CRYSTALLOGRAPHY. 33 MT + and PM + . The only correction which it is necessary to make in this view, is to take the tangent instead of the cotangent of the acute angle of 12, and use that sum in expressing the'relation of the axes. The tangent of 12 is 0.212556, so that the form PM + becomes P_M, and must be termed P . 213 M, instead of PM 47 . The combination ex- hibited by Model 83, will then be expressed by MT 13 , -JP^^M Zn,Ns, or, in general terms, MT_|_, -J- P_M Zn,Ns. To this must be added, the relation of the axes of the crystal = p| mfo t^, because the dimensions of the crystal are not told by the symbols of the planes. 100. Model 85. Obtuse Rhombohedron. MT + . ^PM + Zn,Ns. The angle formed by the vertical plane MT + nw upon the adjoining vertical plane MT + ne, measured across the vertical edge at M, is about 1 28, but it ought, according to a measurement of Haiiy's, to be 1 34 26'. The half of this angle is 67 13', the tangent of which is 2.3808444 ; so that the symbol for the prismatic planes is MT 2 . S8 ' The angle formed by the plane PM + Zn upon the vertical edge at n, is about 130 26'. Deducting 90, we have 40 26' for half the acute angle of the rhombus PM + . The tangent of this is 0.8520704, which gives the symbol P . 852 M, and which bears a very simple relation to the symbol P . 213 M of Model 83 ; as 213 x 4 = 852. Hence the symbol for Model 85 is MTg^.^Po^M Zn,Ns. To which is to be added the dimensions of the axes of the crystal pi nig t a ^. 101. Of the Planes PT There are four of them. First, the plane shown by the lines M p| M 1 9 t 3 in the diagram in page 6; secondly, a plane passing from 9 t 3 to M 2 p M 3 ; thirdly, a plane passing from M 2 p M 3 to 6 t a 12 ; and fourthly, a plane passing from 6 t a 12 to M pz M 1 . Each of these planes cuts the two axes p a and t a in the same manner. They are all INCLINED. They are parallel to the axis m% they are parallel two and two among themselves, and they are all equivalent to one another. They constitute together a HORIZONTAL PRISM whose edges bisect the two axes p a and t a , and whose axis coincides with the axis m a . The polaric position of the four planes is Ze, Zw, Ne, Nw. The symbol for the complement of four planes is PT. The symbol for half the complement is ^ PT. The planes are denoted individually by the symbols PT Ze, PT Zw, PT Ne, PT Nw. The cross section of the horizontal prism PT takes place upon the plane p| t a p t^. It consequently coincides with the east meridian. The form of it depends upon the relative lengths of the two axes p a and t a . , If both axes are equal, the east meridian is a square, and the symbol for the planes is PT. by If the axis p a is longer than the axis t a , the east meridian is a rhombus with its longer diagonal parallel to the axis p a . The symbol for the planes is then P + T. F 34 PRINCIPLES OF CRYSTALLOGRAPHY. c, If the axis t a is longer than the axis p a , the east meridian is a rhombus with its longer diagonal parallel to the axis t a . The symbol for the planes is in that case PT+. d, The measurement of the two axes of the east meridian is effected, and its general properties are ascertained, by the methods described in 5061. EXAMPLES OF THE PLANES PT, PT + , P + T. 102. The varieties of the form PT differ in no other respect from the varieties of the form PM, than that they are situated at right angles to PM, and have the same relation to the axes p a and t a that the form PM has to the axes p a and m a . Hence the same models afford most of the requisite examples. 103. Model 63. Rhombic Dodecahedron. MT.PM,PT. The four planes PT are situated Ze, Zw, Ne, Nw. They touch one another at the acute points of the rhombuses, and form a zone round the crystal, which gives for the east meridian a square with its angles at Z, e,w,N. See 21 B, 31, 89. 104. Model 12. Obtuse Quadratic Octahedron. PM + ,PT+. This form has been already explained at 90. The exact symbol of the set of planes PT+ is PT U18 , and the model is = PM^najPTi.!^. 105. Model 13. Acute Quadratic Octahedron. P + M,P + T. This form was explained at 91. The exact symbol for the set of planes P + T is P 2 . 55 M, and the model is = P 2 .55M,P 255 T. 106. Models 32, 33, 34. Combinations of the Cube, Rhombic Dode- cahedron, and Octahedron. The planes PT differ in no 'respect from the planes PM, only that they are situated at right angles to the latter. See 33. They form angles of 135 with the planes P and T, and, of course, angles of 90 with the planes M, as do all the planes that cut both p a and t a and not m a . The symbols for the three combinations are given in 69. 107. Model 14. Combination of a Quadratic Octahedron with an Acute Quadratic Octahedron, the latter prevailing, pm, P| M, pt, P| T. The planes pt have the positions Z 2 e, Z 2 w, N 2 e, NV. The planes P|T have the positions Ze 2 , Zw 2 , Ne 2 , Nw 2 . The letter T at the pole t of the model marks the plane Pf T Zw 2 . All these planes are parallel to the axis m a . The model has in other respects been so fully explained in 93, 94, that it is only necessary to add that the east meridian of the present combination has the same properties as the north meridian of the com- bination pm, P|M, which has been described in 94. PRINCIPLES OF CRYSTALLOGRAPHY. 35 108. Model 91. Pentagonal Dodecahedron. MT 2 .PM 2 ,P 2 T. The sets of planes MT 2 and PM 2 have been already fully described in 77, 95. I have therefore to confine this notice to the planes P 2 T, which are the four planes that meet in two pairs so as to form horizontal edges across the poles t| and t^. The angle formed by the plane P 2 T Zw upon the plane P 2 T Nw, measured by applying the goniometer across the edge at t, is 126 52', which, as already explained in the referred to above, gives the relation of 2 to 1 for the axes p a and t a , and authorizes the symbol P 2 T. 109- Hence we see that Model 91 represents a form produced by the intersection of three similar and equal rhombic prisms, each of them hav- ing one of its cross axes longer than the other, and each having its infinite axis coincident with one of the axes of the crystal or combination, so that the three prisms cross one another in the centre at right angles. There are several varieties of this form to be found in the mineral kingdom, the most important of which have the following proportional axes : 1 : J, which gives the symbol MTf .PM|,PJT. 1 : f , which gives the symbol MTf.PMf ,Pf T. *. 1 : 2, which gives the symbol MT 2 . PM 2 , P 2 T. These are all easily discriminated by examining the angles formed by the inclination of their planes one upon another, as shown in the following table, or by their inclination upon the planes of P,M,T, or MT.PM,PT, or PMT, or of any form with which they may occur in combination ; as will be hereafter explained circumstantially : r- Inclination of the plane MT_i_nw upon : MT + ne. ' PM + Zn. P + T Zw. MTf.PM|,PfT 106 16' 118 41' 118 41' MT|.PM|,P|T 112 37' 117 29' 117 29' MT 2 .PM^,P 2 T 126 52' 113 35' 113 35' 110. Model 47. Combination of the Cube and the Pentagonal Dode- cahedron, the latter predominating. p,m,t, MT 2 .PM 2 ,P 2 T. The planes of the form P 2 T, or TP 2 , are essentially the same as the planes of the forms MT 2 and PM 2 , already fully described in 83, 96, The 4 planes of P 2 T are those situated between the planes marked P and T on the model. The east meridian of the form p,t,P 2 T is an octagon, exactly similar in form and angles to the octangular north meridian of the planes p,m,PM 2 , and to the octangular equator of the planes m,t,MT 2 . The method of determining the value of the angles that are formed by the inclination of the plane p or t upon P 2 T, has been described in 83. 111. Model 68. Tetrakishexahedron, MT 2 ,M 2 T.PM 2 ,P 2 M,PT a ,P 2 T. This model has twenty-four planes, of which I have already described sixteen in 78 and 97, where I have also explained the principles upon which we proceed in determining what are the properties of the com- bination that is exhibited by the model. It seems to be necessary only 36 PRINCIPLES OF CRYSTALLOGRAPHY. to add, that the rest of the 24 planes, namely, the sets PT ,P 2 T, are those which form the bounds of the east meridian of the model and occupy the following positions : PT 2 == Z 2 e,Z 2 w,N 2 e,NV. P 2 T = Ze 2 ,Zw 2 ,Ne 2 ,Nw 2 . 112. We perceive from the account that has been given of Model 68, that it represents a combination of six similar and equal rhombic prisms, each of them having one of its cross axes longer than the other, and con- sisting of three pairs of prisms, each pair cutting two axes unequally and inversely. This combination is therefore a sort of double of that repre- sented by Model 91- See 109- The mineral kingdom presents several varieties of the tetrakishexahedron, particularly the following : Axes. Resulting Planes. 1 : | = MTf,M|T.PM|,P|M,PT|,P|T. 1:2 = MT 2 ,M 2 T.PM 2 ,P 2 M,PT 2 ,P 2 T. 1 : | = MTf,MfT.PMf,PfM,PTf,PfT. 1:3 = MT 3 ,M 3 T.PM 3 ,P 3 M,PT 3 ,P 3 T. ^1:5 = MT 5 ,M 5 T.PM 5 ,P 5 M,PT 5 ,P 5 T. 1 1 3. These combinations may be all discriminated by the difference in the angles at which their planes incline upon one another, as shown in the annexed table, or upon the planes of any other forms with which it is possible for them to occur in combination. And we have in our know- ledge of the geometrical properties of the equator, and the east and north meridians of the different varieties, an easy and efficient power of control over the accuracy of the mechanical measurements which may seem to distinguish any one of these combinations from the others. Suppose a form of this kind to be given and the angle at the pole n, formed by the incidence of the plane MT + n 2 w upon the plane MT+n 2 e, to be stated to be = 136. We test the accuracy of this measurement as fol- lows : The half of 136 is 68, the tangent of which is 2.4750869, or, when doubled, 4.9501738. This is very nearly the same as the angle of the form J, but the angle formed by the form J is not exactly 136 but 136 24', for the half of the last named angle = 68 12' has a tangent = 2.5001784. Hence we conclude that the given form is in reality that known by the term , and that the given angle of 136 was erroneous. Inclination of the plane MT + n 2 w upon: COMBINATIONS. MT+ n , e> M+T nwi> p+M Zn 2 MT|,M|T.PM|,P|M,PT|,P|T, 112 37' 157 23' 133 49'. MT 2 ,M 2 T.PM 2 ,P 2 M,PT 2 ,P 2 T, 126 52' 143 8' 143 8'. MT|,M|T.PM|,P|M,PTf,P|T, 136 24' 133 36' 149 33'. MT 3 ,M 3 T.PM 3 ,P 3 M,PT 3 ,P 3 T, 143 8' 126 52' 154 9'. MT 5 ,M 5 T.PM 5 ,P 5 M,PT 5 ,P 5 T, 157 23' 112 37' 164 3'. 114. Model 89. Acute Rhombohedron. MT + . JPT + Zw,Ne. It will in this case, as in that described in 99, be proper to consider as unity the axis that is common to both forms. At present, this axis is t'. PRINCIPLES OF CRYSTALLOGRAPHY. 37 The angle formed by the incidence of the plane MT + nw upon the plane MT + sw is 78 28', the half of which is 39 14', and the tangent of 39 14' is 0.8165493 = axis m a when t a is = 1. This gives the symbol M .8i 7 T for the four vertical planes of the combination represented by the model. The angle formed by the incidence of the plane PT + Zw upon the w vertical edge is about 110. I deduct 90 for the right angle, and have a remainder of 20, the tangent of which, = p a , is 0.3639702. This gives the symbol P ^T. Hence the formula for Model 89 is M . 8 i r T. JPo^T Zw,Ne. 115. Of Rhombohedrons in General. The method of describing rhombohedrons given in 99, 100, 114, affords a constant mark of dis- tinction between the two kinds that are commonly called obtuse and acute rhombohedrons. The general formula for all obtuse rhombohedrons is MT + .^P_M, Zn,Ns. The general formula for all acute rhombohedrons is M_T. JP_T Zw,Ne. Both kinds are right rhombic prisms, having the planes MT+, but the obtuse rhombohedrons are terminated by the in- complete complement of planes JP_M and the acute rhombohedrons by the incomplete complement of planes ^P_T. 116. It is however usual with crystallographers to hold rhombohedrons in such a position as to give formulae very different from the above. Thus the obtuse rhombohedron, Model 85, is held with the two obtuse solid angles at the poles Z and N, and the planes turned towards Zn, Zse, Zsw, Ns, Nne, Nnw. This gives the formula iPM + Zn,Ns, JPJfl^T Zse, Zsw, Nne, Nnw. On the other hand, the acute rhombohedron, Model 89, is held with its two acute solid angles at the poles Z and N, but with its six planes also in the positions Zn,Zse,Zsw,Ns,Nne,Nnw. This produces the formula \ iP+M Zn,Ns, |P + M_T Zse, Zsw, Nne, Nnw. 117. Of Oblique Rhombic Prisms. The two models numbered 84 and 87 exhibit examples of the forms commonly called oblique rhombic prisms. They have a certain resemblance to the rhombohedrons, but the resemblance does not hold in all points. If you suppose all the oblique rhombic prisms and the rhombohedrons to have three axes crossing one another in the centre, but situated in directions parallel to the planes of each form, then, in the rhombohedrons the three axes will be alike, but in the oblique rhombic prisms there will be always one axis shorter or longer than the other two. The rhombohedron is the point of unity between a short oblique rhombic prism and a long one, just as the cube is the point of unity between a short quadratic prism and a long one. See Models 1, 2, 3. The consequence of the inequality of the length of the axes of oblique rhombic prisms, is, that they have always two rhombic planes and four rhomboidal planes, whereas the whole six 38 PRINCIPLES OF CRYSTALLOGRAPHY. planes of every rhombohedron are rhombuses. It is however the analogy betwixt the oblique rhombic prisms and the rhombohedrons, as compared with the analogy between quadratic prisms and the cube, that has in- duced me to reject the commonly received method of holding the rhombohedrons with two angles on the axis p* and to consider them to be examples of oblique rhombic prisms. 118. In making use of the term oblique rhombic prism, I employ the commonly received language of crystallography ; but in subjecting the solids known by that term to symbolic description according to the prin- ciples laid down in this work, I do not consider them to be OBLIQUE prisms. The definition that I have given of the word axes, in 2, namely, that they are " three imaginary lines which pass through the centre of a crystal, cross one another there at right angles, and terminate at its surface," does not permit of the assumption of any other than right or vertical prisms ; nor does it appear to me to be either necessary or advantageous to consider any of these prisms to be oblique, since the fact is that the prisms are really straight, and the terminations alone are inclined. All the prismatic planes of such forms can be easily designated by symbols derived from the general formula M,T,MT,MT X , ( 86) ; and all the inclined, pyramidal, or terminal planes, whether the termina- tions of each combination be monofacial or multifacial, can be equally well denoted by some term of the series PM X , PT X , P X M X T X , as will be satisfactorily proved in a subsequent section. 198. 119. Of the rhombic or rhombo-rectangular prisms that are terminated by single oblique planes, there are the following three varieties : 1. Those terminated by the planes ^PM X ; as Model 84. 2. Those terminated by the planes ^PT X ; as Model 87. 3. Those terminated by the planes PMT; as Model 108. 120. Model 84. Oblique Rhombic Prism. MT + 4P_M, Zn,Ns. The form is to be held so as to place the four short edges situated be- tween the rhomboidal planes, in a vertical position. The angle formed by the incidence of the plane MT + ne upon the plane MT + nw, at the n pole, is 124 34'. The half of this is 62 17, the tangent of which is 1.9033738. This gives the formula MT 1<9 . The incidence of the planes P_M upon the n vertical edge is 104 57'; or rather, this ought to be the measure, for some of the models are incor- rect and measure nearly 10 more than this. 104 57' 90 =14 57'. The tangent of this is 0.2670141, which gives the formula Po.^M. The lengths of the axes of the model are about p| m^ t,*. The symbol for Model 84 is therefore MT^. ^Po^M Zn,Ns, p^ m^ t, a ,. 121. Model 87. Oblique Rhombic Prism, M_T.JP_T Zw,Ne. As the last form was the counterpart of the obtuse rhombohedrons, so is this the counterpart of the acute rhombohedrons ; possessing the same planes in the same polaric positions, and only differing from them in the comparative length of the axis p a . PRINCIPLES OF CRYSTALLOGRAPHY. 39 The angle formed by the incidence of the plane MT+ nw upon the plane MT + sw is 87 42'. The tangent of the half of this angle is 0.9606421. Hence the symbol for the prismatic planes, the axis t a being taken for unity, is M . 96 T. The angle formed by the incidence of the plane P_T Zw upon the w vertical edge is 106 6'. Deducting 90 for the right angle, we have 16 6' for the value of the half of the acute angle of the set of planes P_T. The tangent of 16 6' is 0.2886352. Hence the symbol is Po. 289 T. The axes of the crystal are about pft m 3 a 2 1^. The symbol for Model 87 is therefore M . 96 T. ^P^gT Zw,Ne. pf m 3 a 2 t 3 a 3 . RETROSPECT. 122. We have now examined all the varieties of planes that cut either one axis or two axes, and we may here very properly consider what de- gree of power these planes possess of producing complete crystals by their combinations with one another. The complements P,M,PM X , produce a zone of planes round the north meridian, or an infinite prism upon the axis t a , but they produce no com- plete form. The complements P,T,PT X produce a zone of planes round the east meridian, or an infinite prism upon the axis m a , but they produce no complete form. The complements M,T,MT X produce a zone of planes round the equator, or an infinite prism upon the axis p a , but they produce no complete form. And no alteration of the relative lengths of the three axes can make any one of these zones of planes produce closed or complete crystals. The order in which the planes that belong to the zone P,M,PM X , dispose themselves upon a complex combination, may be exemplified by reference to the figure at page 174 of PHILLIPS'S Miner alogy-> which I have already referred to in 73. I shall take only the Zn quarter of the north meridian as an example. The arrangement of the planes is similar in the other three quarters of the crystal. P PM*. PMJ PM + PM P+M PjM P-frM M Z Z 4 n Z 3 n Z 2 n Zn Zn 2 Zn 3 Zn 4 The order in which the planes that belong to the zone P,T,PT X , dispose themselves upon a complex combination, may be exemplified by reference to the figure at page 174 of PHILLIPS'S Mineralogy^ which I have already referred to in 73. I shall take only the Zw quarter of the east meridian as an example. The arrangement of the planes is similar in the other three quarters of the crystal. P PT. PTJ PT + PT P+T PJlT Z 3 w Z 2 w Zw Zw 2 Zw 3 Zw 4 w 40 PRINCIPLES OF CRYSTALLOGRAPHY. The order in which the planes that belong to the zone M,T,MT X , arrange themselves upon such a combination, has been explained in 73. Those who may not have an opportunity of referring to PHILLIPS'S Mineralogy, may compare this tabular arrangement with the marked planes upon Model 32. The planes P,PM,M, and P,PT,T, have the same positions upon the figure and the model. The planes PM_jtj_, PJVTJI, PM + , lie upon the combination parallel with the plane PM, and between PM and P, the plane PM^ij. being nearest of them all to the plane P. And the rest are arranged correspondingly. 123. It is evident that no single zone, however numerous its planes may be, can ever make a complete crystal. On the other hand, many complete forms or crystals are produced by such complements of these planes as are so situated on the different axes as to cross or cut one another. It is of no consequence under what angle of inclination this takes place the act of crossing is the main point ; and any two endless prisms, of any dimen- sions, that cross one another at any angle, may produce a closed form or complete crystal. The only necessary condition to produce this end is that the two principal axes of the combining prisms cross one another at the same level. Each of the following combinations of complements produces a closed form or crystal : P 5 M,T. P,MT. P,M,IMT. P,T,IMT. PM,T. 6, 7, 8, 9, 10, PM,MT. PM, iMT. PT,M. PT,MT. PT,MT. And a great variety of different crystals may be produced by the com- bination of these forms with one another, or with other sets of planes. 124. Of the Planes PMT There are eight planes denominated PMT. One of these is shown in the diagram in 20 by the triangle Pi nin t, which cuts the three axes p a m a t a and is parallel to the inverse triangle MT*3. This latter triangle is produced by the ideal removal of the corner or solid angle marked 2 on the parallelopepidon depicted in the diagram. Now there are eight similar solid angles on this form, from which it follows that there can be placed upon the axes p a m a t a precisely eight triangular planes similar to the one marked p| m a t a . Namely, Poles cut by the Planes. Above the Equator. Below the Equator. pi m n a t- pS m a C p a z m a t a Pi m* t a t p a N Planes produced. PMT Znw. PMT Zsw. PMT Zse. PMT Zne. PMT Nnw. PMT Nsw. PMT Nse. PMT Nne. PRINCIPLES OF CRYSTALLOGRAPHY. 41 The polaric positions of the individual planes are denoted by the signs of the three poles which each plane cuts. The eight planes are all INCLINED. They are equivalent to one another. They all cut the three axes p a m a t a and they are parallel to no axis. They consist of four pair of parallel planes so connected as to constitute TWO HORIZONTAL RHOMBIC PRISMS which cross one another and produce a COMPLETE FORM, or CRYSTAL. This is a property not possessed by any of the sets of planes described previously for as the planes of every other set cut at most only two axes, and have always an axis of their own which coincides either with p a m a or t a , every prism which these sets form runs to infinity upon the coinciding axis, and only produces a complete (closed) form when it is met and crossed by a prism that runs in a different direction. Thus, the complement M,T, or the complement MT, runs to infinity upon the axis p% producing the four vertical sides of a prism, but never forming a complete figure until it is crossed by the complement P, PM, or PT, or by something equivalent. See 122. 125. The symbol for this complement of eight planes is PMT. The symbol for half the complement is JPMT. The symbol for the fourth part of the complement is JPMT. All the planes denoted by these symbols bisect the three axes p a m a t a . They comprise four pair of parallel planes, but the planes which constitute the half and quarter complements very frequently do not occur in parallel pairs. It is easy however to indicate the particular planes which may be present in any fractional part of the complement PMT, by adding to the symbol PMT the polaric positions of those particular planes. 126. The two horizontal prisms which concur to produce the comple- ment P X M X T X sometimes cross one another at right angles, and some- times obliquely. THE REGULAR OCTAHEDRON, PMT. 127. When the prisms cross one another at right angles, they cut the horizontal axes m a and t a equally and at an angle of 45. If this occurs with an equiaxed crystal, or, to speak more precisely, with prisms of such dimensions as can produce an equiaxed crystal, the resulting form is the geometrical regular octahedron, which is figured in the margin, and represented by Model 15. This form has the following pro- perties : The two rhombic prisms by whose intersection it is produced, have angles of 70 32' and 109 28', and consequently have diagonals that are nearly equal to the num- bers 10 and 7, for the tangent of 35 16', the half of 70 32', is 0.7071664. The longer of these diagonals coincides with the axis p a of the resulting crystal. The incidence of any two planes measured over the pole Z is 70 32', and measured across the equator is 109 28'. The axes m a and t a being cut at equal distances from the centre, the G 42 PRINCIPLES OF CRYSTALLOGRAPHY. form of the equator is a square, with the angles at n, e, s, w. The north meridian is a square, with the angles at Z, n, N, s ; the east meridian a square with the angles at Z, w, N, e ; while the north-east and north-west meridians, which cut through the planes of the crystal, are necessarily rhombuses of the same dimensions as the prisms by which the octahedron is produced. The shape of the planes is that of an equilateral triangle, and the angle of their incidence upon one another, measured across any edge, is 109 28'. These characters are sufficient to distinguish the regular octahedron from all other varieties of the octahedron ; and this is the form which is intended to be represented by the symbol PMT when written without addition. ISOSCELES OCTAHEDRONS, P X MT,PM X T,PMT X . 128. When the two rhombic prisms which produce an octahedron by crossing one another at right angles, have other dimensions than those which produce the regular octahedron, then the octahedron which is produced must necessarily have different dimensions from the dimensions of the regular octahedron. Only two different variations can however occur. The rhombic prisms may be either more obtuse or more acute at the angles which come upon the axis p a than the prisms which produce the regular octahedron. In the former case the resulting octahedron will have the perpendicular axis shorter than the horizontal axes. In the latter case, the resulting octahedron will have the perpendicular axis longer than the horizontal axes. One of these forms would require the symbol P_MT, the other, the symbol P+MT. 129. But this explanation of the possible variations that can take place in this operation, rests upon the supposition that, in the three cases par- ticularised, the rhombic prisms always cross one another at the level of the horizontal axes m a and t a , and this supposition is necessary so long as we confine our attention to the consideration of the production of octahe- drons that have a square equator. 130. When however we extend our observation to octahedrons which can be produced by the crossing of two rhombic prisms, still at right angles to one another, and still in such a manner as to cut two axes at angles of 45, but at other levels than that of the equator, or in different positions as regards the longer diagonals of the cross sections of the prisms, we arrive at the following results. OBTUSE ISOSCELES OCTAHEDRONS, P_MT, PM_T, PMT_. P_MT is referred to in 128. 131. Let a rhombic prism cut the system of three axes in the direction of a line passing from Zw to Ne. Let a similar and equal rhombic prism cut the three axes in the direction of a line passing from Ze to Nw. In both cases the shorter diagonal of the rhombic prism is to be situated parallel to the axis m a . The two prisms will cross one another at right angles, and they will cut the axes p a and t a equally at an angle of 45. The resulting combination will require the symbol PM_T. PRINCIPLES OF CRYSTALLOGRAPHY. 43 132. Let a rhombic prism cut the system of three axes in the direc- tion of a line passing from Zn to Ns. Let a similar and equal rhombic prism cut the three axes in the direction of a line passing from Zs to Nn. Let the shorter diagonal of the rhombic prism be, in both cases, parallel to the axis t a . The two prisms will cross one another at right angles, and and they will cut the axes p a and m a equally at an angle of 45. The resulting combination will require the symbol PMT_. EXAMPLES OF P_MT,PM_T, AND PMT_. 133. Hold Model 12 in such a position that its two obtuse solid angles shall be at Z and N, and its four acute solid angles at n, e, s, w. The equator will then be a square, and the north and east meridians will both be rhombuses with their acute angles on the four poles of the equator. This is the Octahedron P_MT described in 128. 134. Hold the same model in such a position that its two obtuse solid angles shall be at n and s, and its four acute solid angles at Z,e,N,w. The equator will then be a rhombus with its acute angles at e and w. The north meridian will be a rhombus with its acute angles at Z and N. The east meridian will be a square, with its angles at Z,e,N,w. This is the octahedron PM_T described in 131. 135. Hold the same Model in such a position that its two obtuse solid angles shall be at e and w, and its four acute solid angles at Z,n,N,s. The equator will then be a rhombus with its acute angles at n and s. The north meridian will be a square with the angles at Z,n,N,s. The east meridian will be a rhombus with its acute angles at Z and N. This is the octahedron PMT_, described in 132. 1 36. I have shown in 67, and in other places, that when the planes of a form that has a short axis and a long one, enter into combination with other planes, it is generally those parts of the planes which, in the uncombined form, project to the poles of the longer axis, that cannot be seen on the combined form ; the most projecting portions of thejm- equiaxed form being replaced or cut off by the planes of combination which replace them. 137. If, with this consideration in mind, we examine the forms indi- cated by the symbols P_MT, PM_T, and PMT_, we shall perceive that when these forms enter into combination with other forms whose axes are shorter than the axis that is considered unity in the form P_MT, it is necessarily those portions of the planes which extend so as to form four acute solid angles, and yield the square section, that must be cut off by the planes of replacement, because these are the portions of the planes that are farthest removed from the centre of the crystal. It follows also of necessity, that when these forms have been cut in this manner by combining planes, the portions which remain upon the combinations must be the following : 138. Of P_MT, there will remain four planes surrounding the pole Z, 44 PRINCIPLES OF CRYSTALLOGRAPHY. and having the positions Z 2 nw, Z 2 ne, Z 2 sw, Z 2 se, and four planes sur- rounding the pole N, and having the positions N 2 nw, N 2 ne, N 2 sw, N 2 se. 139- Of PM_T, there will remain four planes surrounding the pole n, and having the positions Zn 2 w, Zn 2 e, Nn 2 w, Nn 2 e, and four planes sur- rounding the pole s, and having the positions Zs 2 e, Zs 2 w, Ns 2 e, Ns 2 w. 140. Of PMT_, there will remain four planes surrounding the pole e, and having the positions Zne 2 , Zse 2 , Nne 2 , Nse 2 , and four planes sur- rounding the pole w, and having the positions Znw 2 , Zsw 2 , Nnw 2 , Nsw 2 . 141. It is of no consequence by what means by what description of planes the parts of planes that disappear in such a case are removed. The positions of the remaining portions of the planes is in no respect altered by any accidental circumstance that may have attended the re- placement of the absent portions, neither are their positions at all changed by the nature, number, or positions of the superinduced planes the planes of replacement or combination. The portions of the planes of an unequiaxed form which touch the shorter of the two unequal axes of the form, are never replaced, never driven from their polaric positions, or altered in any respect, except that of being diminished in size, by the abstraction of those portions of the planes that extended towards the longer of the two axes of the form, excepting when the unequiaxed form is cut by another form whose three axes are all shorter than the shortest of the axes of the unequiaxed form. This happens, for example, when the form PMT is cut by the form P,M,T, or by PJ,MJ,T, or by any form whose planes fall nearer to the centre of the combination than do any of the poles of the forms P_MT, PM_T, PMT_. From the above positions we may safely draw the inference, that if the three forms described as P_MT, PM_T, and PMT_, were to cut one another, and come together upon one crystal, that crystal must exhibit planes in all the posi- tions described in 138, 139, 140, and be a combination of no less than 24 equal and similar planes. It is indeed easy to perceive that the result cannot be otherwise. The uncombined planes of P_MT have four acute solid angles at the poles m*, m*, t, t*. Let this form be taken as = P 1 M 2 T 2 . If it combines with the form P2M!T2, it is evident that the second form will cut off all those parts of the planes PiM 2 T 2 which extend along the axis m a beyond the distance m? measured from the centre towards m* and m*, while simultaneously the form PjM^, will cut off all those parts of the planes of P^T^ which extend along the axis p a beyond the distance pf measured from the centre towards p z and p*. If the combination be then cut by the third form P 2 M 2 T, , the latter will cut off all those parts of both the preceding forms which extend along the axis t a beyond the distance t* measured from the centre towards t a and t, while it will itself suffer a deprivation of all those parts of its planes that extend along the axes p a and m a beyond the points p? and m^. By this threefold operation, the six poles of the combination are fixed at the points p|l, PN!, m*l, m*l, tl, tl, and thus an equiaxed crystal is pro- duced by the combination of three equal and similar unequiaxed forms. PRINCIPLES OF CRYSTALLOGRAPHY. 45 142. Model 22. Icositessarahedron. P_MT, PM_T, PMT_. This model represents the form alludud to in the last paragraph. It is in fact the result of the combination of the three sets of octahedral planes P_MT, PM_T, PMT_. The four planes round the pole p and the four round the pole p, are those described in 138. The four planes round the pole n, and the four round the pole s, are those described in 1 39- The four planes round the pole e, and the four round the pole w, are those described in 140. Altogether there are 24 planes, in 6 sets of 4 each, if considered in relation to the planes of the cube, or in 8 sets of 3 each, if considered in relation to the planes of the octahedron. The form of the planes is that of a symmetrical trapezium. The positions of the planes have been already described. 143. Several varieties of this form have been found among minerals, the two principal of which have the following symbols : PIMT, PMJT, PMTi, P^MT, PMJT, PMTi. The first of these forms is represented by Model 22. The other has nearly the same shape. Its planes are the same in number and have the same positions, and consequently its edges and solid angles are the same in number. But in general appearance the other form approaches more nearly to that of a cube, whereas Model 22 bears a resemblance to the octahedron. The reason of the difference is sufficiently obvious upon an examination of the relative lengths of the axes of the two forms, for as the three axes of one of the forms have to the three similar axes of the other, the relation of ^ to ^, it follows of the form ^ that its 6 solid angles at Z, N,n,e,s,w, must be more obtuse, and the 8 solid angles at Znw, Zne, Zsw, Zse, Nnw, Nne, Nsw, Nse, must be more acute than those of model 22. Now the 6 angles in both forms represent the positions of the angles of the octahedron, and the 8 angles represent the positions of the angles of the cube; so that one form necessarily approxi- mates to the octahedron and the other to the cube ; both forms being consequent upon the difference in the primary dimensions of the rhombic prisms whence they are derived. The two forms may be discriminated as follows : 144. P1MT,PM1T,PMTJ The angle formed by the two horizontal edges that meet at m*, is 126 52'. The half of this is 63 26', whose tangent is 1.999859 or 2.0. Equal to mf t|. The angle formed by the two inclined edges that meet at m^is the same as that formed by the hori- zontal edges, and has the same tangent, and is therefore equal to m? p. This gives the formula PMJT for the eight planes that meet at the poles m* and m*. From similar measurements made across the poles p| and t, we derive the formula P^MT for the 8 [planes that meet at the poles p| and p^, and the formula PMTJ for the 8 planes that meet at the poles t* and t. 145. PiMT,PMiT,SPMT. The angle formed by the meeting of 46 PRINCIPLES OF CRYSTALLOGRAPHY. any two edges at any one of the 6 poles of the crystal is 143 8', from which is derived the tangent 3.000282, which gives the formula P,M 3 T 3 or P^MT, &c. The measurements are alike at all the six poles. 146. Angles that characterise these two forms : PlMTZ 2 nwupon: PJMTZ 2 se = 109 28' PJMTZ 2 ne = 131 49' PM^TZnV = 146027' PMTi Znw 2 = 146 27' Z 2 nw upon : PMT Z 2 se = 129 31' PJMT Z 2 ne = 144 54' PMJT Zn 2 w = 129o 3 1' PMT^Znw 2 = 129 31' ACUTE ISOSCELES OCTAHEDRONS, P+MT, PM + T, PMT+. 147. I now return to the position quitted in 130, in order to explain a case different from that taken up in 131, but already alluded to in 128. IK Let the two rhombic prisms which we suppose to cut the system of three axes when in the act of producing the octahedral forms, with the consideration of which we are now occupied, be acute, not, as tacitly admitted in 131, obtuse prisms. Let them be, for example, prisms whose cross sections would somewhat resemble the form of the north meridian of Model 13. See 91. With this understanding, we will now retrace the steps that were taken in 128 and subsequently. 148. Let the two rhombic prisms be situated in a horizontal position with their acute edges at the points Z and N, or, what comes to the same thing, with their longer diagonals parallel to the axis p a . Let them cut the axes m a and t a equally at an angle of 45, and cross one another at right angles. The result of this process, as already stated in 128, will be a combination whose symbol must be P + MT. 149. Let one of the rhombic prisms cut the system of three axes in the direction of a line passing from Zw to Ne ; and the other prism cut the axes in the direction of a line passing from Ze to Nw both prisms having their longer diagonals situated parallel to the axis m a , and both cutting the axes p a and t a equally at an angle of 45. The resulting octahedron will be such as to require the symbol PM + T. 150. Let one of the same prisms cut the three axes in the direction of a line passing from Zn to Ns, and let the other prism cut the axes in the direction of a line passing from Zs to Nn both prisms having their longer diagonal parallel to the axis t a , and both cutting the axes p a and m a equally at an angle of 45. The resulting octahedron will be such as to require the symbol PMT+. EXAMPLES OF P+MT, PM+T, AND PMT+. 151. Hold Model 13 in such a position that its two acute solid angles shall be at Z and N, and its four obtuse solid angles at n, e, s, w. The PRINCIPLES OF CRYSTALLOGRAPHY. 47 equator will then be a square, and the east and west meridians will both be rhombuses, with their acute angles at Z and N. This is the octahe- dron P+MT described in 148. 152. Hold Model 13 in such a position that its two acute solid angles shall be at n and s, and its four obtuse solid angles at Z, e, N, w. The equator will then be a rhombus with its acute angles at n and s. The north meridian will be a rhombus with its acute angles at n and s. The east meridian will be a square with its angles at Z, e, N, w. This is the octahedron PM+T described in 149. 153- Hold the same model in such a position that its two acute solid angles shall be at e and w, and its four obtuse solid angles at Z, n, N, s. The equator will then be a rhombus with its acute angles at e and w. The north meridian will be a square with its angles at Z, n, N, s. The east meridian will be a rhombus with its acute angles at e and w. This is the octahedron PMT+ described in 150. 154. Let us put Model 13 into the position described in 151, and denoted by P_j_MT, and let us examine the effects that must be conse- quent upon the combination of such a form with other planes. For the reasons that I have stated in 67, 136, 141,1 form the^opinion that, when P + MT occurs in combination with any form whose axis p a is less than the sum indicated by the term + in P + MT, there must be a replacement of those parts of the planes P+MT, that touch the poles p| and p^. The shape of the planes substituted for the portions of the planes that are removed, depends upon the particular form by which the replacement is effected. Thus, if the planes of combination are the complement P,MT, there will be only a single horizontal plane substituted for each solid angle removed, as shown by Model 80. If the planes of replacement are the complement P_M+T + (equivalent to Model 12), the two four- sided acute pyramids cut off from the apices of P+MT, will be replaced by two four-sided obtuse pyramids, such as are exhibited by Model 14. In all cases of this sort, the planes of P + MT will remain untouched where they meet to form the equator ; so that, however the form may be cut away about the poles pz and p^,, we shall still find about the equator eight planes that have the following polaric positions : Zn 2 w 2 , Zn 2 e 2 , Zs 2 e 2 , Zs 2 w 2 . Nn 2 w 2 , Nn 2 e 2 , Ns 2 e 2 , Ns 2 w 2 . The multiplication by 2 of the poles around the equator, shows that the residual portions of the planes are equally near to any two poles at that level, but at a greater distance from the poles p| and p^, which is the necessary result of the equality of the equatorial axes and the compara- tive greater length of the vertical axis. 155. Let us next place Model 13 in the position prescribed in 152, to make it represent the form PM_j_T. It is evident that in this case, the portions of the planes most liable to be removed from the form in the event of combination are those attached to the poles n and s, and that 48 PRINCIPLES OF CRYSTALLOGRAPHY. the portions most likely to remain upon the combination are the following eight planes attached to the edges of the east meridian : Z 2 ne 2 , Z 2 nw 2 , Z 2 se 2 , Z 2 sw 2 . N 2 ne 2 , N 2 nw 2 , N 2 se 2 , N 2 sw 2 . 156. We shall now place Model 13 in the position which is repre- sented in 153, as being peculiar to the form PMT+. When this form enters into combination, the portions of its planes that are most liable to be replaced are those that extend to the poles t* and t, and the rem- nants of planes most likely to be found upon the combination, are those attached to the north meridian, which are the eight following : Z 2 n 2 e, Z 2 n 2 w, ZVe, ZVw. NVe, N 2 n 2 w, N 2 s 9 e, NVw. 157. Upon comparing these three results we find that the 24 remnants of planes that may be exhibited by the forms P + MT, PM + T, and PMT + , in combination, have all different positions ; that every plane has a basis attached to two poles of the combined form; that there are twenty-four of these bases joined two and two together at the base ; that the twelve edges between these twenty -four bases have the positions of the twelve edges of the octahedron ; and that lines drawn perpendicu- lar to the centres of these twelve bases, will all unite in the positions Zne, Znw, Zse, Zsw, Nne, Nnw, Nse, Nsw, which are coincident with the centres of the planes of the regular octahedron, or with the solid angles of the cube. It follows that if the three forms P_j_MT, PM + T, and PMT_|_, were to combine to form one crystal, the combination must have all the twenty-four planes described in 154, 155, 156; that these planes must meet in pairs at each of the four sides of the equator, of the north meridian and of the east meridian ; that they must meet again in lines that run from the six poles of the crystal towards the points that coincide with the centres of the eight planes of the octahedron ; and that, meeting in this manner, they make a complete form by cutting each other into twenty-four equal and similar obtuse isosceles triangles. 158. Model 17. Triakisoctahedron. P + MT,PM+T,PMT + . This model exhibits the result of the threefold combination described in the last paragraph, embracing all the planes of the forms P + MT, PM + T,PMT_j_. There are consequently 24 planes arranged in 8 sets of 3 each as compared with the octahedron, or in 12 sets of 2 each as compared with the rhombic dodecahedron, with which this form has some analogy. All the planes are obtuse isosceles triangles, the bases and apices of which meet in the points described in 157. It is easy to discriminate the planes belonging to each of the three complements. The 8 planes whose bases are attached to the equator, and which have the positions Zn 2 w 2 , &c., being those that are farthest removed from the axis p a , are consequently those that have the formula P+MT. The 8 planes whose bases are attached to the east meridian, and which are therefore PRINCIPLES OF CRYSTALLOGRAPHY. 49 farthest removed from the axis m a , are the set PM+T. The 8 planes whose bases are attached to the north meridian, and which are therefore farthest removed from the axis t% are the set PMT+, 159- It will be useful to compare the forms and positions of the planes of Model 22 with those of Model 17. We perceive that on the former model, the Icositessarahedron, every plane touches one of the six poles of the crystal, and thence proceeds towards two other poles, which it does not reach ; and that on the latter model, the triakisoctahedron, every plane touches two poles equally, and thence stretches towards one other pole, which it does not reach. Hence we infer that the axes of the fundamental octahedrons of which these two combinations are consti- tuted, are essentially different from one another ; and that the axes of the fundamental forms of Model 22 have the relation of -, +, +, arid those of Model 17, the relation of +, -, -. The observation of the polaric positions of the planes that we find upon complex combinations, serves thus to guide us in forming an opinion of the comparative lengths of the axes of the simple forms to which the planes belong. The following two rules are of special service to us when we are making observations of this nature : 1. When the polaric position of a plane is such as to prove that it must cut two axes if extended all ways till it meets the axes, the plane will be found upon the combination nearest to that one of the two axes which it cuts nearest the centre of the crystal, and the more unequal the length of the two axes, the nearer will the plane be found to the shorter axis. 2. When the polaric position of a plane is such as to show that it must cut three axes when extended in all directions till it meets the axes, then the comparative lengths of the three axes may be determined from the position which the plane occupies upon the combination, , If it is placed close to one axis, and passes thence equally towards two other axes but without touching them, the relation of the axes will be p, m a , t a , or -, +, +. See Model 22. b, If it is placed equajly close to two axes, and thence proceeds equally towards a third axis but without touching it, the relation of the axes will be pjj_ m a t% or +, -, -. See Model 17. c, If the plane is placed close to one axis, proceeds thence chiefly towards the second axis, and only slightly towards the third, thus having an unequal relation to all the three axes, then these axes must necessarily be denoted by p_|. m a t. See Model 25, which will be more fully described here- after. 177. 160. The varieties of the Triakisoctahedron which have been found in the mineral kingdom, are the three following : 1. P|MT, PMfT, PMT|. 2. P 2 MT, PM 2 T, PMT 2 . 3. P 3 MT, PM 5 T, PMT 3 . H 50 PRINCIPLES OF CRYSTALLOGRAPHY. They are discriminated by attention to the following angles : PMT+ ZVw upon PfMT, &c. P 2 MT, &c. P 3 MT, &c. PMT + ZVe = 129 31' 141 3' 153 28' P + MT ZnV 2 = 162 39^ 152 44' 142 S f PM+T Z 2 nw 2 = 162 39^ 152 44' 142 8' The combination represented by Model 17 is the second of these three, or that denoted by the symbol P 2 MT,PM 2 T,PMT 2 . 161. These six varieties of the isosceles octahedron, namely, the three obtuse varieties P_MT, PM_T, PMT_, and the three acute varieties P+MT, PM + T, PMT + , are all that can occur under the essential condi- tion of having one short axis and two long axes. All other changes in the axes have relation to the variable value of the quantities represented by the signs - and +, and do not affect the unmarked axes. For this reason, I propose, when all the three permutations of one kind occur upon the same crystal, to abridge the symbols as follows : 3 P_MT instead of P_MT,PM_T,PMT_. 3 P+MT instead of P+MT,PM4.T,PMT + . in which examples the term 3 signifies the three regular permutations of the three axes. When only one or two similar forms occur upon a com- bination, they must be particularized. The six forms never occur as single uncombined crystals, because all the obtuse forms presented in substance would, according to this system of crystallography, be de- scribed as P_M, P_T, and all the acute forms as P+M, P+T, in the idea of their being quadratic (square-based) octahedrons pertaining to the series P X M,P X T. The isosceles octahedrons formed by the permutations of P X MT are to be regarded universally as forms peculiar to complex combinations. SCALENE OCTAHEDRONS. PJVI y T z . 162. I return now to 126, in which it will be seen that we have still to investigate the nature of the octahedrons that are produced when the combining rhombic prisms cross one another obliquely. ] 63. I shall proceed to consider the nature of the different symmetrical scalene octahedrons that can be produced by the combination of two similar and equal prisms whose cross section is an acute rhombus having angles of 143 24' and 36 36', and whose diagonals are consequently nearly as the numbers 1 and 3, because the tangent of 71 44', the half of 143 24', is 3-029632. You will find the angle of 143 24' to be nearly that of the incidence of the Zenith planes of Model 21 upon the Nadir planes, the measurement being taken with the goniometer right across the horizontal edge that surrounds the equator of the model. 164. It may be useful, in the first place, to show the nature of the principal isosceles octahedrons which caji be produced by the combina- PRINCIPLES OF CRYSTALLOGRAPHY. 51 tion of two rhombic prisms of the above-named dimensions. These forms will serve as points of comparison. a, If one of the prisms passed along the axis t% with the longer diagonal of its cross section parallel with the axis p% it would produce the planes P 3 M. If the other prism passed along the axis m a , with its longer diagonal parallel with p a , it would produce the planes P 3 T. This com- pletes the acute quadratic octahedron P 3 M, P 3 T. ft, If one prism passed along t a with its longer diagonal parallel with m% it would produce the planes PM 3 . If the other prism passed along m a with its longer diagonal parallel with t a , it would produce the planes PT 3 . This completes the obtuse quadratic octahedron PM 3 , PT 3 . c, If one prism, having its longer diagonal parallel with the axis p a , and striking the system of axes at the level of the equator, passed mid- way between m a and t a , cutting both axes equally at an angle of 45, and if this prism was crossed at right angles by the other similar prism, similarly situated, the combination resulting from the operation would be the acute quadratic octahedron P 3 M h4 i 4 T li4U . d, If the two prisms crossed each other under the circumstances just recited, with the single difference of having the shorter instead of the longer diagonal of the prisms parallel with the axis p% the resulting form would be the obtuse quadratic octahedron PiM^^T^^- , I have here to explain the cause of the apparent variations in the lengths of the axes represented in examples a, ft, c, d. I shall give the explanation with reference to the lower part of the diagram in page 6. The lines T 2 T 3 and M 2 M 3 show the course of the prisms as described in cases a and ft, and the square 45 11 10 may be considered to be the base of the resulting octahedrons. But the course of the prisms described in cases c and d is represented by the lines 411 and 510, and the base of the resulting octahedrons is shown by the square M 2 T 2 M 3 T 3 . In the first two cases the axes of the crystal coincide with the axes of the rhombic prisms and are of the same length, but in the last two cases the axes of the crystal are represented by the lines M 2 M 3 and T 2 T 3 , while the axes of the prisms are represented by so much of the lines 411 and 5 10 as lies between the centre p^ and the lines M 2 T 2 M 3 T 3 ; one set of axes crossing the other set at an angle of 45. The consequences are, that the axes of the crystal bear to the axes of the prisms, in cases a and ft the ratio of 1 to 1, but in cases c and e?, a ratio equivalent to that which the secant of an arc or angle of 45 bears to* the radius of that arc; i. e. the ratio of 1.414 to 1.000. Hence the axes of these crystals are equal to the axes of the prisms x 1.414, which number being, in case c, multiplied by 1, gives 1.414, and in case d, multiplied by 3, gives 4.242. If it is requisite to consider the axes m a and t a of these octahedrons to be unity, the above symbols may be converted into others, as follows : 1.414 : 3.0 :: 1.0 : 2.121 4.242 : 1.0 :: 1.0 : 0.236 These proportions give the symbols P 2 ; 121 MT for P 3 M,. 414 T,. 414 ; and P 236 MT for P.M^TMU. 52 PRINCIPLES OF CRYSTALLOGRAPHY. f, By certain processes of derivation which have been already fully explained, we can account for the production of the following isosceles octahedrons, which are equivalent in dimensions to those just described : PM. 836 T, as described in 131. PMT 2S6 , as described in 132. PM 2<121 T, as described in 149. PMT 2 .i2M as described in 150. g, The relation which the axes of an isosceles octahedron P_MT or P_I_MT, bear to the axes of the rhombic prisms that produce it, being necessarily a constant character, and one that is easily examined, is, in consequence, a character of great utility ; for it enables us to determine the ratio of the axes of such octahedrons with very little trouble, either of measurement or calculation, as I shall show by a few examples. h, The angle of incidence of the planes of the regular octahedron that meet at the equator is 109 28'. The tangent of the half of this angle is 1.414, which is equal to the axis p a , the corresponding radius being a line passing from the centre of the crystal towards the north-west, and cutting the equator at the point nw. This line is =r 1.0. It forms with either of the axes m a or t a an angle of 45, and on the principle laid down above (letter e), either m a or t a is equal to the secant of this angle ; therefore =r 1.414. Hence the three axes of this crystal p a m a t a are each equal to 1.414. a, It is stated in 146, that the angle of incidence of the plane P^MT Z 2 nw upon the plane PJMT Z 2 se is = 109 28'. The angle of incidence of two planes of the same form across the equator must there- fore be equal to the supplement of this angle, or to 70 32'. The tangent of the half of this angle is 0.707, which is equal to p a . On comparing this with the constant value of the axes m a or t a , namely, 1.414, we see at once the relation of 0.707 to 1.414, rz \ to 1, upon which relation the symbol P^MT is founded. j, It is also stated in 146, that the incidence of two upper planes of the form P^MT over the apex of the crystal, is 129 31'. I again take the supplement of that angle as the measure of the incidence of two planes of the same form across the equator, and find the tangent of its half = 25 14^', to be 0.4714. On comparing this with the constant value of the axes m a and t a =r 1.414, I see the foundation of the symbol PMT, for 1.414 divided by 0.4714, gives 3. k, You may have to solve a problem that is the converse of those above described. Suppose, for example, that you have the symbol PMT 2 .i2i, and desire to know the value of the angle across the Zn edge, that is to say, the angle of incidence of the plane PMT 2 . 121 Z 2 n 2 w upon the plane PMT 2 .i 2 , Z 2 n 2 e. What you have to do in this case is to multi- ply all the axes of the crystal by 1.414, so as to make p a and m a each = 1.414, and to increase t a in a corresponding ratio. This multiplication produces P|.<^t.4uT s ;9 W . The angle which is desired lies in the direc- tion t a Zn t a . Zn is the middle of this term, and Zn is situated midway PRINCIPLES OF CRYSTALLOGRAPHY. 53 between p| and m a . A line drawn from Zn to the centre of the crystal, which line I will call the polaric line Zn, is equally inclined to the axes m a and p a , and forms an angle of 45 with both of them, because the axes p a and m a cross one another at an angle of 90 and Zn divides this angle into two equal angles. The length of the polaric line Zn is limited at one end by the centre of the crystal, and at the other by the straight line or edge that connects p| with m a . Hence the polaric line Zn is the radius of an angle of 45, and the axes p a and m a are both secants of that angle. Consequently, the length of the polaric line Zn is 1.0, because p a and m a are each = 1.414. Finally, the polaric line Zn = 1.0, is the radius of an angle whose tangent is the axis t% already found to be = 2.999, and the angle that corresponds to this tangent, or to 3.00028, which is nearly the same, is 71 34', the double of which = 143 8' is the desired value of the angle across the Zn edge. 165. Let the aforesaid two rhombic prisms, 163, cross one another at the level of the equator with their longer diagonals parallel with the axis p a . Let the line of their direction be closer to the axis t a than to the axis m% and let them cut both m a and t a so as to produce angles of 50 58' and 39 2' with those axes. This process will produce a scalene octahedron similar to Model 21, the measurements of which are as follow 7 : The equator is a rhombus with angles of 101 56' at m a and m a , and of 78 4' at t a and t a . The north meridian is a rhombus with angles of 133 53' at m a and m*, and of 46 7' at p| and p^. The east meridian is a rhombus with angles of 124 36' at t a and t a , and 55 24' at p| and p^. The model agrees pretty closely with these measurements, as will be found upon applying the goniometer to its edges, in the indicated directions. 166. The scalene octahedron is so called in consequence of its planes being scalene triangles. It has 8 planes that occupy the same polaric positions as the 8 planes of the regular octahedron. It has three unequal axes which are always p_|L m t a . That is to say, it is to be made a rule that every simple scalene octahedron is to be held with the longest axis in the place of p% and the shortest axis in the place of m a . I have called m a the minor axis, with a view to the establishment of this general rule as to the polaric position of any simple scalene octahedron. In all exam- ples which follow, whenever a scalene octahedron, or a rhombic prism occurs, I shall make the shorter diagonal of the rhombic base of the octa- hedron, or the shorter diagonal of the rhombic prism, coincide with the axis m a . And, for the sake of producing uniformity in the mode of cal- culating formulae for the simple scalene octahedrons, I shall always consider the axis t a as unity, which will give them the general symbol P_I-M_T. The scalene octahedrons of other dimensions than P_j_M_T, which only occur upon combinations, and not as substantive uncombined crystals, are of course not embraced in this regulation. 167. Let us now determine the comparative length of the three axes 54 PRINCIPLES OF CRYSTALLOGRAPHY. of the scalene octahedron Model 21. The obtuse angle of the equator at m* is = 101 56'. The half of this is 50 58', and the tangent of 50 58' is 1.2334, which gives the ratio of m a t a . i334 ; but as I have resolved to make the axis t a = unity, I take, with that purpose, not the tangent of the angle of 50 58', but its cotangent, which is 0.8107, and this supplies the symbol m a . 81 t a .oo. I proceed next to ascertain the relation of the axis p a to the axes m a and t a , and as I propose again to consider t a = unity, I perceive that the investigation is one that relates to the form of the east meridian of the crystal, because the diagonals of that meridian are the axes p a and t a . The obtuse angle of the east meridian at t is = 124 36', the half of which is 62 18'. In this case I must take the tangent of the angle and not the cotangent, because I have to estimate the longer of the two axes, and not, as in the last paragraph, the shorter. The tangent of 62 IS' is 1 .9047, which gives the symbol p a . 9 t a >0 . The three axes of the crystal have consequently the ratio of p a . 9 mo 81 t a , and the symbol which designates the complete form is P l 9 M 081 T, which is a variety of the general symbol P + M_T, 1 66. Every other variety of the simple scalene octahedron can be expressed by a similar numeri- cal variation of the same general symbol ; and the numerical value of the three axes of such an octahedron can be always ascertained in the above manner. The cotangent of half the north angle of the equator is equal to the axis m a ; and the tangent of half the west angle of the east meridian is equal to the axis p a : the axis t a being considered as unity in both cases. Having thus fully explained the derivation of a scalene octahedron from two right rhombic prisms, by their intersection at a certain degree of obliquity, and having also explained the method that is to be employed to determine the properties of the resulting form, I shall content myself with giving a very brief description of the several other scalene octa- hedrons that may be derived from the same two rhombic prisms by pro- cesses so entirely analogous to the foregoing, that the mere description of the forms is of itself sufficient to point out the mode of derivation. 168. THE Six VARIETIES or THE SCALENE OCTAHEDRON. 0. P_MT_j_, or P. 81 MTi. 9 . Hold model 21 in such a position that its three axes, which are held to terminate in the solid angles of the Model, become plm'tf. ; that is to say, place the two most acute solid angles at t a and t* , the two most obtuse solid angles at p| and p, and the two inter- mediate solid angles at m a and m a . The Model then represents the form whose symbol is P. 81 MT 1>9 . This form differs in no respect from the form that has been designated, Pi. 9 M .8iT, except in its position. If it were given to me to be described as a substantive form, I should term it Pi.oM. S iT, making its longest axis equal to p a and its shortest equal to m a , agreeably to the rule laid down in 166. The form P^MT^ is one that occurs only in combination with other sets of planes, and this re- mark applies to all the varieties of octahedral forms that follow. It is PRINCIPLES OF CRYSTALLOGRAPHY. 55 with these scalene octahedrons, as with the isosceles octahedrons described in 128 161, they are described individually that they may be cor- rectly descriminated when they are found in combination with one another, but not with a view to their being considered and described as so many substantive forms or complete crystals. The three obtuse isosceles octahedrons = P_MT, PM_T, PMT_, 144, must all, as separ- ate crystals, be denominated P_M, P_T, in which case they would be three equal and similar obtuse quadratic octahedrons; and the three acute isosceles octahedrons P + MT, PM + T, PMT + , 145, must all, as separate crystals, be denominated P+M, P_j_T, being considered as three equal and similar acute quadratic octahedrons. In like manner, the six varieties of the scalene octahedron, which I have to describe here, and which are all similar and equal to the one already described, must, as separate crystals, in comformity with the general rule, 166, be denomin- ated P_j_M_T, as six similar and equal scalene octahedrons. This ex- planation is given to show the importance of paying attention to the positions of these and of all varieties of the octahedron, since it is the positions, and not the shape, or the number of the planes, which consti- tutes that difference in the form which requires a difference in symbols to convey the idea of it. b. P + M_T, or P^M^T. Hold Model 21 in such a position that its axes become pjpnlt*. This is the form already fully described in 165. c. PM_j_T_, or PMi.9T.8i Hold Model 21 in such a position that its axes become p a m+t_! . It then represents the form PM_j_T_. d. P_M + T, or P 81 Mj. 9 T Hold Model 21 in such a position that its axes become pm4.t a . It then represents the form P_M + T. e. PM_T + , or PM. 81 Ti. 9 Hold Model 21 in such a position that its axes become p a in.!t_|_. It then represents the form PM_T_j_. / P + MT_, or P 19 MT 81 Hold Model 21 in such a position that its axes become p_jLm a tl. It then represents the form P_j_MT_. 169- These six are all the symmetrical scalene octahedrons that can be produced by any method of altering the position of the simple form repre- sented by Model 21. We can hold it in positions that will produce various mixed or incomplete forms, but we cannot hold it so as to exhibit any other variety of symmetrical scalene octahedron. The reason is, that six is the greatest number of permutations to which three dissimilar axes can be subjected. But there may be a great many varieties of each of these six kinds of the scalene octahedron, each variety depending for its peculiarites, partly upon the original proportions of the rhombic prisms by whose intersection it is produced, and partly upon the degree of the obli- quity of their crossing. All of them, however, are capable of discrimination by reference to the distances from the centre of the crystal at which the three axes p a m a t a are cut by the planes which constitute each form. The shape of the planes of every scalene octahedron depends upon the distance of each corner of the triangular face from the centre of the crystal. These distances, however, are not different in every individual crystal, 56 PRINCIPLES OF CRYSTALLOGRAPHY. but constant for all the crystals of the minerals of one species ; so that the shape of the planes, or the lengths of the axes of a scalene octahedron, is a character which serves to discriminate one mineral species from another, among those that are in common subject to assume the form of a scalene octahedron, 170. What befalls the planes of the scalene octahedrons when they com- bine with other planes. When the scalene octahedrons combine with other forms they suffer replacement at one or other of their poles according to the ratio which their axes bear to the axes of the forms which combine with them. They are in this matter subject to the same laws as the isosceles octahedrons, and much of what I have written in relation to the latter applies equally to the forms that are now under consideration, more especially what is contained in 136143, 154159- EXAMPLES. 171. Model 80. p + . P li9 M. 81 T. p4_mt a . This is the scalene octahedron P + M_T,* 165, combined with the planes P, the latter subordinate. If, in such a combination, the planes P had a very short axis, the combination would have the appearance of a rhombic table with its terminal edges replaced. The symbol would then have to be P_. p^m.^t. p.lm a t.|.. 172. Model 70. M_. P^M.^T. p4.m!.t a . This is the scalene octahedron I\ 9 M. 81 T, combined with the planes M, the latter subordinate, but still not very small, and therefore to be repre- sented by a small capital letter. If, in such a combination, the planes M were very large, or, what comes to the same thing, if the axis m a was very short, the combination would be a vertical rhombic table having its edges replaced, and its symbol would be M_. pi. 9 m. 81 t. p+mlt*. 173. Model 66. M. 81 T.P lt9 M 81 T. pjf.mjlt a . This is the scalene octahedron P 19 M. 81 T combined with the rhombic prism M. 81 T, neither form predominating. If the vertical planes had been much smaller in proportion to the inclined planes, the symbol must have been m. 81 t. P li9 M. 81 T. p^mjlt' 1 . If they had been much larger in pro- portion to the others, the symbol would have been M. 81 T. p,. 9 m. 81 t. p+mlt 1 . 174. Model 120. p 19 t. P 1>9 M. 81 T. p.|.m a t. This is the scalene octahedron P li9 M. 81 T, combined with the planes P 1>9 T, the latter subordinate. 175. The planes of replacement on this form, and those on the form represented by Model 66, have the same proportional axes as the planes of the scalene octahedron, upon which they are superinduced. The equator of the prismatic planes M__T, Model 66, is similar to the equator of the octahedral planes P + M_T, Model 21. And the east meridian which PRINCIPLES OF CRYSTALLOGRAPHY. 57 bisects the planes P+T, Model 120, is similar to the east meridian that passes through the edges of the octahedron P+M_T, Model 21. In these two cases, there would be no replacement if all the forms were equal as well as similar, but not being equal, there is a displacement of that portion of the form represented by Model 21 that is larger than the equator of M_T in Model 66, or than the east meridian of P+T in Model 120. And the forms M_T and P + T take the place of the portions of P,. 9 M 81 T that are displaced in each example. We have in this series of Models, 21, 80, 70, 66, 120, a good illustration of the statements made in 67, respecting the replacement by the planes of small crystals, of portions of planes cut off from large crystals. 176. What befals the planes of the scalene octahedrons when they com- bine with one another. The case that is to be explained under this title, is of the same impor- tance in respect to the scalene octahedrons as the cases explained in 133145, and 151 160, were in respect to the isosceles octahedrons for, just as the complex octahedrons which are represented by Models 22 and 17 are formed by the combination of certain groups of isosceles octahedrons, so there are other complex octahedrons which are formed by the combination of certain groups of scalene octahedrons, and it is to the consideration of these complex scalene octahedrons that I now pro- ceed. 177. Model 25. Right Hemihexakisoctahedr on with parallel faces. P_MT + ,P+M_T,PM + T_. The reader is in the first place referred to what is said in 159, 2 c. respecting the partial replacement of the planes of scalene octahedrons, or those whose axes are p* m* t*, when they combine with one another. Place Model 25 in an upright position for examination. 178. Hold Model 21 in the position described in 168 , which brings its axes to the ratio of pi m a tj[_. Compare the four planes that touch the pole p| of Model 25 with those parts of the planes of Model 21 that touch the same pole. You will see that their inclinations and posi- tions are nearly the same. The two models have not the same axes, or they would be exactly alike at the pole referred to. They represent different minerals, and were not intended to be used for a comparison of this nature; yet they serve to show that if Model 21 was ground down at the four acute poles, the residual planes left about the pole p would resemble the four upper planes of Model 25. If you next examine the 4 planes that touch the pole p of Model 25, you will find that they agree, in a similar manner, with the planes that touch the same pole of Model 21. The inference to be drawn from the result of this examination is, that the eight planes that surround the poles p and p^ of Model 25 belong to a form such as is represented by Model 21, when held in the 58 PRINCIPLES OF CRYSTALLOGRAPHY. position described in 168 a, so as to require the symbol P_MT + to designate it. 179- Hold Model 21 in the position described in 186 b, so as to make its axes agree with p.|_ ml t a . Then compare the four planes that touch the pole m* of Model 25, with those parts of the planes of Model 21 that touch the same pole, and observe that the two models agree with one another. Make the same comparison between the planes that are situated about the pole m| of both models. You will thus perceive that Model 25 contains also the eight planes that are denoted by the symbol P+M_T. 180. Hold Model 21 in the position described in 168 c, so as to make its axes = p a m_ t. Then compare the eight planes that touch the poles t* and t of Model 25 with those portions of the planes of Model 21 that touch the same two poles. The similarity of their posi- tions and inclinations will satisfy you that Model 25 contains the eight planes which are peculiar to the form denoted by the symbol PM + T_. 181. The twenty-four planes of Model 25 are consequently those which are produced when the three forms P_MT + , P + M_T, PM + T_, cut one another and come together upon one crystal. The combination is of the same nature as that which takes place when the three obtuse isosceles octahedrons P_MT, PM_T, PMT_, combine to produce the form represented by 3 P_MT, Model 22 ; or, as that which takes place when the three acute isosceles octahedrons P + MT, PM+T, PMT + com- bine to produce the form represented by 3 P_|_MT, Model 17. 182. The comparison that I have instituted in 159 between Models 22 and 17, may now be usefully extended to Model 25. I have shown, 159: 1,2, a,6, that we can judge of the comparative lengths of the axes of the forms that are exhibited on Models 22 and 17, from the information that is to be gained by observing the polaric positions of their respective planes, and in 159, 2, c, I have stated that a similar procedure in respect to such forms as are represented by Model 25, would give similar information regarding their axes. It is easy to test the accuracy and utility of this statement by an examination of the models. 183. With Models 22, 17, and 25 placed before us at the same level, and supported in upright position by the mouths of three wine glasses, we can, by directing our notice to the Znw octant of each form, readily make a comparative examination of all three. 184. Model 22. We observe that the plane marked P, which occupies the position Z 2 nw, touches the pole p| and thence proceeds equally towards the poles m* and t which it does not touch. This is the char- acter of a plane of the form P_MT. The two planes that are between the plane Z 2 nw and the equator must both have a longer perpendicular axis than the plane above them, else they would cut the axis p a nearer the centre of the crystal than it is cut by the plane P_MT ; but as in that case the plane P_MT would not appear upon the model at all, but would be entirely displaced by the other two planes, this is a considera- tion that demonstrates the truth of the assumption that the plane which PRINCIPLES OF CRYSTALLOGRAPHY. 59 occupies the position Z 2 nw is a plane belonging to the form P_MT. By a similar train of reasoning it may be proved that the plane which touches the pole m* and is marked M, is a plane of the form PM_T, and that the plane which touches the pole t and is marked T, is a plane of the form PMT_. And that, as each of these three planes has seven counter- parts in the other seven octants of the crystal, the model must necessarily represent the combination PJMT, PM_T, PMT_, or 3 P_MT. 185. Model 17. The plane marked P and M which touches the poles p| and m* equally, and proceeds thence towards the pole t which it does not touch, and which therefore occupies the position Z 2 n 2 w, is a plane of the form PMT + . The plane which touches the poles p| and t equally, and proceeds thence towards the pole m* which it does not touch, and which consequently occupies the position Z 2 nw 2 , is a plane of the form PM+T. It is evident from the direction of the line of com- bination between these two planes, which is from the point Z to the point Znw, that they both cut the axis p a at the same distance from the centre of the crystal. If either of these planes had cut the axis p a nearer to the centre than the other plane, it would have prevented that other plane from touching the pole p|. The plane that touches the poles m* and t v a equally, and proceeds thence towards the pole p| which it does not touch, and which consequently occupies the position Zn 2 w 2 , is a plane of the form P+MT. It is evident that this plane touches the axis p a at a greater distance than either of the planes that occupy the positions Z'n 2 w and Z 2 nw 2 , for it is entirely separated from the pole p| by their intervention. It is also evident that the plane P+MT cuts the axis t a shorter than that axis is cut by the plane PMT_|_, because the latter is entirely separated by it from the pole t ; and finally it is evident that P + MT cuts the axis m a shorter than that axis is cut by the plane PM + T, for it entirely separates by its intervention that plane from the pole m*. All these examples concur in proving that the planes found upon a complex equiaxed combination which is composed of unequiaxed forms, are those parts of the planes of the unequiaxed forms that are situated upon the poles of the shorter or shortest of their unequal axes. 186. Model 25. The above illustrations are intended to be preparatory to the examination of Model 25, and if I have succeeded in conveying my ideas, the reader will find little difficulty in comprehending the nature of the combinations of which this model is the type. The three planes situated in the Znw octant of Model 25, partly resemble those contained in the same octant of Model 22 and partly those of Model 17. That is to say, they resemble those of Model 22 in being each attached to one pole, while they resemble those of Model 17 in being each directed more upon two poles than upon the third. For example, the plane marked P touches the pole p and proceeds thence towards the poles m* and t, but principally towards the pole m*, although it touches neither of them. The plane marked M touches the polem*, and proceeds thence towards the poles t* and p|,but principally towards the pole t, yet touches neither. The plane marked T touches the pole t, 60 PRINCIPLES OF CRYSTALLOGRAPHY, proceeds thence towards the poles p| and m* , but principally towards the pole pi, yet without touching either of them. This is the relation described in 159> 2, c. It is evident from these positions, that each of the three planes that are situated in the Znw octant of Model 25, must, if extended till they cut all the three axes, cut them all unequally ; that the axes of the plane P must have the ratio of p m a t_f. ; that the axes of the plane M must have the ratio of p_|_ ml t a ; that the axes of the plane T must have the ratio of p a m+tl; and that consequently the com- bination must consist of the planes P_MT + , P + M_T, PM + T_, or be a combination of three equal and similar scalene octahedrons ; for that the simple forms must be equal and similar is demonstrated by the exact symmetry of the sets of planes in every one of the eight octants of the combination, and by the precise equality of the three planes in each octant. It is also evident from the positions of the planes, that the three octahedral forms that are present on the combination, are those described in 168 , , C) and not those described in 168, d, e,f. 187. The foregoing observations all refer to an octant^ or one eighth part of the combination; but this limited observation is quite sufficient for all practical purposes, since all the eight octants of an octahedron are equal and similar. It is worthy of remark, that the octants are the divisions into which an octahedron is divided by the equator, and the east and west meridians of the form. The equator separates the zenith from the nadir portion of the combination. The north meridian separates the east from the west portion. The east meridian separates the north from the south portion. The whole form is thus equally and very con- veniently divided into eight portions, and we are enabled to simplify our observations and calculations by referring them to one of these octants, and assuming it as an axiom that what is true of one of them is true of the whole. I therefore always refer to the Znw octant of a combination, as that is the octant which is commonly depicted in figures of crystals, and as indeed it is that which it is generally most convenient to examine. For the same reason, the Models of Crystals have all been marked with P,M,T, at those poles of the axes which denote the limits of this particu- lar octant, namely, at the poles pj, m*, and t. Method of denoting the polar ic positions of planes on the combinations of scalene octahedrons. 188. The unequal bearings of the planes of the Hemihexisoctahedron upon the poles of the crystal, renders it necessary to mark the polaric position of each of the twenty-four planes by symbols so distinct as not to be readily misunderstood. I propose to do this as follows : The symbol Znw denotes, as a general term, the Zenith-north-west octant of an octahedral form, or any plane that bears equally upon the three poles which limit that octant. Z 2 nw, ZnV, Znw 2 denote a plane in that octant which bears more upon one pole than upon the two others. Z 2 n 2 w, Zn 2 w 2 , Z 2 nw 2 , denote a plane in that octant which bears more upon any two poles than upon the third. Following out this system of notation, the symbol ZVw may be held to denote a plane that bears principally upon the pole PRINCIPLES OF CRYSTALLOGRAPHY. 61 p|, less upon the pole m*, and still less upon the polet; which bearings would indicate the symbol P_MT + as the formula of the set to which the plane belongs. Secondly, the symbol Zn 3 w 2 may indicate a plane that bears principally upon the pole m* , less upon the pole t, and still less upon the pole pi? which bearings serve to indicate a plane the sym- bol for whose complement would be P+M_T. Thirdly, the symbol Z 2 nw 3 may indicate a plane that bears principally upon the pole t, less upon the pole p|, and still less upon the pole m*, which bearings point out the symbol PM+T_ as characteristic of the set to which this plane belongs. These three planes are all that belong to the Znw octant of the combination, and being thus denoted by three different signs, the whole combination is in fact denoted, for every other octant can be denoted in a similar manner ; retaining always the measures of prox- imity 2> 3> and changing Z for N, n for s, w for e, and so on, as the differ- ent octants require. In this manner the following twenty-four symbols may be produced to distinguish the twenty-four planes of the combina- tion represented by Model 25. For the convenience of subsequent reference, I have added the value of the axes of the component octa- hedrons of the combination which the model represents. The proofs of this value will be given afterwards. Planes at pj. Z 3 n 2 w Z 3 n 2 e ZV 2 w Z 3 s 2 e Planes at p^. N 3 n 2 w N 3 n 2 e NVw N 3 s 2 e Planes at m* . Zn 3 w 2 Jnw" Nn 3 e 2 Planes at m* . Zs 3 w 2 Zs 3 e 2 Ns 3 w 2 Ns 3 e 2 Planes at t. Z 2 nw 3 2 sw 3 N 2 nw 3 N 2 sw 3 Planes at t*. Z 2 ne 3 Z 2 se 3 N 2 ne 3 189- The mineral kingdom has furnished three varieties of the Right Hemihexakisoctahedron with parallel faces: two of them as substantive crystals, and the third in combination with other planes. The symbols for these three varieties are as follows : i, P^MIT, PMJTJ, 2, 3, They may be discriminated from one another by the difference in the angles at which their planes incline upon one another, as shown in the following table: Inclination of the plane P_MT + Z 3 n 2 w upon : COMBINATIONS. P_MT + P_MT + P + M_T PM+T_ Z 3 n 2 e. Z 3 s 2 w. Zn 3 w 2 . Z 2 nw 3 . 149 O x . 115 23'. 141 47'. 141 47' 154 47'. 128 15'. 131 49'. 131 49 r 160 32'. 118 59'. 131 5'. 131 5' 62 PRINCIPLES OF CRYSTALLOGRAPHY. It is the first of these three combinations that is represented by Model 25, as may be proved by measuring with the goniometer the angles of the incidence of its planes. The second form is distinguished from the other two by having trapezoids and not trapeziums for its planes, the three edges that radiate from the point Znw being parallel with the three edges that join the poles p|, m a , and t. 190. The method of proceeding to determine the value of the axes which belong to the component octahedrons of a combination of this kind, is as follows. I take Model 25 as an example. a. As respects the planes that touch the poles m a and m a . The goni- ometer is applied to the edges of the equator across the pole m a . The angle is found to be 112 38'. The half of this is 56 19', of which arc the tangent is 1.5004. Hence the axes m a and t a have the ratio of m a t a .5004, or ml t% or m a f t a f , or m a t a j. The goniometer is next applied to the edges of the north meridian, across the pole m*. The angle is found to be 143 8'. The half of this is 71 34', the tangent of which is 3.0003. Hence the axes m a and p a have the ratio of mtpa.owm or m i Pa> or m a ^ p a l, or m a f p a f, or m a p a . Hence the ratio of all the axes is = pg m 3 . t or p a l m a j t a j, and the symbol for the planes is PMJTJ, as quoted in the above tables. The measurement of the edges at the pole m a gives the same results as the measurements at the pole m 3 ,. b. As respects the planes that touch the poles t a and t The angle formed by the meeting of the two edges of the equator at the pole t. is 143 8'. That formed by the meeting of the two edges of the east meridian at the pole t is 112 38'. The two angles formed by the edges at the pole t a are precisely similar. These measurements give the ratio of mt a and p^ t a , orp a ^ m^t a , which requires the symbol PMT. c. As respects the planes that touch the poles p| and p The value of the angle of the north meridian at the pole p| is 112 38'. This gives the ratio of p a n%. The value of the angle of the east meridian at that pole is 143 8'. This gives the ratio of p a t a . The planes therefore re- quire the symbol PJM^T. The measurements at the pole p are exactly the same as those at the pole pj. d. Control over the accuracy of these measurements. The equator of the combination is an octagon, wherefore all its angles should be together equal to 1080. See 82. The angle at m a has been found to be (a) = 1 12 38'"] m * () = 112 38' 1 _ 511 o 32 , C (b)= 143 8' [ t a (b) =143 8') 511 32' deducted from 1080, leaves 568 28' for the value of the four remaining angles of the equator, which angles are those produced by the incidence of the planes of the form PM^TJ upon the planes of the form P^MT^, and which are all equal to one another. This aggregate PRINCIPLES OF CRYSTALLOGRAPHY. 63 sum divided by 4 gives 142 7' as the value of each of these four angles, and the application of the goniometer to any one of them shows this to be correct. The north meridian and the east meridian have each eight angles of the same value as the angles of the equator, as can be easily proven, either by direct measurement, or by calculations similar to the foregoing. 191- These examples show how very useful it is to examine, in all cases, the geometrical relations of those three sections of a crystal, which I have named the north meridian, the east meridian, and the equator; for as the poles of every simple octahedron rest in these three sections, and as their axes lie always two in one section, and the third in another section, at right angles to that which contains the two axes, we have in general several easy methods of measuring two of the axes and calculating the length of the third axis of every simple octahedron that forms part of a combination. If, for example, we have access to the north pole of a crystal, the angle of the equator gives the ratio of the axes m a to t% and the angle of the north meridian gives us the ratio of the axes m a and p a . If we have access to the west pole, the angle of the equator gives the ratio of m a to t a , and the angle of the east meridian gives the ratio of t a to p a . If we have access to the zenith pole, the angle of the north meridian gives the ratio of p a to m a , and the angle of the east merid- ian gives the ratio of p a to t a . If we cannot measure the angles at these three poles, but have access to the poles m a , t a , or p^, we obtain the same results. If the apices of the forms are cut from all the poles, but replaced by planes that are perpendicular to the axes, we can still obtain exact results by measuring the edges of combination, and calculat- ing the value of the displaced apices, according to the method that I have explained in treating of the properties of the form that is represented by Model 47. 83. 192. Left Hemihexakisoctahedron with parallel faces. P_M + T, PM_T+, P + MT_. Place Model 25 in upright position, and then turn it round a quarter of a revolution on its principal axis, so as to bring the letter M to the pole t a , and the letter T to the pole m a . It will then exhibit the follow- ing forms : P_M + T, or PJMT1, described 168 d. The planes touch the poles p| and p. PM_T + , or PJMJT, described 168 e. The planes touch the poles m and m*. P + MT_, or PM1TJ, described 168 /. The planes touch the poles t* and t. The combination contains altogether 24 planes, which individually occupy the following positions: 64 PRINCIPLES OF CRYSTALLOGRAPHY. Planes at p| Planes at m* Planes at P^MTJ ZW Z 3 ne 2 P^MTJ Z 3 sw 2 Z 3 se 2 PiMJTZVw Z 2 n 3 e N 2 nV N 2 n 3 e Zn 2 w 3 Zs 2 w 3 Nn 2 w 3 Planes at p Planes at m * Planes at t* N 3 nw 32 N 3 ne ! P^MTJ N 3 sw 2 N 3 se 2 ZVw Z 2 s 3 e P'-M^T NW NVe Nn NsV 1 93. Of the six symmetrical scalene octahedrons that are described in 168, the Right Hemihexakisoctahedron with parallel faces, comprehends the three first, and the Left Hemihexakisoctahedron with parallel faces, the three last. The two combinations are precisely alike if considered as substantive crystals, but they differ altogether in their positions, and consequently in the denominations of the planes that compose them. These differences are fully explained in the above table, and it does not appear to me to be necessary to dwell at all upon the examination of the Left Hemihexakisoctahedron, since what has been said of the means of investigating the properties of the Right form applies equally to the Left. The former combination sometimes occurs in the mineral kingdom as a substantive crystal, the latter never. This, with the differences of posi- tion, are the only grounds of distinction between them. 194. Model 23. Hexakisoctahedron. P_MT + , P + M_T, PM + T_, P_M+T, PM_T + , P + MT_. This is a combination of the six symmetrical scalene octahedrons, which, when combined in two groups, form the Right Hemihexakisocta- hedron with parallel faces, and the Left Hemihexakisoctahedron with parallel faces. The following description of the positions of the 48 planes upon Model 23 is intended to show that no other inference than the above can be legitimately drawn from the information afforded by the examination of these positions. The model bears somewhat the appearance of a cube that has a low eight-sided pyramid upon each of its six planes, and 8x6 = 48 planes ; or of a rhombic dodecahedron that has a four-sided pyramid upon each of its twelve planes, and 12 X 4 = 48 planes. It also has some resem- blance to an octahedron that has a six -sided pyramid upon each of its 8 planes, and 6 X 8 = 48 planes. There exists in the mineral kingdom some forms of this kind, in which the octahedral form is more prominent than it is in the model, which represents a variety that has a tendency to the form of the cube. All these varieties agree in having 48 planes dis- posed in the same relative positions, and they differ only in the angles at which the planes incline one upon another, the cause of which difference may be ultimately traced to the difference in the dimensions of the PRINCIPLES OP CRYSTALLOGRAPHY. 65 rhombic prisms which formed the simple scalene octahedrons that are the components of this complex combination. Planes of the Znw octant of the Hexakisoctahedron. 195. It will be sufficient for our purpose to examine the relations of the six planes that constitute the Znw octant of the combination, because these six planes comprehend one belonging to each of the six combining octahedrons ; and the knowledge of the position of one plane of each complement leads to a knowledge of the whole. These six planes are divided into three groups, namely, two planes that touch the pole pf, two that touch the pole m*, and two that touch the pole t. Then, of the two planes that touch the pole p|, one is attached to the north meridian and the other to the east meridian of the crystal; of the two planes that touch the pole m*, one is attached to the north meridian and the other to the equator ; and of the two planes that are attached to the pole t * , one is attached to the east meridian and the other to the equator. a. The plane that touches the pole p| and is attached to the north meridian is in the position Z 3 n 2 w, and therefore is a plane belonging to the complement P_MT+. b. The plane that touches the pole p| and is attached to the east merid- ian, is in the position Z 3 nw ? , and therefore is a plane belonging to the complement P_M + T. c. The plane that touches the pole m* and is attached to the equator, is in the position Zn 3 w 2 , and therefore is a plane belonging to the comple- ment P + M_T. d. The plane that touches the pole m* and is attached to the north meridian, is in the position Z 2 n 3 w, and therefore is a plane belonging to the complement PM_T_p e. The plane that touches the pole t. and is attached to the east mer- idian, is in the position Z 2 nw 3 , and therefore is a plane belonging to the complement PM + T_. / The plane that touches the pole t and is attached to the equator, is in the position Zn 2 w 3 , and therefore is a plane belonging to the comple- ment P + MT_. , c, e, are the three complements of planes that produce the combina- tion called the Right Hemihexakisoctahedron with parallel faces. b, d, f, are the three complements of planes that produce the combina- tion called the Left Hemihexakisoctahedron with parallel faces. Consequently, the Hexakisoctahedron is a combination that contains all the six regular permutations of the symmetrical scalene octahedron. The two tables given in 188, 192, show the symbols and positions of all the forty-eight planes that belong to this combination, upon the assumption that P_MT + signifies PJMJT; and although the numeral values of the symbols are changed when other varieties of the form are to be indicated, the signs of the positions are correct for all. 196. As the symbols for the three combinations of the scalene octa- hedrons are very long, I propose to abridge them* as follows: K 66 PRINCIPLES OF CRYSTALLOGRAPHY. 6 P_MT + , instead of P_MT + , P + M_T, PM + T_, P_M + T, PM_T+, P + MT_, the symbol for the Hexakisoctahedron. 3 P_MT + Z 3 nV, instead of P_MT + , P + M_T, PM + T_, the symbol for the Right Hemihexakisoctahedron with parallel faces. 3 P_M + T Z 3 nw 2 , instead of P_M + T, PM_T+, P+MT_, the symbol for the Left Hemihexakisoctahedron with parallel faces. 197. Varieties of the Hexakisoctahedron, 6 P_MT + . The mineral kingdom has afforded the following varieties of this com- plex combination: 6 P^MJT. 6 6 6 Pf MJT. These five combinations can be discriminated by attention to the fol- lowing angles of the incidence of their planes upon one another. Inclination of the plane P_MT+ Z 3 n 2 w upon : P_MT + Z 3 n 2 e. P_M + T Z 3 nw 2 , PM_T + Z 2 n 3 w- 6 P^MJT. 149 & 158 13' 158 13' 6PMT 157 23' 147 48' 164 3' 6 P^M^T. 154 47' 162 15' 144 3' 6 P T i r MT. 152 7' 166 57' 140 9' 165 2' 158 47' 136 47' The combinations that are at the beginning of this list approximate to the form of the octahedron, those at the end rather to the form of the cube. By some crystallographers, the former are termed Hexakisocta- hedrons and the latter Octakishexahedrons, a variation in nomenclature which is of little importance. Model 23 is intended to represent the combination 6 Pf MT, but its angles are not made sufficiently correct to afford measurements strictly justificatory of those contained in the above table. They are however exact enough to distinguish this variety from all the others, and to show the positions and general bearings of the different sets of planes. The figure of the Hexahisoctahedron that is commonly given in books on Mineralogy, as the diamond form, is the combination denoted by 6 PJMJT. All the octahedral forms, simple and complex, occur in combina- tion with the different prismatic forms. The suite of models presents numerous examples of such combinations, and this is perhaps the proper place for the description of them ; but as I propose to add a Second Part to this work, treating of the application of Crystallography to Min- eralogy, I shall reserve these details till I come to speak of the Minerals which afford examples of each particular combination. PRINCIPLES OF CRYSTALLOGRAPHY. 67 198. I have now described the properties of all the varieties of octa- hedrons which can possibly occur. They are as follow : = PM X , PT X . 104, 105. PMT. 124, P_MT PM_T }- = 3 P_MT. 142. PMT - ! = 6P X MT. P+MT PM+T }- = 3 P+MT. 158. PMT. - Ik U- >_M+T ) >M_T + I = > + MT_ j P_MT + P+M_T V = 3 PJMT+. 177. PM+T ~ ? ^ = 6 P_MT + . 194. P_M. ^ ' PM_T + J- = 3 P_M+T. 192. P. Every other octahedral or pyramidal form is a combination of two or more of these complements of planes or of their fractions. Hence the following formula embraces every variety of pyramid that can occur, either alone or in combination with prismatic planes : PM X , PT X , PMT, 6 P X MT, 6 P x M y T z . 199. Forms of the Equators of Pyramids. An examination of the forms of the equators of Pyramids, gives the same results as a similar examination of the equators of Prisms. What these results are, I have shown in the table contained in 87. It is only necessary to explain here a few terms that will be employed hereafter to designate the different varieties of equator. A Square Equator is one whose sides are parallel to the axes m a and t% or cut these axes only at angles of 45, and are all of equal length when extended till they meet. Models 1, 4, 12, 17. A Rectangular Equator is one whose sides are parallel to the axes m a and t a , and equal to one another two and two, but not all equal. Model 19- A Rhombic Equator is one whose sides, when extended till they intersect one another, form one or more rhombuses. Models 6, 21, 22. A Rhombo- Quadratic Equator is one whose sides, when extended till they meet, form both a square and a rhombus. Models 47, 91. A Rhombo-Rectangular Equa- tor is one whose sides, when extended till they intersect one another, form both a rhombus and a rectangle. Models 50, 51, 8, 7. To complete the square in Model 91 and the rectangle in Models 8, 7, lines are sup- posed to be drawn parallel to the axis t a , through the poles m a and m*. 68 PRINCIPLES OF CRYSTALLOGRAPHY. SYNOPSIS OF PLANES. 200. Before concluding the present section, it will be useful to take a general survey of the mutual relations of the different sets of Planes and their Axes. SYNOPSIS OP PLANES. Planes that produce PRISMS. Planes that produce PYRAMIDS. Planes that cut Equal Axes, P M T MT PM PT PMT P_ M_ T_ MT+ PM+ PT, P_MT P+ M+l T+ M+T P+M P+T PM_T PMT_ * P+MT PM+T Planes that cut Unequal PMT+ Axes, P-MT+ P+M_T PM+T_ P_M+T PM_T+ P+MT_ Number of Planes to each set, 2 2 2 4 4 4 8 Axes cut by the Planes of each set, P a m a t a m a t a p a m a p a t a p a m a t a Axes parallel to the Planes of each set, m a t a P a t a P a m a p a t a m a none. | 1 So 3 i * *s *: i- *i . Position of the Planes of each i s 1 | 1 i set, when the point of view of the crystal is in the pro- 53 t 8 i> 1 1 i 1 longation of the axis m a ,... a s. if 'g t 'S ^ jj Is 9 .B .S a > & p NH a Abridged Symbols for Octahedral Combinations. See 161, 196. 3 PJMT = P_MT, PM_T, PMT_. 3 P+MT = P + MT, PM + T, PMT+. 3 P_MT + Z 3 nV = P_MT + , P + M_T, PM + T_. 3 P_M + T Z 3 nw 2 = P_M + T, PM T+, P + MT_. 6 P_MT + = P_MT +J P + M_T, PM + T_, P_M + T, PM_T +> P + MT_. PRINCIPLES OF CRYSTALLOGRAPHY. 69 SECTION III. OF PRISMS AND PYRAMIDS AND THEIR COMBINATIONS WITH ONE ANOTHER. 201. A PRISM is a solid contained by three or more vertical planes of the same altitude, and two equal, similar, and parallel horizontal planes. The vertical planes are those whose symbols are M,T, MT, MT_j_, M + T. The horizontal planes are the set P. These are the only planes that be- long to a prism. See 86. 202. A COMPLETE PRISM must have the two horizontal planes P, and at least three vertical planes, which number, and not less, will make up a complete form. A complete prism may have any number of vertical planes greater than three, but it cannot have less, because with less than three vertical planes we have not a complete form. It is seldom how- ever that prisms have less than four vertical planes. 203. An INCOMPLETE PRISM may have any number and combination of the prismatic planes P,M,T,MT,MT + ,M + T, other than suffices to produce a complete prism. For instance, it may have the two horizontal planes without any of the vertical planes, or it may have all the vertical planes without the horizontal planes, or finally, it may have the two horizontal planes with any two, but not more than two, of the vertical planes. It is obvious, from this description of the Incomplete Prism, that it cannot be a self- existing form, and that its planes can only appear upon a crystal in combination with the planes of some other form. 204. A PYRAMID is a solid contained by eight, twelve, twenty-four, forty-eight, or more, Inclined Planes, generally triangular, but some- times quadrangular, the symbols of which planes are PM X ,PT X ,PMT, 6P X MT, 6P x M y T 2 . These are all the planes that belong to a pyramid. There are no verti- cal and no horizontal planes. See 198. The inclined planes must have such positions upon every crystal as to form a solid angle at the pole Pz and another at the pole p. 205. The GEOMETRICAL Pyramid is a piano-facial solid contained by three or more plane triangles which have a common vertex, and whose bases are the sides of a plane rectilineal figure which forms the base of the solid. The common vertex of the plane triangles is called the vertex of the pyramid, and the remaining face is called the base. 206. The CRYSTALLOGRAPHICAL Pyramid consists of two geometrical pyramids joined base to base. By some crystallographers this form is called the double pyramid. But as crystallographical pyramids are always double, it is better, because it simplifies notation, to consider the double pyramid as a single form. On this account I call the lower pyramid the complement of the upper, and consider the double pyramid to be one pyramid. 207. A COMPLETE PYRAMID is one that has at least six, but may 70 PRINCIPLES OF CRYSTALLOGRAPHY. have any greater number of inclined planes, in two sets of equal number ; one set situated above the equator, and the other below it. There must be a solid angle at the pole p| and another at the pole p^. 208. The Complete Pyramids of eight or twelve planes have com- monly triangular planes that taper gradually from the equator to the poles pz and p^; as witness Models 15, 12, 13, 26. But there are other complete pyramids which do not taper gradually from the equator to the poles, and others of which the planes are not triangular. See Models 22, 17, 25, 23. These forms are produced by planes belonging to com- binations that are described by a formula expressing several permutations of the complement P X MT or P x M y T z . But all Complete Pyramids agree with one another in three essential particulars : They have none but inclined planes ; they have the zenith portion similar and equal to the nadir portion of the form ; and they are terminated by solid angles at the two poles p| and p^. 209. An INCOMPLETE PYRAMID is a form that has inclined planes, but is without solid angles at the poles p| and p, where it is terminated by oblique planes, or by straight edges, or by the horizontal planes P. An Incomplete Pyramid may have any number of inclined planes. THREE KINDS OF COMBINATIONS OF PRISMS WITH PYRAMIDS. 210. A COMPLETE PRISM COMBINED WITH AN INCOMPLETE PYRAMID. This is a crystal that contains the planes of the Complete Prism, 202, combined with the planes of the Incomplete Pyramid, 209. A. It must have, a. The two horizontal planes P. b. At least three of the vertical planes M,T, MT X . c. Some, and it is of no consequence how many, of the inclined planes PM X , PT X , P X MT, P x M y T z . And these are necessarily situated upon the crystal between the horizontal and vertical planes, in accordance with the polaric positions proper to each form that enters into the combination. B. This combination must have no solid angles at the poles p| and p^,. 211. AN INCOMPLETE PRISM COMBINED WITH A COMPLETE PYRAMID. This crystal exhibits the vertical planes of the Incomplete Prism, 203, combined with the planes of the Complete Pyramid, 207. A. It must have, a. Eight or more inclined planes forming two equal and similar pyramids, whose bases are parallel with the equator of the combination, and whose apices form solid angles at the poles p| and p^. 6. Some, and any number or combination, of the prismatic planes M,T, MT,MT_|_,M_|_T, which necessarily appear round the equa- tor, replacing part of the base of the pyramid. B. This combination must not have the horizontal planes P. PRINCIPLES OF CRYSTALLOGRAPHY. 71 212. AN INCOMPLETE PRISM COMBINED WITH AN INCOMPLETE PYRAMID. This crystal contains the planes of the Incomplete Prism, 203, combined with the planes of the Incomplete Pyramid, 209 The characters of this combination are as follow : a. It may have the horizontal planes P, but then it must not have more than two vertical planes. b. It may have the horizontal planes P, without any vertical planes. c. It may have any number of vertical planes, provided it is without the horizontal planes. d. If it has the horizontal planes P, it may have any number and combination of the inclined planes to form the incomplete pyramid. e. If it is without the horizontal planes P, the inclined planes of the incomplete pyramid must be such as do not form solid angles at the poles p| and p?,. SECTION IV. OF THE CLASSIFICATION OF CRYSTALS. 213. The arbitrary definitions of PRISMS, PYRAMIDS, and their COM- BINATIONS, which are given in the preceding SECTION, 201 212, provide six distinct terms explanatory of different crystallographic com- binations. The Table of the FORMS OF THE EQUATORS OF PRISMS AND PYRA- MIDS, explained in 87 and 199, provides five other characters adapted to distinguish different crystallographic combinations. I propose to employ these two sets of characters as the foundation of a CLASSIFICATION OF CRYSTALS, as follows : CLASS I COMPLETE PRISMS. Definition 202. Order 1. Square, . . EXAMPLES. Models 1 to 4 2. Rectangular, . 5 3. Rhombic, . 6 4. Rhombo- Quadratic, . 5. Rhombo-Rectangular, . 7 to 11 The title of each Order indicates the form of the Equators of the Crys- tals which it comprehends. 199- CLASS II. COMPLETE PYRAMIDS. Definition 207. Order 1. Square, . . . EXAMPLES. Models 12 to 18 2. Rectangular, . . 19 to 20 3. Rhombic, . . ~~ 21 to 25 4. Rhombo- Quadratic, . ~~ 5. Rhombo-Rectangular, . 26 72 PRINCIPLES OF CRYSTALLOGRAPHY. CLASS III. COMPLETE PRISMS COMBINED WITH INCOMPLETE PYRAMIDS. Definition 210. Order 1. Square, . . EXAMPLES. Models 27 to 44 2. Rectangular, . ^^ 45 3. Rhombic, . _ 46 4. Rhombo- Quadratic, . ~~* 47 to 49 5. Rhombo-Rectangular, . ~~. 50 to 58 CLASS IV. INCOMPLETE PRISMS COMBINED WITH COMPLETE PYRAMIDS. Definition 211. Order 1. Square, . . EXAMPLES. Models 59 to 65 2. Rectangular, . . 3. Rhombic, . . ~~ 66 to 68 4. Rhombo- Quadratic, . 69 5. Rhombo-Rectangular, . .~~ 70 to 75 CLASS V. INCOMPLETE PRISMS COMBINED WITH INCOMPLETE PYRAMIDS. Definition 212. Order 1. Square, . . EXAMPLES. Models 76 to 78 2. Rectangular, . 79 3. Rhombic, . 80 to 90 4. Rhombo-Quadratic, . 91 to 95 5. Rhombo-Rectangular, . 96 to 116 CLASS VI. INCOMPLETE PYRAMIDS. Definition 209. Order 1. Square, . . EXAMPLES. Models 117, 118 2. Rectangular, . . 3. Rhombic, . 119 4. Rhombo-Quadratic, . ~~- 5. Rhombo-Rectangular, . 120 214. DIRECTIONS FOR PUTTING A CRYSTAL INTO A PROPER POSI- TION FOR EXAMINATION AND DESCRIPTION. As the foregoing Classification of Crystals is founded upon the differ- ences that exist in the positions of the planes of crystals when they are held in upright position, it follows- that one and the same crystal may be made to belong to different classes of crystals by merely holding it in different positions. Thus, Model 1, the cube, P,M,T, which belongs to Class I., Order 1, Square Prisms, may be held so as to exhibit the planes M. PT ; or the planes T. PM ; and in either of the latter cases, PRINCIPLES OF CRYSTALLOGRAPHY. 73 the crystal belongs to Class V., Order 2, being " an Incomplete Prism combined with an Incomplete Pyramid, and having a Rectangular Equator." This single example proves the necessity of having a method agreed upon for putting crystals into a proper upright position, so as to establish uniformity in notation. I shall therefore give a few rules that may be observed on this point, but I give them with the confession that they are too vague to be rigidly adhered to, and that I am at present unable to make them so exact as to be quite satisfactory. General Rule. a, The first and principal rule has been given in 8 and 166, and may be here recited. The point of view of a crystal is in the prolonga- tion of the minor axis m a . Hence the observer of a crystal has to hold it before him with the longest axis in a perpendicular position, and the broadest side exposed to his eye. The axes then become p+ ml t a . Rules drawn from the distinction between Prisms and Pyramids. b, When the crystal is a prism, the planes M,T,MT X are to be held in a vertical position. A prism is known by the characters given in 86, 201, 220. c, When the crystal is a pyramid, it is to be held with the two princi- pal solid angles, namely, the two sharpest or two bluntest where any two equal and opposite solid angles are different from the other solid angles, or at any rate, with two similar solid angles, upon the poles pi and p^. The characters by which a pyramid is known, are given in 198, 204, 224. dy Combinations of Prisms with Pyramids, such as are described in 210 212, are to be held with the prismatic planes in a vertical position. Rules drawn from the form of the Equator. See 87, 199. e, If the equator is a square, and the crystal is a prism, or an unequi- axed pyramid, the angles of the equator are to be placed at nw, ne, sw, se. / If the crystal is an equiaxed pyramid, as PMT, the angles of the equator are to be placed at n, s, e, w. g, If the crystal is a combination of a prism with a pyramid, and the pyramid or pyramidal combination is such a one as forms an equiaxed crystal when alone, the crystal must be so placed as to bring the planes of the pyramid or pyramidal combination into the polaric positions which are proper to them as the planes of a self-existing combination. This rule refers specially to the combinations of the cube with the different octahedral forms. h } If the pyramid which is found upon such a combination is one that produces an unequiaxed pyramid when it is separate from the prism, the combination must be placed so as to make the planes of the pyramid agree with the symbols PM, PT, and not with PMT. 74 PRINCIPLES OF CRYSTALLOGRAPHY. ", If the equator is a rectangle, its longer diameter is to be placed upon the axis t a . j 9 If the equator is a rhombus, its obtuse angles are to be placed at the poles m a , mf. k, If the equator is a combination of a rhombus with a square or of a rhombus with a rectangle, it is to be placed with the obtuse angles of the rhombus at the poles m* and m a . The last three rules are however merely repetitions of a, the first rule. SECTION V.OF THE POSSIBLE LIMIT TO THE VARIETY OF PLANES THAT CAN OCCUR UPON CRYSTALS. 215. I have shown that with the exception of the planes denoted by PMT, no single set contained in the synopsis 200, can of itself produce a complete crystal. Yet all the planes which occur upon crystals belong to one or other of the varieties described in this synopsis, whence it fol- lows, that all crystals consist of combinations of these sets of planes, and may be denoted by combinations of their symbols. As the letters of our alphabet, which are not words alone, serve to form all written words by combination, so these planes, few in number and simple in their relations, but incomplete of themselves, produce by combination all the immense variety of perfectly crystallised forms which is presented to our notice, not only in the mineral kingdom, but in the factitious productions of the chemist's laboratory. Our Synopsis of Symbols, 200, is, therefore, a CRYSTALLO GRAPHIC ALPHABET. It represents not simply a few of the planes which are found here and there upon particular crystals, but it shows ALL the planes that can occur upon crystals considered collectively. That this is a true proposition will be manifested by an examination of the general relations of the planes which these symbols serve to denote. 216. Of the Plane P Its essential characters are to be horizontal; to be parallel to the axes m a and t a ; and to cut the axis p a . It cannot lose any of these characters without ceasing to be the plane P. If you imagine it to pass ever so little out of the perfectly horizontal position, it no longer remains parallel to the axes m a and t a . If one edge of it descends in the front towards the axis m% it becomes the plane PM + . If it descends side wise towards the axis t% it becomes the plane PT + . If it descends cornerwise, it falls upon the two axes m a and t a , and be- comes the plane PM + T + . And it cannot pass out of the horizontal position without falling either upon the axis m a or the axis t% or upon m a and t a jointly. The position of the plane P is therefore unalterable. Its characters are distinct and cannot be confounded with those of any other plane ; and the lengthening or shortening of any of the three axes of the crystal has no effect upon the essential characters of the plane P. PRINCIPLES OF CRYSTALLOGEAPHY. 75 217. Of the Plane M. Its essential characters are to be vertical; to be parallel to the axes p a and t a ; and to cut the axis m a . It cannot lose one of these characters without ceasing to be the plane M. If it in- cline ever so little towards the axis p a , it becomes the plane P + M. If it incline towards the axis t% it becomes the plane MT + . If it incline corner wise, it becomes P_f_MT_j_. 218. Of the Plane T. Its essential characters are to be vertical; to be parallel to the axes p a and m a ; and to cut the axis t a . It cannot pass out of its strictly vertical and parallel position, without ceasing to be the plane T, and becoming M + T, P + T, or P + M + T. 219- Of the Plane MT. -Its essential characters are to be vertical ; to cut the axes m a and t a ; and to be parallel to the axis p a . It may cut the axis m a and t a either at equal distances from the centre of the crystal, or at unequal distances, and may consequently produce many varieties of the plane, as MT, MT + , and M_j_T. But it must never cease to be vertical, to cut the two horizontal axes, and to be parallel to the axis p a . 220. The foregoing are all the planes that occur upon prisms, and their characters are perfectly distinct from those of the planes that are found upon pyramids. The planes P are distinguished by their horizontal position, and the planes M,T, MT by their vertical position. 221. Of the Planes PM Their essential characters are to be in- clined from the middle of the top and bottom towards the middle of the front and back of the crystal, so as to cut both the axis p a and the axis m a , and to be parallel to the axis t a . No other planes than PM can have these properties, and no variety of the planes PM, produced by extension of the axes p a or m a , can cease to possess these essential characters. All the varieties of PM + and P+M, like PM, cut the axes p a and m% and are parallel to the axis t a . 222. Of the Planes PT. Their essential characters are to be in- clined from the middle of the top and bottom towards the middle of the left and right sides of the crystal, so as to cut the axes p a and t% and to be parallel to the axis m a . No alteration in the relative lengths of the three axes p a m a t a can have any effect upon these characters, and no planes which are without these characters can be denominated PT. 223. Of the Planes PMT Finally, the planes PMT are inclined; they cut all the three axes, and they are parallel to none. They may cut all the axes equally or all unequally, or two of them equally and one otherwise. But they must cut all the three axes and be parallel to none. These characters are perfectly distinctive, since none of the other sets of planes cut above two axes, and each of them is parallel to at least one axis. The thirteen varieties of the complement PMT contained in the 76 PRINCIPLES OF CRYSTALLOGRAPHY. eighth column of the Synopsis of Planes, 200, all possess these essential characters of PMT, and only differ among themselves in respect to the relative distances from the centre of the crystal at which the edges of their planes cut the three axes. 224. These are all the planes that occur upon pyramids. They are distinguished from the planes of prisms by being inclined to the equator, whereas the prismatic planes are either perpendicular to, or parallel with, the equator. 225. Being now fully acquainted with the positions and mutual rela- tions of the planes contained in the Synopsis, so as not to be liable to mistake one for another, or to confuse them with any thing different, let us turn to the Diagram contained in paragraph 20, and endeavour to trace upon it a plane DIFFERENT from those whose properties we have examined. I have depicted in this Diagram the system of three axes crossing one another at right angles in the centre, and round about it, what we will at present assume to be an amorphous mass of crystallizable matter. This mass is bounded by the lines numbered 12345678 9 10 11 12. 1st, If we propose to trace a plane parallel to any one of the surfaces of this mass, that is to say, a plane that shall cut any one of the three axes, and be parallel to the other two, we merely repeat the planes P, M, or T. 2dly, If we propose to trace a vertical plane through the points 1 5 10 8, or through the points 2 4 11 7, or through any part of the solid so as to be parallel to the vertical axis p a , whatever the possible direction of the plane may be as respects the two axes m a and t% we do in every such attempt produce but different varieties of the planes MT, MT + , and M + T. 3dly, If we propose to trace a plane that shall cut off the edge 1 M 2 at any possible degree of inclination towards the axes p a or in a , or if we attempt to cut through the mass in any direction from the side 24108 to the side 1 5 117? retaining a parallelism to the' axis t a , we shall, in every case, produce planes that are nothing else than varieties of PM, PM + , or P + M. 4thly, If we propose to trace a plane through the points 2 5 1 1 8 or through the points 1 4 10 7, or through any other part of the mass so as to bisect at any angle the axes p a and t a and to be parallel to the axis m a , we produce in every attempt nothing but varieties of PT, PT + , and P+T. 5thly, If, abandoning our attempts to produce new planes by cutting off any face, or any edge, of the mass 1 to 12, we try what can be done by operating upon its corners, we shall find our exertions to be equally fruitless, For, if we cut off a portion which leaves a triangular surface of three equal sides, we simply produce the plane PMT. PRINCIPLES OF CRYSTALLOGRAPHY. 77 If the surface produced by the section has two sides short and one long, it depends only upon the direction of the section, whether we pro- duce the plane P_MT, PM_T, or PMT_, but one of these it must be. If the section has two sides long and one short, it again depends only upon the direction of the section whether we produce the plane P + MT, PM + T, or PMT_j_, but it must be one of these planes. If the surface produced by the section has three unequal sides, then, exactly according to the direction of the section, we have the plane P_MT + , P + M_T, PM + T_, P_M + T, PM_T + , or P + MT_, and we can- not have anything else. Repulsed at all these points, we shall find it to be impossible to attack the solid in any new direction. We cannot imagine a section that shall act otherwise than upon a face, or an edge, or a corner, of the mass figured in the diagram. We have therefore exhausted all the possible cleavages, and are warranted in coming to the conclusion that 226. There cannot occur upon any crystal a plane different from those denoted by the symbols P,M,T, MT. PM, PT, PMT, either written alone, or subscribed by numbers, or by the signs + and -, as represented in the Synopsis, 200. This is the limit to the variety of planes that can possibly occur upon crystals. SECTION VI. OF CRYSTALLOGRAPHIC NOTATION. 227. Admitting the correctness of the proposition contained in 226, that " there cannot occur upon any crystal a plane different from those indicated by the symbols P,M,T, MT X . PM X , PT X , P X M X T X ," then CRYSTALLOGRAPHY the Art of Describing Crystals consists simply in enumerating the symbols which designate the planes that we observe upon the crystals that we wish to describe. Thus A crystal upon which we find the planes P,M,T, is described by the repetition of the symbols P,M,T. A crystal which has the planes MT.PM,PT, is described by the repi- tition of the symbols MT.PM,PT. A crystal which has the planes PMT is described by the repetition of the symbol PMT. A crystal which has the planes P,M,T,MT.PM,PT,PMT, is described by the repetition of the symbols P,M,T,MT.PM,PT,PMT. And when the crystals are not equiaxed, we have only to add the signs + or - or x or a number, as the case may demand, to the letters which refer to the axes whose peculiarities we desire to describe. 228. The symbols are arranged in the order in which they appear in the Synopsis, 200. Those which denote prismatic planes take prece- dence of those which denote pyramidal planes, and those in the left hand columns take precedence of those in the right hand columns ; so as to produce the following series of symbols : 78 PRINCIPLES OF CRYSTALLOGRAPHY. P, P_, P + , M, ML., M+, T, T_, T + , MT, MT + , M + T. PM, PM+, P + M, PT, PT+, P+T, PMT,P_MT,PM_T,PMT_,P + MT,PM + T,PMT + , PJMT+,P+M_T, PM + T_, P_M + T, PM_T + , P+MT_. This is a universal formula, which comprehends every possible variety of planes, and of which every other symbol must be an abridgement. In making these other symbols by abridgement, the rule to follow is, to omit the signs of all absent planes, and write the signs of the planes that are present, in the order in which they stand in this universal formula. Every symbol from P to M + T, and from PM to P + MT_ inclusive, is separated from every other by a comma (,) but every series of prismatic planes is separated from every series of pyramidal planes by a period (.). The method of denoting the halves and fourths of sets of planes, has been explained in 19, 22, 23, 24, 26, 88, 101, 125. The method of denoting the comparative sizes of the planes of differ- ent sets which occur together on complex combinations has been described in 67 to 69. The method of describing the lengths of the axes of crystals, as dis- tinguished from the axes of their complements of planes, has been explained in 1 to 14, and 76. 229- TWIN CRYSTALS There is a variety of crystallized form, com- monly called a twin crystal, or double crystal, or made, which consists of two crystals piercing one another, or of two halves of a crystal joined together in a position more or less inverted. This variety can be denoted by adding the sign x 2 to the symbol of the complete single crystal, or by enclosing the latter symbol when complex within parentheses and then adding the sign x 2 Examples: PMT x 2, Model 16. (P X ,T + ,MT + ) X 2. Model 9- There are many different kinds of these mixed crystals, which could perhaps all be denoted by as many different symbols ; but I question whether it is worth while to burthen our books and memories with different symbols for such a purpose. 230. The following catalogue of the observed forms of the crystals of FLUORSPAR will serve to illustrate the use of this notation, while it also shows the application of the principles of classification explained in 213. CLASS I. Complete Prisms. ORDER 1. Square Equator. P,M,T. The CUBE, Model 1. CLASS II. Complete Pyramids. ORDER 1. Square Equator. PMT. The REGULAR OCTAHEDRON. Model 15. 3 P 3 MT. The TRIAKISOCTAHEDRON. A form of the same kind but not having the same angles as Model 17. See 160. The symbol 3P 3 MT is equal to P 3 MT, PM 3 T, PMT 3 . See 200. PMT, 3p 3 mt. The OCTAHEDRON, PMT, combined with the Triakis- octahedron, 3p 3 mt. The symbols and words that are written in capital letters dis- tinguish the forms that predominate. PRINCIPLES OF CRYSTALLOGRAPHY. 79 CLASS III. Complete Prisms combined with Incomplete Pyramids, ORDER 1. Square Equator. a) The Cube predominant. P,M,T,mt.pm,pt. The CUBE, P,M,T, combined with the rhombic dode- cahedron, mt.pm,pt. Model 27. P,M,T,mt.pm,pt,pmt. The CUBE, P,M,T, combined with the rhombic dodecahedron, mt.pm,pt, and the octahedron, pmt. Model 32. Every plane on this model is marked with the symbols of its dif- ferent forms. P,M,T,3pJmt. The CUBE, P,M,T, combined with the icositessarahedron, pimt,pmlt,pmti. Model 40. P,M,T,mt.pm,pt,3p^mt. The CUBE, P,M,T, combined with the rhombic dodecahedron, mtpm,pt, and the icositessarahedron, pjmt,pmjt, pmtj. P,M,T. 6pjmlt. The CUBE, P,M,T, combined with the Hexakisocta- hedron, Pjmlt,pmjti,p^mtl,pjmti,plmlt,pmitl. Model 41. P,M,T,MT.PM,PT,3pJmt,6pimit,6p I Vmiti. The CUBE, P,M,T, com- bined with the Rhombic Dodecahedron, MT,PM,PT; the icosites- sarahedron, 3p^mt; the hexakisoctahedron, Gp^mjt; and the hexakis- octahedron, P,M,T. PMT. The middle crystal betwixt the cube and the octahedron, in which neither form predominates. Model 29. b) The Octahedron predominant. p,m,t. PMT. The cube p,m,t, combined with the OCTAHEDRON, PMT. -Model 30. p,M,T,mt,pm,pt,PMT. The cube, p,m,t, combined with the rhombic dodecahedron, mt.pm,pt, and the OCTAHEDRON, PMT. Model 33. p,m,t,MT.PM,PT,PMT,3p^mt. The cube, p,m,t, combined with the rhombic dodecahedron, mt.pm,pt; the OCTAHEDRON, PMT; and the icositessarahedron, 3pjmt. ORDER 4. Rhombo-Rectangular Equator. p,m,t,MT 3 ,M 3 T.PM3,P3M,PT 3 ,P 3 T. The cube, p,m,t, combined with the TETRAKISHEXAHEDRON, MT 3 ,M 3 T.PM 3 ,P 3 M,PT 3 ,P 3 T. P,M,T,mt 3 ,m 3 t.pm 3J p 3 m,pt ,p 3 t. The CUBE P,M,T, combined with the Tetrakishexahedron, mt 3 ,m 3 t.pm 3 ,p 3 m,pt 3 ,p 3 t. Model 39- P,M,T,mt,mt 3 ,m 3 t.pm,pm 3 ,p 3 m,pt,pt 3 ,p 3 t. The CUBE, P,M,T; combined with the rhombic dodecahedron, mt.pm,pt; and the tetrakishexahe- dron,mt3,m 3 t.pm 3 ,p 3 m,pt 3 ,p3t. P,M,T,mt,mt 3 ,m 3 t.pm,pm 3 ,p 3 m,pt,pt 3 ,p 3 t,3p^mt. The foregoing com- bination with the addition of the icositessarahedron, pjmt,pm^t,pmt^. p,m,t.6PiMiT. The Cube, P,M,T, combined with the HEXAKISOCTA- HEDRON, 6P1M1T. P,M,T,mt 3 ,m 3 t.pm 3 ,p 3 m,pt 3 ,p 3 t,6pJmJt. The CUBE, P,M,T, combined with the tetrakishexahedron, mt 3 ,m 3 t.pnj 3 ,p 3 m,pt 3 ,p 3 t, and the hex- akisoctahedron, Gpjmjt. 80 PRINCIPLES OF CRYSTALLOGRAPHY. P,M,T,MT,int|,m|t.PM,pmf,p|m,PT,pt|,pft,PMT. The CUBE, P,M,T, combined with the rhombic dodecahedron, MT.PM,PT ; the tetrakis- hexahedron, mtf,m|t.pmf,pfm,ptf,pjt; and the octahedron, pmt. p 10 t,pmt,3pjmt,5(6p x m y t z ). The CUBE, P,M,T, combined with the rhombic dodecahedron, mt.pm,pt; the tetrakishexahedron, mt 3 ,m 3 t,pm3,p 3 m,pt3,p 3 t; the tetrakishexahedron, mt 10 ,m 10 t.pm 10 , p 10 m,pt 10 ,p 10 t; the octahedron, pmt; the icositessarahedron, pimt,pm Jt,pmt J ; and five different varieties of the hexakisocta- hedron, 6p x m y t z . This crystal possesses 338 planes. CLASS IV. Incomplete Prisms combined with Complete Pyramids. ORDER 1 . Square Equator. MT.PM,PT. The RHOMBIC DODECAHEDRON. Model 63. mt.pm,pt,PMT. The rhombic dodecahedron, mt.pm,pt, combined with the OCTAHEDRON, PMT. Model 64. ORDER 3. Rhombic Equator. MT 3 ,M 3 T.PM3,P3M,PT 3 ,P3T. The Tetrakishexahedron. General form of Model 68, but with different angles. ORDER 4. Rhombo- Quadratic Equator* mt.pm,pt,PMT,3pJmt. The rhombic dodecahedron, mt.pm,pt, com- bined with the OCTAHEDRON, PMT; and the icositessarahedron, Splint. mt,mt 3 ,m3t.pm,pm 3 ,p3m,pt,pt3,p 3 t,PMT. The rhombic dodecahedron, mt.pm,pt, combined with the tetrakishexahedron, mt3 5 m 3 t.pm 3 ,p 3 m, ; and the OCTAHEDRON, PMT. SECTION VII. OF CLEAVAGE AND PRIMITIVE FORMS. 231. Certain minerals can be cleaved or mechanically divided in par- ticular directions, which vary with different substances. The cleavages, or planes produced by this mechanical division of crystals, have proper- ties similar to those of their superficial planes. They are horizontal, vertical, or inclined. They cut one, two, or three of the axes of the crystals ; they have definite polaric positions ; and they can consequently be denoted by the same symbols which serve to denote the external planes. 232. The cleavages of crystals are sometimes parallel to their external planes and sometimes not so. Thus, when the external planes of a crystal are P,M,T, the cleavages maybe P,M,T; MT.PM,PT; or PMT. There is no known connection between external form and internal cleavage, so that it is necessary to discover the clearvage or cleavages PRINCIPLES OF CRYSTALLOGRAPHY. 81 of every particular mineral and of every different form by mechanical division. Some minerals afford no cleavage ; others afford two or three different kinds of cleavage, in respect of quality or perfection ; such as 1, very distinct ; 2, less distinct ; and 3, indistinct. I propose to denote these varieties by altering the size of the letters that indicate the cleavage; as P,M,T a very distinct cleavage. MT.PM,PT a less distinct cleavage, pmt an indistinct cleavage. Numerous examples of the application of these symbols are given in the second part of this work, between pages 16 and 94, where the rela- tions of the cleavage planes to the external planes of crystallised minerals may be seen at a glance. 233. When the cleavages of a mineral are so numerous and so arranged as to constitute two or more prisms of four sides which cross or cut one another, they cleave or cut out from the crystal, particular geometrical solids, as cubes, octahedrons, rhombohedrons, and so forth. These pro- ductions of cleavage have been called primitive forms, and it has been assumed by some mineralogists, that they represent the forms of the ultimate molecules of which the whole mass of the crystal from which they are cleaved is composed. But this is an assumption which is entirely incapable of proof, and which is of no use except in so far as it served to render intelligible the systems of crystallography which first brought the term into use. In reality, the primitive form of a mineral shows the direction and the number of its cleavages, and nothing more. The use which Rome de LTsle and Haiiy made of the term was, first to give names to a variety of " primitive forms," and then to name all other forms according to certain degrees of resemblance which they bore to the forms which had been assumed to be primitive forms. The doctrine of primitive forms, therefore, can only be considered as an ingenious artifice of Rome de L'Isle and Haiiy, adopted for the purpose of generalising their views of the relation borne by the secondary forms of crystals to one another, which, indeed, is what Haiiy expressly admits: " The primitive form (noyau) of a crystal," he says, " is merely a theoretical datum,' taken to facilitate the determination of the different crystalline forms belonging to the same substance." Traite de Cristallographie, t. i. p. 65. The doctrine of primitive forms was a use to which these crystallographers turned their knowledge of the facts derived from the observation of cleavage. They built their theories upon these facts ; but although the facts remain true, it does not follow that the theories are true also; and although the doctrine of primitive forms is the part of Haiiy's System of Crystallography which has received the most general approbation of mineralogists, I am inclined to believe that the assumption is more injurious than useful, and accordingly I have recommended a system of notation which entirely dispenses with it. The very common use of the term primitive form, may, however, induce M 82 PRINCIPLES OF CRYSTALLOGRAPHY. some to demand a proof of its alleged injurious tendency: to which I reply by pointing out a few of its practical results, which I think are sufficient to condemn it. One of these results is, that it leads mineralo- gists to describe forms of minerals which do not exist, namely, the ideal primitives, and to omit to describe the forms which do exist, namely, the forms of actual occurrence, which, being nick-named secondary, are treated as if they were of secondary or of no importance; hence, when a student of mineralogy begins to compare crystallized minerals with the descriptions in his books, he finds the two not to agree, most of the commonly occurring crystals of minerals being summarily dismissed as " secondary forms derived from such or such a primitive form." The student then frequently abandons the study of crystallography in despair. I think it better to describe, as the crystals of each mineral, the forms that really occur in nature, and not the forms which, to serve the pur- poses of a scientific hypothesis, are assumed to be contained in the natural forms. In justice to Haiiy, it is right to add, that he described in his work bn Mineralogy the " secondary " or real, as well as the "primitive" or assumed forms of minerals; but many of his disciples content themselves with describing the primitive or non-existing forms alone, by which they save trouble to themselves but not to their readers. Another result is, the difference of opinion produced among mineralo- gists, and the controversies which result, as to the true primitive form of particular minerals, Thus, the primitive form of iron pyrites is, accor- ding to Haiiy, the cube, according to Leonhard the pentagonal dodeca- hedron, and according to Phillips the regular octahedron. This difference of opinion respecting the primitive form of a mineral so well known as iron pyrites, shows how little the character is to be depended upon; while it is fatal to any attempt to introduce a systematic nomenclature, or systematic symbols, to represent complex crystals. Even the mineralogists who employ the term " primitive form," do so with a constant reliance upon its uncertainty. Thus, Phillips says, " If a mineral can be mechanically divided or cleaved in directions which produce only one particular form, that form is denominated its primary or primitive crystal. But some minerals are not so circum- stanced... Fluorspar cleaves in four directions, and affords three different forms, a regular octahedron, a regular tetrahedron, and an acute rhomboid ; of these, the first has arbitrarily been selected as the primary crystal, and convenience may be assigned as the reason for the preference... Other substances are cleavable in a still greater number of directions ; for instance, blende, from which may be extracted a rhombic dodecahedron, and from this an obtuse rhomboid, an octahedron, an acute rhomboid, and an irregular tetrahedron; in this mineral also, the choice of a primary crystal has been arbitrary, the rhombic dodecahedron having been selected... The arbitrary selections just noticed will suffice to induce the suspicion, that in this department mineralogy has not yet attained per- fection ; and also to lead the pupil to investigate as he advances in the science, rather than to take for granted what is asserted without proving PRINCIPLES OF CRYSTALLOGRAPHY. 83 the facts.... Other circumstances also exist, sufficient to make us ex- tremely cautious on this point. Some minerals to which primary forms have been assigned, do not yield, or have not yet been found to yield, to regular cleavage in more than one direction, or even not in any direction. In these determinations, one of two modes has been resorted to : In the first, thin fragments of the substance have been held up between the eye and the light; and by this means the extraordinary sagacity of the Abbe Haiiy has enabled him, in several instances, to declare the probable form of the primary, from the directions of the crevices, or appearances of natural joints which may be observed in the fragment; and, in many, these have afterwards proved to be correct. By the other mode, the primary form is determined by analogy, that is, by a comparison of the forms of the crystals of a mineral with those of other known substances ; but this may in some cases prove a source of error." R. PHILLIPS, Introduction to Mineralogy^ article Structure. The inferences which may be fairly drawn from the above statements are these : 1.) The term primitive form sometimes indicates the direction and number of the cleavages of a mineral ; 2) sometimes the direction and number of only a portion of the cleavages ; arid 3) sometimes merely gives the crystallographer's opinion of what the cleavage form of a mineral would be if the cleavages were more numerous and more distinct than nature has happened to make them. It appears from this, that the doctrine of primitive forms has nothing to recommend it, beyond its presumed convenience as an indicator of cleavage, and that when it pretends to indicate any thing beyond cleavage, it is perpetually liable to lead us astray. But the planes produced by cleavage having properties similar to those of the external planes of crystals, and being capable of indication by methods similar to those employed to represent the external planes, it follows that we do wrong to adopt a doctrine so uncertain and so unsafe as this, merely to gain an end which can be better gained without it ; for which reason I have dispensed altogether with the use of the term primitive form, and should have dispensed even with the mention of it, but that I consider it due to those who may wish to know " how I distinguish the primitive from the secondary forms," which question has several times been put to me, to give my reasons for making no such distinction. By dispensing with the doctrine in question, we are left at liberty to describe every crystal presented to us by nature, according to its real aspect, and are not constrained to trim our descriptions to make them suit the limited range of a preconceived hypothesis. 234. MAUY'S PRIMITIVE FORMS are as follow: 1. The Cube. Model 1. P,M,T. 2. Octahedron. 15. PMT. 3. Rhombic Dodecahedron, 63. MT.PM,PT. 4. Tetrahedron. 117. 5. Pentagonal Dodecahedron. 91. 84 PRINCIPLES Or CRYSTALLOGRAPHY. Hatty's Primitive Forms, Continued: {2 P~ TVT T ' PS'M'T (12. P^MT. 13 P^MP^T 8. Right Rectangular Prism. 5. P X ,M_,T. 9. Rhombic Octahedron. 21. PlgM-^T. 10. Rectangular Octahedron. 82 b . MfT.PJT. 11. Rectangular Ditetrahedron. 82 a . M-^T. P T 7 (T T. 12. Right Rhombic Prism. 6. P x ,Mf T. 13. Oblique Rectangular Prism. 79. M_,T. |Pf M Zn Ns. 14. Right Rhomboidal Prism. 79 b . M_,T. |P|M Zn Ns. T84. M|T. |P T 4 T M Zn Ns. 15. Oblique Rhombic Prism. ^ g ^ M^T iP- 6 -T Zw Ne. 16. Oblique Rhomboidal Prism. 105. T, JMJf Tnesw. |P|M ZnNs. f26 a . iPTZw,iPMf|T 2 : or R t . 17. Rhombohedron. \^Q\ ip 2 T Zw, |P 2 M^T 2 : or R,. 18. Regular Six-sided Prism. 7. P X ,T, M|f T 2 : or P X ,V. 19. Bipyramidal Dodecahedron. 26. Pjf- T, Pjf Mff T 2 : or 2R|f . The combination represented by Model 32, which exhibits the Cube, with its edges and solid angles replaced, contains all the planes of three of the above-mentioned primitive forms, namely, the cube, the rhombic dodecahedron, and the octahedron. This is also the case with Models 33 and 34. In one of these combinations the cube predominates, in another the dodecahedron, in another the octahedron. Which of these three is the primitive form of the minerals that the combinations represent ? Is it that which predominates ? If so, some minerals must have several primitive forms, since it is common to find many combina- tions of one mineral in which different forms predominate. Is the primi- tive form to be determined by cleavage ? If so, does not this admit, that the term " primitive form " expresses merely the direction and number of the cleavages of minerals ? All the combinations represented by these three models occur among the separate crystals of arsenical grey copper, the cleavage form of which is the dodecahedron, which combination also occurs among its separate crystals. But the dodecahedron also occurs among the separate crys- tals of antimonial grey copper, and among other crystals of the same variety we find the tetrahedron predominant, but never the cube nor the octahedron. What is the cleavage of antimonial grey copper? There is none ! We find a third variety of grey copper, the arsenical-antimonial grey copper, a chemical compound or mixture of the other two kinds. The forms predominant among the crystals of this third variety are the dodecahedron, the tetrahedron, and the hemiicositessarahedron, but never the cube nor the octahedron. What is the cleavage of this variety ? No man could foretell it: it is imperfect octahedral! But neither pre- dominant form nor cleavage has served to guide mineralogists in assign- PRINCIPLES OF CRYSTALLOGRAPHY. 85 ing primitive forms to these minerals, for although we find a tetrahedron assigned to that kind whose cleavage is imperfect octahedral, yet we are told that the arsenical variety with a perfect dodecahedral cleavage, has for primitive form an octahedron. See PHILLIPS. 235. MR. BROOKE'S " primary forms" consist of the same variety as Haiiy's " primitives," with the exception of Nos. 5, 11, 13, and 19. The Models comprehend examples of all these primitive forms, and I have added the symbols by which they are designated in the present system. In the works of Haiiy and Brooke, the planes of secondary forms are considered to indicate decrements, or portions cut off from the edges or angles of the primitive forms, and they are indicated by formulae which show the comparative length of the portions of different edges supposed to be removed by each secondary plane, so that the symbols represent, not a form existing upon the resulting crystal, but one that is supposed to have been cut off and removed. 236. According to the present system of crystallography, every set of possible secondary planes that can occur by replacing any set of similar angles or similar edges on either of the above 19 combinations, must belong to one of the seven forms which are represented by the following symbols : P,M,T, M X T. P X M, P X T, P x M y T z , or to one of those hemihedral or tetartohedral modifications of these forms, which are described in SECTION X. SECTION VIII. OF FORMS AND COMBINATIONS. 237. OF FORMS. The word " Form," separated from its mischievous qualifier " primitive," may be employed very usefully in crystallography. In all the preceding sections, when I have had to speak of the planes indicated by the symbols P,M,T,M X T.P X M,P X T, or P x M y T 2 , I have used one of the expressions, set of planes, or complement of planes. Now, instead of either of these expressions, I propose to employ the word " form," to denote all the planes that a given symbol can indicate. In that case, the form P, for example, will signify two planes ; the form MT, four planes; the form PMT, eight planes; the form JMT, two planes; and the form |PMT, four planes. Every "set of planes" exhibited in the synopsis of planes, 200, page 68, is therefore a " Form" ; and, let it be particularly remarked, a " form" is only one set of planes. With this precise and limited signification, I shall employ this term in the following sections. 238. The definition of the term " form" given in Professor MILLER'S lately published Treatise on Crystallography, is as follows : " A form is the figure bounded by a given face and the faces which, by the laws of symmetry of the system of crystallisation, are required to co-exist with it. A form will be denoted by the symbol of any one of its faces inclosed 86 PRINCIPLES OF CRYSTALLOGRAPHY. in braces. Thus, the symbol {hJd\ will be used to express the form bounded by the face (hkt) and its co-existent faces." In this case, a person can only know the number of co-existent faces belonging to any given " form," when he knows also three other things : 1 .) What system of crystallisation it is to which the " form" belongs. 2.) What are the laws of symmetry of that system of crystallisation. 3.) How those laws affect the " form" in question. When he knows all this, he can tell the number of planes belonging to a given " form" ; that is to say, the number of times that (Jikl) is contained in {hkl}. But this comprehensiveness of meaning appears to me to make the term difficult of use to students ; and for this reason I have set narrower limits to its application, and, I think, rendered its meaning less liable to mis- conception. The symbols given in the present work always denote all the planes belonging to a form, except when they have the vulgar fraction 1 or I prefixed, or when, as occurs in a few rare cases, the polaric position indi- cative of a single plane is added to the symbol of its form, as PZ, which means the zenith plane of the form P without the nadir plane. 239. Or COMBINATIONS. Professor MILLER'S definition of the term combination is this : " The figure bounded by the faces of any number of forms, is called a combination of those forms." In this definition I concur, for although it was meant to refer to the term " form," as defined by the Professor himself, it serves equally well to denote combi- nations of the " forms" that are represented by the symbols of the seven sets of planes P,M,T,M x T.P x M,P x T,P x M y T z . Therefore, A " Combination" is a crystal whose faces consist of two or more " Forms" It is impossible to fix, theoretically, a limit to the number of forms that may occur upon a combination. It is the business of the mineralo- gist and the chemist, in applying crystallography to their respective sciences, to point out the combinations that really occur, either in the mineral world or among the products of the laboratory. 240. The following is a free translation, adapted to the illustrations which accompany the present work, of GUSTAV ROSE' s account of the terms " form" and " combination" : " The forms of crystals differ essentially in this, that their faces are either all alike, or (leaving parallelism out of the question) partly or entirely unlike. The first are called simple forms, the last compound forms. The octahedron, Model 15, or the figure on page 41, which is bounded by eight equilateral triangles ; the cube, Model 1, or the figure on page 10, which is bounded by six squares ; or the six-sided pyramid, Model 26, which is bounded by twelve isosceles triangles, are conse- quently simple forms ; while the ordinary crystal of Galena, Model 29, which is bounded by eight equilateral triangles and six squares ; and the quartz crystal, Model 73, which is bounded by twelve isosceles triangles and six rectangles, are both compound forms. PRINCIPLES OF CRYSTALLOGRAPHY. 87 " The simple forms differ from one another in the number, the figure, and* the relative inclinations of their faces, and they arc consequently very different in aspect. Yet the position of their faces, considered in relation to the middle point of the crystal, is always regulated by a deter- minate law of symmetry. All faces, edges, and corners, with very few exceptions, have faces, edges, and corners parallel to them ; and in most cases, one end of a crystal possesses exactly the same faces, edges, and corners that appear on its other end ; so that the crystallographer has commonly occasion to study only one end of the crystal. But it does not always follow, that when a simple form has similar faces, it also has similar edges or corners. The contrary has indeed been shown in the examples already adduced, since the octahedron and cube have similar edges and corners, whereas the six-sided pyramid has edges and corners of two different kinds. Hence the term ' simple form' has not the same meaning in crystallography as in geometry. Many simple forms have dissimilar corners and similar edges, as the rhombic dodecahedron, Model 63, or the upper figure on page 13 ; others have both edges and corners dissimilar, as is the case with the six-sided pyramid; still, in the latter case, the corners are generally symmetrical. " The simple forms are named after the number and figure of their faces, or after other characteristic peculiarities. The faces which bound the forms are denoted by the same names as the forms themselves ; that is to say, the faces of the octahedron are called octahedron faces, the faces of the dodecahedron, dodecahedron faces, &c. In notation, the faces are distinguished by letters or figures; the faces of one and the same simple form receive the same letter or figure, the faces of different forms different letters or figures. " If we take a compound form and imagine any one set of its similar faces to be enlarged till they alone bound the inclosed space, and obliter- ate all the dissimilar faces, we then perceive a simple form. If, for example, we take the Galena crystal, Model 29 3 and in imagination thus enlarge the three-sided faces till they meet one another on all sides, we thereby produce the octahedron, Model 15. If, on the contrary, we enlarge the four-sided faces, we produce the cube, Model 1. Hence, we perceive that a compound form consists of two or more simple forms, or, generally speaking, of as many simple forms as it possesses dissimilar faces. None of these simple forms can naturally appear perfect on the compound, but only portions of each can be visible, and these separated from the other similar portions by intervening portions of dissimilar forms. The faces of each particular form bear a certain relation in respect of magnitude to the faces of the other forms that are present ; in some cases being greater or predominant, in other cases being smaller or subordinate. 241. " As the compound form is a combination of simple forms, it is for that reason generally called a ' combination. 1 A given combination is denoted by the names of the simple forms which it comprises, in the writing of which we begin with the name of the simple form which is 88 PRINCIPLES OF CRYSTALLOGRAPHY. predominant, and place the subordinate forms afterwards ; and when it is particularly necessary, the difference of magnitude is explicitly explained. Thus (P,M,T.pmt*), Model 29, and Model 30, are three different combi- nations of the cube, Model 1, with the octahedron) Model 15 ; and among these, Model 29 is the combination in which the two forms are equipoised ; Model (P,M,T. pmt) is that in which the cube predominates, and Model 30 that in which the octahedron predominates." Elements der Krystallo- graphie, page 2. 242. It will be immediately perceived, that, in consequence of the restricted meaning that I have given to the word " form," many crystals commonly called " forms," or simple forms, must on the proposed system be termed combinations ; as examples of which I may notice the cube, which contains the three forms P,M,T, and the rhombic dodecahedron, which contains the three forms MT, PM, PT, both of which combina- tions are among the simple forms of Rose and Miller. It is generally of no consequence whether crystals such as these are called forms or combinations ; but on the other hand, it is of infinite consequence that our technical terms should have meanings which can be easily found, easily understood, and easily remembered ; and such I hope are the meanings given in this section to the terms " form" and " combination." METHODS OF INDICATING THE GENERAL ASPECT OF COMBINATIONS. 243. The method of distinguishing the relative sizes of the planes of different forms, which ROSE has described, 241, appears to me to be far less effective than the method which I have described in 69- Models. Rose's Symbols. Proposed Symbols. 31.* (a : oo a : oo a) + (a : a : ). P,M,T. pmt. 29. (a : OD a : B oo" a) + (a : a : a). P,M,T. PMT. 30. (a : a : a) + (a : oo a : o> a). p,m,t. PMT. 244. But, indeed, the symbolic description of the relative magnitudes of the planes of different " forms" upon a given " combination," has never been attempted by any crystallographer, as I shall show, by quoting a few specimens of notation from different works. HAUY. i i PAB provided the " primitive form" is the cube. i PBBA'A 1 provided the " primitive form" is the octahedron. 3 'A^P^ 1 provided the " primitive form" is the dodecahedron. i * A cube, with its corners very slightly replaced by the planes of the octahedron. Equivalent to Model 31, without the twelve rectangular planes, or Model 38, with four additional triangular planes. PRINCIPLES OF CRYSTALLOGRAPHY. 89 According to Haiiy's method of notation, any one of the above three symbols can be used to indicate any one of t the following three com- binations : Model 32. P,M,T, int. pm, pt, PMT *\ T3 T> M T mt nm nt PMT \ Al1 containin S the Cube > the Rhombic 64. P 5 M,T, mt. pm, pt, FM J , Dodecahed and the Octahedron. 34. p,m,t, MT. PM, PT, pmt } Haiiy's symbols, take which of the three you "will, do not indicate the general appearance, the aspect) of the crystal. They only tell the dimensions of certain solids presumed to be cut off, by the planes of the combinations, from the corners or edges of the IDEAL SOLIDS which Haiiy supposed to be contained in the combinations, and which he deno- minated their " primitive forms." MOHS The symbol employed by this crystallographer to indicate any one of the three combinations in question, is H.D.O. in which H. signifies the hexahedron or cube, D. the dodecahedron, and O. the octahedron. His symbols cannot discriminate one of the com- binations from either of the two others, nor does he give any method of doing so except in words at length. Another Example of comparative Notation: Model 36. P,M,T, mt. pm, pt, Jpmt C In this case > half the planes of pmt 34. p,m,t, MT. PM, PT. Iprot 1 sh T n upon model 34 are sup P os ? d * n ~JL, I to be su PP resse d, vlz -> those of the - 37. p,m,t, mt.pm,pt, iFMi (. Zne Zsw Nnw Nse octants. Here are three combinations which differ essentially in aspect, although they comprise the same suite of forms. The difference is owing to the predominance in one combination of the Cube, in another of the Dode- cahedron, and in the third of the Tetrahedron, as is very well shown by the models and distinctly indicated in the above symbols. Let us see how these three combinations are described by other methods of notation. WHEWELL. 2 (3) (1, 0, 0) + 2 (6) ( 1, 1, 0) + (4) ( 1, 1, 1) which symbol answers for every one of the three combinations. MILLER. {100}, [HO], /c (111}. which symbol answers for every one of the three combinations. G. ROSE. Model 36. (a : oo a : oo a) -f (a : a : oo a) -f- | r (a : a : a). 34. (a : a : oo a) -f (a : oo a : oo a) -|- 1 r (a : a : a). 37. i r (a : a : a) -f (a : a : oo a) -f (a : oo a : oo a). The difference consists only in the priority of position which is given to the symbol that indicates the predominant form. MOHS. O H. .D. which symbol answers for every one of the three combinations. 90 PRINCIPLES OF CRYSTALLOGRAPHY. HAUY. 1 1 1.0 1 1 . PBA a e E provided the " primitive form" is the cube. i PBBA 2 A 2 provided the " primitive form" is the tetrahedron. 1 2 1 A 1 ! E'P provided the " primitive form" is the dodecahedron. Any one of these three symbols may be used to indicate any one of the three dissimilar combinations just as the primitive form permits. MR. BROOKE'S symbols for combinations are of the same character as Haiiy's, and, like Haiiy's, are based upon the suppositions primitive forms of minerals ; so that a symbol for any combination of two or three forms, is not founded on crystallographic relations, but depends entirely upon the figure of the ideal primitive form of the mineral. 245. The quotations from Haiiy appear to me to show very signifi- cantly the mischief that proceeds from making such a doctrine as that of " primitive forms" the basis of a system of notation. Here we have six examples of crystals, all easy of accurate description, for every one of which Haiiy has three different and very complicated symbols, none of which give more than a vague idea of the aspect of the given crystals. Indeed the symbols seem to be contrived mainly for the purpose of showing the author's opinion respecting the ideal " primitive forms " of the minerals. This is a complete sacrifice of crystallographic precision upon the altar of theoretical mineralogy. 246. With respect to the subject we have immediately under discus- sion, namely, the power of denoting the relative magnitudes of different forms existing on the same combination, it may be questioned by some, especially when they see that the most eminent crystallographers have paid little or no attention to it, whether it is a subject that deserves attention? The answer to which question is, that any previous want of attention to accuracy in observing facts, or want of power to record them, forms no excuse for continued want of attention, when the value of the facts is made known and the power of recording them provided ; and I believe that I may safely add, that no one will glance over the tables in the Second Part of this work without becoming assured that differences in the relative magnitude of forms is a crystallographic character of considerable value, and one which can be indicated with such extreme facility that it would be absurd to neglect it. The silence of former crystallographers respecting the use of symbols capable of indicating differences in the aspect of such combinations as contain forms of similar quality but of unlike magnitude, may be accounted for by supposing them to have been without the symbols, and not without, the will to use them ; particularly as we find crystallographers who make no distinctions in their symbols, giving special instructions for distin- guishing the different combinations by words at length. PRINCIPLES OF CRYSTALLOGRAPHY. 91 SECTION IX. ZONES. 247. The planes of crystals are ranged upon them in such a manner as to form circular bands or Zones, of which there are Jive varieties par- ticularly worthy of attention. a.) The Prismatic or Equatorial Zone. The planes belonging to this zone are those of the forms M,M X T,T. Their position is vertical; they are parallel to the axis p a ; arid they surround the equator. They are crossed by a brown line on the models of crystals. Their polaric posi- tions are described in 70 to 74. b.) The North Zone The planes belonging to this zone are those of the forms P,P X M,M. They are all parallel to the axis t a , and therefore pass from left to right parallel to the observer. They surround the north meridian, and they are crossed by a blue line on the models of crystals. Their polaric positions are described in 122. c.) The East Zone. The planes belonging to this zone are those of the forms P,P X T,T. They are all parallel to the axis m a ; and they sur- round the east meridian. They are crossed by a purple line on the models of crystals. Their polaric positions are described in 122. d. e.) The North-east Zone and the North-west Zone. The planes belonging to these zones are those of the forms P,M X T,PMT, P X MT, P x M y T z . Of the forms P and M X T, I have already spoken. The other planes are parallel to no axis; one half of every given form surrounds the north-east meridian, and occupies the ne and sw quadrants of a crystal ; the other half surrounds the north-west meridian, and occupies the nw and se quadrants. The polaric positions of planes belonging to these zones are described in 124. Besides these five zones, many others are commonly described, but it does not appear to me that they are of sufficient importance to merit any particular explanation. 248. Common Properties of the Planes of Zones. 1.) Every zone is a many-sided endless prism, that cannot produce a crystal, or closed form, until cut or crossed by a form belonging to some other zone. 2.) All the planes of a zone are connected by edges that are perfectly parallel to one another, and to the axis of that zone. 3.) In three of these zones there are two sets of four planes, which all meet one another at an angle of 90, so as to have a square cross-section. These planes are as follow : In the Equatorial Zone : M,T and MT. Where the cross section is the equator, and the axis is p a . In the North Zone : P,M and PM. Where the cross section is the north meridian, and the axis is t a . In the East Zone : P,T and PT. Where the cross section is the east meridian, and the axis is m a . 4.) The North-east Zone and North-west Zone also contain two sets of four planes of remarkable properties, namely, the combination P,MT, the 92 PRINCIPLES OF CRYSTALLOGRAPHY. cross section of which is a square, and the form PMT, the cross section of which is a rhombus, having angles of 109 28' at the opposite sides of the equator, and of 70 32' at the Z and N poles. The cross sections of these zones are the north-east and north-west meridians. 249. The forms and combinations just cited are of great importance, since they serve to guide us in finding the positions of all other forms which can occur with them upon complex combinations. Model 32 repre- sents the whole of them, and its symbol is P,M,T, mt. pm,pt, PMT. If we examine the vertical or prismatic zone, the planes of which are indi- cated on the model by a brown line, we find the form M,T, which make up a square prism, and the form mt, which is a second square prism, both parallel to the axis p a . On the north zone, which is indicated by a blue line, we find the forms P,M, which make up a square prism, parallel to the axis t a , and pm, which is a second square prism, parallel to the same axis. On the east zone, indicated by a purple line, we find P,T, pro- ducing a square prism, parallel to the axis m a , and pt, a second square prism, parallel to the same axis. Finally, in each of the octahedral zones, that is to say, in the north-east zone and north-west zone, we find the square prism made up of P,mt, and the rhombic prism of 109 28' and 70 32', which characterises the form pmt. 250. The iriterfacial angles of these planes are as follow : The planes of one square prism incline upon those of the other square prism in the same zone at an angle of 135, while the angle of the inclination of planes of the square prisms upon planes of the rhombic form, pmt, are respec- tively equal (see 5961) to the half of 70 32' plus 90 (= 125 16'), and the half of 109 28' plus 90 (= 144 44'). Therefore, the angle of P upon M is 90 ; of P upon PM, 135 ; P upon T, 90 ; P upon PT, 135; M upon T is 90; M upon MT, 135; P upon PMT is 125 16'; P upon MT, 90 ; MT upon PMT is 144 44'; M upon PMT is 125 16'; T upon PMT is 125 16'; which several measurements may be proved by means of the goniometer. 251. All the planes which can be contained upon a combination either belong to one of these five zones, or are symmetrically disposed in the spaces situated betwixt the four meridians. The planes which belong to the equatorial, the north and the east zones, are those of forms which cut either one or two axes, and therefore take the symbols P,M,T,MT,PM, PT. The planes of the two octahedral zones (with the exception of P and M X T) are those of forms which cut three axes. The forms which have the symbol P X MT always fall in the direct line of the Znw and Zne zones. The planes of very complex combinations, which do not lie in any of these zones, are the planes of octahedrons that answer to the symbols PM x T,PMT x ,P x M y T 2 . 252. The object of attending to these zones, is, as I have already said, to facilitate the orderly examination and description of crystals. Instead of beginning the description of a combination with an account of a " primitive form," or a " fundamental form," or with any other hypo- thetical matter, depending upon an accidental property of a mineral, it is PRINCIPLES OF CRYSTALLOGRAPHY. 93 better to proceed on grounds strictly crystallographical. It was there- fore prescribed in SECTION VI., and with reference to the present nomenclature and distribution of zones, that the symbols employed to denote the forms present upon any combination, should be ranged in a certain order. This, then, is a law of the present system, established to promote uniformity in nomenclature. You begin the description of a crystal with the horizontal form P. You proceed to the vertical equa- torial zone and you mark down, M,T,MT, M_T, M+T, or any of these forms that happen to be present ; and this completes the description of the prismatic portion of the crystal. You next take the forms of the north zone, PM, P X M, then those of the east zone, PT, P X T, and finally the forms of the octahedral zones and of the open spaces betwixt the zones, arranging the latter forms in the order of the synopsis 200, as PMT, P_MT, P+MT, P x M y T z . You need only open the second part of this work at random, to find numerous examples of the method that is to be followed. SECTION X. THE LAW OF SYMMETRY. 253. In order to be enabled to explain the term " Law of Symmetry" in a satisfactory manner, I shall treat of it in reference to a classification of the forms of crystals under three heads, namely: 1 .) HOMOHEDRAL FORMS, or whole forms. 2.) HEMIHEDRAL FORMS, or half forms. 3.) TETARTOHEDRAL FORMS, or quarter forms. 254. The following is GUSTAV ROSE'S account of these different forms : " Most of the simple forms occasionally suffer a peculiar altera- tion, which is, that the half of their planes, or more rarely the fourth part of them, become so large that they obliterate all the rest. This enlarge- ment and obliteration takes place according to determinate laws, which can be best explained in treating of the forms separately. The result is the production of forms which have only the half or the fourth of the number of planes belonging to the original forms, whence, in contradis- tinction to the latter, they are called Hemihedral or Half Forms, and Tetartohedral or Quarter Forms; while the original forms are called Homohedral Forms." Elemente der Krystallographie, p. 5. 255. Professor MILLER'S account of these forms is this: " The * Holohedral Forms' of any system are those which possess the highest degree of symmetry of which the system admits. ' Hemihedral Forms ' are those which may be derived from a Holohedral Form, by supposing half of the faces of the latter omitted according to a certain law." Treatise on Crystallography, page 21. 256. Both of these sets of definitions leave the matter entirely open for subsequent explanation, which these authors give in treating of the particular forms comprehended in each of their systems of crystallisation. 94 PRINCIPLES OF CRYSTALLOGRAPHY. In the present case, however, I propose to treat this subject with some detail under the general head, and I begin with the following explana- tions of the three chief terms: a.) The Homohedral Forms are such as contain whole forms or com- plete sets of planes, and are designated by the symbols contained in the synopsis, 200. Examples: P,M,T, MT.PM, PT, PMT. b.) The Hemihedral Forms are such as contain half sets of planes or half forms, and are designated by the symbols contained in the synopsis, with the prefix J put to each. Examples: |P,irM,iT,MT.|PM,iPT, 1PMT. c.) The Tetartohedral Forms are such as contain quarter sets of planes or quarter forms, and are designated by symbols which have the prefix i. Examples: iMT.iPM, JPT,1PMT. Hemihedral and Tetartohedral Forms of different kinds are distin- guished by adding to their respective symbols the polaric positions of the planes belonging to each of them, as | PMT Znw Zse Nne Nsw. This symbol shows that the deficient planes are those situated on the homohedral form at the poles Zne Zsw NnwNse. 257. One important point which I beg of the reader to keep in view, in examining the following arguments, is, that many crystals which ROSE, MILLER, and other crystallographers would call " forms" or " simple forms," are called by me " combinations." Thus, the icosi- tessarahedron, Model 22, is with them a " form" of the octahedral system of crystallisation; but with me it is a " combination" of the three " forms," P_MT, PM_T, PMT_. The ultimate difference is, that they would call such a crystal a " Homohedral Form," while I would call it a " Combination of Homohedral Forms." This explanation will, I hope, prevent ambiguities from obscuring the meaning I wish to convey. A " Homohedral Form," therefore, is a " Form," according to the definition of that term given in 237- 1.) HOMOHEDRAL FORMS. 258. There is a peculiarity common to the planes of all homohedral crystals, which demands our special observation. It is, that every plane of such a crystal is a portion of a four-sided endless prism, the axis of which either coincides with the axis p a , or m a , or t% or else touches the central point of the crystal where these three axes cross, and cuts them there at an angle which is peculiar to, and characteristic of, every dif- ferent crystallised substance. 259. Thus, the planes M and T form together a square prism, the axis of which coincides with the vertical axis p a . MT is another square prism; M_,T and M_j_,T are rectangular prisms; and M_T and M + T are rhombic prisms; all having axes that coincide with the same vertical axis p% and all being prisms of four sides. In the same manner, P,M and PM form square prisms; P_,M and P+,M, rectangular prisms; and P_M and P+M, rhombic prisms; all PRINCIPLES OF CRYSTALLOGRAPHY. 95 four-sided, and the axes of all of which are coincident with the trans- verse axis t a . Again, the symbols P,T and PT ; P_,T and P + ,T; P_T and P + T, denote square, rectangular, and rhombic prisms, all four-sided, and the axes of all of which agree in position with the minor axis m a . And finally, all the varieties of the form PMT, consist of two such prisms, square or rhombic, which have axes that are not coincident with either p a , m a , or t a , but which cut those axes in the centre of the form at various angles, according to the particular nature of the crystallised substance, by which particular nature, as influencing the power of crys- tallisation, the dimensions, the angles, and the direction of the cutting prisms appear to be controlled. 260. No single four -sided prism can produce a closed crystal, but any two such prisms which cross one another at any angle, produce at once a closed form or crystal ; and all homohedral crystals are produced in this manner. I have given an example of the result of this crossing of prisms in the formation of the series of octahedrons, described in 162 199; where I have shown that the same two cutting prisms, if the expression is allowable, acting at the same level, and with the same centre, upon the same set of axes, produce either isosceles or scalene octahedrons, and each of these of several different kinds, according as the line of action of the cutting prisms is more or less removed from parallelism with the axis p a or m a or t a . It is because no single four-sided prism can produce a closed crystal, that none of the symbols of forms, except PMT, ever indicate a com- plete crystal, 124. But there are innumerable complete crystals which are entirely free from the form PMT. These crystals, therefore, consist of combinations of other forms, and are so many various examples of the closed crystals produced by the crossing of two or more four-sided prisms. 261. The most complicated crystals of minerals exhibit the same traces of these four-sided cutting prisms, as do the simplest geometrical solids, and we are led by this observation to adopt the following crystallo- graphic hypothesis : The planes of homohedral crystals have been formed by cutting prisms of four sides square, rectangular, or rhombic, which have some- times acted in a line parallel to one of the three rectangular axes of the crystal, and sometimes across them, but always with their centres upon the central point of the crystal, where the three rectangular axes cross one another. 262. This property of the planes of homohedral forms, of being refer- able to cutting prisms which produce planes in sets of four fixed equally around the centre of the crystal, has given rise to what crystallographers have termed the LAW OF SYMMETRY. They mean to intimate by this term that whenever you find upon a crystal one plane of a form that usually occurs in a set of 4 or 8 planes, you will probably find upon the same crystal all the other planes of that form. In virtue of this law it is 96 PRINCIPLES OF CRYSTALLOGRAPHY. held that the top of a crystal is similar to the bottom, the front to the back, the left side to the right side, &c. And it is upon this principle that I have divided the planes into sets of 8, 4, and 2, constituting the series of Forms contained in the Synopsis, 200. In order to place the matter clearly before you, I shall re-arrange these Forms below. Table of all the possible kinds of Homohedral Forms. UNIAXIAL FORMS: BIAXIAL FORMS: TRIAXIAL FORMS. 1. P. 10. MX. 19. PMT. 2. P_. 11. M_T. 3. P + . 12. M + T. 20. P_MT. 21. PM_T. 4. M. 13. PM. 22. PMT_. 5. M_. 14. P.M. 6. M+. 15. P+M. 23. P+MT. 24. PM+T. 7. T. 16. PT. 25. PMT+. 8. T_. 17. P_T. 9. T+. 18. P+T. 26. P.MT+. 27. P+M_T. 28. PM + T_, 29. P_M + T. 30. PM_T + . 31. P+MT_. I mean by Uniaxial Forms, such as cut one axis and have two planes to the set ; by Biaxial, such as cut two axes and have four planes to the set ; and by Triaxial, such as cut three axes and have eight planes to the set. As the polaric position of every plane of all the above thirty-one Forms has been distinctly described in the foregoing Sections, it is easy to determine, by a slight examination, whether any of the planes of a given Form are absent from a combination or not. 2.) HEMIHEDRAL FORMS. 263. The Law of Symmetry is a rule which has many exceptions. The planes which commonly occur in sets of four, sometimes occur in sets of two only; and those which should be found in sets of eight, are frequently reduced to four or two. When this is the case, the forms belong to those that are denominated Hemihedral or Tetartohedral, to the former when they are half forms, and to the latter when they are quarter forms. I propose, in the first place, to take the Hemihedral forms into consideration. 264. The general law which establishes the coincidence of the central points of the axes of four-sided cutting prisms, when they cross one PRINCIPLES OF CRYSTALLOGRAPHY. 97 another to produce an octahedron, is subject to a very remarkable ano- maly. The two rhombic cutting prisms, whose intersection at the same level (considered irrespectively of the angle at which they intersect one another) produces an octahedron, sometimes cross one another at dif- ferent levels, and produce a form which represents only the half of an octahedron ; that is to say, a form which possesses only four planes instead of eight. 265. The Tetrahedron, or Hemioctahedron, |PMT, Model 117, is an example which illustrates this peculiarity. When this model is placed in upright position, its solid angles are at the poles Zne Zsw Nnw Nse, its edges are at the poles Z N n e s w, and its planes are at h. poles Znw Zse Nne Nsw. The angle formed by the plane Znw upon the plane Zse, over the pole Z, is 70 32'. The equator, the north meridian, and the east meridian, are all squares, which have their angles at the same poles as are occupied by the angles of the same sections of the regular octahedron. There are no planes belonging to the prismatic zone, the north zone, nor the east zone. There are two planes on the north-west zone, and two planes on the north-east zone ; being together equal to \ PMT. These relations are seen at a glance, by placing the model in position, and examining it in relation to the coloured lines marked upon it. The north-west and north- east meridians of the model are both triangles, having precisely the same angles that the north-west and north-east rhombic meridians of the regular octahedron would have, if they were each divided into two triangles by the shorter diagonal ; that is to say, every triangle has one angle of 70 32', and two equal angles of 54 44'. There being four planes upon the tetrahedron, and its properties being as described, it is evident that this form is the half of the regular octahedron, and that it consequently is properly denoted by the symbol ^PMT. 266. I proceed to notice the mode of derivation of this form, to which the foregoing description of it is but preliminary. I assume, in the first place, that the two Zenith planes of Model 117 were produced by a rhombic cutting prism of 70 32' and 109 28', whose shorter diagonal was equal to the Nadir edge of the model, and whose longer diagonal was twice the length of the axis p a . Secondly, I assume that its two Nadir planes were produced by a cutting prism of precisely the same dimensions as the prism that produced the Zenith planes. Thirdly, I assume that these prisms crossed one another at the same angle as they must have done to produce a regular octahedron ; but that, when they crossed, instead of having both their axes at the same level, they were at such different levels that the axis or centre of one prism was at the level of the outer edge of the other prism. Conse- quently, the two prisms, instead of cutting through each other from edge to edge, only cut the half of each other, or from one edge to the centre ; so that but two sides instead of four from each prism were disposed upon the octahedral, or rather the hemioctahedral, combination. Hence the result- ing crystal is the same that would be formed by the intersection at a o 98 PRINCIPLES OF CRYSTALLOGRAPHY. right angle of two triangular prisms, one of them having a side, and the other an edge, uppermost. The former would represent the lower half of the upper rhombic cutting prism ; the latter the upper half of the lower cutting prism. 267. Or HEMIHEDRAL FORMS WITH INCLINED PLANES. One of the most striking characters of the form produced in the manner which I have described, is, that it has no parallel planes. Now, there are several combinations of hemihedral forms which have the same property. They are commonly termed Hemihedral forms with inclined planes, and, without any exception, their formation may be explained by the hypo- thesis just applied to the tetrahedron, namely, by the supposition that their individual forms are generated by the intersection of two rhombic prisms, which cut one another at a difference of level equal to half the length of one of the diagonals of the cross-sections of the cutting prisms, which diagonal is sometimes the longer, sometimes the shorter, according to the nature of each form. 268. The hemihedral forms with inclined faces, or rather, the " com- binations " produced by such forms, are as follow : The Tetrahedron PMT. Model 117. The Hemiicositessarahedron i(3P_MT). Model 119. The Hemitriakisoctahedron |(3P + MT). Model 18. The Hemihexakisoctahedron |(6P_MT+). Model 24. We see from this list that every variety of octahedral form, which is particularised in the Synopsis of Forms, 200, is subject to this curious anomaly, and liable to produce combinations of hemihedral forms with inclined planes. Each of the above-cited hemihedral combinations contains half of the eight planes of all the simple octahedrons that combine to produce the corresponding homohedral combination ; and what makes the anomaly still more remarkable, is, that the several octahedrons do not produce single hemihedral forms similar to the tetrahedron, such as |P_MT, or IP+MT, or iP x M y T z , but only combinations of three or six such hemi- hedral forms ; and that we find every hemihedral combination to contain invariably four entire octants out of the eight that complete the homo- hedral combination, and that these four selected octants are always the same. Thus, on comparing Model 117, 1PMT, with Model 15, PMT, Model 119, |(3P_MT), with Model 22, 3P_MT, Model 18, |(3P + MT), with Model 17, 3P+MT, Model 24, |(6P_MT + ), with Model 23, 6P_MT + , ' it will be found, that every particular hemihedral form (with the excep- tions to be stated in 269) contains all the planes of the corresponding homohedral form that belong to the following octants: Znw ZseNneNsw. .and none of the planes that belong to the octants: ZneZswNnwNse. PRINCIPLES OF CRYSTALLOGRAPHY. 99 Now, as every simple octahedral form, equiaxed or unequiaxed, contains one plane in every octant, it follows that the hemihedral combinations can contain only half the planes of each of the forms that belong to the differ- ent homohedral combinations. There is, consequently, a loss of one half of the planes of each complete set or form ; and this, as above described, I explain by supposing that the rhombic prisms, by whose intersection the forms are produced, cut one another at different levels, the difference being in all cases exactly equal to one half the diameter of each of the two cutting prisms. A result which this hypothesis would indicate, and which a practical examination of crystals shows to be true, is, that all these hemihedral combinations bear a general resemblance to the hemiocta- hedron or tetrahedron. Thus, Model 119 is a tetrahedron, with a low three-sided pyramid upon each face ; Model 18 is also a tetrahedron, with a low three-sided pyramid upon each face; and Model 24 is a tetra- hedron, with a low six-sided pyramid upon each face. The general resemblance to the tetrahedron which runs through all these combina- tions is a parallel to the general resemblance of the respective homohedral combinations to the regular octahedron ; and it greatly simplifies our calculations of angles, and references to polaric positions, and other matters, to bear these resemblances in mind. 269. Right and Left, or Direct und Inverse, Hemihedral Forms. It sometimes happens that two hemihedral forms, or combinations, of the same kind, occur upon one crystal. In such a case, one of them contains the planes of the octants Znw Zse Nne Nsw ; and the other, the planes of the octants Zne Zsw Nnw Nse. The planes of the first series of octants are large, and those of the last series small; and this cannot be otherwise ; for if the planes of both sets of octants were equal as well as similar, they would constitute a homohedral form, and not two hemihedral forms. Thus, Model 118 contains two Hemioctahedrons, which, however, do not make up an Octahedron, like Model 15, although the angles of the inci- dence of all the planes are the same. These two sets of half-forms are called the Right (or Direct) and Left (or Inverse) Hemihedral forms. They can be discriminated in symbols, by writing the Right form in capital letters, and the Left form in small letters, as ^PMT, Jpmt, and by adding Znw to the symbol of the Right form, and Zne to the symbol of the Left form, as |PMT Znw, ipmt Zne. In all cases where only one hemihedral form with inclined faces occurs upon a combination, it is assumed to be the right or direct form, and has assigned to it the positions Znw Zse Nne Nsw, which are indicated in the symbol by the single sign Znw. When both the direct and inverse forms occur upon the same combination, the positions of the first are indicated by Znw, and of the last by Zne. 270. There is no single hemioctahedron with inclined faces besides the regular tetrahedron. All the other hemioctahedrons are triple or six- fold, being rather hemihedral combinations than hemihedral forms. It is sufficient to excite our surprise, to see the action of the cutting prisms in forming the scries of complex octahedrons which I have described in 100 PRINCIPLES OF CRYSTALLOGRAPHY. 124 199, a series of combinations so extensive, so diversified, and yet so regular. It is matter of still greater astonishment, to find these cutting prisms intersecting one another after a different law, but with precisely the same degree of regularity, to form another series of com- binations, the hemioctahedrons. And finally, it is no less wonderful to perceive, as we do upon a close examination of the facts, that all those combinations which appear to contain a right and left hemioctahedron of the same kind, are produced by the identical cutting prisms which form the homohedral and the single hemihedral forms ; acting still at the same angle, but not at the level which produces the homohedral form, nor at the level which produces the hemihedral form, but at a level INTER- MEDIATE between these two levels! Hence, two cutting prisms of 109 28' and 70 32' which intersect one another at a right angle, may, although precisely of the same magnitude, produce three very different forms according to the level at which they act. !._) If they act at the same level, they produce the regular octahedron, Model 15, PMT. 2.) If they act at a difference of level equal to one half the length of the longer diagonal of their cross section, they produce the regular tetra- hedron, Model 117, ^PMT. 3.) If they act at any level intermediate between these two levels, they produce the combination of two tetrahe- drons, Model 118, |PMT,ipmt. 271. OF HEMIHEDRAL FORMS WITH PARALLEL FACES The hemi- hedral forms with parallel faces are of quite a different character from the hemihedral forms with inclined faces. The latter, generally speaking, belong to equiaxed, and the former to unequiaxed, combinations. I believe the only exceptions are a few unimportant unequiaxed tetrahedrons. They do not, therefore, occur together on the same crystals, nor are they ever characteristic of the same mineral or chemical substance. All the hemihedral forms with inclined planes belong to the octahedral zones, with the exception of the remarkable three-sided or hemi-rhombic prisms peculiar to the mineral called tourmaline ; whereas, the hemihedral forms with parallel planes occur not only in the octahedral zones, but in the north, the east, and the prismatic zones ; as I shall show by examples drawn from each of the zones mentioned. 272. Hemioctahedrons of the Oblique Prisms The hemioctahedrons of the oblique prisms are formed when a single rhombic cutting prism has acted on a combination so as to produce four planes of a scalene octahedron, but has not been cut by a second prism so as to supply the other four planes requisite to complete the octahedral form. Hemiocta- hedrons of this kind are extremely numerous, but all the varieties are capable of reduction to four classes : a.) In this class, the axis of the rhombic cutting prism has passed in the direction of the plane of the north meridian, and from the Nn pole towards the Zs pole. The planes formed on the combination occupy the positions Znw Zne Nsw Nse. See Model 115, which is terminated by PRINCIPLES OF CRYSTALLOGRAPHY. 101 the hemioctahedron |P x M y T z Znw Zne Nsw Nse; see also Model 103, which contains two dissimilar hemioctahedrons, one of them formed by a cutting prism, whose line of action was nearly parallel to p a , the other by a cutting prism/whose line of action was nearly parallel to m a ; the planes of the latter being in consequence thrown towards the east zone, and of the former towards the prismatic zone. b.) The axis of the rhombic prism has again passed in the direction of the plane of the north meridian, but from the Zn pole towards the Ns pole ; hence, the planes left upon the combination occupy the positions Zse Zsw Nne Nnw, or precisely the positions left unoccupied by the planes of the forms belonging to class a. See Models 57, 71, and 112, all of which exhibit the hemioctahedron 1P X M X T Z Zse Zsw Nne Nnw. The crystal represented by Model 75 contains the hemioctahedrons of both class a. and class b. ; but they are so nearly of a size and agreement in angles as to appear like a homohedral octahedron. This is an error of the model, because the two hemihedral forms are often very distinct on the mineral which it is intended to represent. c.) The axis of the rhombic cutting prism has passed in the direction of the plane of the east meridian, and from the Nw pole towards the Ze pole ; so that the four planes occupy the positions Znw Zsw Nne Nse. See Model 98, which is terminated by the hemioctahedron |P x M y T z Znw Zsw Nne Nse. d.) The axis of the rhombic cutting prism has again passed in the direction of the east meridian, but from the Zw pole towards the Ne pole, and consequently has formed four planes, which occupy the posi- tions Zne Zse Nnw Nsw, or the inverse positions of the planes of class c. See Models 26% 26 b , 26 C , 26 d , 72, 114, and 114% all of which have the forms !P x M y T z Zne Zse Nnw Nsw. e.) The hemioctahedrons of class a occur in combination with those of class b. ; and those of class c. with those of class d.; but the hemioctahe- drons of the first two classes are never found combined with those of either of the last two classes. 273. Scalene Hemioctahedrons which have all their four Planes on one Meridian This class of hemihedral forms is not so abundant nor so important as the others. The examples of it are generally considered to be irregular or misshapen crystals, rather than as belonging to a particu- lar class of combinations. As such I may mention the octahedron |PMT Znw Zse, ipmt Zne Zsw, having four large arid four small planes on different zones, and which is produced by two cutting prisms of similar angles, acting at the same level, and crossing at a right angle ; but being, at the time of crossing, of very different relative magnitudes. Also many varieties of Topas, which contain such combinations as P,M_,T, m x t. P x M y T z Znw Zse, Jp x m y t z Zn 2 e Zs 2 w, p x m y t z Zne 2 Zsw 2 ; and many other octahedrons belonging to the prismatic class of crystals, which are liable to present two hemioctahedrons of different magnitude, but having the same relations in respect to their axes. These variations 102 PRINCIPLES OF CRYSTALLOGRAPHY. depend either upon the absence of one of the two cutting prisms neces- sary to produce the homohedral form, or upon the difference in magni- tude of the two cutting prisms, by the intersection of which the form was produced. A different example of hemioctahedrons having all their four planes upon one meridian is afforded by the Scalenohedron. 274. Model 26 f , the Scalenohedron, is a combination of three hemioc- tahedrons, and represents the planes of three similar and equal cutting prisms. The axis of one of tfyese prisms passes from Zw 2 to Ne 2 , nearly parallel to t a , and throws four planes into the positions Z 2 n 2 e Z 2 s 2 e N 2 n 2 w Z 2 s 2 w, so that they are intersected by the north meridian. The axis of a second prism passes from Zn 3 e 2 to Nn 3 w 2 , and throws four planes into the positions Z 2 nw 3 Z 2 se 2 N 2 nw 2 N 2 se 2 . The axis of the third prism passes from Zs 3 e 2 to Nn 3 w 2 , and throws four planes into the positions Z 2 ne 2 Z 2 sw 2 N 2 ne 2 N 2 sw 2 . The three cutting prisms, therefore, produce the hemihe- dral forms of three very dissimilar scalene octahedrons. 275. The indication of the polaric positions of the different kinds of hemihedral forms described above, is effected by naming the positions of the two zenith planes of each form. Thus : Zne Znw, Znw Zsw, Znw Zse. The planes that are not indicated are parallel to those that are indicated. 276. Hemihedral Forms of the North Zone. The forms P and M become hemihedral, in so far as they frequently appear on a combina- tion differing in size, and therefore in distance from the centre of the crystal, so as, for accurate description, to require the notation : P_Z and p+N ; M_n and m + s ; and sometimes one of the two planes is absent altogether. Hemihedral forms of this kind may be seen in abun- dant variety on the crystals of commercial alum. Models 57, 71, 79 b , 79, 84, 103, 105, 109, 112, all represent combina- tions which include the hemihedral form JP X M, and which occupies in all of them the positions Zn Ns. 277. Models 101, 101% and 106, represent combinations which contain the hemihedral form |P X M Zn Ns, and in addition another hemihedral form occupying the corresponding inverse positions, as |P x MZsNn. The direct and inverse forms in this, as in every case, must be of dif- erent magnitudes, else the two hemihedral forms would constitute a homohedral form. 278. Hemihedral Forms of the East Zone. Leaving out of question the forms P already spoken of in 276, and the form T, which sometimes occurs as Tw alone, or as T_w, t_j_e, the only other forms belonging to the east zone are the varieties of P X T, with the hemihedral forms of which we have now to deal. Models 26 a , 26 b , 2G C , 26 d , 114, 114 a , 72, and 87, all represent combinations which contain the form ^P X T, and in all of which it occupies the positions Zw Ne. Model 26 e represents a combination which contains the forms |P X T PRINCIPLES OF CRYSTALLOGRAPHY. 103 Zvv Ne, and also two varieties of the form ^P X T Ze Nw, in addition to three hemioctahedral forms. 279. Hemihedral Forms of the Equatorial Zone The hemihedral forms JM and IT being already explained, I have only to speak of the hemihedral form |M X T. Model 105 exhibits the form |M X T holding the positions ne sw. Model 107 exhibits three varieties of the hemihedral form |M X T, which occupy the positions n' 2 w s 2 e, nw 2 se 2 , ne sw. The mineral named Topas frequently exhibits ^m_t, Jm + t, and similar forms occur on many prismatic minerals. But the hemihedral forms of this zone are, gener- ally speaking, much less numerous, and much inferior in importance to the hemihedral forms of the four other zones. 280. Polaric Positions of Hemihedral Biaxial Forms. The two planes of each hemihedral form of the three last mentioned zones, namely, |P X M, JP X T, and JM X T, are always parallel to one another, and there- fore are on opposite sides of a crystal. They are never in contact, nor inclined to one another. It is sufficient, in writing the symbols of these forms, to indicate the position of only one of the planes, since the other plane is always parallel to it. Thus: P X M Zn, |P X T Zw. When there are two dissimilar hemihedral forms in the same zone, the one that has the largest planes has assigned to it the positions just recited ; and the one that has the smallest planes, is placed in the oppo- site quadrants. Thus: JM x Tnw, Jm x tne. |P x MZn, |p x mZs. |P x TZw, ip.tZe. 281. OF CERTAIN FORMS THAT ARE COMMONLY, BUT ERRONEOUSLY, CALLED HEMIHEDRAL. Several of the forms which I have described in the foregoing paragraphs, 271 280, are not in general considered to be examples of hemihedral forms ; while, on the other hand, there are two octahedral combinations which crystallographers commonly denom- inate hemihedral forms, but, as it appears to me, without good reason. The Pentagonal Dodecahedron, Model 91 This combination contains the three forms, MJT. P^M, Pf T, presenting four planes on the prismatic zone, four on the north zone, and four on the east zone, all of which planes are perfectly symmetrical. The combination, therefore, contains none but homohedral forms. It is, nevertheless, said to be the hemi- hedral form of the tetrakishexahedron, Model 68, which combination contains the six homohedral forms M|T, Mf T. P^M, Pf M, P^T, Pf T. If the pentagonal dodecahedron consisted of i(MJT, Mf T. PM, PfM, 104 PRINCIPLES OF CRYSTALLOGRAPHY. PIT, PfT), instead of M|T.PM, Pf T, it would be perfectly correct to call it the half of the tetrakishexahedron ; but there is so great a dif- ference between half the number of the forms, and half the planes of all the forms of a combination, that I think the application of the term hemihedral is, in this instance, quite erroneous. The combination M|T. PJM, Pf T, is a complete crystal, a combination of homohedral forms, and is by no means dependent for its characters upon the other forms which occur upon the tetrakishexahedron. For example, M^T is not the half of MJT, Mf-T, but one of two different forms, either of which may, and frequently does, occur in combination without the other, and without being held to be hemihedral. In like manner, we may say that PM is not the half of PiM, Pf M, nor Pf T the half of PJT, Pf T ; but that each is a homohedral form in its own right, and must be so considered, whether it occurs in combination with PfM and P|-T, or without them. And if it be admitted that the forms M|T, PJM, and Pf T are individually homohedral, then it must also be admitted, that the pentagonal dodecahedron is homohedral, inasmuch as a hemihedral com- bination cannot be produced by homohedral forms. 282. The Right Hemihexakisoctahedron with parallel faces, Model 25, which contains the forms PIMJT, PMJT, P|MTJ, (or 3P|M|T). This combination is said to be the half of the hexakisoctahedron, Model 23, which contains the forms PiMiT,PMJTi,PlMT|,P|MTiPiM|T, PM|Ti, (or 6PMJT), a different hemihedral form of which has been already explained, namely, the hemihexakisoctahedron with inclined faces, Model 24, which contains the forms IPMT, JPMiTi, IPJMTi, iPJMTi, ^PiMIT, IPMiTJ, or in the abridged symbol, i(6PMiT). This also appears to me to be a piece of needless theoretical complexity. Model 24 represents the true and the only hemihedral form of Model 23, namely, it exhibits a combination of half the planes of all the forms that belong to the homohedral combination; whereas Model 25 exhibits a combination of half the number of homohedral forms that belong to the homohedral combination. Four of the octants of the hemihedral form with inclined faces, are exactly similar to four of the octants of the homohedral form, and other four are entirely wanting, whereas all the octants of what is called the " the hemihedral form with parallel faces," are totally different from the octants of the homohedral form. The planes of Model 25, if separately enlarged, would make three complete octahedral forms, or three complete scalene octahedrons, namely, the forms PiMIT, PMJT^, and P|MT|, while the separation and enlarge- ment of the planes of Model 24, would produce six scalene tetrahedrons, namely, mMiT,iPMTUPiMTi,andmMTiiP|MiT,|P|MTi. This last consideration places in the most striking point of view, the difference in the characters of the two combinations, one of them being decidedly homohedral, and the other decidedly hemihedral. PRINCIPLES OF CRYSTALLOGRAPHY. 105 283. It is singular enough to observe, that, although the pentagonal dodecahedron and the hemihexakisoctahedron with parallel faces, are so little entitled to the character of hemihedral forms, yet crystallographers have made of these two combinations a distinct class of crystals, under the denomination of " Hemihedral Forms with parallel faces." They probably did not know what else to do with them ; and, certainly, so long as their hemihedral nature is insisted on, they will continue to be difficult of disposal. On the other hand, it is only necessary to admit that " combinations of homohedral forms" are " homohedral combinations," to get immediately quit of the difficulty. But this would render it necessary to cease to apply the term " form" to any solid of more than eight sides, and to substitute for it, in the descriptions of all the equiaxed crystals except PMT, the term " combination." It has often been mentioned as an extraordinary circumstance, that minerals which possess the form of the pentagonal dodecahedron, such as Iron Pyrites, should not, like other hemihedral forms, be pyro-electric. Perhaps the wonder will not appear so great, when it is understood that this form is not essentially, but only theoretically, hemihedral. 3.) TETARTOHEDRAL FORMS. 284. The forms P,M,T, which consist of only two planes each, can- not occur in tetartohedral forms. But these are, I believe, the only forms exempt from this irregularity. I have seen a crystal of Wernerite, which required the symbol P + ,M,T, iMT, and examples of the forms JPM, |PT, though still uncommon, can no doubt be produced, as in Alum and Tourmaline; while the tetartohedral varieties of the form P x M y T 2 are an essential part of the combinations that are called doubly oblique prisms. The forms JMT, iPM, PT, consisting only of a single plane, are necessarily unsymmetrical and out of all rule. The form JP x M y T z is, however, to a certain extent symmetrical. In the greater number of cases, it forms a pair of parallel planes, unlike any other planes on the same combination, and always appearing on c^stals which have no right angles. See the Minerals which compose the class of doubly oblique prismatic combinations, Part II. page 91. The regular octahedron sometimes occurs with one pair of parallel faces very large, and three other pair very small = |PMT, fpmt. The only example that I recollect of tetarto-octahedrons that have the two planes on the same end of the combination, occurs in the case of the crystals of Tourmaline, which are frequently terminated by rhombohe- drons that differ at each end, and which consequently present examples of iP x M, 1P X T and iP x M y T z . See Part II. pages 59, 60. None of these cases are of sufficient importance to merit a special investigation. 106 PRINCIPLES OF CRYSTALLOGRAPHY. Discrimination of Homohedral, Hemihedral, and Tetartohedral Forms. 285. PROBLEM: It is demanded, whether the forms on a given com- bination are homohedral, hemihedral, or tetartohedral. Place the crystal in position, and imagine it to be divided into octants by the equator, the north meridian, and the east meridian. If the octants are all alike, none but homohedral forms are present. If there are two kinds of octants, a hemihedral form is present. If there are four kinds of octants, a tetartohedral form is present. 286. It only remains to be added, that the planes of all hemihedral and tetartohedral forms incline upon one another, and upon the planes of any homohedral forms with which they occur in combination, at the same angles as do the planes of the corresponding homohedral forms ; so that the calculations which serve for homohedral forms, serve equally well for all the fractional forms. SECTION XL A THEORY OF CRYSTALLISATION. 287. The use which I have made of the term cutting prism in explain- ing the Law of Symmetry, will probably meet with objections. It may be represented as absurd to suppose that nature employs a sort of working tool in the formation of crystals, or it may be demanded, what evidence I find of the separate existence of these " cutting prisms," other than the appearance of the crystals that they are said to produce ? It was not my intention to enter, in this work, into any discussion relative to the theory of crystallisation; and it was to prevent any expectation that I should discuss that subject, that, at the beginning of the work, I defined crystallography to be merely " the art of describing crystals." I am still of opinion, that a method of crystallography and a theory of crystallisation are things so very different, that they need not be treated of together, and that I might be permitted to employ, arbitrarily, the term " cutting prism," as a convenient method of ex- plaining certain facts relating to the symmetry of crystals of showing the regularity which prevails in the grouping of their planes, without being constrained to plunge into the details of a theory of crystallisation, of being forced to try to explain, not the derivation of some of the planes of crystals, but the mode of the production of the crystals themselves. Nevertheless, as I have broken ground by giving a hypothetical explana- tion of the derivation of certain forms, and as many persons may expect, in a work on crystallography, something more than a mere technical description of the figures of crystals, it will not be going much farther out of the way, if I explain just so fully what I mean by cutting prisms, PRINCIPLES OF CRYSTALLOGRAPHY. 107 as will render the preceding section intelligible. Whatever comparisons and assumptions I make, having merely this end in view, are only there- fore to be considered as a figurative method of describing crystals. To pretend to give a true theory of crystallisation, would be greatly to over- leap the present bounds of physical science. 288. The planes of crystals which we may conceive to be produced by the action of four-sided prisms, crossing or cutting one another at diiferent angles and in different directions, are enumerated in 259 as those which constitute the forms P,M,T, M X T, P X M, P X T, and P x M y T z . Now these forms are represented in 200, as all the forms that are known in crystallography; and, in SECTION V., I have shown that they represent all the forms that can possibly occur upon crystals. It follows thence, that these FOUR-SIDED CUTTING PRISMS are the generators of all the forms that can appear upon crystallised combinations in other words, that they are the " primitive" or " original forms" whence the planes of all crystals are derived. For this reason, and in order to have a convenient term for common use, I venture to propose for these gener- ating prisms the name of EIDOGENS, from the words, eidos, " a form," and genomai, " to generate." I should be better pleased with the term PRIMITIVE FORM, but unfortunately that term has acquired a meaning which unfits it for my present purpose. By EIDOGEN, then, is meant a four-sided prism of indefinite length. 289- Let us proceed to investigate the origin of the " eidogen" and the manner in which it works. 290. I assume that the particles of all crystals, originally, and at the period of the production of the crystals, were in a state of mobility; being dissolved in a liquid, fused by heat, or suspended in some kind of gas; that the movements of the particles depended on the directing power of electricity ; and that each particle or molecule, whether chemi- cally simple or compound in its nature, whether a single physical atom or a group of atoms, was endowed with polarity, having what we may conveniently denote, in the terms proposed by Mr. Graham, a CHLOROUS pole and a ZINCOUS pole, which poles had severally the power of inducing polarity in adjacent mobile particles, and of attracting dissimilar poles and repelling similar poles. I proceed to investigate the phenomena of crys- tallisation in reference to the case of a saline solution. 291. A saline solution, at a certain state of concentration, begins to yield crystals. Before the crystals appear in the solid state, it is probable that they are completely formed in the liquid state; for the smallest visible crystal is as perfect in its forms as is the largest crystal that ever existed. At different temperatures, any given salt may combine with different quantities of water, or be subject to other changes in com- position, and produce different crystallisable compounds. The forma- tion of each of these compounds may give rise to the production of a certain amount of electrical power, which is now known to be atten- dant both on the exercise of the power of chemical combination and on the act of crystallisation. The crystallisable particles set in motion by 105 PRINCIPLES OF CRYSTALLOGRAPHY. electricity, arrange themselves in the order prescribed by their polarity, so that a single row of particles presents the following arrangement : 1.) CZCZCZCZCZCZCZCZCZ But longitudinal arrangement is not the only one of which particles thus acted upon by electricity are capable. The attractive and repulsive forces must act laterally as well as longitudinally; and may, therefore, produce a lateral arrangement of particles somewhat in this order: C 2.) CZC where the central Z represents the zincous pole of one of the electrified particles of the longitudinal series. But the induced polarity would not be at an end, when the grouping of particles had proceeded thus far. The central zincous pole would now have a chlorous pole on each side of it in the longitudinal series, and four chlorous poles about it in the lateral series. The strength of this attraction would immediately bring the zincous poles of a number of other particles into the combination, and produce a series which may be conceived to increase by accession of particles as follows : 3.) 4.) Z 5.) C ZCZ CZC Z ZCZCZ CZCZC ZCZ ZCZCZCZ CZCZCZC ZCZCZ ZCZCZCZCZ CZCZCZCZC ZCZ ZCZCZCZ CZCZCZC Z ZCZCZ CZCZC ZCZ CZC Z C in which 3.) represents the next stage to 2.), while 4.) and 5.) repre- sent different views of a subsequent stage where 5.) is the next plane of particles to 4.) in a longitudinal series of similar lateral extent. In this manner a single row of electrified particles may be supposed to become connected with a multitude of other particles ; every particle in the longitudinal series inducing polarity in all the mobile particles around it, and becoming the centre of a lateral plane of particles. 292. But there may also be a power which regulates the extent to which the polarity induced by any given particle of a crystallising substance may proceed laterally, and this power I take to be magnetism. The same power which changes the poles of a magnetic needle when an electrical current is passed across it, may rationally be supposed to regu- late the disposition of the particles of a crystallising salt which have been set into motion by an electrical current; and there are many reasons to believe that this is the case. By way of giving an explicit view of this matter, I shall assume, that when an electrical current has dis- posed a series of crystallising particles in the order shown by the figures 1.) to 5.), that there is produced a series of lateral magnetic currents, often but not always at right angles to the electrical cur- PRINCIPLES OF CRYSTALLOGRAPHY. 109 rent, and tending to regulate ike limits within which the power of crystallisation shall operate. Thus I suppose that across each longi- tudinal electrical series, two magnetic axes are formed; the one ex- tending from e to w, the other from n to s in figure 6.), and that c z c 6.) e ZCZCZCZCZCZCZCZCZ w C Z C n the magnetic currents pass from Z in the centre, which is a pole of the longitudinal series, towards the poles n and s, the extremities of one axis, and spread thence towards e and w, the extremities of the inverse axis, in order to complete the circle and return to Z. Will it be said that this is an extravagant assumption ? Is it a whit more extravagant than the assumption so commonly agreed to, that the magnetic effluvia passes from the two ends of a bar magnet to re-enter the magnet by other poles situated in intermediate parts of it ? But the truth of the latter assumption, it will be urged, can be proved by experiment. When iron filings are agitated on a sheet of paper laid over a bar magnet, they arrange themselves in the order of the magnetic currents. Good, I reply ; and when a slice of a transparent crystallised substance, cut properly from an eidogen, is examined by a polariscope, the particles of the mineral are seen to be arranged in an order that wonderfully corresponds with the figure of the magnetic curves, and which renders visible, in the most surprising and most beautiful manner, the two magnetic axes for whose existence I contend. No one who attentively considers the black crosses and curves exhibited by minerals when examined in polarised light, can, I think, hesitate to ascribe them to the effects produced by the arrangement of the particles of the crystallised mineral, if not in the order of the magnetic curve, at least in an order which bears a great resemblance to it, But here is an experiment which appears to throw light on this sub- ject : II x 45 90 135 180 Take a piece of transparent calcareous spar of the size of figure R, and place it over a black cross made with thin lines and of the form of figure x. The crystal may either be placed with a plane flat on the paper, or held with four sides in a vertical position, so that when you look down upon the paper through the upper plane, the point of sight shall be in a line with the axis of the prism. Place the longer diagonal 110 PRINCIPLES OF CRYSTALLOGRAPHY. of the terminal plane of the crystal upon the horizontal line of the cross, figure x. Then look through the crystal at the cross : it will appear like fig. 0. Turn the crystal horizontally 45: the cross will then appear like fig. 45. Turn the crystal other 45: the cross will appear like fig. 90. Turn the crystal other 45: the cross will appear like fig. 135. Turn the crystal other 45: the cross will appear like fig. 180. All these changes take place in turning the crystal through half a circle. If you turn it through the other half circle, a similar set of changes take place, and at 360, the final view of the cross is exactly like the first, fig. 0. The following experiment is equally curious : D If you hold the rhombic prism of calcareous spar over a double cross making angles of 45 at the centre, as shown by figure A, keeping as before the longer diagonal of the terminal plane over the horizontal line of the figure, the cross appears as in figure B. If you turn the crystal round horizontally, changes take place in the figure of the cross similar to those described above, and which are very decided at every 45, until at 90 the figure is exactly the reverse of what it is at 0, the broad double band being then vertical. If you hold the crystal over a figure containing lines which make angles of 30 round the centre, as shown by figure C, the appearance produced at 0, that is, when the longer diagonal of the terminal plane of the crystal is placed over the horizontal line of the figure, is exactly like figure D. The phenomenon exhibited in these experiments is commonly known under the appellation of the double refraction of light. But the particu- lar point to which I wish to draw attention is, the proof afforded by these experiments, that the particles of the crystal which are situated in the direction of the longer diagonal of the terminal plane, have different properties from the particles which are situated in the direction of the shorter diagonal, and that the particles which are situated at intermediate points have properties in agreement with the ratio of their proximity to one or other of these diagonals ; in other words, that the properties of the particles of the crystal depend directly upon their magnetic rela- tions, and that double refraction is one of these properties, and crystalline form another. The fact that electricity and magnetism are directly concerned in the operation of crystallisation, is satisfactorily made out by numerous ex- periments, such as the production of perfectly regular crj^stals of metallic PRINCIPLES OF CRYSTALLOGRAPHY. Ill copper, metallic silver, red oxide of copper, and other insoluble sub- stances by means of simple voltaic circles of very low power. 293. I conceive, then, that an EIDOGEN is formed thus, and has the properties here recited : 1.) There is an arrangement of particles in longitudinal order, of greater or less extent, according to the mass of matter present, and to the degree of electrical excitement produced by the temperature, the circumstances attendant on the act of crystallisation, and the particular properties of the matter under operation, 2.) That there is a contemporaneous exertion of magnetic power, which regulates the arrangement of the particles laterally, and restricts the indefinite combination that would result from the continuous and unchecked propagation of electric polarity by induction among an in- finite number of crystallisable particles. 3.) That the two magnetic axes, ns and ew, fig. 6), are of variable length relative to one another, and are individually regulated by the special nature of the crystallising substance, and by the accidents which modify each act of crystallisation. 4.) That the passage of the magnetic current from pole to pole situ- ated at the ends, and at proportional distances throughout the length, of the magnetic axes, regulates the form of the eidogen. 5.) That innumerable eidogens may exist in a given liquid, without the appearance of a single solid crystal. 6.) That when an eidogen is completely formed, its electricity becomes latent. 7.) That the eidogens may be pierced by one another, and crossed in all directions, without being necessarily destroyed. 8.) But that the formation of some one eidogen, produced by the action of a very powerful electrical force, may take place at the expense of several other eidogens. 9.) That crystals are formed by the crossing of different eidogens, which thus cut out closed forms or crystals, bounded by planes of a determinate figure, which crystals may still remain liquid, and be sub- ject to farther truncation or intersection by other eidogens, previous to their solidification. 10.) That crystals become solid in virtue of the exercise of cohesive attraction, which, on the separation of the solvent occasioned by any sufficient physical cause, gives coherence to the solid particles, and ter- minates for a period the electric action and the mobility which results from it. 11.) That different chemical substances, or possibly different physical groups of particles, regulate, not only the relative dimensions of the mag- netic axes, but also the direction, intensity, and variations of the electrical current ; so that from a given centre there may proceed numerous eido- gens, of determinate dimensions, in determinate number, and proceeding in determinate directions, varying with the physical or chemical properties of a given substance. 112 PRINCIPLES OF CRYSTALLOGRAPHY. 12.) That the plane in which the magnetic axes lye, may sometimes be inclined, and not at right angles, to the electrical current. 13.) That a solid crystal, apparently consisting of one eidogen, is, in fact, a congeries of eidogens, which can often be mechanically separated into numerous smaller masses of eidogens, by planes passing parallel to some of the faces of the eidogens, or to the magnetic plane, or to one of the magnetic axes ; which planes of separation are commonly called " planes of cleavage." 14.) That although the solidification of a crystal suspends, it does not annihilate, the electric properties of its eidogens, which caft be recalled into action by change of temperature and other physical forces. 15.) That the alterations which take place in the size or figure of a crystal, after its solidification, depend upon the joint action of the chemi- cal and mechanical forces to which the crystal is subject. a.) Chemical Action. If the crystal is placed in a solution which, owing to continuous evaporation, reduction of temperature, or other cause, is in a crystallisable state, new eidogens will form around the crystal, corresponding in figure, number, and direction to the nature of the substance, the figure of the solid crystal, and the intensity of the electrical excitement. The origin of these new eidogens is the chemical action which takes place between the particles that constitute the crystal and those that exist in the solution, which several particles I conceive to have the same disposition to combine with the liquid of the solution that two separate quantities of any base have to combine with a qantity of acid incompetent to saturate the two quantities of base. The commencement of this chemical action on the liquid by the outer par- ticles of the crystal, induces polarity in the surrounding mobile particles, and originates the new eidogens. Every plane of the solid crystal being part of an eidogen, is immediately coated by an addition to that eidogen, and this coating is followed by a second or third according to the con- tinuance of the action. Thus, if figure a represents the form of a cross- section of the solid crystal supposed to be submitted to action, figure b may represent it with one cleavage plane added, and figure c with two such planes. c.} Z b.) C ZCZ a.) Z CZC ZCZCZ ZCZ CZCZC ZCZCZCZ Z CZC ZCZCZ C ZCZ Z The solidification of these coatings, or additional layers of particles, is produced by the same force that effects the solidification of the original crystal itself. If, on the contrary, the solid crystal is placed in a solution which is not saturated with the same substance, or which is exposed to increase of temperature, then no increase in the size of the crystal is produced, for the chemical action which takes place between the crystal and the solution, overcomes the cohesive attraction which binds the particles of PRINCIPLES OF CRYSTALLOGRAPHY. 113 the crystal together. The crystal dissolves in the liquid, and the crys- tallisation is detroyed. b.) The mechanical forces act differently. When a crystal is freely suspended in a crystallising solution, the eidogens generated by the resulting chemical action take those positions around the crystal which the electric and magnetic currents direct. But where any mechanical obstruction comes in the way, the progress of crystallisa- tion ends ; for, as the liquid eidogens have no power to overcome mecha- nical obstructions, they proceed no farther in the direction of such obstructions. Hence, the face upon which a crystal lies in a pan, or by which it is affixed to an insoluble substance, can 'receive no addition. Hence also the diversity of incomplete and irregular crystals produced by the crystallisation both of salts and minerals between masses of inert solid matter, or in holes and confined situations, where the liquid eido- gens had no opportunity to extend and arrange themselves freely and symmetrically. Mr. Spencer's very curious experiment of producing, by voltaic agency, veins of metallic copper amidst a porous mass of stucco, shows, in my opinion, very decidedly, the influence of mere mechanically ob- structions in retarding the arrangement of eidogens, and so preventing the formation of regular crystals. 294. Let us now examine a few cases in crystallography with reference to the foregoing hypothesis. #.) EHRENBERG, in examining crystallising solutions under a powerful microscope, saw no commotion in the liquid, and no appearance of any arrangement of particles, preceding the actual formation of the solid crystal (PoGG. Ann. Bd. 35) ; but I do not conceive this to militate against the hypothesis of the previous formation of liquid eidogens, because in examining a transparent solution with common light, it was a priori unlikely that he should see the solid particles in movement. It is probable that a similar examination of a crystallising solution with polarised light, would show the existence of the eidogens, b.) EHRENBERG, in observing the crystallisation of common salt, first saw flat six-sided tables, which, on the subsequent appearance of cubical crystals, melted away. This is a phenomenon something like the sup- positions destruction of one eidogen by another, noticed in 293, No. (8). c.) This hypothesis readily accounts for the formation of prisms of all dimensions, their lateral dimensions or angles being determined (1) (4) by the electric or magnetic properties of each substance ; the word sub- stance meaning either a chemical substance, or a group of physical atoms ; it being impossible to determine whether the phenomenon is chemical or physical. d.) The formation of prisms with oblique terminations is accounted for by (12), where the plane of the magnetic axes is supposed to be inclined to the electric current. It is, of course, alike rational to suppose the inclination of this plane to be in the direction of its shorter as of its Q 114 PRINCIPLES OF CRYSTALLOGRAPHY. longer diagonal j so that this hypothesis explains equally well the pro- duction of the two different kinds of oblique prisms. e.) The hypothesis also readily explains the production of pyramids or octahedrons formed by the intersection of two, three, six, twelve, or more eidogens ; forming pyramids of eight, twelve, twenty-four, forty- eight, or any greater number of planes ; four planes being the quantity produced by every eidogen that acts upon the same centre. See No. (11)' 293. Am I asked for evidence in support of the hypothesis laid down in (1 1), namely, that different substances have an inherent power to regulate the number, direction, and intensity of the electric and magnetic currents ? The evidence is, that exactly such combinations of eidogens exist, as would be produced if the hypothesis were true. You read and interpret a cypher, and you naturally infer that you have found the key to it. Besides, what extravagance is there in the supposition that, at a given centre, where an electric current is admitted to exist, such a system of vibrations may come into play as suffices to turn the electric current several times at right angles, or at equal angles, and simultaneously to change the magnetic poles ? Suppose the phenomenon to become visible to the eye, would it appear more astonishing than is the symmetrical arrangement of iron filings by magnetic power, or the symmetrical arrange- ment of common dust upon a plate of glass, which takes place when you apply a fiddle-bow to the edge of the glass ? f.) The production of hemihedral forms with inclined faces, such as the tetrahedron, may be explained by the intersection of two eidogens having similar proportions and magnitudes, but situated at different levels ; the amount of the difference of level being always equal to one half of the width of the intersecting eidogens. A hemihedral form with inclined faces contains, therefore, planes derived from the half of two eidogens. As it is reasonable to imagine that there is a systematic cir- culation of the electric and magnetic currents in every complete eidogen, one might suppose that a form containing incomplete eidogens would have peculiar electrical relations ; and this supposition is proved by expe- riment to be well grounded, for very nearly every crystal that contains hemihedral forms with inclined faces, possesses the singular property of pyro-electricity; that is to say, it becomes electric with polarity upon every change of temperature, as witness tourmaline, boracite, and the like. The phenomena described in 270, the singular relations which hold between the octahedron, the tetrahedron, and the combination of two tetrahedrons, are easily explained according to this hypothesis. If the reader will merely read 270, and substitute throughout the word eido- gen for the term " cutting prism," the theory of the formation of the crystals in question will be instantly comprehended. ) sin B sin C where cos x cos B cos C \ A,B,C a cosa=: 39. A,B,c a . 6)=Un i c |2!Mz|l . sini(A-B) 6)=tan^C' . Then, a = ( + &) + \(a - b) 40. A,B,c 6 Same as No. 39 Then, 6 = (o+6)- J(o-6) Logarithmic Equations. Log cos x = log cos B + log cos C - 10. Log cos a = log 2 + log cos | (A + x) + log cos ^ (A - x} - (log sin B + log sin C) + 10 Log tan % (a + b) = log tan g c + log cos ^ (A^- B) - log cos (A + B) Log tan % (a - 6) = log tan i c + log sin i(A-B)-log*ii(A + B) 41. A,B,c C sin C = sin c sin A sin a a is first found by No. 39. 42. A,B,c C sin x where cot x tan A cos c cot c cos (B - a?) 43. A,B,c a cot a = - - -- ' cos x where cot x = tan A cos c Log sin C = log sin c + log sin A - log sin a Log cot x = log tan A + log cos c - 10 Log cos C == log cos A + log sin ( B - a 1 ) - log sin x Log cot x = log tan A + log cos c - 10 Log cot a = log cot c + log cos (B - a?) - log cos x 46. o,6,C B Same as No. 45. 47. o,6,C 44. A,B,c b Similar to No. 43, B and A changing places. 4ot4r cos^(a -6) Log tan i (A + B) = log cot ^ C + log co* 45. AC A tani(A+B)=cotiC a ( a _ ft) _l og cos i (a + 6) rai(a - 6) Log tan $ ( A - B) = log cot \ C + log sin i ( - ) - log sin J ( + 6) Log sin c = log sin C + log sin a - log sin A Log tan x = log tan a + log cos C - 10 Log tan A = log tan C + log sin a? - log sin (b - ,r). Log tan x = log tan b + log cos C - 10. Log cos c = log cos b + log cos (a - x) - log cos x Log cos x = log cos b + log cos c - 10 Log cos A = log 2 + log sin ^ (x + a) + log sin (a? - a) - (log sin b + log sin c) + 10 sin C sin a sin c = : . sin A A is first found by No. 45. 48. o,6,C A tan A tan C sin sin (6 - a?) where tan x = tan a cos C 49. o,6,C B Similar "to No. 48, a and b changing places. 50. 0,6.0 51. 52. cos b cos (a - x) cos c = s - cos x where tan x = tan b cos C A cosA = 2sin^(ar+g)8inji(a?-o) where cos x = cos 6 cos c s-ft)sin(s-c) A sini A where s == | ( + b + c) ;sinA = ^[log sin (s-b) + logsin(s-c) - (log sin"& + log sin c) + 20] PRINCIPLES OF CRYSTALLOGRAPHY. 123 C.) QUADRANT AL SOLID TRIANGLES. Where side c = 90. A solid triangle which has one of its sides a quadrant, is called a quadrantal solid triangle. For the solution of the several cases of quadrantal solid triangles, only two quantities are required to be given, besides the side of 90. The equations which contain the sign may give either the desired angle or its supple- ment. See 330, 331. No. Given. Sought. Equations. Logarithmic Equations. tan A 53. A, B sin B = tan ft log sin B = log tan A + 10 log tan a. cos a 54. A, b sin b = CQg ^ log sin b = log cos a + 10 log cos A. sin A 55. A,a C sin C = ^n^" lo S sin C = log sin A + 10 - log sin a. tan A 56. A,B tan a = g - n jj log tan = log tan A + 10 log sin B. tanB 57. A,B b tan b = gin A log tan 6 = log tan B + 10 - log sin A. 58. A,B C cosC = -cos A cos B log cos C = log cos A + log cos B - J 0. 59. A,6 a cos a = cos A sin 6 log cos a = log cos A + log sin 6-10. 60. A,b B tan B = tan 6 sin A log tan B = log tan b + log sin A - 10. 61. A,6 C tan C = - CQ3 ^ log tan C = log tan A + 10 - log cos b. 62. a,B A tan A = tan a sin B log tan A = log tan a + log sin B - 10. 63. a,B 6 cos b = cos B sin a log cos b = log cos B + log sin a 10. tan B 64. a,B C tan C = - ^-^ log tan C = log tan B + 10 - log cos a. cos a 65. a ,6 A cos A = ^-^ log cos C = log cos a + 10 log sin b. cos b 66. ap B cos B = . _ log cos B = log cos b + 10 log sin a. 67. a,6 C cos C = cot a cot b log cos C = log cot a + log cot b 10. sin A 68. A,C sin a gm Q log sin a = log sin A + 10 log sin C. 69. A,C B C os B =-- log cos B = log cos C + 10 - log cos A. 70. A,C b cosb = - tan ^ log cos 6 = log tan A + 10 - log tan C. 71. a,C A sin A = sin C sin a log sin A = log sin C + log sin a - 10. 72. ,C B tan B = - tan C cos a log tan B = log tan C + log cos a - 10. 73. ,C b cotb=- ^-^- Jog cot b = log cos C + 10 - log cot . tan B 74. B,/, A sin A = ton 6 log sin A = log tan B + 10 - log tan b. cos b 75. B/> a sin = ^~g log sin a == log cos b + 10 log cos B. 76. B/ C sin C = ^y log sin C = log sin B + 10 - log sin b. 124 PRINCIPLES OF CRYSTALLOGRAPHY. cos C 77. B,C A cos A = cosB 78. 79. B,C B,C a b sin b -=. tanB tanC sin B sin C 80. 6,0 A tanA = tan C cos b 81. 6,0 a cot a = cos C "cot b 82. i,C B sin B = sin C sin b log cos A = log cos C + 10 log cos B. log cos a = log tan B + 10 log tan C. log sin b =3 log sin B + 10 log sin C. log tan A = log tan C + log cos b 10. log cot a = log cos C + 10 log cot b. log sin B = log sin C + log sin 6-10. D.) RIGHT-ANGLED PLANE TRIANGLES. These are printed with the other formulae chiefly for the sake of con- venience in reference. The subjoined figure represents lineally several of those important functions of an angle, which the Table of Sines, &c. gives numerically. They have been already explained in 37 62, but may still be advantageously recited here. Let the arc a a be part of a circle which has w for its centre, and the angle cwn for its limits. Then en is its sine and cw its cosine. Let ci be an arc parallel to the arc a a, and having the same centre, w, and the same limits. Then c n is its tangent, c w its cotangent, and n w its secant. Let c c be an arc with n for its centre, and the angle cnw for its limits. Then c w is its sine and c n its cosine. Let c o be an arc parallel to the arc c c, and having the same centre, n, and the same limits. Then c w is its tangent, c n its cotangent, and n w its secant. The sines of the angles of -a triangle are proportionate to the opposite sides. The figure new represents a i / a right - angled plane triangle ; it also represents the quadrant of an /: equator, c or new is the right / ; angle and the centre of the equa- tor. M and T are axes perpen- dicular to one another. M = m a T = t a . The angles n, or cnw, and w, or cwn, are of variable magnitude; but one of them is always the complement of the other, so that if, in any right-angled triangle, a * either n or w be given, then n, c and w are all given; because the sum of the three angles of a triangle is equal to two right angles. Hence: c = 90; n = 90 w ; and w = 90 n. PRINCIPLES OF CRYSTALLOGRAPHY. 125 No. Given. Sought. Equations. Logarithmic Equations. 83. c,M,T n tan n = Log tan n = log T + 10 - log M JM M 84. c,M,T n cot n = Log cot n == log M + 10 - log T, 85. c,M,T w tan w = Log tan w = log M + 10 - log T T lif ;. c,M,T w cot w = Log cot w = log T + 10 - log M [Put T or t a =. 1 (unity), then M or m* = { JjJ ^ Jj or 87. n m a ni 3 = cot n -N 88. w m a m a = tan w I 89. m> n n = cot m a f Axrs t a being 1 .0. 90. m w w = tan m a J E.) OBLIQUE-ANGLED PLANE TRIANGLES. Let the three angles be called A, B, C, and the three sides, situated respectively opposite to the three angles, a, J, c. There must always be three quantities given, and one of these a side. If two of the angles A and B be given, the third C is found thus: C = 180 - ( A + B). No. Given. Sought. Equations. Logarithmic Equations. 91. A,B, C C = 180 - (A + B) 92. A,a,6 B sin B = log sin B = log b + log sin A - log . When A is less than 90 and a less than 6, the angle B may be either that given in the Table of Sines, or 180 - B. 93. A,a,6 C Find B by 92; then C by 91 . 94. A,a,6 c Find C by 93; then c by 95. 95. A,a,C c c = g Sm log c = log a + log sin C - log sin A sm A 96. A,B,6 a a = . log a = log b + log sin A - log sin B 97. A,B^> c c = ? n _ log c = log b + log sin C - log sin B sm B 98.AAc a log rin - log 2 + J log 6 + j log c + log - b + c * cos ^ A- log (6 + c) a = (6 + c) cos x log a = log (b + c) + log cos x - 10 99. A,6,c B tanl(B-C) = jlcot4A logtan | (B - C) = log(6 -c) +log cot^ A o + c - log (6 + c) Now,^(B + C) = 90-^A; because A + B + C = 180 and ^ (A + B + C) = 90. Then, 126 PRINCIPLES OF CRYSTALLOGRAPHY. 100. A,b,c C Same as No. 99. Then, 101. -A- 6 c j - log b - log c] + 10 where s = (a + b + c) Another metfiod, 102. a,b~c A cos^A = .COO 6^ r-H CO tO rH CO Oq Oq Oq Oq" rH rH i I rH rH rH rH rH rH rH ? ' OOCOtoOt>-rHl>.rHOCOOOCOCOI>.at>OqtoOOtOrHCOOq ^tOCOb~OOr-Hl>.COi IClt^tOrHCOOqr-Hr-HOOtOOCOOqOS o ocooo t^ o co o oq co rH 00 tO rH CO Oq Oq Oq" r-3 r-H rH rH rH rH r-H rH ^ QO OO r-H t>l rH co c. CO O CO CO OO CO O cocooooco Cioooqi>.t>.t>.cpcpcptoo O O CO O _ O O CO O O CO tO j^ tO CO Ol CO r-H GO r-H CO rH CO Co r-H o co oi oq oootooooooq Ci rH CO Oq r-H r-H r-H r-H !" OCOtrHrHrHOCOrHOOOCOGOCOO COOOCOrHOOOOI>.COiOr-HCOOOrHr-HOr-HCOt>.COO oocoococor-HOooooqcor ir>.cooi>"rHoqooocorHcooq QQ^^^^^^^OOOqi>.COCOiqtqiOrHrHrHrHCOCOCOCOCO o co o <_ O CO to O CO O to CO !>. rH r-H ooo tot>. t> b^ ' COOO GOCO COt>.r-H COCOi I COOOrH COrHO O O rH O b-O tO O O t>- O CO < toco r-H OO CO CO Oq r-H i-H Oi ^^ fe J222:i CO CO CO CO rH !>. CO !>. l>. Oi to oq CO O < CO O O CO r-H co o o co t>. ,_} ,_; oo co to OOCOOCO(Mr-H iO OO O5 I>.O O'r-H CO CO O Oi OO 1>.CO CO o o o o o o o o o o o o o to o to o o CO rH rH *>T J ^ OOCOOOCOOOiOOlOOOCOiO^COC^r i OCOOOI>.O_ , OCOOOCOC^lOr-HOOCOCOr-l co^Hc^r-H 140 PRINCIPLES OF CRYSTALLOGRAPHY, given, Sulphate of Barytes, Part II., page 68, combination P, Model 6; required, the equatorial angle at the pole n. Referring to the Table of Indices, and in the horizontal line of 5ths, and below the numerator 4, we find .8000. This is the cotangent of 51 20', or half the angle of incidence of the plane ne upon the plane nw of the Form MJT, which angle is 51 20' X 2 = 102 40'. The Table of Indices presents no great advantages in the cases of such simple vulgar fractions as those which I have quoted, but it will be found to be very convenient when the fractions are such as J^, |^, T 5 T , or B other odd numbers which cannot be readily converted into decimal fractions by mental calculation without the help of the pen. 326. The axes of Forms belonging to a given zone are multiples of one another, for the same Mineral. The Table of Indices presents an- other peculiarity, which it is useful to remember in examining complex crystals of minerals. Whenever a combination is found, which con- tains several rhombic forms in one zone, as, for example, Mf T, mjt, mf t, Muriate of Copper, then a single horizontal line of the table contains the length of the variable axis of all the forms of the given zone pecu- liar to that mineral. Thus, if we take the mineral just quoted, and refer to the line of thirds among the denominators, then under numerator 2, we find the cotangent which shows the angle of a triangle of MfT; under numerator 4, the cotangent which shows the angle of a triangle of m|t; under numerator 6, the cotangent which shows the angle of a tri- angle of mft. These cotangents are .6667, 1.333, and 2.000, and the corresponding angles are 56 19', 36 52', and 26 34'; so that the inter- facial angles of the plane ne upon nw of these respective vertical forms, peculiar to Muriate of Copper, are 112 38', 73 44', and 53 8'. This view of the mutual relations of the axes of all the forms belonging to any single zone of a particular mineral, is frequently of important service in correcting imperfect measurements, and supplying data when defective. 327. Construction of Symbols for Biaxial and Triaxial Forms. It will be seen, that besides giving a Table of Indices, I have added two columns of Indices to the Table of Sines and Tangents. These are in- tended to be used in the construction of symbols, as I shall show by an example.' Given, model 82% with the symbol M X T. P X T, and the mea- surements ne on nw = 118 4', Ze on Zw = 110. Required, the value of x and x expressed in vulgar fractions. a.) Divide 118 4' by 2, and find the cotangent of the result: ^-i' = 59 2', cot. .6000. This is the value of m a , when t a is 1.0; and against this angle you find, in the outer column of the Table of Sines, the vulgar fraction f , which is the value of x in M X T. b.) Divide 110 by 2, and find the cotangent of the result: ^ = 55, cot .7002. This is the value of p% when t a is 1.0, and against this angle in the Table of Sines you find the vulgar fraction j 7 T) , which is the value of x in P X T. The symbol M X T. P X T is, therefore, equal to MfT, P/oT. But as PRINCIPLES OF CRYSTALLOGRAPHY. 141 T is of the same length in M X T and P X T, it is better to change f for its synonyme T %, and write M^T. P-f^T. c.) In forming symbols for Triaxial Forms, it is necessary to bear in mind that there are three relations to be shown, namely, the length of p a , m a , and t% or the distance of the poles Z,n,w, from the centre of the triaxial form. Let the general symbol be P x M y T 2 , and the value of x, y, z be 1,2, 3. Then we have P : M P : T M : T 1 : 2 1 : 3 2 : 3 This is equal to p^in^tf or P { M 2 T 3 ; but as it is desirable that t a should, as frequently as possible, be made unity, it is better to write P^M|T, which indicates the same relations. 328. CALCULATION OF OBLIQUE-ANGLED SOLID TRIANGLES. 5.) To find the angle across the edge Znw of Model 82 a , or the in- clination of a plane of M^T to a plane of Py^T. Take the Zn pole of the combination a,s the vertex of an oblique- angled solid triangle. This pole is at the point where the two upper planes of P X T meet the two front planes of M X T. The solid triangle which I shall take consists of the following parts : the north meridian, which is a rectangle, with an angle of 90 at the pole Zn. Call this side c of the solid triangle. Then side a will be the plane P X T Zw, and side b will be the plane M x Tnw. Agreeably to this arrangement, angle A will be the inclination of M x Tnw on the north meridian; angle B will be the inclination of the plane P X T Zw on the north meridian ; and angle C will be the inclination of M X T nw upon P X T Zw across the edge Znw, which is the angle required in the problem. Of the parts here named, we know the value of c 90, of A = 59 2', of B = 55, the two latter being found by problems 1) and 3), 322. The present problem is, consequently, given, side c and angles A, B; required, angle C, and this can be solved either by Formula 41, or Formula 42. We have therefore a choice between two methods of calculation, and we naturally ask, what is to guide us in choosing betwixt them? If we compare formulae 41 and 42 with one another, we perceive that the former is considerably the simpler of the two; but that it requires a given quantity, which is to be found by a preliminary calculation. This preliminary calculation is, as directed in the Formula, to be effected by means of Formula 39. Upon referring to this Formula, we find that the preliminary calculation is much longer than either 41 or 42; so that Formula 41 becomes altogether considerably longer than Formula 42; and for this reason the results by each being the same we should pre- fer to work our calculation by Formula 42. But there is yet another thing to be taken into consideration, in judging of the comparative merits of these two Formulae. Although by problem 5.) we seek only to find angle C of the oblique-angled solid triangle that we have described, 142 PRINCIPLES OF CRYSTALLOGRAPHY. there is afterwards another calculation to make, problem 6.), in order to learn what are the plane angles of the external planes of Model 82 a . Now, it appears from a comparative examination of the model and the solid triangle, that two of the plane angles of the model are equivalent to side a and side b, of the same solid triangle whose angle C is equiva- lent to the angle required in problem 5.) It appears, also, that these two sides are given by the solution of the equation contained in Formula 39> with its subordinate equation, Formula 40 ; consequently, if we re- solve the equations contained in Formulae 39> 40, and 41, we solve prob- lems 5.) and 6.) together; whereas, if we begin by solving problem 5.) with the short Formula No. 42, we shall still have to resolve the equa- tions in Formulae 39 and 40, or in the two additional Formulae, Nos. 43 and 44, in order to be able to solve problem 6.) It is therefore advisable to solve problem 5.) by means of Formulae 39 and 41, instead of employing the short Formula No. 42. 329. Formula 39- Given, A, B, c; Sought, a. First Equation : Log tan } (a + &)= log tan |c + log cos i (A B) log cos|(A + B). Second Equation : Log tan i (a b) = log tan ic + logsin i(A B) logsini(A + B). Then, a =-. i (a + 6) -f \ (a ) A is given = 59 2' ; B = 55 ; c = 90. It is advisable to commence a calculation of this nature by drawing out a full plan of it in accordance with the Formula, and in the manner represented in the first of the two opposite columns. It is always pos- sible to do this, because every step of the calculation is set forth in the Formula, and it is proper to do it with the greatest care, since this is the most important part of the process. When the plan is ready, you resort to the Table of Sines and Tangents, and extract the functions, as shown by the second of the two columns opposite. In the present example, you take first the log tan of angle 45, and write it both in the first and second equation. Then you look for the log cos of angle 2 1', and write it in the first equation, and while the Table of Sines is still open at the place of 2 1', you extract its log sin, and write it in the second equation. Finally, you turn to angle 57 1' in the Table of Sines, and extract its log cos for the first equation, and its log sin for the second equation. You already perceive one of the uses to be derived from beginning a calculation of this kind by drawing out a full plan of it, which use is, to be enabled to avoid the trouble of turning up repeatedly the same angle in the Table of Sines, by extracting at once all the functions of the same angle which may be required at different stages of the calculation. Another use of drawing out a plan of the calculation is, that it promotes an orderly method of working, and prevents the accidental substitution of multiplication for division, or other confusion of the quantities, which is apt to occur when this precaution is neglected. PRINCIPLES OF CRYSTALLOGRAPHY. 143 The two equations being prepared in the manner above described, the two additions and two subtractions are readily made in agreement with the Formula. You have then only to look in the Table of log tangents for the angles answering to 10.2638 and 8.6227, which you find to be 61 25' and 2 24/, and to add these together, as prescribed in the final equation, to find the result of the equation, and the desired quantity of the problem, which is, a = 63 49'- Hence, the calculation is the simplest thing imaginable ; the entire difficulty of proceeding consisting, in fact, in properly choosing the parts of the solid triangle, and carefully drawing up the plan of the operation. Details of t/ie First Equation: Details of the First Equation: log tan ^ c = 45 = log tan I c = 45 0' = 10.0000 ~A -= 59 2' A = 59 2' B = 55 B = 55 (A - B) = 4 2' (A - B) = 42 + log cos g (A - B) = 2 1' = + log cos (A - B) = 2 1' = 9.9997 19.9997 A = 59 2' A = 59 2' B = 55 B = 55 (A + B) =114 2' (A + B)=1142' - log cos i (A + B) = 57 1' = - log cos $ (A + B) = 57 1' = 9.7359 log tan $ (a + b) = log tan J (a + 6) = 6125'= 10.2638 Details of the Second Equation : Details of the Second Equation : log tan i c = 45 = log tan J c = 45 . sin B Now, as A and B are alike, this formula is equivalent to the following : cos A cos a . sm A But by Formula 104, we find that ^~ == cot A, so that Formula 4 can be abridged to cos a = cot A. We look, therefore, in the Table of Natural Cotangents for 54 44', the value of A, and we find its cotangent to be .7072, which is precisely the product of ^^ when worked at length, as in 310. The abridged Formula thus obtained, is one that is very extensively used in crystallography, as will appear in the sequel. 336. The manner in which I have brought this subject under the reader's attention, will, I hope, not only enable him to make considerable use of the Formulae empirically, but put it in his power to consult a mathematician with advantage, because the latter can adjust his instruc- tions to the object which this statement places in view. PRINCIPLES OF CRYSTALLOGRAPHY. 149 SECTION XIII. AN INQUIRY INTO THE VARIETY OF FORMS AND COMBINATIONS WHICH OCCUR UPON THE CRYSTALS OF MINERALS. 337. I take as the basis of this inquiry, a work entitled, " Elemente der Krystallographie, nebst einer tdbellarischen Uebersicht der Miner alien nach den Krystallformen, von GUSTAV ROSE. Zweite Auflage, Berlin, 1 838." The reasons which induce me to make choice of this work are these : It contains a full account of a system of crystallography, which, in contradistinction to Haiiy's system, I may call the German system of crystallography. This system was invented by WEISS of Berlin, and has been adopted by NAUMANN, MOHS,. and ROSE in Germany, by MIL- LER in England, and by other distinguished mineralogists. It is the system of crystallography, which, slightly modified by different writers, is now in most general use throughout Europe. ROSE'S work is, more- over, the latest published continental treatise on this science, and it is the production of one of the most eminent of living mineralogists. It may therefore be held to contain a fair representation of the crystallo- graphy of the present age. 338. ROSE'S, or rather WEISS'S, SYSTEM, is founded on the assump- tion of what are termed, " Six Systems of Axes of Crystallisation" which systems bear the following titles : G. ROSE : Literal translation : 1.) Das regulare System. 1.) The regular system. 2.) Das zwei- und einaxiage. 2.) Two-and-one-axed. 3.) Das drei- und einaxige. 3.) Three-and-one-axed. 4.) Das ein- und einaxige. 4.) One-and-one-axed. 5.) Das zwei- und eingliedrige. 5.) Two-and-one-membered. 6.) Das ein- und eingliedrige. 6.) One-and-one-membered. Professor MILLER'S names for these six systems are as follow : 1.) The Octahedral System of Crystallisation. 2.) The Pyramidal System. 3.) The Rhombohedral System. 4.) The Prismatic System. 5.) The Oblique Prismatic System. 6.) The Doubly Oblique Prismatic System. In describing these six systems, I shall employ Professor MILLER'S titles in preference to those afforded by the literal translation of ROSE'S. 339. There are three topics to handle in the following inquiry: First, I have to show, after ROSE, what forms and combinations con- stitute the crystals of the mineral world, and how they are classified, both according to his method and to mine. 150 PRINCIPLES OF CRYSTALLOGRAPHY, Secondly, I have to give the mathematical proofs of the separate identity of the several forms, and to show how the different combinations are to be mathematically analysed. Thirdly, I have to prove that the System of Crystallography which is recommended in the present work, is suitable for the exact and con- venient description of every combination and form thus analysed and identified, and, therefore, suitable for all the purposes of the mineralogist. Let it not be supposed, however, that I am about to give a full account of Rose's book : that is not my object. What I purpose to do is, to recite the forms and combinations which he adduces as constituting each system of crystallisation, and then to explain and illustrate these forms and combinations according to the methods developed in the foregoing sections of this treatise. I shall describe the same objects that Rose describes, but in entirely different terms. 340. The principles upon which the classification of crystals into the above-named six systems is effected, are as follow: All natural crystals may be divided into two classes the EQUIAXED and the UNEQUIAXED crystals. The equiaxed are those whose axes are all alike, as p a m a t a ; the unequiaxed, those whose axes are dissimilar, as p a m a t a , or plm^t*. This distinction refers to Combinations, 239, not to Forms, 237. 1.) Now, it is an ultimate fact in mineralogy, that when a mineral produces an equiaxed combination, it produces no combinations that are unequiaxed. This may be considered a universal rule, subject to a few trifling exceptions. The equiaxed minerals, or the crystals whose axes are p a m a t a , are therefore made to constitute the regular system, or Octa- hedral System of Crystallisation, which is called octahedral, because the regular octahedron is one of its most important crystals. This is the first of the six systems of crystallisation. 2.) When the unequiaxed crystals are compared with one another, we find that we can select a class whose axes are p a m a t a ; that is to say, a class of combinations whose equatorial axes m a and t a are both alike, but different from the vertical axis p a . It is again an ultimate fact in miner- alogy, that a mineral which produces a single combination of this two- and-one-axed kind, never produces a combination whose axes have any other relation than this. The two-and-one-axed class of crystals is therefore a second system of crystallisation, and as the crystals com- monly called Square-based Pyramids, Models 12 and 13, are very im- portant forms of this system, it has thence been termed the Pyramidal System of Crystallisation. 3.) All the other unequiaxed crystals have the relation of p*m a t a ; that is to say, their three rectangular axes are all different from one another. In this peculiarity, the crystals of the remaining four systems agree ; but, at the same time, they disagree in several other very important par- ticulars. For example, we can select from them a class of crystals, whose axes have always the relation of p a m a 5 t a 3 or p x m a 4 t a 3 , and which, if they are PRINCIPLES OF CRYSTALLOGRAPHY. 151 prisms, have 6, 1 2, or 24 vertical planes, or if they are pyramids, have 3, 6, or 3 X planes meeting at the poles Z and N. The rhombohedron, Model 26 a ; the hexagonal prism, Model 7; the twelve-sided prism, Model 10; and the six-sided pyramid, Model 26, present examples of this class. We are again guided by the observance of an ultimate fact in mineralogy, which is, that a mineral which exhibits any one of the forms here named, may exhibit any of the others, but never can produce a combination whose axes are different from p* m* 5 t* 3 or p.m^t?* This is the third system of crystallisation, and as the rhombohedron is a very important crystal of this system, it has thence been termed the Rhombo- hedral System of Crystallisation. Its other designation of the three- and-one-axed system, is due to the circumstance that WEISS and his followers describe the forms of this system in reference to a system of three similar equatorial axes which cross in the centre of the equator at an angle of 60, instead of a system of two equatorial axes which cross at an angle of 90. In this particular, the present system of crystal- lography differs from theirs essentially. 4.) Pursuing the examination of the unequiaxed crystals, we find it impossible to classify them any farther by reference to their axes, which are always p*niyt*, but not plmutfa nor pim*^. We therefore shift our ground, and classify them upon a different principle. All the rest of the unequiaxed crystals whose north zone, east zone, north-east zone, and north-west zone, present only Homohedral forms, 257 262, constitute the fourth system of crystallisation, which, because it contains the minerals that produce the extensive family of rhombic and rectangular prisms, is called the Prismatic System of Crys- tallisation. 5.) The unequiaxed crystals whose north zone, east zone, north-east zone, and north-west zone, present Hemihedral Forms, 272 280, constitute the fifth system of crystallisation, which is called the Oblique Prismatic System of Crystallisation, because it contains the minerals that produce prisms with single terminal planes of the east zone or north zone, set obliquely on the vertical prisms. 6.) The unequiaxed crystals whose north-east zone and north-west zone present Tetartohedral Forms, 284, constitute the sixth system of crystallisation, which is called the Doubly Oblique Prismatic System of Crystallisation, because all the planes, both prismatic and pyramidal, appear as if they were parallel to three axes that cross each other obliquely in every direction. The classification of the minerals of the last three systems depends upon ultimate facts in mineralogy similar to those which lead to the classification of the minerals of the first three systems. A mineral which presents a combination belonging to the 4th, 5th, or 6th system, never presents a combination belonging to any other system. This is a broad statement, liable, like all general rules, to particular exceptions, yet not to such exceptions as render the general statement untrustworthy. Such are the " six systems of axes of crystallisation." They all 152 PRINCIPLES OF CRYSTALLOGRAPHY. rest upon what I have called an ultimate fact, that a mineral which pro- duces a combination belonging to one of the six systems, never produces a combination belonging to any of the other systems. This curious phenomenon cannot be accounted for: the fact is as inexplicable as extraordinary; but it affords a very excellent basis for the classification of crystallised minerals. Each of these six systems contains a certain number of characteristic forms and combinations, which I shall now proceed to examine in detail, taking the name of each system of crystallisation as the title of a separate chapter. I. THE OCTAHEDRAL SYSTEM OF CRYSTALLISATION. 341. The character of the Forms belonging to this system, as given by ROSE, is this, They have three Axes, which are all equal, and placed at right angles to one another. ROSE'S enumeration of the Forms belonging to this system of crys- tallisation, is as follows : A. Homohedral Forms: 1. The Octahedron, Model 15. PMT. 2. The Cube, 1. P,M,T. 3. The Rhombic Dodecahedron, .- 63. MT.PM,PT. 4. The Icositessarahedron, - 22. 3P_MT. 5. The Triakisoctahedron, 17. 3P+MT. en T* ,-u 1,1 r Q /M_T,M + T.P_M, 6. The I etrakishexahedron, 08. < T> mr oW-n T, ^ Jr+M, JP_ 1 , r_|_ 1 , 7. The Hexakisoctahedron,' 23. 6P_MT + . B. Hemihedral Forms: 1. The Tetrahedron, Model 117. i PMT. 2. The Hemiicositessarahedron, 119. i (3P_MT). 3. The Hemitriakisoctahedron, 18. \ (3P+MT). 4. The Hemihexakisoctahedron with inclined faces, 24. | (6P_MT + ). 5. The Pentagonal Dodecahedron,. . . 91 . M_T. P_M, P + T. 6. The Hemihexakisoctahedron with parallel faces, 25. 3P_MT + . I have added to ROSE'S name of each crystal the number of the Model, and the new symbol which is intended to represent it. These crystals are " Forms," according to ROSE'S explanation of that term, inasmuch as each of them contains none but similar and equal planes; but, with the exception of Models 15 and 117, they are all " Combinations," accord- ing to the new definition of that term given in 239? since, with the two exceptions named, every one of them contains several of the " Forms" that are enumerated in 200. See SECTION VIII. PRINCIPLES OF CRYSTALLOGRAPHY. 153 NORMALS. 342. Besides the three rectangular axes, which I call p*m a t% but which ROSE calls a, a, a, there are other four lines, or axes, of great impor- tance in the consideration of the combinations that belong to this system. These are the lines that pass through the centre of the crystal, and con- nect the eight corners of the cube, and which are perpendicular to the eight faces of the octahedron. These are sometimes called Hexahedron axes, sometimes Normals to the Octahedron faces, and sometimes Tri- gonal axes. The POLES in which these lines terminate are described in 21, pages 6, 7, as Znw Zne Zse Zsw Nnw Nne Nsw Nse. The line which passes from any one of these poles to the centre of the crystal will, throughout this section, be called the NORMAL of that pole ; hence, a line betwixt c and 2, figure in page 6, will be the Znw normal. Less important than the four lines which connect the corners of the cube, but still of considerable utility, are the six lines which connect the centres of the edges of the cube, and which, therefore, terminate at the poles nw ne se sw Zn Zs Ns Nn Zw Ze Ne Nw. See 21. These lines will also, with a view to avoid the use of the word axis, be termed normals, and the length of each normal will be a line from the pole at the surface of the crystal, to c, its centre. Thus, a line betwixt c and M, figure in page 6, will be the Zn normal. It will probably be occasionally convenient to consider all the 26 poles that are marked on the figure in page 6, and enumerated in 21, to be the terminations of normals, and, in order that they may be referred to when necessary, either individually or in groups, I shall give them the following names: Unipolar Normals. The normals that touch the poles Z N n e w s. They meet the centres of the planes of the cube, the corners of the octahedron, and the four-faced angles of the rhombic dodecahedron. Bipolar Normals. The normals that touch the poles nw ne se sw Zn Zs Nn Ns Ze Zw Ne Nw. They meet the centres of the edges of the cube, the centres of the edges of the octahedron, and the centres of the planes of the rhombic dodecahedron. Tripolar Normals. The normals that touch the poles Znw Zne Zse Zsw Nnw Nne Nse Nsw. They meet the corners of the cube, the centres of the planes of the octahedron, and the three-faced angles of the rhombic dodecahedron. Relation of these Normals to the Angles of the Homohedral Forms of the Octahedral System of Crystallisation. In Models 22 and 23, all the 26 normals terminate in solid angles. In Model 17 and 68, only the unipolar and tripolar normals terminate in solid angles, while the bipolar normals terminate in edges. In Model 63, the unipolar and tripolar normals terminate in solid angles, and the bipolar normals in planes. In Model 15, the unipolar normals terminate in solid angles, the bipolar normals in edges, and the tripolar normals in planes. And in Model 1, the unipolar normals terminate in planes, the bipolar normals in edges, and the tripolar normals in solid angles. 154 PRINCIPLES OF CRYSTALLOGRAPHY. ROSE'S Catalogue of the Minerals that belong to the octahedral sys- tem, is given in Part 1L, pages 3 12. A symbolic catalogue of the Forms and Combinations which are pre- sented by the crystals of each of these minerals, is given in Part II., pages 1532. The Table in Part II., page 15, is a synopsis of the Forms and Combinations which belong to the system. 1. THE OCTAHEDRON. Model 15. PMT. 344. This form is figured and fully described at page 41. ROSE'S symbol for it is (a : a : ), which intimates that each face cuts the three rectangular axes in a similar manner. It is indispensable to an octahedron, that its equator, north meridian, and east meridian, shall be squares, and that the angles of these sections shall be at the poles Z N n e s w. It follows that the angle of inclination of any edge to either of the axes p a m a t a is ~ = 45. The north-east and north-west meridians are rhombuses of 109 28'. Therefore, the angle over any edge is 109 28', and the angle of incidence of two planes over a solid angle is 70 32'. The plane angles of the faces are all 60. According to the principles of classification explained in SECTION IV. the octahedron is a complete pyramid with a square equator. The minerals which occur in this form are quoted at page 100, Part II., being in number no fewer than thirty-six. ROSE gives the following method of indicating single planes of the octahedron:" It is useful in many cases to denote each of the eight faces of the octahedron by a particular mark. With this intent, we indicate the front half of the horizontal axis, or the part turned towards the observer, by p the back part by a',, the rijfht half of the horizontal axis which is parallel to the observer by a /; , the left half by a' /; , the upper half of the ver- tical axis by a lin the lower half of itbya'^ ; the signs for the eight faces of the octahedron are then as follow," (column 1) : I have added, in a separate column, the signs by which I propose to effect the same end. ROSE'S SIGNS. NEW SIGNS. M (a, <*// a//,) PMT Znw. 2.) (<>', a // a///) PMTZsw. 3.) M a! tl a///) PMTZse. 4.) (, ei a a///) PMTZne. 5.) (, a // a> M ) * PMT Nnw. 6.) K a // '///) PMT Nsw. 7.) K a 'u a' //; ) PMTNse. 8.) K a> tl a' //; ) PMTNne. There is an error in Rose's sign, No. 8.), where a', should be a y . I copy it as it stands in the original, because it shows the difficulty of getting such marks correctly printed. It is still more difficult for any body to remember them. It was the sight of this Table that induced me to introduce into crystallography the use of the astronomical terms Zenith, Nadir, north, east, south, west, equator and meridian, which I have found to aid the memory so much in remembering polaric positions, and so greatly to facilitate the description of particular sections and zones, that I should be very unwilling to give up the use of them. PRINCIPLES OF CRYSTALLOGRAPHY. 155 N 345. PROBLEM. Given, the symbol PMT ; sought, the interfacial angle of the plane Znw upon the plane Nnw. Divide the form PMT into octants, 301, take the Znw octant, and find, by the follow- ing methods/the inclination of the plane Znw to the equator, which is half the required angle of Znw or Nnw. The axes of PMT are p a m a t% or P = 1, M = 1 , T = 1 . The given octant is, therefore, the simplest form of a right-angled solid triangle. Take pole n for its vertex. Then you have given, the following three quanti- ties: M = the right angle = C; the plane en w = 45 = side a; and the plane cnZ = 45 = side b. The angle required is that across the edge E, which, being opposite to the given part called side b, is angle B. Hence, this problem is the same as Given, sides a, b> sought, angle B: Formula 14. log tan B = log tan b -f 10 log sin . 10 + log tan b = 45 = 20.0000 log sin a = 45 = 9.8495 log tan B = 54 44' = 10.1505 Therefore, 54 44' is the angle across the edge marked E in the figure of the octant, or the inclination of the plane Znw to the equator. Then, 54 44' x 2 = 109 28', is the interfacial angle of Znw on Nnw. 346. PROBLEM. Given, the symbol PMT; sought, the interfacial angle of the plane Znw upon the plane Zne. This problem is the same as the foregoing, except that the required angle is that formed by two planes meeting at the north meridian instead of the equator. .) Proceed as before, but instead of Formula 14, take Formula 13; for the problem now is, Given, sides a, b; sought, angle A. Formula 13. log tan A = log tan a -f- 10 log sin b, 10 + log tan a = 45 = 20.0000 log sin b = 45 = 9.8495 log tan A = 54 44' = 10.1505 Therefore, 54 44' is the inclination of the plane Znw to the north meridian ; and twice that angle, = 54 44' X 2 = 109 28', is the inter- facial angle of Znw on Zne. b.) Another method of working this problem, is by means of the quad- rantal solid triangle, No. 58. Put c = 90 = angle of the equator at the north pole, measured from the edge ne upon the edge nw. Then, 156 PRINCIPLES OF CRYSTALLOGRAPHY. A = 54 44' and B = 54 44', will be half the inclination of tlie plane Zne on Nne and of Znw on Nnw ; and C will be the inclination of Zne on Znw demanded in the problem. Now, according to the principles explained in 331, angles A and B, being both under 90, are both positive, and the sum produced by their multiplication together is also positive, but since this product is prefixed by the sign in the Formula, it must be changed from positive to nega- tive. Hence, the angle found by resolving this equation will not be the angle contained in the table, but its supplement. These considerations remove the ambiguity which rests upon Formula 58. Formula 58. log cos C = log cos A -f- log cos B 10. log cos A = 54 44' = 9-7615 + log cos B = 54 44' = 9-7615 log cos supplement of C = 70 32' = 9-5230 Therefore, C = 109 28'; because 180 70 32' = 109 28. See 330, 331. 347. PROBLEM. Given the symbol PMT, sought the value of each of the three plane angles of the face Znw. a.) Take the same solid triangle as in problem 345, and observe that the part now required is the side opposite to the right angle, that is to say, side c. Hence the problem is: Given, sides a,b; sought, side c, namely : Formula 15. log cos c = log cos a + log cos b 10. log cos a = 45 = 9-8495 4- log cos b = 45 = 9-8495 log cos c = 60 = 9-6990 Therefore, each of the plane angles required is 60. b.) Take the quadrantal solid triangle described in 346, b.), and employ Formula 56, log tan a = log tan A -f- 10 log sin B. 10 + log tan A = 54 44' = 20.1505 log sin B = 54 44' == 9-91 19 log tan a = 60 = 10.2386 c.) Here it may be noticed, that when the three axes, p a m a t% of the form PMT are all alike, the three plane angles of an external face of the form are also all alike; when the axes are p*m a t a , or p a m*t a , or p a m a t|, that is to say, two of them alike but different from the third, then the plane angles are also two alike but different from the third; and when the three axes are p*m a t*, or all unlike, then the three plane angles are also all dissimilar. Hence the distinction of regular, isosceles, and scalene octahedrons. PRINCIPLES OF CRYSTALLOGRAPHY. 157 N 348. PROBLEM. Given, Model 15 with, the symbol P X MT; required, the value of the characteristic x . Take, with the goniometer, the inclina- tion of the plane Znw upon the planes Nnw and Zne. Both are 109 28'. Then take the Znw octant of the form as a solid tri- angle with pole n for its vertex. The parts measured of the solid triangle are the edge M or right angle = C; the edge N = ^^ 540 44/ _ angle A; and the edge E = ^^' = 54 44' = angle B. The part re- quired is the plane angle Znc, which, being a side of the solid triangle, and opposite to angle B, is side b. We require the plane angle Znc, because the natural tangent of that angle is the value of x in the problem. Hence the problem to be worked is this: Given, angles A,B; sought, side b: Formula 5 ; namely, cos b = j^-J which Formula can be simplified by Formula 104, (see 335), into cos b = cot A. As we have here only to compare two quantities with one another, we do not need the aid of logarithms, but content ourselves with referring to the table of natural numbers, where we find: nat cot A = 54 44' = .7072 nat cos b = 45 00' = .7072 That is to say, we first look in the column of natural cotangents for the angle 54 44' and find its natural cotangent to be .7072. Then we look in the column of natural cosines for the number .7072, and find its angle to be 45. This, then, is the value of angle Znc, and the natural tangent of this angle is 1.0 or unity; so that PJMT means PMT. You will observe, that you can, if you please, use the logarithmic functions, but it is without gaining any advantage. Thus : log cot A = 54 44' = 9.8495 log cos b = 45 00' = 9.8495 349. PROBLEM. Given, Model 15, with the symbol PMT. Required, the inclination of the Znw normal, to the axis p a , to the equator, to the Znw plane, and to the other normals. The Znw quadrant of the north-west meridian of the form PMT is a right-angled triangle, having an angle of 54 44' where the plane Znw meets the equator, and an angle of 35 16' where it meets the pole Z. A perpendicular dropped from the hypothenuse or longest side of this triangle, upon the right angle at the centre of the crystal, divides the triangle into two triangles having angles similar to those of the undivided triangle. Put Zc = axisp a ; EC = diagonal of the equator; ZE = Znw quadrant of the north-west meridian. Then ZcE = 90; cZE = 35 16'; ZEc 158 PRINCIPLES OF CRYSTALLOGRAPHY. = 54 44'. Let nc be the perpendicular dropped from the side ZE. Then the triangles Zcw and cEn will have similar angles, namely, 54 44', the angle at n being always 90. But the line nc will have the same relation to m a and t a as to p a , as- suming the line Zc to represent either p a , m* a , or t a . Therefore, n is the centre of the Znw plane of the form PMT. Therefore, it is the Znw pole, and the line nc is the Znw normal. Consequently, the plane Znw is perpendicular to the Znw normal, and the inclination of this normal to *E axis p a is the angle Zen = 54 44' or the comple- ment of 35 16'; the inclination of it to the equator is the angle Ecn = 35 16' or the complement of 54 44'; and its inclination to the Nnw normal (or to the normal of any adjoining plane of PMT) is twice the angle ncE } or 35 16' x 2 = 70 32'. It follows from these results, that the inclination of every one of the eight tripolar normals is : To an adjacent UNIPOLAR normal 54 44', To an adjacent BIPOLAR normal 35 16', To an adjacent TRIPOLAR normal = 70 32'. 2. THE CUBE. Model 1. P,M,T. 350. This combination is described in 21 25. ROSE'S symbol for it is (a : oo a : oo a). According to the principles of classification explained in SECTION IV., the cube is a complete prism with a square equator. The minerals which occur in this shape are quoted at page 97, Part II., being thirty-five in number. The method of proving trigonometrically the value of the plane angles of Model 1, and of finding the inclination of its planes and edges to the Znw normal, and the inclination of that normal to the axis p a , is given in 363. 351. COMBINATIONS OF THE CUBE, P,M,T, WITH THE OCTAHEDRON, PMT. a.) Model 29. P,M,T. PMT. Rose's symbol for which is (a : oo a : oo a) -|- (a : a : a). The middle crystal between the cube and the octahe- dron. A complete prism combined with an incomplete pyramid. The minerals which occur in this shape are quoted at page 105, Part II. b.) Model 31.* (See note page 88). P,M,T, pmt. Rose's symbol is (a : QO a : oo a) -J- (a : a : a), The cube with its corners slightly trun- cated by the planes of the octahedron. A complete prism with an incomplete pyramid. The minerals which occur in this form are quoted at page 104, Part II. c.) Model 30. p,m,t.PMT. Rose's symbol is (a : a : a) -f- (a : oo a : co a). The octahedron with its angles replaced by the planes of the cube. A PRINCIPLES OF CRYSTALLOGRAPHY. 159 complete prism with an incomplete pyramid. The minerals which occur in this shape are quoted at page 106, Part II. 352. PROBLEM. Given, Model 29, with the symbol P,M,T, P X MT; required, the value of the index x . When the apex of an octahedron is truncated by the horizontal plane PZ, the inclination of every terminal edge and terminal plane upon the horizontal plane is equal to 90, added to the inclination of the terminal edge or terminal plane to the axis p a . The inclination of a terminal edge or terminal plane to the axis p a , is half the inclination of two opposite terminal edges or terminal planes measured over the pole Z. The inclination of a plane of the form PMT to the axis p a is 35 16 / (= 70 32' -4- 2). And 35 16' -f- 90 = 125 16'. Take now, with the goniometer, the inclination of one of the triangular planes of Model 29 upon each of the square planes which surround it. The measurement is in every case 125 16'. Therefore, P,M,T, P X MT, contains the planes of the cube and the regular octahedron, and the value of x is unity. A difference of magnitude in the planes P,M,T, and PMT, makes no difference in the angle at which they incline upon one another. Thus, the angles of P,M,T upon pmt, are the same as those of p,m,t upon PMT, as will be found on measuring Models 31 and 30. 353. Another method of showing the interfacial angles of the planes of P,M,T upon those of PxMT It will be proved in 363, 6.), that the planes of PZ, Mn, Tw, incline upon the Znw normal at an angle of 35 16'. But the plane PMT Znw is perpendicular to the Znw normal, 349. Therefore, 61, the inclination of the planes PZ, Mn, Tw, to the planes PMT Znw which they surround, must be 90 -{- 35 16' - 125 16'. 3. THE RHOMBIC DODECAHEDRON. Model 63. MT. PM, PT. 354. This combination is described in 21 B, 31, 89, and 103. Rose's symbol for it is: (a : a : oo a). The rhombic dodecahedron is an incomplete prism with a complete pyramid. The minerals which occur in this shape, 28 in number, are quoted at page 110, Part II. The measurements over the poles Z,N,n,e, w,s, are as follow: The angle formed by two opposite planes = 90, by two opposite edges = 1 09 28' ; as may be proved by the goniometer. We may consider the rhombic dodecahedron to be a cube, having a four-faced pyramid upon each of its planes, the shorter diagonals of the planes of the dode- cahedron showing the form of the base of the pyramid ; or we may con- sider it to be an octahedron with a three-faced pyramid upon each of its planes, the longer diagonals of the planes of the dodecahedron showing the form of the base of the pyramid. The six apices of the four-faced pyramids are united by p a m a t 3 : the eight apices of the three-faced pyra- mids, by the tripolar normals, Znw, &c. 160 PRINCIPJLES OF CRYSTALLOGRAPHY. 355. PROBLEM. Given, the combination MT. PM, PT. Model 63; required, the angle of incidence of the plane PM Zn upon the plane PT Zw ; that is to say, the angle across an edge of the combination. Suppose the Znw octant of Model 63 to be divided vertically into two equal portions by the north-west meridian, and one of these portions, namely, that which contains part of the plane PM Zn, to be taken as a solid triangle with pole Z for its vertex. A side of this triangle will be the Zn quadrant of the north meridian, which, as the meridian is a square, gives 45 for the value of this side. Call it side a. As the north meridian is at right angles to the form PM, that edge or angle of the solid triangle produced between plane PM Zn and side a, will be angle C == 90. Then angle B will be the edge or angle at axis p a where the north meridian and the north-west meridian intersect one another at an angle of 45; and angle A (opposite side ) will be half the inclination of the plane PMZn upon the plane PT Zw, which is the unknown quan- tity sought. The quantities given are, therefore, side a = 45, and angle B = 45; and we have to find angle A. This case is answered by Formula 10; namely, cos A = cos a sin B: or log cos A = log cos a -f- log sin B 10. log cos a = 45 = 9-8495 + log sin B = 45 = 9-8495 log cos A = 60 = 9.6990 Angle A being half the inclination of plane PM Zn upon plane PT Zw, it follows that the angle demanded is twice 60 = 1 20. 356. PROBLEM. Given, the combination, MT. PM, PT, Model 63; required, the inclination of the edges between PM and PT, that is to say, the oblique edges of the four -faced pyramid, to the axis p a . Take the solid triangle made use of in the last problem, and with the given data, find side c (opposite angle C), which is the Znw quadrant of the north-west meridian. The problem is : Given, side a and angle B; sought, side c; Formula 12: tan c = ^~, or log tan c = log tan a -f- 10 log cos B. 10 + log tan a = 45 = 20.0000 log cos B = 45 = 9.8495 log tan c = 54 44' = 10.1505 This product, 54 44', is the desired angle, showing the inclination of the terminal edges to axis p a . It is proved in the same manner, that the edges between MT and PM incline upon the axis m a at an angle of 54 44'; and that the edges between MT and PT incline upon the axis t a at the same angle. Of course, the inclination of any two of these edges to one another over the pole Z, n, or w, is 54 44' X 2 = 109 28', as was assumed in 354. PRINCIPLES OF CRYSTALLOGRAPHY. 161 357. PROBLEM. Given, the combination MT.PM, PT, Model 63; required, the plane angles of its external faces. Take the solid triangle made use. of in the foregoing two problems, and with the given data, find side b (opposite angle B) ; which angle is the half of the acute plane angle of a face at the pole Z. The problem therefore is: Given, side a and angle B; sought, side b. Formula 11. log tan b = log tan B -{- log sin a 10. log tan B = 45 = 10.0000 + log sin a = 45 = 9.8495 log tan b = 35 16' 9-8*95 Twice this product, or 35 16' X 2 == 70 32', is the value of the acute plane angle at the pole Z, and the supplement of 70 32' (= 180 70 32') = 109 28', is the value of the obtuse plane angle at the pole Znw. 358. Another method of solving the foregoing problem. Take the same solid triangle as before, with the same vertex, pole Z, but employ angle B = 45, and angle A, found by the problem in 355, or by measurement of the model, to be = 60. Then seek side b t using Formula 5 : log cos b = log cos B -|- 10 log sin A. 10 + log cos B = 45 = 19-8495 log sin A = 60 = 9-9375 log cos b = 35 16'= 9.9120 The product is necessarily the same as that of the last problem. 359. PROBLEM. Given, the combination MT. PM, PT, Model 63; required, the plane angles of the faces of the combination, and the inclin- ation of the edges and planes to the tripolar normals, or specially, the inclination of the three edges and of the three planes contained in the Znw octant, to the Znw normal. CALCULATION OF THREE-FACED PYRAMIDS. Hold Model 63 in such a position that the pole Znw is placed in the position of pole Z, and then look from Zenith down upon the model. You will perceive a six-sided prism terminated by a three-faced pyra- mid, the Znw normal being in the position of the axis p a . Now, sup- pose the model to be divided into three sections, by vertical planes pass- ing through the three terminal edges, and meeting at the Znw normal, and suppose also that each of these sections is divided into two smaller sections, by other vertical planes passing in the direction of the shorter diagonals of the three rhombic terminal planes, and also meeting at the Znw normal. Take one of these sixths of the crystal as a solid triangle, having the pole Znw for its vertex, and observe what parts of the tri- angle are known, and what are unknown. In the first place, the bisec- Y 162 PRINCIPLES OF CRYSTALLOGRAPHY. tion of the rhombic terminal planes, by a plane perpendicular to them, produces an edge of 90, which we will call angle C; secondly, the divi- sion of the crystal into six equal portions, by planes, which all meet at a vertical axis under the pole Z, namely, the Znw normal, produces a vertical edge of 60, as part of each section of the crystal, which edge we will call angle A ; and, thirdly, as the angle across each terminal edge of the three-faced pyramid is known to be 120, 355, the half of it = 60, is the value of the third edge of the triangle, which may be called angle B. We know, therefore, the three angles of the solid triangle : A and B being each = 60, and C = 90; and with these data we can find the three sides of the solid triangle, which are respectively opposite to the three angles. Now, side a of this solid triangle is half the obtuse plane angle of an external plane of Model 63 ; side b is an inner vertical plane, which shows the inclination of an external plane to the Znw normal; and side c is another inner verticle plane, which shows the inclination of a terminal edge to the Znw normal. We have therefore three equations to resolve: a.) Given, angles A, B ; sought, side a. formula 4. cos a = ^ B which, (as A and B are both = 60), Formula 104 transforms to cos a = cot A. nat cot A = 60 = .5774 nat cos a = 54 44' = .5774 Hence the obtuse plane angles of the faces of Model 63 are 109 28', and, consequently, its acute plane angles are 70 32', (= 180' 109 28') as was found by the Problem in 357. b.) Given, angles A,B; sought, side b, Formula 5. This Formula admits the same abbreviation as the last, and gives the same result, namely, 54 44', which is the inclination of a plane of Model 63 upon the Znw normal. c.) Given, angles A, B ; sought, side c. Formula 6. log cos c = log cot A + log cot B 10. log cot A = 60 = 9.7614 + log cot B = 60 = 9-7614 log cos c = 7Qo 32' = 9.5228 This product, 70 32', is the inclination of an edge of Model 63, to the Znw normal. The following is a check on the correctness of this calcu- lation: The inclination of a plane to the vertical axis, added to the in- clination of an edge to the vertical axis, must be equal to the inclination of a plane to an edge measured over that vertical axis. Now, 54 44/ + 70 32' = 125 16', which, on applying the goniometer to Model 63, will be found to be the exact inclination of a plane to an edge measured over the trisolid angle at Znw. 360. There is a peculiarity in the results afforded by Equations b.) and c.)> which is extremely important in respect to certain calcula- PRINCIPLES OF CRYSTALLOGRAPHY. 163 tions, which will have to be considered in another Section, regarding the crystals that are called rhombohedrons. When Model 63 is held in the position described in 359, with the pole Znw in the position of the pole Z, the Model has the same planes, and occupying the same polaric position as the planes belonging to Model 71, which is a combination of the unequiaxed rhombohedral class. But the peculiarity to which I wish to direct your attention is the relation which holds between the inclina- tion of A PLANE and of AN EDGE of a three-faced pyramid to the AXIS which is perpendicular to the trisolid angle where the three planes and three edges meet. In the case of Model 71, and of all the rhombohe- drons, this perpendicular axis is p a ; but in the case of Model 63, this axis is the Znw normal. This difference, however, does not affect the relations which the edges and planes in question bear to the line perpen- dicular to the trisolid angle, where the edges and planes all meet. 361. Suppose now that the plane of the pyramid, which we have found to incline on the vertical axis, (Equation b,) at an angle of 54 44', in- clines also on the equator of the combination, it must do so at an angle of 35 16'. Suppose, also, that the edge of the pyramid which (Equa- tion c,) inclines upon the vertical axis at an angle of 70 32', inclines at the other end upon the equator, the angle formed there must be 19 28'. If we call the vertical axis p a , and the line along the equator, which is touched by the edge and plane in question, t a , and if we put p a = 1, then the distance from the centre of t a to the point where it touches the inclined plane will be tan 54 44'= 1.4141, and the distance to the point where it touches the inclined edge will be tan 70 32' = 2.8291. These relations may be expressed by the symbols P}J?^T, and P|gg^T. From this observation we draw the following conclusion : 362. When a pyramid consists of three equal and similar planes, separated by three equal and similar edges, the point where one of the inclined edges touches the equator is TWICE AS FAR FROM THE FOOT OF THE VERTICAL AXIS as is the point where the equator is touched by the diagonal of one of the inclined planes. In other words, the cotangent of the inclination of a plane of a three-faced pyramid to the vertical axis, is twice the cotangent of the inclination of an edge to the same axis. Thus: Inclination of plane, 54 44', cot .7072 Inclination of edge, 70 32', cot .3535 363. PROBLEM. To find the inclination of the planes PZ, Mn, Tw, of the Cube, Model 1, and of the three edges which separate them, to the Znw normal. The principle laid down in 362 will be found to apply even to the cube, if it is placed in such a position as to resemble a rhombohedron. Hold Model 1, with the trisolid angle Znw, in the place of the pole Z, and then suppose the model to be divided into six similar vertical portions, by six vertical sections, proceeding from the central axis out- 164 PRINCIPLES OF CRYSTALLOGRAPHY. wards. See 359- Take one of the resulting sixths as a solid triangle, having the pole Znw for its vertex. You will then have, as given parts, angle C = 90 = inclination of a terminal plane to a section perpen- dicular thereto; angle A = 60 = inclination of two adjacent vertical sections at the vertical axis ; and angle B = 45 = half the inclination between two external planes, across an external edge. With these data you can find side a = half the plane angle at Znw of an external face ; side b = inclination of an external plane to the Znw normal ; and side c = inclination of an external edge to the Znw normal. .) Given, angles A, B ; sought, side a. Formula 4. log cos a = log cos A -f- 10 - log sin B. 10 + log cos A = 60 = 19.6990 log sin B = 45 == 9-8495 log cos a = 45 = 9-8495 Hence the plane angles of Model 1 are each twice 45, or 90". b.) Given, angles A, B ; sought, side b. Formula 5. log cos b = log cos B -j- 10 log sin A. 10 + log cos B = 45 = 19.8495 log sin A = 60 = 9-9375 log cos b = 35 16' = 9-9120 This is the inclination of the external planes to the Znw normal. Then, as PZ inclines to this normal at an angle of 35 16', the normal itself must incline to the nw vertical edge of the model, and also to the axis p a , at an angle equal to 90 35 16', or 54 44', because PZ is at right angles to p% and also to the vertical edge nw. These relations are cor- roborated by the following equation : c.) Given, angles A, B ; sought side c. Formula 6. log cos c = log cot A + log cot B 10. log cot A = 60 = 9.7614 -f log cot B = 45 = 10.0000 log cos c = 54 44' = 9.7614 This is the inclination of any edge of the form P,M,T, to the Znw normal. The product of equation b.) = 35 16', and that of equation c.) = 54 44^ are together equal to 90, which the goniometer will show to be the angle of inclination of any plane of Model 1, upon any edge, mea- sured over a solid angle. As the tangent of 35 16' is 0.7072, and the tangent of 54 44' is 1.4141, we may express the products of equations 6,)andc.) in symbols as follows: Pjj$gT and piofioo/f, i n w hi c h expression the reader will again observe the curious relations pointed out in 361, 362; since 07072 is the half of 14141, as 14141 is the half of 28291. PRINCIPLES OF CRYSTALLOGRAPHY. 165 364-. COMBINATIONS OF THE RHOMBIC DODECAHEDRON, MT.PM,PT, WITH THE OCTAHEDRON, PMT. Model 64. mt. pm, pt, PMT. Model 65. MT. PM, PT, prat. Both of these combinations are incomplete prisms combined with com- plete pyramids. The minerals which occur in these forms are con- tained in Class 4, Order 1, Genus 1, in Part II., page 110. 365. PROBLEM. Given, the combination MT.PM,PT,PMT; required, the inclination of a plane of PMT upon a plane of MT, PM, or PT. The edges of PMT have the same polaric positions, and the same relations to p a m a t% as have the planes of MT. PM, PT; consequently, the planes of MT.PM, PT, replace the edges of PMT evenly; and, 61, the incidence of a plane of PMT upon a plane of MT, PM, or PT, is the half of 109 28' + 90 = 144 44', as will be found on applying the goniometer to Models 64 and 65. 366. Another method of solving this problem^ -The plane PMT Znw, which replaces the trisolid angle at the Znw pole of MT. PM, PT, is exactly perpendicular to the Znw normal, 349. Now the planes of MT. PM, PT. incline upon this normal at an angle of 54 44', 359, equation 2.) Hence they must incline upon PMT at an angle of 54 44' + 90 = 144 44'. 367. COMBINATIONS OF THE CUBE P,M,T, WITH THE RHOMBIC DO- DECAHEDRON, MT. PM, PT. Model 27. P,M,T, MT. PM, PT. Model 28. p,m,t, MT. PM, PT. Model 36.* P,M,T, mt. pm, pt. (*This supposes the hexagonal planes of ipmt shown on four of the corners of the cube to be away.) These combinations are complete prisms with incomplete pyramids. The minerals which occur in these shapes are placed in Class 3, Order 1, Genus 1, Groups a and b, Part II., pages 104, 105. Analysis of these Combinations : As any two opposite planes of MT. PM, PT meet at the poles Z, N, n, e, s, or w, at an angle of 90, and as the planes of P,M,T are perpen- dicular to the axes which connect these poles, it follows, that the planes of P,M,T cut those of MT. PM, PT at an angle of ~ + 90 = 135, which measurement with the goniometer shows to be correct. 368. COMBINATIONS CONTAINING P,M,T, WITH MT. PM, PT, AND PMT. Model 31. P,M,T, mt. pm, pt, PMT. 32. The same with the Forms marked. 33. P,M,T, mt. pm, pt, PMT. 34. p,m,t, MT. PM, PT, pmt These combinations are complete prisms with incomplete pyramids. 166 PRINCIPLES OF CRYSTALLOGRAPHY. See the minerals described in Part II., pages 105, 106. Class 3, Order 1, Genus 1, Groups a, b, c. The planes of these combinations show the inclinations of the planes of every one of the three combinations upon the planes of both the others. The reader may take instrumentally the angles round the equator and the four meridians of each of these three models, and see that they agree with the angles of the different forms, and also that their aggre- gate value is in every case such as it should be, according to the princi- ple explained in 79 85. 4. THE ICOSITESSARAHEDRON, P_MT, PM_T, PMT_: or 3PJMT. Varieties of this combination : PiMT, PM|T, PMT: or 3P|MT. PMT, PMT, PMTi : or 3PMT. Model 22 is 3PMT. 369. This combination is described in 128 146. ROSE'S symbol for it is: (a : a : m a) or for the variety exhibited by Model 22, and described as 3P|MT, his symbol is: (a : a : a). The icositessarahe- dron is a complete pyramid with a rhombic eqtfator, and falls into Class 2, Order 3, Genus 1. The minerals which occur in this shape are quoted in Part II., page 102. The icositessarahedron may be considered as a rhombic dodecahedron, having a flat scalene pyramid with a rhombic base, resting upon each plane. The edges on Model 22 are of two kinds as respects their length. Those which meet at the tripolar normals are short, and will be called s in the following problems. Those which meet at the unipolar normals are long, and will be called /. 1 370. PROBLEM. Given, Model 22, with the angle across a long edge, I. Required, the inclination of the edge I to the axis p a , and the value of the index _ in the symbol P_MT. a.) Let the angle across the edge / = 131 49'. Take the Znw octant of Model 22 as a solid triangle with pole Z for its vertex. Then the vertical edge at the junction of the north and east meridians will be angle C = 90 ; angle A will be J / = ^^ = 65 54|'; angle B will be the same; and the part required is side a, which is the Zn or Zw portion of one of the meridians, and the inclina- tion of a long edge to the axis p a . As angle A and angle B are similar quantities, the Formula to be used is No. 4 modified by No. 104, or cos a = cot A. nat cot A == 65 54i' = .4471 nat cos a = 63 26' = .4472 Therefore, the inclination of the edge / to p a is 63 26'. This is the measurement which proves that the four upper and four lower planes of Model 22 require the symbol PJMT; for the cotangent PRINCIPLES OP CRYSTALLOGRAPHY. 167 of 63 26', the inclination of an edge to the axis p% is .5000. It is proved in the same manner, that the eight planes at the poles n, s, require the symbol PM|T, and that the eight planes at the poles e, w, require the symbol PMT^ or Pf Mf T. The accuracy of these calculations is checked by the direct application of the goniometer to any two edges of the model which meet at any one of the six poles Z, N, n, e, w, s, where the angle must be 63 26' X 2 = 126 52'. 5.) Let the angle across the edge / ===== 144 54'. The solution of this problem gives 71 34' for the required angle, and J for the value of the index _ in 3P_MT. See 145. 1 leave the working of this problem to the reader. 371. PROBLEM. Given, the symbol P|MT, PM^T, PMT|; required, the angle across a long edge. This problem is the reverse of the preceding problem. Take the Znw octant of the combination as a solid triangle, with the pole Z for its vertex. The symbol P|MT shows the axes to be plm a lo t^. Look, therefore, for the angle of which ^ or .5 is the cotangent. This angle is 63 26', and it represents the inclination of the north and east meridians, that is to say, of two different terminal edges of the pyramid, to the axis p a . These parts are sides a and b of the solid triangle, which have angle C = 90 between them. The part of the triangle to be found is, there- fore, angle A or B, one of which represents the inclination of a plane to the north meridian, and the other, the inclination of a plane to the east meridian. Hence, the problem is as follows: Given, sides a, b; sought, angle A. Formula 13. log tan A = log tan a -{- 10 log sin b. . 10 + log tan a = 63 26' = 20.3010 log sin b = 63 26' = 9-9515 log tan A = 65 54*' = 10.3*95 Twice this product, or 65 54^' x 2 = 131 49', is the required angle across a long edge. 372. PROBLEM. Given, the symbol PJMT, PMT, PMT$; required, the angle across an edge I. Answer, 144 54'. This problem is left for the reader to work. 373. PROBLEM. Given, Model 22, with the angle across a short edge, s ; required, the inclination of the three short edges, and of the three planes between them, to the Znw normal ; also, the plane angle of the faces at the pole Znw. The principle upon which the solution of this problem depends, is explained in 359 363, which sections relate to the analysis of three- faced pyramids. 168 PRINCIPLES OF CRYSTALLOGRAPHY. Let the angle across the edge s = 146 27'. Form a solid triangle, with th,e pole Znw for its vertex, and having an angle C = 90, an angle A = 60, and an angle B = ^^ = 73 13*'. With these data, find the following three parts of the solid triangle : side a = half the plane angle at Znw of one of the external faces of the crystal, side b = inclination of the external planes of the Model to the Znw normal, side c = inclination of the external edges of the Model to the Znw normal. .) Given, angles A, B ; sought, side a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 60 = 19.6990 log sin B = 73 13|' = 9-9811 log cos a == 58 31' = 9-7179 Twice this product, or 58 31' X 2 == 117 2', is the plane angle of each face of Model 22 at the pole Znw. b.) Given^ angles A, B ; sought, side b. Formula 5. log cos b = log cos B -f- 10 log sin A. 10 4- log cos B = 73 13|' = 19-4603 log sin A = 60 = 9-9375 log cos b = 70 32' = 9.5228 This product, 70 32', is the inclination of the external planes to the Znw normal. c.) Given, angles A, B ; sought, side c. Formula 6. log cos c = log cot A + log cot B 10. log cot A = 60 = 9.7614 + log cot B = 73 13|' = 9.4792 log cos c = 79 58J' == 9-2406 This product, 79 58*', is the inclination of the external edges to the Znw normal. There is a direct and easy check over the accuracy of these calcula- tions. According to the principle stated in 362, the tangent of the product of equation c.) should be a line twice the length of the tangent of the product of equation b.) Now the tangent of 70 32' is 2.8291, arid the tangent of 79 58*' is 5.6569 5 which is its double within a fraction. If the calculations were made with tables reckoning seconds, the products would be exact, instead of merely approximate, as in this and many other examples of brief calculation in this work. PRINCIPLES OF CRYSTALLOGRAPHY. 169 374. PROBLEM. The same as the preceding, but with the angle across the short edge s = 129 31'. See 146. I leave the solution of this problem as an exercise for the reader. . 375. PROBLEM. With the information contained in 369 373, to find all the external plane angles o/Model 22, P|MT, PM|T, PMT|. a.) The plane angle at the pole Znw is found, by 373 a.), to be 117 2'. Call this a. .) To find the plane angle at the pole Z, take the solid triangle described in 370 a.), in which are given, C = 90; A = 65 54|'; and B = 65 54|'. The part required is side c. Hence : Formula 6. log cos c = log cot A -f- log cot B 10. log cot A = 65 54|' === 9.6505 + log cot B = 65 54' = 9.6505 log cos c = 78 28' = 9.3010 This product, 78 28', is the plane angle at the pole Z. Call this b. c.) The plane angles at the poles Zn and Zw are both alike. Call each of them c. I have shown in 82, that the four angles of a plane of four sides, which description applies to the faces of Model 22, are together equal to 360. But we have found one angle, , to be 117 2', and another angle, b, to be 78 28'. Therefore, each of the remaining angles, c, must be i [360 (117 2' + 78 28') = 164 30'] = 82 15'. Thus: Angle at pole Znw =117 2' Z = 78 28' Zn = 82 15' Zw = 82 15' All the four angles = 360 00' d.) Problem. The plane angle c can also be found by a direct opera- tion, when you know the angle across both the external edges s and /. Example: The angle across s is 146 27'. Call the half of it B = 73 13|'. The angle across / is 131 49'. Call the half of it A = 65 54|'. Take a solid triangle with the pole Zn for its vertex, and let planes pass through the edges s and /, and intersect each other at the Zn normal. Then C will be the angle produced by the junction of these sections at the Zn normal ; A will be half the Z 2 n edge /; B half the Z 2 n 2 w edge s; and c, the plane angle required. This problem can be solved by Formula 6. log cos c = log cot A -f- log cot B 10. log cot A = 65 54|' = 9.6505 + log cot B = 73 13i' = 9-4792 log cos c = 82 15' =9.1297 This product is confirmatory of the accuracy of the calculation given in c.) 170 PRINCIPLES OF CRYSTALLOGRAPHY. 376. PROBLEM. Given, the symbol 3P|MT; required, the inclination of the planes to the axis p a . a.) First find half the angle across an edge /, by means of problem 371. Call this angle B = 65 54|'. b.) Take the Znw octant of Model 22, and divide it into two vertical portions by the north-west meridian. Choose the portion adjoining the north meridian as a solid triangle, with the pole Z for its vertex. Then B is the inclination of the external plane to the north meridian = 65 54|'; C is the inclination of the Z 2 nw plane of the model upon the north-west meridian; c is the Zn quadrant of the north meridian = 63 26', see 371 ; and b is the Znw quadrant of the north-west meridian, or the inclination of the plane Z 2 nw to the axis p% which is the quantity demanded in the problem. Therefore, we have given, B, c; to find, b. formula 27. log sin b = log sin c + log sin B 10. log sin c = 6326' =9-9515 + log sin B = 65 54' = 9.9604 log sin b = 54 44' = 9.91 19 Therefore, 54 44' is the inclination of a plane of PJMT to p a , and 54 44' x 2 = 109 28' is the inclination of the plane Z 2 nw to Zrse over the pole Z, as given in 146. 377. PROBLEM. Given, the symbol 3PJMT; required, the angle across a short edge, s. a.) The equator of this combination is eight-sided, or is shaped like n e s w in the annexed figure, where the three angles of the triangle ncE PRINCIPLES OF CRYSTALLOGRAPHY. 171 are together equal to 180; namely, angle ncE = 45, or half the right angle new; angle cnE, found by problem 370 = 63 26'; and angle cEn = 180 (45 + 63 26') = 71 34'. The north meridian and east meridian of the combination 3P1MT, are exactly like the equator. Compare the octagon in this figure with the three principal sections of Model 22. Find, by problem 371, the inclination of an external plane upon the plane figured nEc, which is the same angle as the inclination of an external plane to the north meridian = 65 54J'. Then take a solid triangle, having the pole nw, marked E in the figure, for its vertex; the plane angle nEc = 71 34' for side ; the inclination of the north-west meridian to the plane nEc = 90 for angle C; and the inclination of the external plane PM|T Zn 2 w to the plane nEc = 65 54 J'. for angle B. With these data we can find angle A, which is the inclina- tion of the plane PM|TZn 2 w to the north-west meridian, or half the angle across a short edge, the quantity demanded in the problem. Hence the problem for solution is: Given, #,B; to find A. Formula 10. log cos A = log cos a -j- log sin B 10. log cos a = 71 34' = 9-5000 + log sin B = 65 54|' = 9-9604 log cos A = 73 13i' = 9.4604 Twice this product = 73 13|' x 2 = 146 27' is the angle across a short edge of Model 22, 3P|MT. >.) Another method of solving this problem. The inclination of the Znw normal to the axis p a , 349, is 54 44'. The inclination of the plane PJMT Z 2 nw to the axis p a , 376, is also 54 44'. Therefore, the inclination of the plane P|MT Z 2 nw to the normal Znw, or the angle Znc in the annexed figure, is 180 (54 .44' -f 54 44 ; ) == 70 32'. Take a solid triangle such as is described in 373, with the pole Znw for its vertex; the angle 70 32', or the inclination of a plane to Znw, for side b', an interior vertical edge = 60, for angle A; the inclination of an external plane to a sec- tion through its shorter diagonal = 90, for angle C; and with these data, find angle B, which is half the angle across a short edge proceeding from the pole Znw, and half the quantity demanded in the problem. You have, therefore, Given, A,b ; to find B. Formula 8. log cos B = log cos b + log sin A 10. log cos b = 70 32' = 9.5228 + log sin A == 60 - 9-9375 log cos B = 73 131' = 9.4603 172 PRINCIPLES OF CRYSTALLOGRAPHY. Twice this product, or 73 13|' x 2 = 146 27', is the required angle across a short edge of 3P|MT, Model 22. 378. PROBLEM. Given, the symbol 3PJMT; required, the angle across a short edge, s. The answer is given in 146. The reader may work the problem according to either of the methods described in 377. 379- PROBLEM. To find the angle across a long edge of Model 22, 3PJMT, when the angle across a short edge is given = 73 131'. a.) First Method. Find by problem 373, equation I.), the inclina- tion of the plane Z 2 nw to the Znw normal. Let this be 70 32'. The inclination of the same plane to the axis~p a is 180 (70 32' -f- 540 44') = 54 44'. See 377, b.) Take a solid triangle consisting of half the Znw octant of Model 22, with the pole Z for its vertex, and the edge where the plane Z 2 nw meets the north-west meridian for its right angle = C. Then, angle A is the vertical edge where the north meridian cuts the north-west meridian, which angle is therefore 45; side b is the inclination of the plane Z 2 nw to the axis p a = 54 44'; and angle B is the inclination of the plane Z 2 nw to the north meridian, or half the desired angle across a long edge of Model 22. You have, therefore, this problem to solve : Given, A, b; to find B. Formula 8. log cos B = log cos b -\- log sin A 10. log cos b 54 44' = 9-7615 + log sin A == 45 = 9.8495 log cos B = 65 54' = 9.6110 Twice this product, or 65 54' x 2 = 131 48', is the angle across a long edge of the model. This product is, however, \' too little, as the true doubled angle is 131 49'. This error is occasioned by the brevity of the logarithmic numbers employed in these calculations, and by want of attention to divisions smaller than minutes. b.) Second Method. Find by problem 373, equation c.), the inclina- tion of a short edge of Model 22 to the Znw normal. Let this be 79 58i'. The inclination of the Znw normal to the equator is 35 16', 349. Therefore, the inclination of the equator to the short edge of the model which connects the poles Znw and nw is 180 (79 58 J' + 35 I6') = 6445i'. Take now the solid triangle described in equation .), 377, with the pole nw for its vertex, and angle C = 90 as there described; side a found as above = 64 45 1'; and angle B, given in the problem (| angle across s) = 73 13|'. With these data, find angle A, which is half the angle across a long edge of the model. PRINCIPLES OF CRYSTALLOGRAPHY. 173 Formula 10. log cos A = log cos a -f- log sin B 10. log cos a = 64 45|' = 9.6299 + log sin B = 73 13' = 9-9811 log cos A = 65 54' =9-6110 This product is the same as that given by the first method of calculation. c.) Third Method. The same as b, and with the same solid triangle, but with a change in the quantities. Take now B = 73 13i' as given in the problem, and 5 = 71 34/, as determined in 377, equation a.) With these data, find A by Formula 22. log sin A log cos B + 10 log cos b. 10 + log cos B = 73 13|' = 19.4603 log cos b = 71 34' = 9-5000 log sin A = 65 54' = 9-9603 380. PROBLEM. With the information contained in 369 379, to determine all the angles of the equator and of the four meridians of Model 22. a.) The equator, the east meridian, and the north meridian are all alike, so that it will only be necessary to examine the equator. Turn to the figure in page 170. The equator is represented by the thick lines in that figure. The four angles marked n e s w are all alike, and the four angles marked a i E o are all alike. The whole angles are together equal to 1080, see 82. It is proved in 370, that half the angle marked n in the figure is 63 26'. Therefore, n is 126 52', and the four angles marked n e sw are together equal to 126 52' x 4 = 507 28'. De- ducting this sum from 1080, we have 572 32' for the value of the four angles marked aj Eo. Dividing 572 32' by 4, we have 143 8' for the value of each of the angles last mentioned. See 79. In problem 377, equation a.), we found the value of half the angle marked E in the figure to be 71 34', and since 71 34' x 2 == 143 8', we have in that deter- mination a proof of the correctness of the present calculation. b.) The north-east and north-west meridians are alike, so that we need only calculate the angles of one of them, namely, the north-west. We have here eight angles in all, but there are angles of three different kinds. Examine the Model. At poles Z and N, the angle is twice the inclina- tion of a plane to axis p a . Therefore, 376, 109 28'. At poles nw and se, the angle is twice the inclination of a short edge of the model to the equator. Therefore, 379, b.), 129 31'. At poles Znw, Zse, Nmv, and Nse, the angle is the inclination of a plane to a short edge over the tripolar normal. Therefore, 373, b.) and c.), = 70 32' + 79 58|' = 150 30y. If these angles are correct, their aggregate sum must be 1080. 174 PRINCIPLES OF CRYSTALLOGRAPHY. Proof. 109 28' x 2 = 218 56' 129 31' x 2 = 259 2' 150 30J' X 4 = 602 2' 1080 00' 381. COMBINATIONS OF THE ICOSITESSARAHEDRON WITH THE RHOM- BIC DODECAHEDRON. MT. PM, PT, 3pimt. mt.pm,pt,3PplT. Model 69. These combinations are incomplete prisms with complete pyramids. The minerals that are found in these forms are quoted at page 112, Part II. in Class 4, Order 4, Genus 1. Analysis. The diagonals of the faces of 3P|MT, which connect the two dis- similar angles, have the same positions as the edges of the rhombic dodecahedron, MT. PM, PT. Therefore, when the two combinations occur together, the edges of the latter combination are replaced by the planes of the former, as shown on Model 69- As the angle across an edge of MT.PM, PT is 120, 355, the inclination of a plane of SP^MT upon any plane of MT. PM, PT, must be = ^ + 90= 150. 382. COMBINATIONS OF THE ICOSITESSATIAHEDRON WITH THE CUBE. Model 39. P,M,T. 3p|mt. This combination is a complete prism with an incomplete pyramid. Minerals, Part II, page 105. p,m,t. 3P|MT. Similar to Model 22, with the solid angles at ZNnesw truncated by p,m,t. Minerals, Part II., page 108. Analysis. The inclination of a plane of P,M,T to an adjoining plane of 3PJMT is 54 44' + 90 = 144 44', in which 54 44' is the inclination of a plane of PMT to p a , as found by problem 376. 383. COMBINATION OF 3P|MT WITH PMT AND P,M,T. P,M,T. PMT, Splint. Minerals, Part II, page 108. Analysis. The diagonals of the planes of Model 22, which connect the poles Zn Zw and nw, have the same position as the edges of the combination P,M,T, PMT, Model 29. Therefore, the planes of 3P|MT truncate the edges of that combination, when all the three combinations occur upon the same solid. The inclination of a plane of PMT to an adjoining plane of 3PJMT is 70 32' + 90 = 160 32', in which 70 32' is the inclination of a plane of 3P|MT to the Zn\v normal, 373, b.) PRINCIP-LES OF CRYSTALLOGRAPHY. 175 384. COMBINATIONS OF 3PJMT WITH OTHER FORMS. The combination 3P$MT oocurs very frequently subordinate, rarely predominant, and scarcely ever in an isolated condition. ROSE quotes the following as its most characteristic combinations : P,M,T. Spimt Minerals, Part II., page 105. MT. PM, PT, 3plmt 111. PMT,3pJmt - 102. pmt, 3PIMT 102. MT.PM, PT.pmt, Spimt 112. MT.PM,PT, PMT,3pJmt 111. P,M,T,MT.pM,PT,3pimt ~ 105. These combinations may be all investigated according to the methods described in 369 383. It is therefore unnecessary to give the details of their analysis. 5. THE TRIAKISOCTAHEDRON, P + MT, PM + T, PMT+: or 3P+MT. Varieties of this combination : Pf MT, PM|T, PMTf : or 3P|MT. P 2 MT,PM 2 T,PMT 2 : or 3P 2 MT. P 3 MT,PM 3 T,PMT 3 : or 3P 8 MT. Model 17 is 3P 2 MT. 385. This combination is described in 147160. ROSE'S symbol for it is ( : a : 2a). The triakisoctahedron is a complete pyramid with a square equator. The Minerals which occur in this shape are described at page 101, Part II,, in Class 2, Order 1, Genus 1. There are three known varieties of the combination, namely, 3P|MT, 3P 2 MT, and 3P 3 MT, most of which commonly occur subordinately, and rarely either predominant or isolated. This combination is that which, in the language of the older crystallographers, is said to bevel the edges of the octahedron. There are two kinds of edges on this combination, namely, long edges which connect the poles of p a m a t a , and short edges, which meet, three at each tripolar normal, and four at each unipolar normal. The equator, the north meridian, and the east meridian, of this com- bination, are all squares. Therefore, the long edges incline to p% m a , and t a , at an angle of 45. 386. PROBLEM. Given, Model 17, 3P 2 MT, with an angle across a long edge = 141 3'. Required, the angle across a short edge. .) Suppose Model 17 to be divided into eight portions by the four meridians, or by vertical sections passing through the eight edges that meet at the pole Z. Take the octant which contains the plane PMT + Z 2 n 2 w for a solid triangle, with the pole Z for its vertex. This plane is stamped with P and M on the model. You have, here, an oblique- angled solid triangle, where C is half the angle demanded across a short edge ; B is an angle of 45 formed by the meeting of the north meridian 176 PRINCIPLES OF CRYSTALLOGRAPHY. with the north-west meridian at the axis p a ; A is angle of 70 311', being- half the angle across a long edge ; and c is a side of 45, being the Zn quadrant of the north meridian, or the inclination of the Zn long edge of the model to p a . The problem is, therefore, Given, A == 70 31-^', B = 45, c = 45; required, C, and the Formula which answers to it is No. 41 with No. 39, as explained in 328; but the problem can also be solved by Formula 42, which I shall employ in preference, for the sake of varying the examples : c log cot x = log tan A -f log cos c 10. Formula 42. j ^ CQS c = log cog A + bg gin ^ __ ^ __ ]og gin ^ First Equation : log tan A = 70 311' = 10.4515 + log cos c = 45 = 9.8495 log cot x = 26 34' = 10.3010 Second Equation: log cos A = 70 3H' = 9.5230 B = 45 x = 26 34' + log sin (B a?) = 18 26' = 9.5000 19.0230 log sin x = 26 34' = 9.6505 log cos C = 76 22' 9.3725 Twice this product, or 76 22' x 2 = 152 44', is the angle across a short edge of Model 17. See 160. b.) Another Method. Half the difference between the angle across a long edge and 109 28', is the complement of the inclination of a plane to the Znw normal of 3 P + MT. Find this first, and then calculate the angle across a short edge, by the method described in 359. That is to say, from the angle across a long edge, take the octahedral angle 109 28'. Divide the residue by 2. The complement of this last product is the inclination of a plane of 3P+MT to the Znw normal. Illustration. Model 17 resembles a regular octahedron with a flat three-faced pyramid fixed upon each plane. Consequently, the angle across a long edge of this model, includes the angle across the edge of the regular octahedron, together with the angles at which two of these flat pyramids incline upon two different faces of the included octahedron. Therefore, the angle across a long edge of the model, minus 109 28', or the octahedron edge, and also minus half the residue, is equal to the inclination of one of the flat pyramid faces upon one of the faces of the included octahedron. Then, again, the Znw normal is perpendicular to a plane of the regular octahedron, so that the complement of the inclina- PRINCIPLES OF CRYSTALLOGRAPHY. 177 tion of the pyramidal plane upon the face of the octahedron, is the inclination of the same plane upon the Znw normal. Put the angle across a long edge = 141 3'. Then 141 3' 109 28' = 31 35'. And ^ = 15 471'. Complement of 15 47 = 74 12|'. This is the inclination of a plane of Model 17 upon the Znw normal. Now, form a solid triangle, containing a sixth of Model 17, with the pole Znw for its vertex, in the manner described in 359. The known parts of the solid triangle are then, angle C = 90, formed by the inclina- tion of a plane of the model to a section perpendicular to that plane ; angle A = 60, formed where two sections meet at the Znw normal ; and side b, the inclination of a plane of the model to the Znw normal, already found to be 74 12V. With these data, you have to find angle B, which is half the desired angle across a short edge of the model. Given, A = 60; b = 74 12j'; to find, B. Formula 8. log cos B = log cos b + log sin A 10. log cos b = 74 12V = 9.4348 + log sin A = 60 = 9.9375 log cos B = 76' 22' = 9-3723 Twice this product, or 76 22' x 2 = 152 44', is the desired angle across a short edge of the model. See 1 60. c.) Another example of Method b Put the angle across a long edge = 129 31', which agrees with the combination 3Pf MT. To find the angle across a short edge. 129 31' 109 28' = 20 3'. The half of it = 10 1 V. Its com- plement is 79 58J', which is the inclination of a plane of 3Pf MT upon the Znw normal. Now, form a solid triangle as directed in b.), and take the same Formula, but change the value of b. Given, A = 60 ; b = 79" 58V; to find, B. Formula, 8. log cos B = log cos b -f log sin A 10. log cos b = 79 58J' = 9-2407 + log sin A = 60 = 9-9375 log cos B= 81 191' = 9.1782 Twice this product, or 81 19!' X 2 = 162 39V is the angle across a short edge of the combination 3P|MT. See 160. d.) The reader may work, according to either of these methods, the following problem : Given, 3P 3 MT, with the angle across a long edge = 153 28'; required, the angle across a short edge. The answer is given in 160. 387. PROBLEM. Given, Model 17, 3P 2 MT, with the angle across a short edge ; to find the angle across a long edge. .) Form a solid triangle, as described in 386, b.), with the pole Znw for its vertex, and calculate the inclination of the planes of Model 17 to 2A 178 PRINCIPLES OF CRYSTALLOGRAPHY. the Znw normal. Then double the complement of the discovered angle, and add to it 109 28'. The product is the desired angle across a long edge of the model. b.) Illustration. Put the angle across a short edge = 152 44'. The known parts of the solid triangle will then be as follows : angle C = 90; angle A = 60; angle B = 1 -^|^ = 76 22'. And the part required, namely, the inclination of a plane to the Znw normal, will be side b. The problem is, therefore, Given, A, B; to find, b. Formula 5. log cos b = log cos B -f- 10 log sin A. 10 + log cos B == 76 22' = 19.3724 -. log sin A = 60 = 9-9375 log cos b = 74 12F = 9.4349 The complement of 74 12J' is 15 47J', twice which is 31 35', and this added to 109 28' produces 141 3', which is the angle across a long edge of Model 17. This problem is the reverse of that given in 386, and the calculation is therefore also~reversed. c.) Put the angle across a short edge = 142 8', as it is found to be in the combination, 3P 3 MT. This gives the equation : log cos b = log cos 71 4' + 10 log sin 60. 10 + log cos 71 4' = 19.5112 log sin 60 = 9.9375 log cos b = 68 = 9.5737 Complement of 68 = 22. Twice 22 = 44. This, added to 1 09 28' is 153 28'* In ROSE'S Krystallographie, page 27, this angle is stated 158 28', in mistake for 153 28'. d.} Put the angle across a short edge = 162 39 &', which is the angle of the combination 3PJMT. Find the angle across a long edge. I leave this problem for the reader to work : the answer is given in 160 and 386 c.) 388. PROBLEM. Given, Model 17, 3P 2 MT, with the angle across a long edge, to find the inclination of a short edge to the Znw normal. Proceed as in 386, b.) to form a solid triangle with pole Znw for its vertex. Then observe that you have this problem to solve. Given, A,B ; to find c, Formula 6 ; or else, Given, A,b; to find, c, Formula 9 ; or else, Given, B, b ; to find, c, Formula 24. Any one of these Formulae will answer the purpose. I shall take Formula 6. log cos c = log cot A -}- log cot B 10. log cot A = 60 = 9.7614 -f log cot B = 76 22, = 9.3848 log cos c = 81 57' = 9.1462 Therefore, 81 57' is the inclination of a short edge to the Znw normal. PRINCIPLES OF CRYSTALLOGRAPHY. 179 Control over the accuracy of this calculation : This control is effected on the principle stated in 362, that the cotangent of the inclination of a plane of a three-faced pyramid to the vertical axis, is twice the cotangent of the inclination of an edge of the pyramid to the same axis. Now, we find by 386, b.), 'that the inclination of a plane to the Znw normal of Model 17, which is the vertical axis in this case, is 74 12|'. The cot of this angle is .2828. Half this sum is .1414, which is the cot of 81 57', proved, by 388, to be the inclination of an edge to the given vertical axis. 389. PROBLEM. Given, a triakisoctahedron, Model 17, 3P_j_MT; required, the plane angles of its external faces. a.) Let the symbol be 3P 2 MT. This problem is solved by a solid triangle having the pole Znw for its vertex. There must, therefore, be given, the angle across one of the edges, or the inclination of a plane or of an edge to the Znw normal. If nothing is given, the angle across a short edge is taken with the goni- ometer. Put this angle = 152 44', as found by problem 386, a or b.) Form a solid triangle, as described in 386, b.), having the following known parts, namely : angle C = 90 ; angle A = 60 ; and angle B = 76 22'. The part of the triangle to be determined is side a, which is half the plane angle of one of the faces of Model 17 at the pole Znw. This problem can be solved by Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 60 = 19.6990 log sin B == 76 22' = 9.9876 log cos a = 59 2' = 9-7114 Twice this product, or 59 2' x 2 = 118 4', is the obtuse plane angle of the faces of Model 17. The two acute angles are 180 118 4' = 61 56', or each separately is 30 58'. b.) Let the symbol be 3P 3 MT. c.) Let the symbol be 3P|MT. First find the angle across a short edge, as described in preceding paragraphs, and then finish the calculation according to the above model. 390. PROBLEM. Given, the symbol 3P 2 MT ; required, a.) the angle across a long edge, b.) the angle across a short edge, c.) the inclination of a plane to the Znw normal, d.) the inclination of a short edge to the Znw normal, e.) the obtuse plane angle of a face, and f.) the acute plane angle of a face. The symbol 3P 2 MT signifies the three forms P 2 MT, PM 2 T, PMT 2 . It is the combination represented by Model 17. The crystals denoted by the symbols 3P 3 MT and 3P|MT, have the same number effaces and the same general aspect as this model, but differ among themselves in all their angles. 180 PRINCIPLES OP CRYSTALLOGRAPHY. e andf) The plane angles of a face are found by the problem in 389, when the inclination of a short edge to the Znw normal is known. d.) The inclination of a short edge to the Znw normal is found by the problem in 388, when you know the inclination of a plane to the Znw normal. c.) The inclination of a plane to the Znw normal is found by the pro- blem in 386 b.), when you know the angle across a long edge. 6.) The angle across a short edge is found by the problem in 386 7 when you know the angle across a long edge. a.) Therefore, all the required angles can be found, if you first find the angle across a long edge, by the method described in the next paragraph. 391. To find the angle across a long edge of 3P 2 MT, when the sym- bol alone is given. The long edges of the combination are those which bound the equator and the north and east meridians. The planes whose longest edges meet at the equator are those of the form P 2 MT. The angle across a long edge may therefore be said to be the inclination of plane Zn 2 w 2 to plane Nn 2 w 2 of the form P 2 M T. Let the annexed figure represent the Znw octant of this form. Then P,M,T, are the axes p a a m?t? of P 2 MT. The line E is the nw edge of the equator, and the inclination of the plane Zn 2 w 2 to the plane new across the edge E, is half the angle across a long edge of the model, demanded in the problem. If you take this octant as a solid triangle with pole n for its vertex, then you have the angle across the edge M = 90 angle C ; the plane angle cnw = 45 = side a ; and the plane angle Znc, which answers to tan 2.0000 = 63 26' = side b. With these data, you have to find angle B, (opposite side b), which is the angle across the edge E. Hence the problem to be solved is, Given, a, b; to find, B. Formula 14. log tan B = log tan b + 10 -log sin a. 10 + log tan b = 63 26' = 20.3010 log sin a = 45 = 9.8495 log tan B = 70 S1J' = 10.4515 Twice this product, or 70 31 j' X 2 = 141 3', is the angle across a long edge of the combination 3P 2 MT. The angle across a long edge of the combination 3Pf MT, or 3P 3 MT, can be found in the same manner. 392. PROBLEM. Given, Model 17, with the symbol P+MT, PM+T, PMT + ; required, the value of the index + . PRINCIPLES OF CRYSTALLOGRAPHY. 181 a.) Let the angle across a long edge be measured and found = 141 3'. Observe, that the combination consists of three acute square-based octahedrons, one resting upon the equator, one upon the north meridian, and one upon the east meridian. In every case, the long edges divide two planes belonging to the same octahedron. Begin with the octahedron whose planes are attached to the equator. This is P + MT. Half the given angle is 70 3H'. This angle expresses the inclination of a plane to the equator. The equator is square, there- fore the inclination of the nw edge of the equator to the axis m a is 45. Form a right-angled solid triangle with pole n for its vertex, and in which angle C = 90, is the inclination of the equator to the north meridian; angle A = 70 31i' the given inclination of the Znw plane to the equator ; and side b = 45, the inclination of the equator to the axis m a . With these data, you can find side or, which is the inclination of the Zn edge of the form P+MT to the axis m a , no part of which edge is visible upon the model, because it is replaced by the form PMT + . This problem, Given, A, b ; to find, a, can be solved as follows : Formula 7. log tan a = log tan A + log sin b 10. log tan A = 70 31J' = 10.4515 + log sin b = 45 = 9.8495 log tan a = 63 26' = 10.3010 This product, 63 26', is the inclination of the Zn edge of P + MT to m a at the pole n. The tangent of 63 26' is 2.0000, which is the length of the axis p a when m a is 1.0. Therefore, the value of the index _j_ in P_I_MT is 2, and the symbol is P 2 MT. In the same manner, it is easy to prove, that the planes which sur- round the north meridian require the symbol PMT 2 , or P|M|T, and that the planes which surround the east meridian require the symbol PM 2 T. b.) Let the angle across the long edge = 129 31'. c.) Let the angle across the long edge = 153 28'. These two problems are to be treated in the same manner as problem a.) The answers are P|MT, PM|T, PMT| : or 3P|MT. P 3 MT, PM 3 T, PMT 3 : or3P 3 MT. 393. COMBINATIONS OF THE TRIAKISOCTAHEDRON WITH OTHER FORMS. MT. PM, PT, SP^MT, 3p|mt. This combination somewhat resembles Model 69? only the rhombic dodecahedron is predominant, and the icosi- tessarahedron subordinate. The planes of the triakisoctahedron replace the remaining edges of the icositessarahedron which meet at the tripolar normals. Minerals, Part II. page 112. PMT, 3p 3 mt. The bevelled octahedron. Minerals, Part II, page 100. 182 'PRINCIPLES OF CRYSTALLOGRAPHY. MT. PM, PT,PMT, 3p 3 mt. The combination represented by Model 65, with the addition of three narrow planes around each face of the regular octahedron. Minerals, Part II. page 110. P,M,T, PMT, 3p 2 rat, 3p 3 mt. The cube, with each corner replaced by the octahedron, and also by six other planes, inclining two on each edge of the cube. Minerals, Part II. page 105. 394. Angles across the edges of the above combinations. The inclination of 3P+MT upon MT.PM, PT, is half the angle across a long edge of 3P+MT added to 90. The inclination of 3P_|_MT upon PMT, is half the inclination of a plane to a tripolar normal of 3P+MT, added to 90. 6. THE TETRAKISHEXAHEDRON, M_T, M + T. P_M, P + M, P_T, P + T. Varieties of this Combination : Mf T, Mf T. PfM, PfM, PfT, P|T. M^T, Mf T. PiM, PfM, P^T, PfT. Mf T, M|T. PfM, PfM, PfT, PfT. MJT, Mf T. PM, PfM, PT, PfT. MJT, Mf T. PM, PfM, PT, PfT. M^T, MVT. PJ M, PLOM, P^T, P' T T. Model 68 is MT 2 ,M 2 T. PM 2 ,T 2 M, PT 2 ,'P 2 T: or MiT, Mf T. PiM,PfM^, P|T, Pf f. 395. This combination is described in 111 113. ROSE'S symbol for Model 68 is (2 a : a : o> a). The tetrakishexahedron resembles a cube that has a four-sided square-based pyramid upon each of its planes. It has two kinds of edges, namely, 12 long edges which connect the tri- polar normals, and pass through the poles of the bipolar normals, and 24 short edges which connect the unipolar with the tripolar normals. It is an incomplete prism with a complete pyramid, and has a rhombic equa- tor. Hence, the minerals which it represents are contained in Class 4, Order 3, Genus 1, Part II., page 112. Of the six varieties enumerated above, those having the indices \ and \ occur in an isolated state, but the rest only subordinately. 396. PROBLEM. Given, M|T, Mf T. PJM, PfM, P|T, PfT, Model 68, with the angle across a long edge ; required^ the angle across a short edge. Let the angle across a long edge be 143 8'. In this case, the equa- tor of the combination resembles the octagon noeasiwE drawn in the following diagram ; so that the equator of Model 22 and 68 are similar. Upon comparing this diagram with Model 68, we perceive that the thin straight lines between a i o and E in the diagram, have the same positions as the long edges on the model, which pass through the ter- minations of the bipolar normals and connect the tripolar normals. They PRINCIPLES OF CRYSTALLOGRAPHY. 183 are, therefore, the edges of the cube, upon the planes of which the six flat square-based pyramids of the tetrakishexahedron are assumed to be superimposed. Now, the angle across a long edge, say, for exam- ple, the angle nEw in the diagram is equal to the right angle oEi, plus the angle oEn, plus the angle iEw. Therefore, if the angle nEw is 1^-10 of ano 143 8', the angle oEn or iEw is 90 = 26 34'. This angle is the complement of the angle cnEin the diagram, which is consequently 63 26', and which represents the inclination of the planes of Model 68 to the unipolar normals, or to the axes p a m a t a . a.) Suppose the uppermost pyramid of Model 68 to be divided by the four meridians into eight sections. Take one of these as a solid tri- angle, with pole Z for its vertex. Then you have angle C = 90, the edge formed by the intersection of the north or east meridian with an external plane ; side a r= 63 26', inclination of a plane to the axis p a ; and angle B = 45, formed by the intersection of the north-west meri- dian with the east or north meridian. With these data, you have to find angle A, which is half the angle across a short edge of the model, or across an oblique or terminal edge of the flat pyramid. Formula 10. log cos A = log cos a + log sin B -10. log cos a = 63 26' = 9.6505 + log sin B = 45 = 9-8495 log cos A = 71 34' = 9-5000 Twice this product, or 71 34' x 2 = 143 8', is the required angle across a short edge of Model 68. 1S4 PRINCIPLES OF CRYSTALLOGRAPHY. This model is, however, not sufficiently well made to demonstrate this fact instrumentally. I mention this circumstance, lest the reader should be puzzled by the non-agreement of the mechanical measurement with the result of the reckoning. b.) Another Method. When you know the inclination of a plane to the base of the flat pyramid, found above = 26 34', take a solid triangle with the pole Znw for its vertex. Then you have angle C = 90, the inclination of the base to the north-west meridian ; the above described angle of 26 34' for angle B ; and the plane angle cEo in the diagram = 45, for side a. With these data, you have to find angle A, which is half the angle across a terminal edge of the pyramid. Formula 10. log cos A = log cos a + log sin B 10. log cos a = 45 = 9.8495 + log sin B = 26 34' = 9.6505 log cos A = 71 34' = 9.5000 Twice 71 34' = 143 8', is the angle demanded. 397. PROBLEM. Given, M|T, MfT.PJM, PfM, P|T, PfT, Model 68, with the angle across a short edge ; required, the angle across a long edge. The given angle across a short edge is 143 8'. The operations described in the foregoing problem are here to be reversed, since we now have given the angle across a terminal edge of a square-based pyramid, and are required to determine thence the angle across the equator. ^~ = 71 34' = half the angle across the terminal edge. Call this angle A. Take the solid triangle described in 396 .), with pole Znw for its vertex. The given parts are now A = 71 34', a = 45, and C = 90. With these data, you have to find B, which is the inclination of a plane of the pyramid to the base. All these quantities are fully described in the preceding paragraph, so that it would be needless to go into farther details. You have, therefore, given, A,a; to find, B. Formula 2. log sin B = log cos A + 10 log cos a. 10 + log cos A = 71 34' = 19.5000 log cos a = 45 = 9.8495 log sin B = 26 34' = 9.6505 This product, 26 34',' is the same as the angle oEn of the diagram in 396. To find the angle nEw in the same diagram, which is the required angle across a long edge, as already explained in 396, it is necessary to add to angle oEn the angles oEi and iEw, or, in other words, we must double the product of the equation and add 90. Thus : 26 34' x 2 = 53 8' + 90 = 143 8'. PRINCIPLES OF CRYSTALLOGRAPHY. 185 The angles across the short edges of MiT,Mf T.PJM,Pf M,PJT,Pf T, are the same as those across the long edges. This is not the case with any other variety of the tetrakishexahedron, the angles of all of which may, however, be calculated in the same manner as those of the variety which I have chosen as an example. 398. PROBLEM. Given, MT, Mf T. PJM, PfM, P|T, Pf T, Model 68, with the angle across a short edge; required, the inclination of the planes to the axes p a m a and t a . The given angle is 143 8'. Divide the uppermost flat square-based pyramid into eight sections, by the meridians which pass through the four terminal edges, and across the four terminal planes. Take one of the resulting eighths as a solid tri- angle with pole Z for its vertex. Then you have given, angle C = 90 = inclination of an external plane to the north meridian; angle B = 45 = intersection of the north meridian with the north-west meridian at axis p a ; and angle A = 5 = 71 34' = half the angle across the terminal edge that is divided by the north-west meridian. With these data, you have to find side a, which is half the upper portion of the north meridian, or the inclination of a plane to the axis p a . Therefore, you have given. A, B ; to find, a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 71 34' = 19.5000 log sin B 45 = 9.8495 log cos a = 63 26' = 9.6505 This product, 63 26', is the inclination of a plane of Model 68 to any one of the axes p a m a t a . See 396, equation .), which is the counter- part of the present problem, and which shows the method of determining the inclination of a plane to an axis, when the given quantity is the angle across a long edge. 399. PROBLEM. Given, MIT, MfT. P1M, PfM, P^T, Pf T, Model 68, with the angle across a long edge 143 8'; required, the inclina- tion of the planes to the axes p a m a t a . a.) One method of solving this problem is contained in 396, where the required angle is shown to be 63 26'. b.) Another Method. As the equator of the model is eight-sided, all its angles must amount to 1080, and since it has four angles of one kind (across the unipolar normals), and four of another kind (across the bipolar normals), one angle of each kind must together be equal to ~- = 270. Therefore, the angle across a unipolar normal is 270* minus the angle across a bipolar normal. Applying this principle to the present example, where the angle across a bipolar normal is given at 143 8', you find the angle across a unipolar normal to be 270 143 8' = 126 52'. 2B 186 PRINCIPLES OF CRYSTALLOGRAPHY. The half of this angle =. =*= 63 26', is the required inclination of a plane of the crystal to an axis. 400. PROBLEM. Given, MT, MfT. P1M, PfM, P^T, Pf T, Model 68, with the angle across a short edge; required the inclination of the short edges to the axes p a m a and t a . The given angle across the short edge is 143 8'. Divide the flat square-based pyramid into four sections, by the north- east .and north-west meridians, which pass through the terminal edges of the pyramid. Take one of these sections as a solid triangle, with pole Z for its vertex. The parts given are angle C = 90, the intersection of the two meridians at axis p a ; angle A = 71 34', half the angle across a terminal edge ; and angle B, the same. The part required is side a or side &, either of which shows the inclination of an edge to an axis. This problem, given, A,B ; to find> a, can be solved by Formula 4. cos a = ^^. But since A and B are similar quantities, the equation is reduced by Formula 104, to cos a = cot A. nat cot A = 71 34' = 9-5228 nat cos a = 70 32' = 9-5228 This product, 70 32', is the inclination of a short edge of Model 68 to any one of the three axes, p a m a or t a . Twice the product, or 70 32' X 2 = 141 4', is the inclination of two opposite edges across an axis. 401. PROBLEM. Given, MT, MfT. P|M, PfM, P|T, Pf T, Model 68, with the angle across a long edge =r 1 43 8' ; required, the inclination of the short edges to the axes p a m a t a . When the inclination of the short edges to the axes of this combina- tion are to be found from the angle across a long edge, proceed as follows : Take the solid triangle described in 396, a), in which are known, C = 90; = 6326'; and B = 45; and with these data, find side c, which is the inclination of a terminal edge of the pyramid to axis p a . Formula 12. log tan c = log tan a -f- 10 log cos B 10+ log tan = 63 26 / = 20.3010 log cos B = 45 = 9-8495 log tan c =70 32' = 10.4515 402. PROBLEM. Given, MJT, Mf T. P|M, Pf M, PiT, Pf T, Model 68, with the angle across a short edge ; required the plane angles of the ex- ternal faces of the combination. The given angle across the short edge is 143 8'. a.) To find the obtuse plane angle at pole Z. Take the solid triangle described in 400, and with the given angles, find side c, which is the obtuse plane angle at pole Z. Given, A = 71 34' j B = 71 34'; C = 90; required, c. PRINCIPLES OF CRYSTALLOGRAPHY. 187 Formula 6. log cos c = log cot A -f log cot B 10. log cot A =71 34' = 9.5228 + >g cot B = 71 34' = 9.5228 log cos c = 83 37' = 9.0456 b.) Another way to find the obtuse plane angle at pole Z. Take the solid triangle described in 396, a.), with pole Z for its vertex, and in which the given parts are C = 90; A = 71 34'; and B = 45; as there described. With these data, you have to find b, which is half the obtuse plane angle at pole Z. Formula 5. log cos b= log cos B + 10 log sin A. 10 + log cos B = 45 = 19.8495 log sin A=7134 / = 9,9771 log cos b =41 481'= 9.8724 Twice this product = 41 48 J' X 2 = 83 37', is the obtuse plane angle at pole Z. c.) To find the acute plane angles of the faces of Model 68. Each face is a triangle, of which one angle is found to be 83 37', and the other two angles are therefore necessarily together equal to 180 83 37' = 96 23', or each of them is equal to 48 11^'. These results are corro- borated by those of the next problem. 403. PROBLEM. Given, M|T, Mf T, PJM, Pf M, P^T, Pf T, Model 68, with the angle across a short edge=. 143 8', and the angle across a long edge = 143 8'; required, the plane angles of the external faces, and the inclination of the short and long edges to the tripotar normals. CALCULATION or Six- FACED PYRAMIDS. a.) To find the acute plane angle of the faces at pole Znw. Put pole Znw, Model 68, in the place of pole Z. Suppose the Model to be divided by vertical planes that pass through the six edges which meet at pole Znw. These vertical planes will intersect one another at the Znw normal, and cut out sections of the crystal, having each an in- terior edge or angle of the value of 60 (= 360 -j. 6). Take one of these sections as a solid triangle, with pole Znw for its vertex, and call the interior angle of 60 angle C. Then angle A will be half the angle across a long edge = 71 34', and angle B will be half the angle across a short edge = 71 34'. In the present case, these two angles have the same value, but in general they have different values; the calculation, however, is performed in the same way, whether the value of A and B is the same or not. With angles A, B, C, given as above, you have to find side c, which is the acute plane angle^f an external face and one of the angles demanded in the problem. The quantities constituting this equation form an oblique-angled solid triangle, on which account the cal- culation is longer than it would be if the given parts formed only a right- 188 PRINCIPLES OF CRYSTALLOGRAPHY. angled solid triangle. It is quite unnecessary to calculate any of the parts of the tetrakishexahedron by means of oblique-angled solid triangles, because, as I have shown, it is possible to divide the six-faced pyramid out of which the oblique -angled solid triangle is taken, into a series of right-angled solid triangles. But I give the equations contained in the present problem, in order to show the reader what can be done with a six-faced pyramid by means of oblique-angled solid triangles, in cases where reduction of the given pyramid to right-angled solid triangles is impossible. The Formula to be employed in resolving this equation is No. 37. To make the given parts suit the terms in this equation, where a instead of c is the part to be found, it is necessary to change the designation f the interior angle of 60 to A, when B will represent half the angle across a short edge, and C half the angle across a long edge, each of them being 71 34'. The equation is then as follows: Formula 37. Sin iq = ^ / "" co ^ os ri ( n s ~ A)> where S = \ (A+ B + C). Log sin ^ a = i [log cos S + log cos (S A) + 20 - (log sin B + log sin C) J A= 60 S = 101 34' 180 B.= 71 34' A= 60 S=10134 / C= 71 34' (S A) = 41 34' Suppt.ofS = 78 26' 2)203 8' S 101 34' log cos S= 78 26' = 9.3021 + log cos (S A) = 41 34' = 9.8740 + 20 = 39.1761 log sin B = 71 34' = 9.9771 | 19 9542 + log sin C = 71 34' =9.9771 J 2)19.2219 sin J a = 24 5f = 9.61095 Twice this product, or 24 5|' x 2 = 48 11|', is the acute plane angle required, which result agrees with that found by problem 402, c). This angle being one of two similar acute angles on each face of the model, they are together equal to 96 23', and the third or obtuse plane angle of each face is 180 96 23' = 83 37', as found in 402, a). As this is one of the cases in which the calculation gives an ambigu- ous result, it is proper to remind the reader of the cautions that were given on this score in 330. The first product obtained in the calcula- tion is S = 101 34'. This angle being greater than 90 is negative. You take therefore the cosine of its supplement, and prefix the sign to it. The next product is (S A) =41 34', which being less than 90 is po- PRINCIPLES OF CRYSTALLOGRAPHY. 189 sitive. The multiplication of the negative and positive quantities together, produces a negative result ; but the sign of this result must be changed ive, because the Formula employed contains the negative sign. Consequently, the product 19.1761 is a positive quantity, and as the quan- tities afterwards employed in the calculation are all positive, so the final result is positive. b.) To find the inclination of the short edges to the tripolar normals. Take the same solid triangle as in equation a), with the given parts, C = 60; B = 71 34'; and A == 71 34'. Then, if B is taken as half the angle across a short edge, the part to find is a. The Formula to be used is No. 37, as before. S = 10134' A= 71 34' (S A)= 30 log cos S = 78 26' = 9.3021 + log cos (S A) = 30 = 9-9375 + 20 = 39.2396 771 1 . , q qH6 log sin C = 60 = 9-9375 J " 2)19.3250 ( log sin B = 71 34' = 9.9771 1 " \ sin \ a = 27 22' = 9.6625 Twice this product, or 54 44', is the inclination of a short edge of the model to the Znw normal. c.) To find the inclination of the long edges to the tripolar normals. Take the same solid triangle as in equation &), but put B equal to half the angle across a long edge, and A equal to half the angle across a short edge. Then a will be the inclination of a long edge to the tripolar nor- mal. But since angle A and angle B are both alike, the result of this equation must come out the same as that of equation b) : in other words, both the short edges and the long edges of Model 68 incline upon the tripolar normal, at an angle of 54 44'. This coincidence is accidental, and does not occur with the other tetrakishexahedrons. d.) Proof of the correctness of the calculations contained in equations b) and c). The inclination of the short edge of Model 68 to axis p a was found by 400, to be 70 32'. The complement of this angle, added to the inclination of plane PZ to the Znw normal, is the inclination of the given short edge to that normal. Now, the complement of 70 32' is 19 28', and the inclination of PZ to the Znw normal was found by 363 b) to be 35 16'. Then 19 28' + 35 16' = 54 44'. Finally, the inclination of a long edge of Model 68 to the tripolar normal, is evidently the same as the inclination of a vertical edge of the cube to that normal; and this angle was found by 363 c) to be 54 44'. The product of 190 PIIINCIPLES OF CRYSTALLOGRAPHY. 54 44' + 54 44', is 109 28', which will be found, by approximate mea- surement with the goniometer, to be the inclination of a short edge to a long edge of Model 68, across the pole Znw. 404. PROBLEM. Given, the symbol M_T, M + T. P_M, P + M, P_T, P+ T, with the angle across a long edge 126 52' ; Required, the value of the two characteristics, and -J-. a.) Find by problem 399, equation b), the inclination of the planes of the combination to the axes p a m a t a . 270 126 52' = 143 8' and ^' = 7y 34'. This is the inclination a plane of M_T to m a , or of M_j_T to t a . Therefore, the cotangent of 71 34' gives the value of the sign , and its tangent the value of the sign -f. The first is = J, and the second = f. Hence the combination is MJT, Mf- T. P^M, Pf M, PT,PfT. 5.) Given, the index; required, the corresponding angle. If, on the contrary, the part given is the symbol, MfT, M|T. P|M, P|M, Pf T, P|T, and you are required to tell the angle across a long edge, you first seek in the Table of Indices, page 139, for the value of the indices f and f , which you find to be : f = .6667 | = 1.500 The first of which is the cotangent, and the second the tangent of 56 18^'. Twice this angle =112 37', is the angle across pole n, from plane M|T ne to plane M|T nw. Hence the angle across a long edge is 270 112 37' = 157 23'. See 113. 405. COMBINATIONS OF THE TETRAKISHEXAIIEDRON WITH OTHER FORMS. MT, mft, m|t. PM, pf in, p|m, PT, pft, pft, 3rfMT. A combination somewhat resembling Model 69, but in which the rhombic dodecahedron predominates, and in which the planes of the tetrakishexahedron replace those edges of the icositessarahedron which lye on the equator, and on the north and east meridians, the edges of combination converging towards the axes p a m a t a . Minerals : Garnet from Friedberg. MT, mH> mft. PM, pirn, pf m, PT, pit, pf-t, 3PJMT. A combination similar to Model C9, but having the edges of the icosi- tessarahedron that meet at the unipolar normals replaced by very narrow tangent planes, which are the planes of the tetrakishexahedron. Minerals : Garnet from Dognatzky. P,M,T, m^t, mft. pm, pf m, pit, pft. Model 45. p,m,t, MiT, Mf T. PiM, Pf M, PJT, Pf T. The bevelled cube : in the first example the cube, and in the second example the tetrakishexahedron predominant. The first occurs most generally. Examples : Fluorspar from Alston Moor. The second is presented by Fluorspar from Bohemia. PRINCIPLES OF CRYSTALLOGRAPHY. 101 P,M,T, MT, mf t, m|t. PM, pf m, pfm. PT, pf t, p|t,.pMT. P,M,T, MT, mjt, mf t. PM, pjm, pfm, PT, p^t, pf t, PMT. P,M,T, MT, mit, mf t. PM, pm, pfm, PT, pt, pf t, PMT. Three combinations similar to Model 31, but having all the edges betwixt the cube and the rhombic dodecahedron replaced by the planes of the tetrakishexahedron. Minerals: Green Fluorspar from Cumberland and Native Copper from Siberia in the first form ; Fluorspar from Cumberland in the second form ; and Red Oxide of Copper from Siberia in the third form. 406. Analysis of the foregoing Combinations. In all these combina- tions, the tetrakishexahedron occurs in company with the cube or the rhombic docecahedron, which renders the analysis extremely easy. The inclination of a plane of the tetrakishexahedron to a plane of the cube, is the inclination of a plane to an axis added to 90. The inclination of a plane of the tetrakishexahedron to a plane of the rhombic dodecahedron, is half the angle across a long edge added to 90. Examples : The angle of M upon m_t, Model 45, is 161 34'. De- ducting 90, we have 71 34' = inclination of a plane of m_t to m a . The cotangent of 7.1 34' is . Hence, Model 45 is P,M,T, mjt, mf t. p^m, pfm, pjt, pf t. 407. The inclination of a plane of m_t upon a plane of MT, in a com- bination containing the rhombic dodecahedron and a tetrakishexahedron, is 146 18|'. Deducting 90 we have 56 18|', which is half the angle across a long edge of the tetrakishexahedron. The difference between this product and 135, say, 135 56 18i' = 78 41^', is the inclina- tion of a plane of m_t of that form to m a . The cotangent of 78 41 J 7 is . Therefore, the tetrakishexahedron contained in the given combina- tion is, mt, mft. p^m, pfm, pit, pf t. 7. THE HEXAKISOCTAHEDRON, or Six-fold Octahedron. P_MT + , P + M_T, PM + T_, P_M+T, PM_T + , P + MT_ : or 6PJV1T+ . Varieties of this Combination : GPiMiT. GPiM^T. 6P T \MfT. Model 23 is 408. The hexakisoctahedron is described in 194 197. It is a complete pyramid with a rhombic equator. The only minerals which present the hexakisoctahedron as a separate crystal, are Garnet = 6PJMJT, and Diamond = 6PiMT. See Part II. page 102. The other varieties occur only subordinately. The edges of this combination are of three kinds as regards their length. The longest edges connect the unipolar and Jripolar normals ; 192 PRINCIPLES OF CRYSTALLOGRAPHY. the middle edges connect the unipolar and bipolar normals ; and the shortest edges connect the bipolar and tripolar normals. The angles across these different edges of all the known varieties of the combination, are given in 197. From the description of this combination given in 194, it will be seen, that its unipolar normals have, in reference to the external planes of the solid, the character of the principal axis of a dioctahedron or eight- sided pyramid ; its bipolar normals, the character of the principal axis of a rhombic pyramid; and its tripolar normals, the character of the principal axis of a scalenohedron, or scalene six-sided pyramid. Advan- tage will be taken of these peculiarities in the following calculations. 409- PROBLEM. Given, Model 23, 6P_MT + , with the angle across a long edge =158 47', and the angle across a middle edge = 165 2' ; required, the inclination of the middle edge to the axis p% and to the Zn bipolar normal. .) Grant the uppermost flat eight-sided pyramid of Model 23, which contains the Zenith forms of the forms P_MT + and P_M_j_T, to be divided into eight sections by the four meridians which pass through its terminal edges. Take the Z 3 n 2 w octant as a solid triangle, with pole Z for its vertex. Name the given parts as follows : Let the angle formed by the intersection of the north and north-west meridians, be angle C = 45; let half the long edge^= JHL= = 79 23i' be angle A; and let half the middle edge = 1 -^-' = 82 31', be angle B. The part re- quired, namely, the inclination of the middle edge to p a , will then be side a. These quantities constitute an oblique-angled spherical triangle, the solution of which requires Formula 37. sin \ a = ^ l^|p, where g _ 4 (A + B+ C). log sin \ a = J {log cos S + log cos (S A) + 20 (log sin B + log sin C) } A = 79 23J' S = 103 27i' B = 82 31' A = 79 C = 45 S A = 24 31' 2)206 S == 103 27 J' log cos Supplement of S = 76 32!' = 9.3667 + log cos (S A) = 24 3f ' = 9.9605 + 20 = 39.3272 log sin B = 82 31 r = 9-9963^ _ 19 8458 + log sin C = 45 = 9-849 5 / " 2)19.4814 sin 1 a = 33 24' = 9-7407 PRINCIPLES OF CRYSTALLOGRAPHY. 193 Twice this product, or 33 24/ x 2 = 66 48', is the required inclina- tion of the middle or Z 2 n edge to the axis p a . b.) When the arithmetical indices are given with the symbol, as 6Pf M^T, instead of 6P_MT_j_, then it is easier to find the inclination of the middle edge to the axis p a by the problem contained in 411, which is worked by a right-angled solid triangle ; but when the symbol con- tains only the signs _ and +, two equations must be solved, 409, 410, for the purpose of finding the arithmetical equivalent of these two signs. c.) The inclination of the middle edge to the bipolar normal Zn, is 135' minus its inclination to the unipolar normal, namely, 135 66 48' .rr 68 12'. See 412, 418. 410. PROBLEM. With the information contained in the preceding problem, to find the value of the indices _ and + in the symbols P_MT + , P_M_j_T, which characterise the dioctahedron or eight-sided pyramid of Model 23. CALCULATION or EIGHT- SIDED PYRAMIDS. This problem is of considerable importance, as it is used, not only in the investigation of the pyramids of the hexakisoctahedron, but also of those of the dioctahedrons of the pyramidal system of crystallisation. a.) We had given, in the last problem, the angle across the middle edge, half of which is = 82 31', and we obtained as the product of an equation, the inclination of that edge to axis p a = 66 48'. Now it will be observed, that this angle is on the north meridian, and that its co- tangent will give the relation of the two axes, p a and m a , to the two planes that lie on each side of the middle edge. This cotangent is .4286 or f , which intimates that the axis p a bears to the axis m a the relation of 3 to 7. b.) To complete our knowledge of the axes of the octahedron to which these two planes belong, we have next to find the relation of p a and m a to t a . With this in view, we take the Znw octant of an assumed octa- hedron, and use it as a right-angled solid triangle with pole Z for its vertex. We know the value of the angle across N, be- cause it is given in the problem = 82 31', and we know the value of the plane angle nZc, because it is the inclination of the edge N to axis p a , found above = 66 48'. Call the first datum angle A, and the second, side b. The edge or angle P, between the planes nZc and wZc, will then be angle C = 90. What we now want to learn is, the value of side a, which is the plane angle wZc in the innexed diagram, since this angle will give is the required relation of axis p a to axis t\ It is evident that this angle can be found by the following calculation : 2c W 194 PRINCIPLES OF CRYSTALLOGRAPHY. Given, A = 82 31' ; b = 66 48'; C = 90; to find, a. Formula 7. log tan a = log tan A + log sin b 10. log tan A = 82 31' = 10.8815 + log sin b = 66 48' = 9.9634 log tan a = 81 52' = 10.8449 This angle, 81 52', is the inclination to the axis p a of an edge which does not appear on Model 23, because it is replaced by an accompanying octahedron ; but it is the Zw boundary of the simple scalene octahedron, whose Zn boundary has been found to incline on p a at an angle of 66 48'. Hence the cotangent of the product of the last equation gives the demanded relation of axis p a to axis t a , and we find in the table that the cotangent of 81 52' is .1429 or f . c.) By .) we found the relation of the axes, p a to m a = 3 to 7 ; and now in b.) we find the relation of p a to t a = 1 to 7 ; or putting p a = 3 in order to agree with the former product, and multiplying -J- by 3, we have / T , or p a to t a = 3 to 21. This makes the relation of the three axes to one another to be p!jm a t a i, and gives for the octahedron the sym- bol P/yM/yT. But the index fa is equal to ^, and the index S 7 T is equal to J, so that the symbol P/ T M 2 7 T T, can be reduced to the simpler syno- nymous expression of P^MJT. This, therefore, is the translation of P_MT + , while the translation of P_M + T, the symbol of the associated form, is PfMTf d.) There are, on Model 23, six pyramids of eight sides, such as that which we have just investigated. All of them converge to a point over a unipolar normal. Their angles are all alike. They can be calculated by the same methods as are employed above, and they give similar results as regards their axial relations. The only difference is, that every pair of pyramids affects the three axes differently from the other two pair. Thus, the pyramids which converge at the poles Z and N, give the symbol Pf M^T, Pf MT^ ; those which converge at the poles n and s, give the symbols PM^TJ, P^M^T ; and those which converge at the poles e and w, give the symbols P^MT^, PM^Tf . These differences being attended to, all the varieties of the hexakisoctahedron may have their indices calculated from their angles in the manner here explained. The calculation of the dioetahedron requires two angles to be given. These are assumed above, to be those across the two terminal edges. I shall presently explain what steps are to be taken when one of the given angles is that across a horizontal edge. 411. PROBLEM. Given, the symbol 6P^M^T; required, the angle across the Z 2 n or middle terminal edge of the combination. Calculation of the Angles from the Symbol of a Hexakisoctahedron. The reader of the foregoing problem will perceive that the solution of this problem merely requires a reversal of the process followed there. PRINCIPLES OF CRYSTALLOGRAPHY. 195 The present operation is as follows : You take one of the octahedrons of 6PM|T, namely the first, PJMT. You begin by finding the rela- tions of the axes; next, the plane angles of the north and east meridians of the simple octahedron ; and finally, the inclination of the north meridian to the ex- ternal plane, which inclination is half the required angle across the middle or Z 2 n is equal to P 2 M 3 T 6 , the axial relations of which are p a to t a == i = .3333, and p a to m a = f = .6667. Taking these fractions for cotangents, you have .3337 cot 71 34', and .6667 cot 56 181'. Consider these angles to be two sides of an octant or right-angled solid triangle; call 71 34' side a, and 56 18i' side b. Then angle A will be half the required angle across the Z 2 n edge of the given hexakisoctahedron. Formula 13. log tan A = log tan a + 10 log sin b. 10 + log tan a= 71 34' =20.4772 log sin b = 56 18|' = 9-9201 log tan A = 74 30' = 10.5571 This product is the inclination of the north meridian to one of the adja- cent planes of the uppermost flat eight-sided pyramid in the given hex- akisoctahedron. Twice the product, or 74 30' x 2 = 149, is the angle across a middle or Z 2 n edge. See 197. This calculation is the first that must be made whenever the symbol of a hexakisoctahedron is known, and the external angles are to be calcu- lated from it. The products afforded are the inclination of a middle edge to axis p a and the angle across the middle edge. With these two data, all the other angles of the hexakisoctahedron can be calculated by means of the various problems given in this section. On the other hand, when- ever any two angles of a hexakisoctahedron are given, it is possible to deduce from them those particular angles which are afforded by the pre- sent problem, and from which, by the process given in problem 410, the value of the characteristics of the symbol of the hexakisoctahedron can be readily determined. These two problems are therefore of great importance as regards the hexakisoctahedron, since one of them serves as a guide from the crystal to the symbol, and the other from the symbol to the crystal. 412. PROBLEM. Given, Model 23, 6Pf M^T ; required, the inclination of the middle edge to the wnipolar normal Z or axis p a , and to the bipolar normal Zn ; required also, the plane angles of the north meridian, the cast meridian, and the equator, of Model 23. 196 PRINCIPLES OF CRYSTALLOGRAPHY. a.) First find, by the process described in problem 411, the inclina- tion of the middle edge to axis p a at the unipolar normal Z. For this purpose, you take the single octahedron PfM|T, two planes of which meet at the middle edge in question, and thus intimate that the inclina- tion of the middle edge to p a is, in fact, the inclination of the edges of these planes to that axis. PfM|T is equal to P 3 M 7 T 21 , because, f = / T , and J =./i ; so that the relation of p a to m a is = f. On looking at the Table of Indices, page 139, you find the cotangent corresponding to f to be .4286, which gives 66 48' for the inclination of the middle edge to axis p a . b.) The unipolar and bipolar normals cut one another in the centre of the crystal at an angle of 45. A line which connects the terminations of a unipolar and a bipolar normal, forms therefore a triangle with these two normals ; which triangle has one angle of 45, and two other angles equal together to 180 45 = 135. Now we have found one of these angles, namely, that at pole Z, to be 66 48'. Hence the other angle is 135 66 48' = 68 12'. This is therefore the inclination of a middle edge of Model 23 to the bipolar normal Zn. c.) Another method of reckoning the inclination of the middle edge to the bipolar normal. Twice the inclination of the midddle edge to the unipolar normal Z, is the inclination of two opposite middle edges over the pole Z. Therefore, 66 48' x 2 = 133 36'. The north meridian of Model 23 is an octagon similar to figure n o e a si wE in page 183, having four similar angles of one kind at the ter- minations of the unipolar normals, and four similar angles of another kind at the terminations of the bipolar normals. For the reasons stated in 82, two of these angles, or one of each kind, are together equal to 270. Now, in the present case, we know that an angle over a uni- polar normal is 133 36'. Wherefore an angle over a bipolar normal must be 270 133 36' = 136 24', half of which sum *. ^i' = 68 12', must be the inclination of the middle edge to the Zn normal. d.) To faid the plane angles of the equator, fyc. It follows from the foregoing investigation, that the equator, the north meridian, and the east meridian, have alternate angles of 133 36' at the unipolar normals, and 136 24' at the bipolar normals, which relations agree with the geomet- rical characters of the given octagonal sections, since (133 36' -f- 136 24') x 4 = 1080, which is the aggregate value of the angles of an octagon. 413. PROBLEM, Given, Model 23, 6P_MT + , with the angle across a long edge =158 47', and the angle across a middle edge = 165 2'; required) the inclination of the long edge to axis p a . Form a solid triangle, similar to that employed in the equation 409> but change the designation of the angles as follows. We have, as before, three angles or edges given, and are required to find a side. In the for- mer example, the required side was part of the north meridian, but in the present example, it is part of the north-west meridian. To make the PRINCIPLES OF CRYSTALLOGRAPHY". 197 problem suit the Formula, this side must nevertheless be called side a ; in which case, angle A will be half the angle across the middle edge = 82 31'; angle B will be half the angle across the long edge = 79 23V; and angle C will be the interior angle of 45, described in 409- Hence the problem is : Given, A = 82 31'; B = 79 23 J/ 5 C == 45 ; to find, a. Formula 37. Sin \ a = V '^^ SCO^S-A^ where S _ | (A + B + C). . sin x> sin. O Log sin \ a == i [log cos S + log cos (S A) + 20 (log sin B + log sin C) ] A = 82 31' B = 79 231' S = 103 27i' C = 45 A = 82 31' 2 ) 206 54V S A = 20 56i' S =103 27 J' Supplement of S = 76 32f ' log cos S = 76 32|' = 9.3667 + log cos (S A) = 20 56f = 9-9703 + 20 = 39.3370 log sin B = 79 23V = 9.9925 \ , q ft49n + log sin C = 45 = 9.8495 / = 2 ) 19.4950 sin | a = 33 59f == 9.7475 Twice this product, or 33 59l x X 2 = 67 59V, is the required incli- nation of the long edge of Model 23 to axis p a . Twice 67 59 V> or 135 59 r > is the inclination of two opposite long edges to one another, measured over the pole Z. 414. PROBLEM. Given, Model 23, SPiMi T ; required, the inclination of the long edge to axis p a . Find by problems 412 a.), and 411, the inclination of the middle edge to axis p a , and the inclination of a plane to the north meridian. Call the first c = 66 48 r , and the second A = 82 31'. Take the oblique-angled solid triangle described in 409> but with different angles and different designations to the angles. Put a, the required side, equal to the inclination of the long edge to axis p a . Then, B will be the interior angle of 45. The given quantities are now two angles and an intermediate side, and you are required to find a side opposite to angle A. These quantities form the following equation : 198 PRINCIPLES OF CRYSTALLOGRAPHY. Given, A = 82 31'; B = 45; c = 66 48'; to find, a. J-, , J log cot x = log tan A -f log cos c 10. *' { log cot a = log cot c + log cos (B x) log cos x. log tan A = 82 31' = 10.8815 B = 45 + log cos c = 66 48' = 9.5954 x = 18 26' log cot a? = 18 26' = 10.4769 B x = 26 34' log cot c = 66 48' = 9-6321 + log cos (B x) == 26 34' = 9.9515 19.5836 -log cos x = 18 26'= 9.9771 log cot a = 67 59J' = 9-6065 This product, 67 59i', agrees with the product of problem 413. 415. PROBLEM. Given, Model 23, 6Pf M|T, with the inclination of a long edge to the axis p a = 67 59|'; required, the inclination of the same edge to the tripolar normal Znw. The tripolar normal Znw, the unipolar normal Z, or semi-axis p a , and the long edge of Model 23, which connects pole Z with pole Znw, form together a triangle. In this, the inclination of the unipolar to the tri- polar normal is 54 44', 349 ; the inclination of the unipolar normal to the long edge is given = 67 59i' ; and the inclination of the long edge to the tripolar normal is necessarily equal to 180 (54 44' + 6759| / ) = 57 16V; because 57 16*' + 54 44' + 67 591' = 180 = the three angles of a triangle. In problem 424, the inclination of the long edge to the tripolar normal is found to be 57 16|' by another process. 416. PROBLEM. Given, Model 23, GPfM'-T; required, the angle across the Z 2 nw or long terminal edge of the combination. First use problems 411, 412 a.), to find the inclination of the mid- dle edge to axis p a , and half the angle across the middle edge. The latter is given in 409 = 82 31'; the former is found by the same pro- blem to be = 66 48' ; of course, the products of calculations by means of the methods given in 412 .) and 411, would be the same as these. Take the oblique-angled solid triangle described in 409, having pole Z for its vertex, and an interior angle of 45. Call this A. Then, B will be 82 31'; the required angle across a long edge will be C; and the above described side of 66 48' between A and B will be c. Hence we have : Given, A = 45 A , B = 82 31'; c = 66 48'; to find, C. This requires Formula 42. log cot x = log tan A + log cos c 10. log cos C = log cos A -f log sin (B x) log sin x. PRINCIPLES OF CRYSTALLOGRAPHY. 199 log tan A = 45 = 10.0000 B == 82 31' + log cos c = 66 48' = 9.5954 x = 68 30' log cot x == 68 30' = 9.5954 B x = 14 1' log cos A == 45 == 9.8495 + log sin (B x) = 14 1' = 9.3842 19.2337 log sin x = 68 30' = 9.9687 log cos C = 79 23V = 9.2650 Twice this product, or 79 23J' X 2 = 158 47', is the required angle across the Z 2 nw or long terminal edge of Model 23. 197. 417. PROBLEM. Given, Model 23, 6P^MJT ; required, the plane angle at pole Z of one of its external faces. Call the required plane angle a. Then the solid triangle employed in 416, with the same given quantities, may be employed to solve the present problem, with the help of Formula 43. Given, A = 45 ; B = 82 31' ; c = 66 48' ; to find, a. Formula 43. log cot x = log tan A -f log cos c 10. Jog cot a = log cot c -f- log cos (B x) log cos x x = 68 30' ; B x = 14 1'. See 416. log cot c = 66 48' = 9-6321 + log cos (B x) = 14 r = 9.9869 19.6190 log cos x = 68 30' = 9.5641 log cot a = 41 23V = 10.0549 This product, 41 23 J', is the required plane angle of a face of Model 23, at pole Z. 418. PROBLEM. Given, Model 23, 6P_MT + , with the angle across a middle edge = 165 2', and the angle across a short edge = 136 47'; required, the inclination of the middle edge to the Zn normal. The Zn normal of Model 23 has the character of the principal axis of a scalene pyramid, as respects the four planes which surround the pole Zn. Divide this pyramid into four sections by planes passing through the terminal edges, and meeting at the Zn normal. Take one of these sections as a right-angled solid triangle, with pole Zn for its vertex. Then you have, angle C = 90 = intersection of two dividing planes at the Zn normal; angle B = 82 31' = half the angle across a middle edge ; and angle A = 68 23 J' = half the angle across a short edge. 200 PRINCIPLES OF CRYSTALLOGRAPHY. With these data, you have to find side a, which is the required inclina- tion of the middle edge to the Zn normal. The problem is, therefore : Given, A = 68 231' ; B = 82 31' ; C = 90 ; to find, a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 68 23*' = 19.5662 log sin B = 82 31' = 9-9963 log cos a = 68 12' = 9.5699 This product, 68 12', is the required inclination of the middle edge to the Zn normal. The inclination of the middle edge to axis p a is = 135 68 12' = 66 48'. 412. Hence the data given in this problem are sufficient to lead to a knowledge of the arithmetical value of the signs _ and _j_ in the symbol P_MT + . See 410. 419. PROBLEM. Given, Model 23, 6PfM^T, with the inclination across a middle edge =165 2', and the inclination across a short edge = 136 47'; required, the inclination of the short edge to the Zn bipolar normal. Take the right angled solid triangle employed in the last problem, and with the same given quantities seek for side b, which is the required inclination of a short edge to the Zn normal. Given, A = 68 231'; B = 82 31'; C = 90; to find, b. Formula 5. log cos b *= log cos B + 10 log sin A. 10 + log cos B = 82 31' = 19.1147 log sin A = 68 23J' = 9.9684 log cos b= 81 57' = 9.1463 This product, 81 57', is the inclination of the short edge of Model 23 to the Zn bipolar normal. 420. PROBLEM. Given, the same data as in the last two problems; required, the plane angle at pole Zn of an external face of the model. Use the same solid triangle as in 419. Given, A = 68 23J'; B = 82 3l'; C = 90; to find, c. Formula 6. log cos c = log cot A -f- log cot B 10. log cot A = 68 231' = 9-5978 + log cot B = 82 31' = 9.1185 log cos c = 87 T = 8.7163 This product, 87 l', is the plane angle of the faces at pole Zn. 421. PROBLEM. Given, Model 23, P^M^T, with the angle across a middle edge = 165 2', and the angle across a long edge = 158 4 7'; required, the angle across a short edge. PRINCIPLES OF CRYSTALLOGRAPHY. 201 Find by the methods given in 409 or 412, the inclination of the middle edge to the Zn normal. Put it = 68 12/. Then take the right angled solid triangle described in 418. Use these quantities : C = 90 = the interior angle; B = 82 31' = half the angle across the middle edge; a = 68 12' = inclination of the middle edge to the Zn normal. Then seek A = inclination of an external face to the section passing through the short edges. Twice A will be the required angle across a short edge of the model. Given, a ="68 12'; B = 82 31'; C = 90; to find, A. Formula 10. log cos A = log cos a -|- log sin B 10. log cos a = 68 12' = 9.5698 + log sin B = 82 31' = 9.9963 log cos A = 68 23|' = 9.5661 Twice this product, or 68 23i' X 2 = 136 47', is the required angle across a short edge of the model. The angle across a short edge can be found, when only the symbol is given. You first find by problem 411, the angle across a middle edge; and the inclination of that edge to p a ; then by problem 412, the incli- nation of the middle edge to the Zn normal ; and finally, by problem 421, the angle across the short edge. 422. PROBLEM. Given, Model 23, P_MT + , with the angle across a short edge = 1 36 47', and the angle cross a middle edge = 1 65 2' ; re- quired, the angle across a long edge. Use the right angled solid triangle described in 418, to find the inclination of the middle edge to the Zn normal, which is 68 12'. The difference between 68 12' and 135 = 66 48', is the inclination of the middle edge to axis p a . The auxiliary angle of 135, introduced here, is the supplement of the inclination of the unipolar normal to the bipolar normal. See 412, 6). You now take an oblique angled solid triangle, formed as described in 416. You have then, A, the interior angle of 45; B = 82 31' = half the angle across a middle edge given in this problem ; and c = QQ 48', he inclination of the middle edge to axis p a . With these data, and the formula quoted in 416, you find C = 79 23i'. This product is half the required angle across the long edge, which angle is 158 423. PROBLEM. Given, Model 23, Pf MJT, with the inclination of a short edge to the Zn bipolar normal, as found by problem 419, = 81 57'; required, the inclination of the short edge to the tripolar normal Znw. The bipolar normal Zn meets the tripolar normal Znw, in the centre of the crystal at an angle of 35 16'. These two normals, in conjunction with a short edge of the model, complete a triangle. It follows, that 2D 202 PRINCIPLES OP CRYSTALLOGRAPHY. since the inclination of the short edge to the bipolar normal is 81 57', its inclination to the tripolar normal must be = 180 (35 16' + 81 570 = 62 4 ?'; because 35 16' + 81 57' + 62 47' = 180. Hence, 180 35 16' = 144 44/, is the inclination of a short edge to a unipolar normal added to its inclination to a tripolar normal. If one of these angles is known, the other is found by taking the known angle from 144 44'. Thus, 144 44' 81 57' = 62 47', as determined above. 424. PROBLEM. Given, Model 23, P_MT + , with the angle across a long edge = 158 47', and the angle across a short edge = 136 47' ; re- quired, the inclination of a long edge to the Znw normal. Put the Znw normal in the place of a principal axis, and suppose the model to be divided into six portions by sections passing through the long and short edges, and meeting at the Znw normal. Each of these six portions will have an interior angle of 60 where two sections meet at the Znw normal. Take one of these portions as a solid triangle, with pole Znw for its vertex. Then the given parts of the solid triangle are C = 60 = interior angle; B = ^ = 79 23J' = half the angle across a long edge ; and A = ~JZ! = 68 23F = half the angle across a short edge. With these data, you can find a, which is the inclination of a long edge to the Znw normal. Given, A = 68 23|' ; B = 79 23i' ; C = 60 : to find, a. The precautions necessary to be observed in using the following For- mula, have been described in 403. Formula 37. sin l a = ^ ^f' whcre S = K A + B + C ) log sin \ a =. cos S + log cos (S A) + 20 (log sin B + log sin C)] A = 68 23i' B= 79 23|' S = 103 531' C = 60 A = 68 23V 2)207 47' S A = 35 30' S = 103 53' Supplement of S = 76 6|' log cos S = 76 6J' = 9.3804 + log cos (S A) = 35 30 ; = 9-9107 + 20 = 39-2911 log sin B = 79 23*' = 9.9925^ + log sin C = 60 = 9.9375 / ' 2)19.3611 log sin \ a = 28 38J' = 9.G8055 PRINCIPLES OF CRYSTALLOGRAPHY. 203 Twice this product, or 28 38J' X 2 = 57 1GJ', is the required incli- nation of a long edge of Model 23 to the tripolar normal Znw. See 415. 425. Check on the accuracy of this Calculation. The north-west meridian of Model 23 is an octagon, with three kinds of angles. The angle at Z and N is 135 59', 413. The angle at the tripolar normals is equal to the inclination of the long edge to the tripolar normal == 57 16^', 424, added to the inclination of the short edge to the same normal = 62 47', 423 ; equal together to 120 3J'. The angle at the bipolar normals is twice the inclination of a short edge to the bipolar normal, or 81 57 X X 2 = 163 54', 419. Consequently, the eight angles are as follows : 135 59' x 2 = 271 58' 120 3 1' x 4 = 480 14' 163 54' X 2 = 327 48' 1080 426. PROBLEM. Given, Model 23, 6Pf $JT, with the angle across a long edge 158 47', and the angle across a short edge = 136 47', required, the inclination of a short edge to the Znw tripolar normal, and to the Zn bipolar normal. a.) Take the same oblique-angled solid triangle and the same Formula, as were used in problem 424, but change the designations of the angles as follows, in order to make angle A fall opposite the required side, which the Formula calls side a. Given, A = 79 23 J' ; B = 68 23i' ; C = 60; to find, a. Refer to 424 for the preamble of the operation. S = 103 531', as found in 424. A= 7 24 30' log cos S = 76 61' = 9.3803 + log cos (S A) = 24 30' = 9.9590 + 20 = 29.3393 C log sin B = 68 23fr' = 9.9684^ \ + log sin C = 60 = 9.9375 / 2)19.4334 log sin 1 a = 31 23J' = 9.7167 Twice this product, or 31 23J' x 2 = 62 47', is the required inclina- tion of a short edge to the tripolar normal. This agrees with the calcu- lation in 423. 204 PRINCIPLES OF CRYSTALLOGRAPHY. b.) The inclination of a short edge to the bipolar normal Zn, is 144 44' 62 47' = 81 57'. See 423. 427. PROBLEM. Given, Model 23, 6PfM^T, with the angle across a long edge = 158 47', and the angle across a short edge = 136 47' ; required, the plane angle of the external faces at pole Znw. Employ the same oblique-angled solid triangle, and the same Formula, as in 424, 426 ; but change the designations of the angles as follows : Given, A = 60; B = 79 23J'; C = 68 23J'; to find, a. S = 103 53|' A = 60 S A = 43 53|' log cos S = 76 6| A = 9.3803 + log cos (S A) = 43 53J' = 9.8577 + 20 = 39.2380 / log sin B = 79 23|' = 9.9925 1 , q QfinQ X + log sin C = 68 231' = 9.9684 J 2)19.2771 log sin i a = 25 47$' = 9.63855 Twice this product, or 25 47 J' X 2 =51 35', is the required plane angle of the external faces at pole Znw. 428. PROBLEM. Given, Model 23, GP^M^T, with the angle across a long edge = 158 47 ', and the angle across a short edge = 136 47' ; required, the angle across a middle edge. First, find by problem 426 .), the inclination of a short edge to the bipolar normal Zn. Call this a = 81 57'. Then take a right-angled solid triangle, such as is described in 418, with pole Zn for its vertex, and having for its angles, half the angle across a short edge, half the angle across a middle edge, and the angle formed by the intersection of planes passing through the middle and short edges, and meeting at the Zn normal. The value of this last angle is 90. Call this angle C. The value of half the angle across a short edge is given in the problem at J2.' = 68 23 f. , Call this angle B. Then the in- clination of the short edge to the Zn normal will be side a = 81 57'- With these data, you can find half the angle across a middle edge, since it will be angle A of the same right-angled solid triangle. The problem is, therefore : Given, a = 81 57'; B = 68 23^'; C = 90; to find, A. Formula 10. log cos A = log cos a + log sin B 10. log cos a = 81 57' = 9.1462 + log sin B = 68 23|' = 9.9684 log cos A = 82 31' = 9.1146 PRINCIPLES OF CRYSTALLOGRAPHY. 205 Twice this product, or 82 31' x 2 = 165 2', is the required angle across a middle edge. 429. PROBLEM. Given, Model 23, 6P^MJT; required, the plane angles of its external faces. The plane angles of the faces of Model 23 have been found to be as follows : At pole Z, 417 = 41 23 j' At pole Zn, 420 = 87 l' At pole Znw, 427 = 51 35' 179 59i' The aggregate sum, 179 59J', is i' less than it ought to be. This arises from inattention to fractions of angles smaller than half minutes, and from the brevity of the logarithmic numbers employed. If the angles were always reckoned to seconds, and the logarithmic numbers carried out to 7 decimal places, these occasional errors would not occur. 430. COMBINATIONS CONTAINING THE HEXAKISOCTAHEDRON. MT. PM, PT, 3PJMT, 6pjmjt. MT. PM, PT, 3P|MT, 6pimjt These combinations are represented by Model 69, with the addition of narrow planes replacing the edges between the planes of the rhombic dodecahedron and those of the icositessarahedron. Minerals: Garnet, &c., Part II., page 112. P,M,T. 6pmjt. Model 40. PjM,T. 6pf mft. P,M,T.6p T 3 T mft. The cube, with the solid angles replaced each by six scalene triangular planes. Minerals : Fluorspar, Part II., page 105. P,M,T, MT. PM, PT, 3p^mt, 6pim|t, 6p 1 5 r mf t. The cube predominant, having the edges replaced by the planes of the rhombic dodecahedron, and the solid angles replaced each by fifteen small planes. Minerals : Fluorspar, Part II., page 105. 431. Analysis of these Combinations. In all these combinations, the planes of the hexakisoctahedron are placed in such a manner that mea- surements can be taken across two of the edges of the hexakisoctahedron. In the combination which Model 69 partially resembles, these edges will be the middle and short edges, because the long edge is replaced by a plane of P_MT. In the combination which Model 40 resembles, the edges to be measured are the long edge and the short edge, since the middle edge is here replaced by the plane PZ. Measurements across any two edges of the hexakisoctahedron, are sufficient to guide us to a knowledge of its symbol, as I have demonstrated in 408 429. When 206 . PRINCIPLES OF CRYSTALLOGRAPHY. the known angles are those across a middle edge and a short edge, we use problems 409, 410; when they are those across a long edge and a short edge, we use problems 428, 409, and 410. I proceed next to the investigation of ROSE'S Hemihedral Forms of the Octahedral System of Crystallisation, the general characters of which have been already fully explained in 263 270. I. THE TETRAHEDRON, or Hemioctahedron. Model 117. IPMT. 432. This form is described in 265. ROSE'S symbol for it is \r (a : a : ). According to the principles of classification explained in Section IV., it is an incomplete pyramid, since it has none but inclined planes, and yet is without solid angles at the poles Z and N. It has a square equator. The minerals which occur in this form are quoted at page 122, Part II. in Class 6, Order 1, Genus 1. 433. The unipolar normals terminate in the middle of the edges of this form ; the bipolar normals, in the middle of the lines drawn on the faces of the model so as to connect the poles of the unipolar normals. The tripolar normals terminate in the centre of each of the four faces, and in each of the four solid angles of the model. The lines which con- nect the unipolar normals of this form, and which are drawn on Model 117 in coloured ink, show the position of the edges of the octahedron. 434. PROBLEM. Given, Model 117, |PMT, with the angle across every edge = 70 32'; required, the plane angle of the faces. a.) Take one of the solid angles of the form as an oblique angled solid triangle, and seek the value of a plane angle by means of Formula 37. As the faces of Model 117 are equilateral triangles, we have given, A, B, C each = 70 32'; to find, a. The answer will of course be 60. b.) A shorter method of calculation is, to divide one of the solid angles of the form into two equal sections, by a plane assumed to pass through one of the edges and across one of the planes of the model. Taking one of these sections as a solid triangle, the problem is this : Given, A = ^ 35 16' ; B = 70 32' ; C = 90 ; to find, a. In this case, a is half the required plane angle. formula 4. log cos a = log cos A -f- 10 log sin B. 10 + log cos A = 35 16' = 19.9119 log sin B = 70 32' = 9.9744 log cos a = 30 = 9.9375 Twice this product, or 30 X 2 = 60, is the required plane angle. 435. PROBLEM. Given, Model 117, |PMT, with the angles across PRINCIPLES OF CRYSTALLOGRAPHY. 207 every edge = 70 32' ; required) the inclination of the tripolar normals ivhich terminate in a plane, to axis p a , and the inclination of those ivhich terminate in a solid angle, to an edge of the model. a.) The inclination of the tripolar normals Znw Zse Nne Nsw to axis p a , is found by the process given in 349. It is 54 44'. The inclina- tion of the four other tripolar normals to an edge is 35 16'. This is evident, from the consideration, that the north-west meridian of Model 117, divided into two portions by the vertical axis p a , is exactly equal to the figure in 349. b.) Another Method. Take the solid angle of the model as a three- faced pyramid, and assume it to be divided into six portions by the pro- cess described in 359. One of these portions, taken as a right angled solid triangle, will have the following known parts : A = 60; B = 35 16'; C = 90. With these you can find, a = half a plane angle of one of the faces ; b = inclination of an external plane to the tripolar normal ; and c = inclination of an edge to a tripolar normal. Formula 4, log cos a = log cos A -f- 10 log sin B. 10 + log cos A = 60 = 19-6990 log sin B = 35 16' = 9.7615 log cos a = 30 = 9-9375 Formula 5. log cos b = log cos B -f- 10 log sin A. 10 -f log cos B = 35 16' = 19.9119 log sin A = 60 = 9.9375 log cos b = 19 28' = 9-9744 Formula 6. log cos c = log cot A -f log cot B - 10. log cot A = 60 = 9.7614 + log cot B = 35 16' = 10,1505 log cos c = 35 16' = 9.9H9 c.) Upon the principle laid down in 359, c.), the products b -f c, should be equal to the inclination of an edge to a plane of Model 117, measured over a solid angle. Now, b + c = 19 28' + 35 16' = 54 44', and as the cross section of Model 107 is a triangle of which one angle is 70 32', and the other two angles are equal to each other, one of the angles must be i(180 70 32' = 109 28') = 54 44'. Hence it is evident that the tripolar normals terminate in the middle of each plane and in the middle of each solid angle of the tetrahedron. 436. COMBINATIONS CONTAINING THE TETRAHEDRON. iPMT, |pmt. Model 118. This combination is described in 269, 270. It is an incomplete pyramid with a square equator, and the mine- rals which it represents are quoted at page 122, Part II. in Class 6, Order 1, Genus 1. The inclination of a plane of |PMT on a plane of Jpmt, is 208 PRINCIPLES OF CRYSTALLOGRAPHY. 109 28', which sufficiently shows the relation of this form to the octahe- dron and tetrahedron. p,m, t. |PMT. P,M,T. ipmt. Model 38. The first of these combinations would be represented by Model 37, if the twelve pentagonal faces were away. These combinations represent the planes of the cube in combination with those of the tetrahedron ; in one of them, the cube predominates, and in the other the'tetrahedron. They represent complete prisms combined with incomplete pyramids. Minerals: page 105, Part II. Class 3, Order 1, Genus 1. Groups, , d. Analysis. The inclination of the planes of the cube to those of the tetrahedron is = ^^ -f 90 = 125 16/. As a single plane of the tetrahedron is exactly similar to a single plane of the octahedron, the same analytical processes serve to discriminate both the tetrahedron and the octahedron. See 364 368. MT. PM, PT, Jpmt. Model 65, if four alternate triangular planes were away. mt. pm, pt, |PMT. Model 78. The rhombic dodecahedron and the tetrahedron. The first is an incomplete prism with a complete pyramid ; the second an incomplete prism with an incomplete pyramid. Minerals : page 110 and 114, Part II. Analysis. See 366. p,m,t, mt. pm, pt, JPMT. Model 37. P,M,T, mt. pm, pt, Jprnt. Model 36. p,m,t, MT. PM, PT, |pmt, Model 34, supposing four alternate triangu- lar planes to be away. Combinations containing the tetrahedron, the cube, and the rhombic dodecahedron, one of them predominant over the other two, in each crys- tal. All of them are complete prisms with incomplete pyramids. Mine- rals: pages 105, 106, Part II. Class 3, Order 1, Genus 1. Analysis, 368. P,M,T, MT. PM, PT, |PMT, ipmt. Model 35. The cube, the rhombic dodecahedron, the right tetrahedron, and the left tetrahedron. A com- plete pyramid with an incomplete prism. Minerals, page 105, Part II. Class 3, Order 1, Genus 1. Analysis, 368. The combinations of which there are models are further explained in the description of the models, Part II. page 123. 2. THE HEMIICOSITESSARAHEDRON. iP_MT,iPM_T,iPMT_: or i(3P_MT). Varieties of this Combination: mMT, iPM|T, IPMTi: or |(3P1MT). iPiMT, IPMiT, iPM-TJ: or .J(3PiMT). Model 119 is PRINCIPLES OF CRYSTALLOGRAPHY. 209 437. This is the hemihedral variety of the combination 3P_MT, Mo- del 22. See 267 269. The varieties are discriminated by the inci- dence of their planes upon one another, which are exactly the same as the corresponding angles of the homohedral combinations. Upon comparing Model 1 19 with Model 22, both being held in upright position, it will be seen that the planes of the half form, Model 1 19, are those which belong to four octants of the whole form, Model 22, namely, the octants Znw, Zse, Nne, Nsw. This is what constitutes the right hemiicositessarahedron. But there is another variety of the half form, which is called the left hemiicositess- arahedron, and which comprises the other four octants of the whole form, namely, the octants Zne, Zsw, Nnw, Nse. In combination together, these two half forms are discriminated by the symbol i(3P_MT), ^(3p_mt), because they are necessarily always une- qual though always similar. When one of them occurs alone, it is always to be considered as the right form, and denoted by |(3P_MT) Z 2 nw, or simply by |(3P_MT); the last symbol is sufficiently precise, because it is a rule of common acceptance, as I have already stated, 269, that every single tetrahedral form, is considered to be the right form, or that which possesses the Znw octant. There are two kinds of edges on the hemiicositessarahedron. Long edges which pass through the poles Z N n e w s and connect the following tripolar normals, Zne Zsw Nnw Nse ; and short edges, which pass through the bipolar normals and connect the eight tripolar normals. The follow- ing are the angles of the two varieties : Across a short edge. Across a long edge. i(3P|MT) 146 27' 109 28' |(3PiMT) 129 31' 129 3V The hemiicositessarahedron is an incomplete pyramid with a rhombic equator. Minerals: page 122, Part II. Class 6, Order 3. 438. Analysis. The planes of ^(3P_MT) have the same form as the planes of 3P+MT. Compare Model 119 with Model 17. The planes of (3P-f MT) have the same form as the planes of 3P_MT. Compare Model 18 with Model 22. I point out these facts, because they present a remarkable example of contrariness, which is apt to puzzle a young crystallographer. The following are the chief points to be observed in relation to the measurements of the hemiicositessarahedron. The angle across a short edge of the hemihedral form, is the same as the angle across a short edge of the homohedral form. The angle across a long edge of the hemihe- dral form is equal to twice the inclination of a plane of the homohedral form to axis p a . I shall give instructions for finding the angle across a long edge when that across a short edge is known, or the angle across 2E 210 PRINCIPLES OF CRYSTALLOGRAPHY. a short edge when that across a long edge is known. This, with problem 370, is sufficient information for the derivation of the symbol of the combination. 439. PROBLEM. Given, Model 119, |(3PMT), with the angle across a long edge = 129 31'; required, the angle across a short edge. a.) First find the inclination of a plane to the Znw normal. Model 119 resembles a tetrahedron having a low three-sided pyramid super- imposed upon each face. Hence the angle across the long edge is equal to the angle across the edge of a tetrahedron, plus twice the inclination of one of the oblique planes to the face of the tetrahedron. Conse- quently, if you take from the angle across the edge of the form = 129 3T, the angle across the edge of the tetrahedron = 70 32', the residue, divided by 2, will give the inclination of one of the oblique planes to the face of the tetrahedron. Say 129 31' 70 32' = 58 59'; and **^ = 29 29V. The complement of this angle = 60 30V, is the inclination of a plane to the tripolar normal, which normal is perpen- dicular to the plane of the enclosed tetrahedron. b.) Another Method. The unipolar normal meets the tripolar normal at an angle of 54 44', 349. The unipolar normals meet the plane of J(3P|MT) at an angle of ^^ = 64 54J'. Therefore, the inclination of that plane to the tripolar normal is = 180 (54 44' + 64 451') = 60^ 30J'. c.) Having now the inclination of a plane to the Znw normal, we can calculate the angle across an edge proceeding from that normal, by dividing the three-faced pyramid of Model 119 into six right-angled solid triangles, on the principle explained in 359. We take one of these triangles with pole Znw for its vertex, and in which the known quantities are C = 90 = inclination of an external face to a plane cutting it through the middle ; A =r 60 = an interior edge where two intersecting planes meet at the Znw normal ; and b = 60 30 J' = in- clination of an external plane to the Znw normal. With these data, we have to find B, which is half the angle across a short edge of the model. Given, A = 60 ; b = 60 30J'; C = 90; to find, B. Formula 8. log cos B = log cos b -f- log sin A 10. log cos b = 60 30' = 9.6922 + log sin A = 60 = 9.9375 log cos B = 64 46' = 9.6297 Twice this product, or 64 46' X 2 = 129 32', is the required angle across a short edge of Model 119. This result is, however, 1' too much, as the angle across a short edge of (3PMT) is the same as that across a long edge, 129 31'. 440. PROBLEM. Given, the combination, |(3PMT), with the angle across a long edge = 109 28'; required, the angle across a short edge. PRINCIPLES OF CRYSTALLOGRAPHY. 2 1 1 Find, by problem 439, .)> tne inclination of the planes to the tri- polar normal. i(109 c 28' 70 32') = 19 28', the supplement of which is 70 32'. The rest of the solution of this problem is contained in 377, I.), where the required angle is proved to be 146 27'. 441. PROBLEM. Given, Model 119, ^(3P^MT), with the angle across a short edge = 129 31'; required, the angle across a long edge. a.) Find the inclination of a plane to the tripolar normal, 439- Call this x. Then half the angle across a long edge will be 180 (54 44' + x). 439, b.) b.) Take pole Znw for the vertex of a right-angled solid triangle, and one-sixth of the crystal to form this triangle, as described in 439? c.) The known parts of the triangle are C = 90; A = 60; and B = 64 451'. With these data, you have to find b, which is the inclination of a plane to the tripolar normal. Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 64 45|' = 19.6298 log sin A = 60 = 9-9375 log cos b = 60 30|' = 9.6923 c.) Now, according to .), half the angle across a long edge of the model is equal to 180 (54 44' + 60 30|') = 64 45 i'. Twice 64 45|' is 129 31/. 442. PROBLEM. Given, the combination |(3P^MT), with the angle across a short edge = 146 27'; required, the angle across a long edge. Proceed as directed in 441. To find x, or the inclination of a plane to the tripolar normal, say, Given, A = 60; B = 73 13i'; C = 90; to find, b. Formula 5. log cos b = log cos B -f 10 log sin A. ^ 10 + log cos B = 73 13i' = 19.4603 log sin A = 60 = 9-9375 log cos b = 70 32' = 9.5228 Then, 180 ( 54 44' + 70 32') = 54 44'. Twice this product, or 54 44' x 2 = 109 28', is the required angle across a long edge of the given combination. 443. PROBLEM. Given, Model 119, (3P|MT), with the angle across a short edge = 129 31'; required, the plane angles of the external faces. Take the right-angled solid triangle employed in 441, and with the same given partsl A = 60; B = 64 45 i'; C = 90; find a, which is half an external obtuse plane angle of Model 1 1 9. 212 PRINCIPLES OF CRYSTALLOGRAPHY. Formula 4. log cos a = log cos A -f- 10 log sin B. 10 + log cos A = 60 = 19.6990 log sin B = 64 45 i' == 9-9564 log cos a == 56 26' = 9-7426 Twice this product, or 56 26' x 2 = 112 52', is the obtuse plane angle of the model. Each acute plane angle is half the difference between 112 52' and 180, or gg'-j" 08 * = 330 34'. 444. PROBLEM. Given, the symbol -J(3P_MT) with the angle across a short edge =93 40', and the angle across a long edge = 176 30'; required, the value of the index _ in the symbol. These are the angles ascribed by Phillips to a variety of the hemi- icositessarahedron, which approaches nearly to the form of a cube. See his Mineralogy, article Arseniate of Iron, page 235, figure 5. a.) First examine the accuracy of these measurements, as follows : To find x, 442, say, Given, A = 60; B = 46 50' (= ^) ; to find, b. Formula 5. log cos b = log cos B 4- 10 log sin A, which gives log cos 37 49' = log cos 46 50' + 10 log sin 60. Then, 180 , (54 44' + 37 49') = 87 27'. Twice this product = 174 54', is the angle across a long edge of the given combination, which differs nearly 2 from Phillips's measurement of 176 30'. b.) To find the index of the symbol, take the right-angled solid triangle described in 376, b.) The parts known in the present example are, A = interior angle of 45, where the north and north-west meridians cross one another ; b = 87 27' = inclination of an external plane to p a ; C = 90 = inclination of an external plane to the north-west meridian. Then c will be the inclination of a long edge of the icositessarahedron to axis p a , the cotangent of which will give the required index of the symbol. Given, A = 45; b = 87 27'; to find, c. Formula 9. log tan c =. log tan b + 10 log cos A. 10 + log tan b = 87 27' = 21.3513 log cos A = 45 = 9.8495 log tan c = 88 12' = 11.5018 The cotangent of 88 12 7 is =. ^g, which gives for the given hemiicosi- tessarahedron the symbol ^(P^-MT). But if the calculation is made on the basis of Phillips's measurement of the long edge, the angle produced is 88 46', the cotangent of which is = -^, affording the symbol 445. COMBINATIONS CONTAINING THE HEMIICOSITESSARAHEDRON. 1PMT, l(3pjmt). ipmt, PRINCIPLES OF CRYSTALLOGRAPHY. 21 3 Combinations of the tetrahedron and the hemiicositessarahedron, some- what resembling Model 94, on the supposition that the twelve rhombic planes on the corners of that model were absent. Minerals: page 122, Part II. mt. pm, pt, 1PMT Znw, l(3p|mt) Znw. Model 94. MT. PM, PT, ipmt Znw, i(3p$mt) Zne. Model 96. Both these forms contain the rhombic dodecahedron, the right hemi- octahedron, and the right hemiicositessarahedron. They are, neverthe- less, very different in appearance from one another, and the difference is principally caused by the predominance in the one form of the dodeca- hedron, and in the other of the hemioctahedron. It commanly happens with the combinations that contain both homohedral and hemihedral forms, that the predominance of the one or the other makes a very striking difference in the general appearance of the combination. The hemiicositessarahedron contained on Model^ 94 is that described by the symbol i(3pimt). The one contained on Model 95 is ^(3pjmt). The proof of this exists in the difference of the angle of incidence of the two planes which form an edge at the pole Z. The hemihedral forms |PMT and |(3P_MT) contained on Model 94, both occupy the same set of octants ; but those contained on Model 95, occupy different sets of octants. The simpler form |PMT is therefore ascribed to the Znw octant, and the more complex form i(3P_MT), to the left set. This occurrence of two sets of hemihedral forms in dif- ferent positions upon one crystal, shows the grounds of the distinction necessary to be made between the right and left hemihedral forms. mt. pm, pt, iPMT, i(3p|MT) Znw, i(3p|mt) Zne. P,M,T, MT. PM, PT, |PMT Zne, |PMT Znw, i(3pjmt) Znw. iPMT, l(Splmt), K 3 Pj mt -) 446. Analysis of these Combinations. A single plane of the hemi- icositessarahedron ^(3P_MT), has the same properties as a single plane of the icositessarahedron 3P_MT. Consequently, the instructions given for the analysis of combinations that contain the latter, apply equally well to the analysis of those which contain the' former. See 381 384. The inclination of a plane of ^PMTto a plane of J(3P_MT) is 90-f-#, in which formula x signifies the inclination of a plane of J(3P_MT) to the tripolar normal. 3. THE HEMITRIAKISOCTAHEDRON. iP + MT, ^PM+T, iPMT + : or |(3P+MT). Varieties of this Combination ?: JPfMT, iPM|T,iPMTf : or J(3P|MT). |P 2 MT, |PM 2 T, JPMT,: or |(3P 2 MT). Model 18. 447. This combination only occurs with other forms, and never in an isolated state. I shall, however, describe it as a complete crystal ; PRINCIPLES OF CRYSTALLOGRAPHY. because calculations founded on measurements of its angles are made exactly in the same way as if the planes could form a separate crystal. It presents two kinds of edges, namely, short edges which meet three together at the tripolar normals Znw Zne Nne Nsw, exactly as on the homohedral crystal 3Pf MT, Model 17 ; also, long edges which meet three together at the tripolar normals Zne Zsw Nnw Nse. The solid angles produced by the meeting of three short edges are obtuse : those pro- duced by the meeting of three long edges are acute. The long and short edges meet together at the unipolar normals, and produce four- faced angles having the character of rhombic pyramids. The lines drawn on Model 18 to connect the unipolar normals, or the similar plane angles, indicate the position of the long edges of the homohedral combination, 3P|MT, Model 17. It follows from this description, that if the angle across a short edge of Model 18 be known, that across a long edge can be found ; or if the angle across a long edge be known, that across a short edge can be found ; or if either be known, the inclination of the planes to the three axes p a m a t a can be determined, and the index for the symbol be thence deduced. ROSE describes the only known variety of this combination as J(3P|MT), with the angles as follow: across a long edge = 82 10', across a short edge = 162 39 J'. MILLER and VON KOBELL describe it as J(3P 2 MT), with the angles as follow: across a long edge = 90; across a short edge = 152 44'. Model 18 agrees with (3P 2 MT). The combination on which the hemitriakisoctahedron occurs in the mineral world, according to ROSE, is Fahlerz from Dillenberg, MT.PM, PT. J(3PJMT), i( 3 Pl mt > The predominant form in this combination is that represented by Model 119> J(3P_MT). Its four acute solid angles are replaced by the twelve rhombic planes of the dodecahedron, which rest on the short edges of the hemiicositessarahedron, as shown by Model 94. The planes of the hemitriakisoctahedron, Model 18, replace the shorter edges of Model 119, appearing as long narrow planes meeting at the tripolar normals. See ROSE'S Krystallographie, fig. 34. 448. PROBLEM. Given, Model 18, l(3P_j_MT), with the angle across a short edge = 152 44'; required, the angle across a long edge, and the value of the index + in the symbol. Find the inclination of the short edge to the tripolar normal Znw; next its inclination to the unipolar normal Z ; and then, by means of a right-angled triangle, with pole Z for its vertex, find the angle across the long edge. a.) To find the inclination of the short edge to the unipolar and tri- polar normals. Form a right-angled solid triangle as directed in 359, with pole Znw for vertex, and the following known parts : A = 60 ; C = 90; B = 76 22' (= ^ii'). Then find c, which is the inclina- tion of an edge to the tripolar normal. PRINCIPLES OF CRYSTALLOGRAPHY. 215 Formula 6. log cos c = log cot A + log cot B 10. log cot A = 60 = 9.7614 + log cot B = 76 22' = 9.3848 log cos c = 81 57' = 9.1462 This product, 81 57', is the inclination of a short edge to the tripolar normal. Then the inclination of the short edge to the unipolar normal is 180 (54 44' + 81 57') = 43 19'. b.) To find the angle across a long edge. Assume Model 1 8 to be divided by planes passing through the long and short edges, and intersecting at axis p a . Take one of the sections as a right-angled triangle with pole Z for vertex. Then you have C = 90 = interior edge of intersection ; B = 76 22' = half the angle across a short edge ; a = 43 19' = inclination of a short edge to p a . With these data, find A, which is half the angle across a long edge. Given, a = 43 19'; B = 76 22'; to find, A. Formula 10. log cos A = log cos a -f log sin B 10. log cos a = 43 19' = 9.8619 + log sin B = 76 22' = 9.9876 log cos A = 45 = 9.8495 Twice this product, or 45 X 2 = 90, is the required angle across a long edge of Model 18. c.) To find the inclination of a plane to a bipolar normal. First, find the inclination of a plane to the tripolar normal by means of the triangle described in ). Say, Given, A = 60; B = 76 22'; C = 90; to find, b, which is the inclination required. Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 76 22' = 19.3724 log sin A = 60 = 9.9375 log cos b = 74 12i' = 9.4349 This product, 74 12J', is the inclination of a plane to the tripolar nor- mal. Its inclination to a bipolar normal is 180 - (35 16' + 74 121') = 70 31V. d.) To find the value of the index + in the symbol. Suppose Model 18 to be divided by the north and east meridian into quadrants. Take one of these as a right-angled solid triangle, with pole Z for its vertex, and the following given parts : C = 90, intersection of the two meri- dians ; A = 70 31 J' = inclination of a plane of Model 18 to the Zn bipolar normal, or to the north meridian ; b = 45 quadrant of the north meridian. With these data, find a, which will be that side of the solid triangle whose cotangent shows the relative value of axes p a and t a of the form under investigation. 216 PRINCIPLES OF CRYSTALLOGRAPHY. Given, A 70 31|'; b = 45; to find, a. Formula 7. log tan a = log tan A -f- log sin b 10. log tan A == 70 3H' = 10.4515 + log sin b = 45 9.8495 log tan a 63 26' = 10.3010 The cotangent of this product, 63 26', is .5000 = J, which proves that the plane whose inclination to the north meridian is 70 31 i' belongs to the form P|MT, or PMT 2 , which is one of the three forms that pro- duce the combination, P 2 MT, PM 2 T, PMT 2 , or 3P 2 MT. See 386. 449. PROBLEM. Given, Model 18, |(3P 2 MT), with the angle across a short edge 152 44', and the angle across a long edge 90; required, the plane angles of the faces. Take the solid triangle described in 448, a), with the same given parts, and find a, which will be half the angle at pole Znw. Given, A = 60; B = 76 22' ; to find, a. Formula 4. log cos a log cos A + 10 log sin B. 10 + log cos A 60 = 19.6990 log sin B = 76 22' = 9-9876 log cos a = 59 2'= 9-7114 Twice this product, or 59 2' x 2 = 118 4', is the obtuse plane angle of Model 18 at pole Znw. Form a similar equation, substituting half the angle across a long edge for half the angle across a short edge, which will give half the plane angle at pole Zne. 10 + log cos A = 60 19.6990 log sin B 45 = 9.8495 log cos a = 45 9-8495 Twice this product, or 45 x 2 = 90, is the plane angle at pole Zne. The angles at pole Z and N are each J {360 (118 4' + 90) 151 56'} =r7558'. Control over this Calculation. With the triangle described in 448, 5.), find the value of the angle at pole Z by a direct process. Given, A = 45 (half angle across a long edge); B 76 22'; to find, c. Formula 6. log cos c log cot A + log cot B 1 0. log cot A = 45 10.0000 + log cot B = 76 22' = 9.3848 log cos c = 75 58' 9-3848 The four plane angles of the face of Model 18 are therefore 75 58' + 11 8 4' + 75 58' + 90 = 360. PRINCIPLES OF CRYSTALLOGRAPHY. 21? Many other problems respecting this combination could be given; but as they would be principally variations or repetitions of those relating to the triakisoctahedron, and as the hemihedral form is not of much con- sequence, I pass them over. 4. THE HEMIHEXAKISOCTAHEDRON WITH INCLINED FACES. _, 1P_M + T, or K6P-MT+). Varieties of this Combination: , IPiMT*, JPJMTJ, JPJMiT, or 6P ^ M T Model 24 - or 1(6PM T). 450. The first of these combinations is the hemihedral variety of the hexakisoctahedron 6PJM^T. The second is a hemihedral combination of which no corresponding homohedral variety has been discovered, and no hemihexakisoctahedrons have been found to correspond with the rest of the known hexakisoctahedrons, 408. Some of the properties of this combination have been detailed in 263 270, 282, 283. There are three kinds of edges on the hemihexakisoctahedron. Twelve short edges which connect the unipolar with the tripolar normals Znw Zse Nne Nsw. These have the same positions as the short edges of the hemitriakisoctahedron, Model 18. Twelve long edges, which connect the eight tripolar normals. These have the same positions as the short edges of the hemiicositessarahedron, Model 119- Twelve middle edges, which connect the unipolar normals with the tripolar normals Zne Zsw Nnw Nse. These have the same positions as the twelve acute edges of the hemitriakisoctahedron, Model 18. Both the known varieties of this combination have the remarkable property, that the measurements across the longest and shortest edges are alike. These measurements are as follow : Combination. Long edge. Middle edge. Short edge. 1(6P1M|T) 158 13' 110 55' 158 13' $(6PfM}T) 152 20' 122 53' 152 20' 451. COMBINATIONS CONTAINING THE HEMIHEXAKISOCTAHEDRON. mt. pm, pt, }PMT, ^(SpjMT), l(6p*mjt). A combination exactly similar to Model 94, with the addition of six small narrow planes replacing the edges between the rhombic planes of the dodecahedron and the rectangular planes of the hemiicositessarahe- dron. Grey Copper. P,M,T, MT. PM, PT, ^PMT, |pmt, l(3p|mt) Zne,i(6pJ-m|t) Znw A combination resembling Model 35, with the addition of six small 2r 218 PRINCIPLES OF CRYSTALLOGRAPHY. planes replacing the angles where P,M,T meet |PMT, at the four cor- ners Znw Zse Nne Nsw ; and with three narrow planes replacing the edges between MT. PM, PT, at the four corners Zne Zsw Nnw Nse. The set of twenty-four small planes constitute the right hernihexakisoctahe- dron, Model 24. The twelve small planes constitute the left hemiicosi- tessarahedron, Model 119. Altogether, there are sixty-two planes on this combination. Boracite. 452. Analytical Processes. As I have illustrated the hexakisoctahe- dron very fully, I think it needless to give many problems regarding the hemihedral form ; which, moreover, is not of much importance, although capable of affording as many analytical processes as are given between 408 431. I shall therefore merely notice a few leading problems, and refer the reader to similar equations, given in the preceding pages, for the details. Let Model 24, (6PiMT) be the subject of inquiry. Put the long edge = /, the middle edge = m, and the short edge = s. If / and s are given, you form a solid triangle containing a sixth of the flat six-faced pyramid at pole Znw. Then you find x = inclination of a short edge to the Znw normal, y = inclination of a long edge to the Znw normal, and z = plane angle of a face at the Znw normal. If / and m are given, you form a solid triangle consisting of a sixth of the acute six-faced pyramid at pole Zne. Then you find, p = inclina- tion of a middle edge to the Zne normal, q inclination of a long edge to the Zne normal, and r = plane angle of a face at the Zne normal. If m and s are given, you form a solid triangle, consisting of a fourth of the rhombic pyramid at pole Z. Then you find f= inclination of a short edge to axis p a , g = inclination of a middle edge to axis p a , and h = plane angle of a face at pole Z. With these data, and with the inclinations of the normals to one another, 349, the forms of the meridians, and the inclination of the external planes to the meridians, you have all the necessary data for eve^ kind of calculation directed in the article on the hexakisoctahe- dron, page 191. 5. THE PENTAGONAL DODECAHEDRON. M_T. P_M, P + T. Varieties of this Combination : MfT. P|M, P|T. MfT. PfM, PJT. MiT.PJM,PfT. Model 91. 453. This combination is described in 108, and again in 281. It has two kinds of edges, namely, six long edges, situated two on the north meridian at n and s, two on the east meridian at Z and N, and two on the equator at e and w; also, twenty-four short edges so situated as to form a three-faced pyramid at the termination of each of the tripolar normals. PRINCIPLES OF CRYSTALLOGRAPHY*. 219 The following are the angles across the edges of the different varie- ties : Long edges. Short edges. M|T. PIM, PfT 106 16' 118 41' MfT.PfM, PfT 112 37' 117 29' M|T. P|M, PfT 126 52' 113 35' The variety represented by Model 91, MJT. PM, PfT, occurs in the mineral world as a complete crystal, particularly beautiful in Cobalt Glance and Iron Pyrites. It is thence sometimes called the Pyritohedron. The other varieties only occur in combination with other forms. The Penta- gonal Dodecahedron is an incomplete prism with an incomplete pyramid, arid has a rhombo-quadratic equator. See page 117, Part II., where it occurs in Class 5, Order 4. Genus 1. 454. PROBLEM. Given, Model 91, M_T. P_M, P+T, ivith the angle across a long edge = 126 52' ; required, the value of the indices _ and _l_ in the symbol. Divide 126 52' by 2 : the cotangent of the remainder is the index of the shorter axis of each form. ^^ = 63 26'. cot = \. That is M_T = M|T, P_M = P|M, and P + T = PT, or taking T for unity = P-f-T- Hence the complete symbol is M|T. PJM, PfT. In many parts of this work the symbol is written MT 2 . PM 2 , P 2 T; but I think it better to keep all the indices between the two letters of the symbol, and always to make axis t a = I. In the same manner, if the given angle across a long edge is 106 16', you sayj ]^L = 530 8 /. cot = f . Then the symbol is M|T. P|M, PfT. And if the angle is = 112 37', then, H^H' = 56 181'. cot -f, which gives MfT.PfM, PfT. 455. PROBLEM. Given, Model 91, M_T. P_M, P+T, with the angle across a long edge = 128 52' ; required, the angle across a short edge. Suppose the model to be divided by the north meridian into two pieces. Take one half of it as a solid triangle, having for its vertex the solid angle where plane P_M Zn meets the two front planes of M_T. Then the known parts of the triangle are, C = 90 = inclination of the north meridian on the plane P_M ; A = 63 26' = inclination of M_T nw on the north meridian; and b = 116 34', the inclination of P_M Zn to the front edge of the model. This last angle consists of the prismatic angle of 90 added to the complement of the inclination of plane P_M to p a , namely, 90 63 26' = 26 34'. With these data find B, which is the angle across the short edge between P_M Zn and M_T nw. Given, A = 63 26' ; b = 1 16 34' ; C =. 90 ; to find B. Formula 8. Log cos B log cos b + log sin A 10. Since angle b = 116 34' is not in the table, you substitute its supplement, 180" 220 PRINCIPLES OF CRYSTALLOGRAPHY. 116 34' = 63 26', of course paying attention to any possible ambi- guity that can arise from that substitution, 330. log cos b = 63 26' = 9-6505 + log sin A = 63 26' = 9-9515 log cos B = 66 25' = 9.6020 The supplement of this product, or 180 66 25' =113 35', is the required angle across a short edge of Model 91. 456. PROBLEM. Given, Model 91, M_T. P_M, P + T, with the angle across a short edge =113 35', and across a long edge = 126 52'; re- quired, the plane angles of the external faces of the model. The solid triangle employed in 455, serves also for this problem. a.) To find the plane angle of P_M at Zn. Given, A = 63 26' ; b = 63 26' ; to find, a. Formula 7. log tan a = log tan A + log sin b 10. log tan A = 63 26' = 10.3010 + log sin b 63 26' = 9.9515 log tan a = 60 48' = 10.2525 Twice this product, or 60 48' X 2 = 121 36', is the obtuse plane angle of P_M at Zn. The product is doubled, because, from the nature of the solid triangle, a is only half the required angle. b.) To find the plane angle of MJT at Zn. Given, A = 63 26' ; b = 63 26' ; to find, c. Formula 9. log tan c = log tan b -f- 10 log cos A. 10 + log tan b = 63 26' = 20.3010 log cos A == 63 26' = 9-6505 log tan c = 77 24' = 10.6505 In equation a), the process afforded the correct angle, but in this case, we have the supplement of the correct angle, which is 102 36', as may be found by approximate measurement with the goniometer. c.) There are in all five plane angles on each face of Model 91, of which one is found by a) to be 121 36', and two are found by b) to be each 102 36', or together 205 12'. The remaining two are similar to one another. According to the principle in 16, t, all the angles of a penta- gon are equal to (180 x 5) 360 = 540. Now, if one of them is 121 36', and two others are 205 12', the two last must be together equal to 540 326 48' = 213 12', or separately, they must be 106 36'. This last product is corroborated by problem, 457. The five plane angles of each face of Model 91, are therefore 121 36' + 106 36' 4- 106 36' + 102 36' + 102 36' = 540. PRINCIPLES OF CRYSTALLOGRAPHY. 221 457. PROBLEM. Given, Model 91, M|T.PM, PfT, with the angle across a short edge =113 35'; required, the plane angle of the faces at pole Znw. Form a solid triangle on the principle explained in 359? having pole Znw for its vertex, and for its given parts, C = 90 ; A - 56 47|' ( = ll^i); B = 60. With these data, find b, which will be half a plane angle at Znw. Formula 5. log cos b = log cos B -f 10 log sin A. 10 + log cos B = 60 = 19.6990 log sin A = 56 47J' = 9.9226 log cos b = 53 18' = 9.7764 Twice this product, or 53 18' x 2 106 36', is the plane angle at the meeting of two short edges at pole Znw. 458. PROBLEM. Given, Model 91, MiT.PJM,PfT, with the angle across a short edge =113 35', and across a long edge = 126 52'; re- quired, the plane angle of M|T at Zn. Take Formula 6. log cos c = log cot A 4- log cot B 10, in which A is the supplement of 113 35'; B, the half of 126 52'; and c the re- quired plane angle of MJT at Zn. log cot A = 66 25' = 9.6400 + log cot B = 63 26' = 9.6990 log cos c = 77 24' = 9.3390 The supplement of this product is the required angle, 180 77 24' = 102 36', 456, b). 459. PROBLEM. Given, Model 91, MJT. PM, PfT, with the angle across a short edge = 113 35' ; required, the inclination of the external planes to the tripolar normal Znw. Take the solid triangle used in 457. Given, A = 56 47 J'; B = 60; to find, a. Formula 4. log cos a =. log cos A -j- 10 log sin B. 10 + log cos A = 56 47 j' = 19.7385 log sin B = 60 = 9.9375 log cos a = 50 46' = 9.8010 This product, 50 46', is the inclination of the external planes to the Znw normal. 460. COMBINATIONS CONTAINING THE PENTAGONAL DODECAHEDRON. p,m,t, MIT. P|M, Pf f. Model 47. P,M,T, m|t. pirn, pft. Rose, figure 53. 222 PRINCIPLES OF CRYSTALLOGRAPHY. These combinations are complete prisms with incomplete pyramids, See page 107, Part II. Class 3, Order 4, Genus 1. Bright White Cobalt and Iron Pyrites from Elba, present these combinations. iM, PfT, pmt. . PJM, Pf T, PMT. Model 92. m|t. pirn, pf t, PMT. Model 93. These combinations are incomplete prisms with incomplete pyramids, and fall into Class 5, Order 4, Genus 1, page 117, Part II. Model 92 is called the middle crystal between the cube and the pentagonal dode- cahedron. p,m,t, MT. PM, Pf T, PMT. Model 48 or 49. P,M,T, m|t. pjm, pf t, pmt. Rose, figure 54. P,M,T, mt, M|T. pm, P|M, pt, pf T. Complete prisms with incomplete pyramids. Class 3, Order 4, Genus 1, page 107, Part II. 461. Analysis of these Combinations. a.} The planes of P,M,T, incline upon adjoining planes of M^T. PJM, Pf T, at an angle of 90 + 63 26' = 153 26', in which formula, 63 26' represents the inclination of a plane of the pen- tagonal dodecahedron to a unipolar normal, 454, to all of which normals, the planes of P,M,T, are perpendicular. b.) The planes of PMT incline upon adjoining planes of M.JT. P|M, Pf T, at an angle of 90 + 50 46' = 140 46', in which formula, 50 46' represents the inclination of a plane of the pen- tagonal dodecahedron to a tripolar normal, to all of which normals, the planes of PMT are perpendicular. c.) The planes of the rhombic dodecahedron incline upon adjoining planes of the pentagonal dodecahedron, that is to say, a plane of MT upon a plane of MJT, at an angle of 135 + 26 34' =161 34', in which formula, 135 represents the inclination of MT upon M, and 26 34' represents the inclination of M| T upon t a , of which two angles it is that the inclination of MT upon M^T is composed. Refer to the figure in 396. If nEw is the required angle, then en E is 63 26' and oEn is 26 34'; while iEw is 45, and oEw is 135. Here, we assume oE = M, Ew = MT, and nE = MJT. 6. THE HEMIHEXAKISOCTAHEDRON WITH PARALLEL FACES. P_MT+, P + M_T, PM + T_ : or 3P_MT + . Varieties of this Combination : PIMiT, PMTO, PIMTi: orSPiMJT. PM|T, PMTO, P|MT|: or 3P1MJT. Pj-MJT, PMJTJ, P^MTi: or 3P}MT. Model 25 is the first of these combinations, or PRINCIPLES OF CRYSTALLOGRAPHY. 223 482. This combination is described in 177 193, and again in 282. The first variety occurs in an isolated state, as Iron Pyrites from Pied- mont. It is a complete pyramid with a rhombic equator. See Class 2, Order 3, Genus 1, page 102, Part II. The other varieties occur only in combination. There are three kinds of edges on the hemihexakisoctahedron with parallel faces ; namely, twelve long edges which meet in pairs at Z and N on the north meridian, at e and w on the east meridian, and at n and s on the equator ; twelve short edges which also meet in pairs, at Z and N on the east meridian, at n and s on the north meridian, and at e and w on the equator ; twenty- four middle edges, which meet three together so as to form a three-faced pyramid at the termination of every tripolar normal, and thence proceed to meet the junction of the long and short edges on the principal sections. It therefore presents three different kinds of solid angles or pyramids. The angles across the different kinds of edges of the several varieties of this combination, are as follow : Long Edges. Middle Edges. Short Edges. 3PJMJT 3P1M1T 3P|MiT 149 0' 154 47' 160 32' 141 47' 131 49' 131 5' 115 23' 128 15' 11 8 59' 463. PROBLEM. Given, Model 25, 3P_MT_|_, with the angle across a long edge = 149, and across a short edge = 1 15 23'; required, a.) the inclination of the short edges to the axes, b.) the inclination of the long edges to the axes, c.) the plane angle of the faces at pole Z, d.^ the value of the indices _ and _|_, and e.) the angles of the equator and the two principal meridians. a.) To find the inclination of the short edges of Model 25 to the axes p a m a t a . Assume the model to be divided into quadrants by the north and east meridian. Take the Znw quadrant as a right-angled solid tri- angle, with pole Z for its vertex. Then C = 90, is the interior angle at p. a formed by the intersection of the two meridians ; A = ^ = 74 30', is half the angle across a long edge; and B = ^-^' = 57 41 1', is half the angle across a short edge of the model. With these data, you can find, .) = inclination of a short edge to an axis, 5.) = inclination of a long edge to an axis, and c.) = the plane angle of a face at pole Z. Given, A == 74 30'; B = 57 41|'; required, a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 74 30' = 19.4269 log sin B = 57 41 J' = 9-9269 log cos a = 71 34' = 9.5000 This product, 71 34', is the required inclination of the short edges of Model 2,5 to the axes p* m a t a . 224 PRINCIPLES OF CRYSTALLOGRAPHY. b.) To find the inclination of the long edges of Model 25 to the axes p a ra a t a . Employ the same triangle and the same quantities as in a.) Given, A = 74 30'; B = 57 41'; required, b. Formula 6. log cos b = log cos B -{- 10 log sin A. 10 + log cos B 57 41 1' = 19.7279 log sin A = 74 30' = 9.9839 log cos b 56 18' = 9.7440 This product, 56 18', is the required inclination of the long edges of Model 25 to the axes. c.) To find the plane angle at pole Z of the external faces of Model 25. Take the triangle employed in .) and b.) Given, A = 74 30'; B = 57 41 J'; required, c. Formula 6. log cos c log cot A + log cot B 10. log cot A = 74 30' = 9.4430 + log cot B = 57 41 J' = 9.8010 log cos c 79 54' = 9.2440 This product, 79 54', is the required plane angle of the faces at pole Z. d.) To find the value of the indices _ and _|_ in the symbol descriptive of Model 25, 3P_MT + . The combination represented by Model 25, contains the three octahedral forms P_MT + , P + M_T, PM + T_. The first named of these forms is that whose planes meet at poles Z and N, and, therefore, those whose dissection affords the solid triangle used in the above calculations* Hence, the cotangent of the inclination of the long edge to an axis shows the relation of p a to m a , and the cotangent of the inclination of a short edge to an axis, shows the relation of p a to t a . These cotangents are as follow : cot 71 34' = .3333 or J = p a t = pjtj. cot 56 18' = .6669 or f = pi mj. This relation is equal to pjjmlte, which gives us the symbol P 2 M 3 T C ; but if we make t a equal to unity, then m a becomes | or J, and p a becomes f or ^, which reduces the symbol to the convenient expression, PJM| T. Hence, the symbol for Model 23 is, briefly, 3PJMJT, or at length, e.) To find the angles of the equator, and of the north and east meri- dians of Model 23 These three sections are all alike, so that the examination of one of them serves for the whole. The equator is an octagon with three kinds of angles. The angles at poles n and s are equal to twice the inclination of a long edge to an axis, or 56 18' X 2 = 112 36'. The angles at poles e and w are equal to twice the inclination of a short edge to an axis, or 71 34' X 2 = 143 8'. The value of the angles between the four poles n e s w is found as follows : PRINCIPLES OF CRYSTALLOGRAPHY. 225 Aggregate value of the angles of the equator = 1080 Value of angles at n, s, 112 36' x 2 225 12'^ Value of angles at e, w, 143 8' x 2 = 286 16'/ = The other four angles are together 568 32' Hence the value of each angle where a long edge meets a short edge is *:= 142 8'. 464. PROBLEM. Given, Model 25, 3P|M|T, with the angle across a middle edge = 141 47'; required, a.) the plane angle of the faces at pole Znw, b.) the inclination of the planes to the tripolar normal, and c.) the inclination of the middle edges to the tripolar normal. Form a right-angled solid triangle, as directed in 359, taking pole Znw for its vertex, and designating the given parts as follows : A = 60 interior vertical edge of the triangle ; B = 141 47 ' = 70 53&'= half the angle across a middle edge ; C = 90 = inclination of an external face to a section. With these data, find the following parts : side a = half the plane angle at Znw ; side b = inclination of a plane to the Znw nor- mal ; side c = inclination of a middle edge to the Znw normal. a.) Given, A = 60; B = 70 531'; to find, a. Formula 4. log cos a = log cos A 4- 10 log sin B. 10 + log cos A = 60 = 19.6990 log sin B = 70 531' = 9.9754 log cos a = 58 3' = 9.7236 Twice this product, or 58 3' X 2 = 116 6', is the required plane angle at pole Znw. b.) Given, A = 60; B = 70 53*'; to find, b. Formula 5. log cos b = log cos B -f 10 > log sin A. 10 + log cos B = 70 53 J' = 19.5150 log sin A = 60 = 9-9375 log cos b = 67 47' = 9.5775 This product, 67 4?', is the required inclination of a plane to the Znw normal. c.) Given, A = 60; B = 70 53i'; to find, c. Formula 6. log cos c = log cot A + log cot B -10. log cot A = 60 = 9.7614 + log cot B = 70 53V = 9-5396 log cos c = 78 28' = 9-3010 This product, 78 28', is the inclination of a middle edge to the Znw normal. 2G 226 PRINCIPLES OF CRYSTALLOGRAPHY. Check on the accuracy of this Calculation. See 362. b.) Inclination of plane, 67 47' cot .4084. c.) Inclination of edge, 78 28' cot .2046. 465. PROBLEM. Given, Model 25, 3P|M1T, with the angle across a long edge = 149, across a short edge = 115 23', and across a middle edge = 141 47'; required, the plane angles of the faces at the point where the three different edges meet, namely, a.) the angle formed by a middle edge and a short edge, and b.) the angle formed by a middle edge and a long edge. a.) Assume the model to be divided into octants, and take the Znw octant as a solid triangle, having for its vertex the point where the long, short, and middle edges all meet. The problem is then as follows : Given, A = ^ = 74o 30' ; B = !^ == 570 41V; C = 141 47' ; to find, a = angle formed by a middle edge and a short edge. Formula 37. sin } a = V/~ C ^^ S C ~ A) , where S = i( A + B + C). Log sin \ a = J (log cos S + log cos (S A) + 20 (log sin B + log sin C)}. A = 74o 30' S -= 136 59 V 180 B= 57041V A= 74030' S =1360591' C = 141o 47' S A= 62o 29 y SupptofS = 43o Of 2 ) 273 58J 7 S = 136o 59F log cos S = 43 Of = 9-8640 + log cos (S A) = 62 291' = 9-6646 + 20 = 39.5286 log sin B = 57 41V = 9-9269 1 {180 ) 10 71 oo ll^l = ~ 9.7914 f ' 38 13' J ' 2)19.8103 log sin \ a = 53 30' = 9.90515 Twice this product, or 53 30' x 2 = 107, is the plane angle formed by the meeting of a middle edge and a short edge. b.) The faces of Model 25 have four angles, equal together to 360. One of these angles was found by 464 a.), to be 116 6'; a second was found by 463 c.), to be 79 54'; and a third by 465 .), to be 107. The fourth must consequently be 360 (79 54' + 116 6' + 107)= 57. 466. PROBLEM. Given, the symbol, 3P$MT; required, the angle across a long edge and across a short edge of the combination. This requires a reversal of the processes given in 463, d and #.) PRINCIPLES OF CRYSTALLOGRAPHY. 227 The angle of which $ is the cotangent, is the inclination of the short edges to p% because I expresses the relation of p a to t a , and the short edges connect the poles of these two axes. The angle of which is the cotangent, is the inclination of the long edges to p a , because expresses the relation of p a to m a , and the long edges connect the poles of these two axes. The fraction expresses the relation of p a to m a , because t a is 1, m a is J of 1, and p a is i of 1 ; or, mul- tiplying all these axes by 6, because t a is = 6, ra a = 3, and p a = 2. These two cotangents, J and , correspond to the angles 71 34/ and 56 IS'. If these are taken as sides of a right-angled solid triangle, and called side a and side b, then angle A and angle B of the same triangle will be half of each of the two angles required by the problem. Given, a 71 34'; b = 56 18'; to find, A. Formula 13. Given, a = 71 34'; b 56 18'; to find, B. Formula 14. On working these problems, the answers will be found to be the angles given in the problem 463, namely, the angle across a long edge = 149, across a short edge 115 23'. 467- PROBLEM. Given, Model 46 ; required, the inclination of a plane 0/*P_MT_j_ upon the plane PZ, to be calculated from the symbol, p,m,t. 3P| MI T. Or, the inclination of a plane of P_MT + upon the plane PZ and the angle across a short edge being given ; required, the value of the indices q/"P_MT + . Model 46 represents one of the compounds of this form of frequent occurrence, and in which, the short edges are entirely replaced. A spe- cial process is therefore required to determine their value, without a knowledge of which we cannot calculate the value of the indices of the combination. a.) From the symbol P^MJT, to find the inclination of a plane of that form to the equator. If the combination 3P^M|T contained no forms but P^MJT, its equator would resemble the rhombic plane PZ seen upon Model 46. Hence, the inclination of PZ to P^M^T, on Model 46, is the supplement of the inclination of PZ to the external plane of the small pyramidal portion of P^M^ T which is assumed to be cut off, or replaced. Take a right angled solid triangle, consisting of one-fourth of the re- placed pyramid, with pole n for its vertex. Then angle C = 90 is the right angle between the equator and the north meridian ; side a is the side bounded by axis m% axis p% and the Zn edge of the north meridian, and as p a is 2 and m a is 3, the value of this side is the angle of which J or .6667 is the tangent = 33 42' ; side b is the side bounded by axis m a axis t a , and the nw edge of the equator, and as m a is J and t a is 1, the value of this side is the angle of which a J or .5000, is the cotangent = 63 56'. We have therefore given, a = 33 42'; b = 63 26'; to find, A, which is the inclination of an external plane to the equator. 228 PRINCIPLES OF CRYSTALLOGRAPHY. Formula 13. log tan A = log tan a + 10 log sin b. 10 -f log tan A = 33 42' = 19.8241 log sin B = 26 34/ = 9.9515 log tan A = 36 43' = 9.8726 This product 36 43', is the inclination of PJM|T to the equator. Its supplement, or 180 36 43' = 143 17', is the required inclination of a plane of P_MT + to the plane PZ, Model 46, calculated from the sym- bol PMjT. b.) With the foregoing data, to find the angle across a long edge of SPJMjT. Use the same triangle as in a), to find angle B, which is half the angle across a long edge of 3P^M^T. Given, a = 33 42'; b = 63 26'; to find, B. Formula 14. log tan B = log tan b -f- 10 log sin a. 10 + log tan b = 63 26' = 20 3010 log sin a = 33 42' = 9.7442 log tan B = 74 30' = 10.5568 Twice this product, or 74 30' X 2 = 1 49> is the angle across a long edgeofSPiMiT. c.) With the foregoing data, to find the angle across a short edge of 3PJMJT. Take the same triangle as in a) and b), but alter the vertex to pole Z. Then you have C = 90 =r intersection of the north and east meridians at p a ; B = 74 30' = half the angle across the long edge ; a = 56 18' = inclination of a long edge to axis p a . With these data, you have to find A, which is half the angle across a short edge. Given, a = 56 18'; B = 74 30'; to find, a. Formula 10 log cos A = log cos a + log sin B 10. log cos a = 56 18' = 9.7442 + log sin B = 74 30' = 9 9839 log cos A = 57 41' = 9.7281 Twice this product, or 57 41' X 2 = 115 22', is the required angle across a short edge of SP^MJT. d.) Given, Model 46, p,m,t. 3P_MT + , with the inclination of pZ vpon P_MT + = 143 17', and the angle across a long edge = 149; required, the inclination of the long edge to axis p*. Take the solid tri- angle described in d) with pole n for its vertex. Then you have given, C = 90 = inclination of the plane PZ to the east meridian of the given triangle ; A = 74 30' = half the angle across the long edge, and B = 36 43' = supplement of the inclination of plane PZ to a plane of P_MT + on Model 46, which supplement is the inclination of an external plane of the replaced pyramid to its equator or to plane PZ of the model. See a). With these data, you have to find b, the complement of which is the inclination of the long edge to axis p a . PRINCIPLES OF CRYSTALLOGRAPHY. 229 Given, A = 74 30' ; B = 36 43' ; to find, b. Formula 5. log cos b = log cos B -f 10 log sin A. 10 -f log cos B = 36 43' = 19.9040 log sin A = 74 30' = 9.9839 log cos b = 33 42' 9.9201 The complement of 33 42' is 56 18', which is the required inclination of a long edge to axis p a . See 463, b. e.) It appears from this investigation, that when you have given, Mo- del 46, and are required to find the angle across the replaced short edge or the inclination of the short edges to axis p a , in order to deduce thence the value of the indices of the symbol, you have in the first place, to find the inclination of the long edge to axis p a , by problem, 467> d), then the angle across a short edge by 467? c), and finally, the value of the indices by problem, 463, d). The fundamental data of these calcula- tions, are the angle across a long edge of Model 46, and the inclination of PZ to a plane of P_MT+, both of which must be taken with the goni- ometer. 468. COMBINATIONS CONTAINING THE HEMIHEXAKISOCTAHEDRON WITH PARALLEL FACES. p,m,t.3PiM|T. Model 46. P,M,T. 3pi m lt. Rose, fig. 53, without o. P,M,T. prat, 3pimjt. Rose, fig. 53 a . MJT. P|M, Pf I, Spimjt. Rose, fig. 51 . M|T. PJM, Pf T, pmt, Spimjt. Rose, fig. 5 l a . p,m,t, MjT. pjM, pf T, 3PiMJT, Spjmit. Rose, fig. 47 a . These combinations present the cube, P,M,T; the octahedron, PMT; the Pentagonal Dodecahedron, MJT. P^M, Pf I] ; and the Hemihexakis- octahedron, grouped in various methods. Iron Pyrites is the mineral which presents all the varieties. Those containing P,M,T, are complete prisms with incomplete pyramids, and the others are incomplete prisms with incomplete pyramids. See Part II. Class 3, Order 1, page 105; Class 3, Order 4, page 107 ; Class 5, Order 4, page 1 17. 469. Analysis of these Combinations. The inclination of a plane of pmt upon a plane of 3P_MT_j_ is 90 + x, in which formula, x is the inclination of a plane of 3P_MT + to the tripo- lar normal. The inclination of a plane of 3p|mit to a plane of MT. PM, PfT is x + #, in which, x is the inclination of a plane of M|T. PJM, Pf T to the tripolar normal, and y is the supplement of the inclination of a plane of 3p|m|t to the tripolar normal. The inclination of a plane of p,mjt, upon a plane of 3PMJT is found by the problem in 467. 230 PRINCIPLES OF CRYSTALLOGRAPHY. 470. THE ASPECT OF COMPLEX CRYSTALS BELONGING TO THE OC- TAHEDRAL SYSTEM OF CRYSTALLISATION, USEFUL AS A MEANS OF DISCRIMINATING THEIR COMPONENT FORMS. As the 13 combinations quoted in 341 are all that belong to the Octahedral System of Crystallisation, it follows that every single crystal of this class must either consist of one of these combinations, or of two or more of them combined together. In every example of such combina- tions, there will be one combination predominant, and all the co-existing combinations will appear upon it subordinately, replacing its angles or edges. Perhaps the combinations called " middle crystals," such as P,M,T.PMT, Model 29, and M_T. P_M, P + T, PMT, Model 92, may be held to be exceptions from this rule ; yet even in these, one of the combinations always predominates over the other. In examining a crystal of this kind, you first fix your attention upon the combination which predominates, and then endeavour to discriminate the co-existing subordinate combinations. Since no combination can occur but such as are indicated in 341, and since these can only occur upon one another in a certain regular order, there is not many forms to choose among, nor much difficulty to be found in discriminating them all from one another, even when several occur upon one crystal. The following table will, however, probably afford assistance in discriminating a variety of forms such as can co-exist on the same crystal. The points attended to in the table are these: 1.) The predominant combination, 2.) the number and inclinations of the planes that replace the edges of the predominant form, and 3.) the number and inclination of the planes that replace the angles of the predominant form. Where the edges and angles are of different kinds, this difference is noted. No respect is paid to hemihedral subordinate forms. No double replace- ments are attended to, because it would make the table too long. The reader will therefore observe, that the cube with 3 planes replacing each edge, is P,M,T, mt, m_t, m + t. pm, p_m, p_f.m, pt,p_t, p_^_t, and that the rhombic dodecahedron with 3 planes replacing each edge, is MT,PM,PT, 3p_mt, 6p_mt_j_. The table contains only such combinations as are of frequent occur- rence in the mineral kingdom. " Among all the occurring forms of the octahedral system of crystal- lisation, homohedral and hemihedral, the most important are the following: The Octahedron = PMT. The Cube = P,M,T. The Rhombic Dodecahedron = MT. PM, PT. The Icositessarahedron ^ = 3PiMT. The Tetrahedron = |PMT. The Pentagonal Dodecahedron J... = M|T. P|M, Pf T. " These forms occur more frequently than any others ; they very often form complete isolated crystals, and when they occur in combination, it is their faces which predominate. As this is not the case with the PRINCIPLES OF CRYSTALLOGRAPHY. 231 other forms, they are of far less importance." ROSE, Elemente der Krystallographie, p. 58. THE OCTAHEDRON predominant. PMT. Angles replaced by : 1 tangent plane = p,m,t. 4 planes, inclining on the edges = m_t, m+t. p_m, p+m, p_t, p+t. 4 planes, inclining on the planes = 3p_mt. 2 planes, inclining on the edges = m_t. p_m, p + t. 8 planes, inclining obliquely, partly on the edges and partly on the planes = 6p_mt+. Edges replaced by : 1 tangent plane = mt. pm, pt. 2 planes, inclining on the planes = 3p+mt THE CUBE predominant. P,M,T. Angles replaced by : 1 tangent plane = pmt. 3 planes, inclining on the planes = 3p_mt. 3 planes, inclining on the edges = 3p_f_mt. 3 planes, inclining obliquely, partly on the planes and partly on the edges= 3p_mt_|_. 6 planes, inclining obliquely, partly on the planes and partly on the edges = 6p_mt+. Edges replaced by: 1 tangent plane = mt. pm, pt. 1 plane with different inclination at each side = m_t. p_m, p + t. 2 planes = m_t, m + t. p_m, p + m, p_t, p+t. THE RHOMBIC DODECAHEDRON predominant. MT. PM, PT. Unipolar angles replaced by : 1 tangent plane = p,m,t. 4 planes, inclining on the planes = m_t, m+t. p_m, p+m, p_t, p+t. 4 planes, inclining on the edges = 3p_mt. 8 planes, inclining obliquely = 6p_mt+. Tripolar Angles replaced by : 1 tangent plane == pmt. 3 planes, inclining on the edges = 3p_mt. 3 planes, inclining on the planes = 3p+mt. 6 planes, inclining obliquely =: 6p_mt+. Edges replaced by: 1 plane = 3p_mt. 2 planes = 6p_mt+. THE ICOSITESSARAHEDRON predominant. 3P_MT. Unipolar angles replaced by : 1 tangent plane = p,m,t. 4 planes inclining on the edges = m_t, m + t. p_m, p+m, p_t, p+t. 232 PRINCIPLES OF CRYSTALLOGRAPHY. Tripolar angles replaced by 1 plane = pmt. Bipolar angles replaced by 1 plane = nit. pm, pt. Long edges replaced by 1 plane == m_t, m + t. p_m, p + m, p_t, p + t. THE TRIAKISOCTAHEDRON predominant. 3P_|_MT. Tripolar angles replaced by 1 plane = pmt. Long edges replaced by 1 plane = mt. pm, pt. Unipolar angles replaced by 1 plane = p,m,t. THE TETRAKISHEXAHEDRON predominant. M_T, M + T. P_M, P + M,P_T, P + T. Unipolar angles replaced by 1 plane = p,m,t. Long edges replaced by 1 plane = mt. pm, pt. Tripolar angles replaced by 1 plane = pmt. THE HEXAKISOCTAHEDRON predominant. 6P_MT+. Unipolar angles replaced by 1 plane = p,m,t. Bipolar angles replaced by 1 plane = mt. pm, pt. Tripolar angles replaced by 1 plane = pmt. THE TETRAHEDRON predominant. ^PMT. Edges replaced by: 1 tangent plane = p,m,t. 2 planes = ^(3p_mt) Znw. Angles replaced by : 1 plane = ^pmt Zne. 3 planes, inclining on the planes = mt. pm, pt. 3 planes, inclining on the edges = J(3p_mt) Zne. 6 planes = (6p_mt + ). THE HEMIICOSITESSARAHEDRON predominant. ^(3P_MT). Obtuse angles replaced by 1 plane = ipmt. Acute angles replaced by: 3 planes, inclining on the short edges mt. pm, pt. 6 planes, inclining on the planes = |(6p_mt + ). Short edges replaced by 1 plane = i(3p + mt). THE HEMIHEXAKISOCTAHEDRON WITH PARALLEL FACES predominant. 3P_MT + . Unipolar angles replaced by 1 plane = p,m,t. 2 planes, inclining on the long edges = m_t. p_m, p+t. 4 planes = 3p_mt_{_. THE PENTAGONAL DODECAHEDRON predominant. M_T. P_M, P+T. Long edges replaced by 1 plane = p,m,t. Tripolar angles replaced by 1 plane = pmt. 3 planes = 3p_mt + . Bipolar angles replaced by 1 plane = mt. pm, pt. "* PRINCIPLES OF CRYSTALLOGRAPHY. 233 471. I have paid no attention to hemihedral replacing forms in the above table, because their planes replace the angles and edges of other forms precisely as do the planes of the corresponding homohedral forms. The reader has only to remember that whenever a hemihedral subord- inate form is present, the predominant combination is affected by its replacements only at four tripolar angles instead of eight, which four angles are invariably either those at the poles Znw Zse Nne Nsw, or at the poles Zne Zsw Nnw Nse. There are in all but four varieties of hemihedral combinations, namely, |pmt; (3p_mt); |(3p_|_mt); |(6p_mt_}_), and these all contain hemioctahedrons, and, therefore, only affect alter- nate octants of a crystal. The pentagonal dodecahedron, and hemihexa- kisoctahedron with parallel faces, do not, as I have shown, 281, properly belong to the hemihedral combinations ; and they cannot, as I shall now proceed to show, cause the crystallographer any perplexity. " Any number of holohedral forms may occur in combination with each other, and with any hemihedral forms with inclined faces, or with any hemihedral forms with parallel faces. IT is SAID, that hemihedral forms with inclined faces have never been observed in combination with hemihedral forms with parallel faces" MILLER, Treatise on Crystal- lography) p. 24. " The different hemihedral forms with inclined faces can occur in combination with one another, as can also those with parallel faces ; and forms out of either class can occur in combination with homohedral forms. The combinations already cited contain many examples of this character. But hemihedral forms with inclined faces have never been observed in combination with hemihedral forms with parallel faces, although WE CAN SEE NO REASON WHY THIS SHOULD BE so." ROSE, Elemente der Krystallographie, p. 58. If you refer to ROSE'S classification of minerals, contained in Part II. of this work, you will find that " the minerals whose crystals present hemihedral forms with parallel faces are distinguished by one star *, and those which present hemihedral forms with inclined faces, by two stars**, page 2 ; and if, farther, you examine the crystals belonging to these minerals, pages 15 to 32, you will observe that every mineral which sometimes presents either M_T. P_M, P+T, or 3P_MT + , never presents any one of the four hemihedral combinations, and vice versa, every mineral which sometimes presents any one of the four hemihedral com- binations, never presents either M_T. P_M, P+T, or 3P_MT_)_. In other words, the presence of either M_T. P_M, P_|_T, or 3P_MT + upon a mineral, proves the mineral to be one of those which presents no kind of hemihedral forms, while the presence of any of the four hemihedral combinations is a guarantee that the mineral does not include M_T. P_M, P+T, or 3P_MT + among its possible combinations. The ques- tion, why any given mineral produces only homohedral forms ? belongs to the theory of crystallisation, and need not be discussed here. I have speculated sufficiently regarding it in SECTION XL In the meantime, we know, from observation, that certain minerals present hemihedral 2H 234 PRINCIPLES OF CRYSTALLOGRAPHY. - forms, and that certain others do not. This simple fact contains an answer to the doubt expressed or implied in the quotations from MILLER and ROSE. The so-called hemihedral forms with parallel faces, are in fact homohedral forms, and the minerals which present them are found by experience to be of that class which never present any hemihedral form whatever. Hence the reason why hemihedral forms with inclined faces never occur in combination with hemihedral forms with parallel faces, is, that the two kinds of forms characterise minerals of entirely different nature. It cannot be said, that I am taking advantage of merely new definitions of the words form and combination, in order to clear up this difficulty; for it must be remembered, that I have shown every one of the four kinds of hemihedral combinations to be produced by the partial intersection of eidogens, or by the intersection of half rhombuses, whereas the homohedral combinations, including M_T. P_M, P_j_T, and 3P_MT + in the number, are all produced by the intersection of eidogens or complete rhombic prisms. This is a tangible ground for the distinction between hemihedral and homohedral combinations. With the expression of these views, I dismiss the Octahedral System of Crystallisation, in the account of which, I think, I have clearly esta- blished the separate identity of nine homohedral and four hemihedral combinations, and shown how the combinations of these varieties with one another can be readily analysed, and by what symbols they can be intelligibly and conveniently described. II. THE PYRAMIDAL SYSTEM OF CRYSTALLISATION. 472. The character of the Forms belonging to this system, as given by ROSE, is this: 'They have three axes, which are all placed at right angles to one another, but of which two are equal and different from the third. Therefore, the AXES = p x m a t% including p!_m a t a and p4.m a t a . ROSE'S enumeration of the Forms belonging to this system of crystal- lisation, is as follows : A. Homohedral Forms: 1. The Quadratic Octahedron: first position, ......... = P X MT. Model 12.] second position, ...... = P X M, P X T. Model 13. 2. The Horizontal Planes, ......... = P. 3. The Quadratic Prism : first position, ......... = MT. Model . .^ J second position, ...... = M,T. Model 3 4. Model 22-.* 5. The Eight-sided Prism ........ = M_T. M + T. * The Dioctahedron, or Eight-sided Pyramid, would be represented by Model 22, pro- vided the upper pyramid PMT was away, and the planes of PM_T and PMT_, were continued from the equator in both directions, till they met in eight-faced solid angles at the poles Z and N. PRINCIPLES OF CRYSTALLOGRAPHY. 235 B. Hemihedral Forms : 1. The Tetrahedron = 1P X MT. 2. The Hemi-dioctahedron, = K P * M - T > PJM+T). ROSE'S Catalogue of the Minerals that belong to the Pyramidal Sys- tem, is given in Part II., pages 3 12. A synopsis of the forms and principal combinations belonging to the system is given at pages 32, 33. A symbolic catalogue of the forms and combinations presented by the crystals of the minerals of this system, is given at pages 33 43. The AXES of every natural crystal or combination belonging to this system, are p|m a t a ; that is to say, pi m a t a , or p+ m a t a , but not p a m a t a . See 340, No. 2), and Part II. page 32. The FORMS of which a crystal or combination is composed, may be either equiaxed or unequiaxed. The latter kinds generally prevail, but the former are not excluded, although they never produce an equiaxed combination. ROSE designates the axes of the crystals of this system as follows: p a by c, m a by , t a by ; whereas, in the octahedral system, all the three axes were designated by the term a, and in the prismatic system they are designated byc,a,&. Hence, p a is sometimes ,and some- times c, and t a is sometimes a and sometimes b. It appears to me, that much is lost and nothing gained by altering the names of the axes, either in this' or any other system. I shall, therefore, always call them p a m a t a . This will enable me to describe the forms belonging to the pyramidal and the other four systems of crystallisation briefly and yet distinctly, but it will oblige me to pay little attention to Rose's descriptions and illustrations, many of which are rendered tedious by two injudicious principles of the German system of crystallography, one of which is the above-mentioned change in the designation of the axes in different sys- tems; the other is the assumption of fundamental forms, and the consequent abandonment of a measure of unity for the value of the axes. See 479. To point out the special evils which flow from these two injudicious principles, would take up too much room, and needlessly interfere with the description of the system. A. Homohedral forms of the Pyramidal System. 1. THE QUADRATIC OCTAHEDRON. PJMT, Model 12. P X M,P X T, Model 13. 473. The quadratic octahedron is a pyramid with a square base, similar to the base of the regular octahedron, but the principal axis of which is either greater or less than the principal axis of the regular octa- hedron. When the principal axis is greater than that of the regular octahedron, the result is an ACUTE quadratic octahedron, similar to Model 13; but when it is less than that of the regular octahedron, the result is an OBTUSE quadratic octahedron, similar to Model 12. 474. The quadratic octahedrons assume two different positions on the crystals of this system, which positions ROSE calls the first position and second position. As the octahedrons which occupy these two different positions require different symbols, I shall treat of them separately. a.) Octahedrons of the first position. These are the octahedrons whose planes occur in the north-east zone and north-west zone, and whose oblique terminal edges form the boundary of the north meridian 236 PRINCIPLES OF CRYSTALLOGRAPHY. and the east meridian. See the coloured marks on Model 12. The octahedrons of this class require the symbol PJV1T; or if acute, the symbol P + MT, and if obtuse, the symbol PJMT. b.) Octahedrons of the second position* These octahedrons have their planes on the north zone and the east zone, and their oblique terminal edges form the boundary of the north-east meridian and north-west meridian. See Model 13. They consist of a rhombic form of the east, zone, and an equal and similar rhombic form of the north zone. Hence,' they require the symbol P X M, P X T ; or, if acute, the symbol P + M, T + T, and if obtuse, the symbol P_M, P_T. 475. Simultaneous occurrence of Octahedrons of the two positions on one crystal. These two kinds of octahedrons often occur together on one crystal, when they necessarily present planes on the north zone, the east zone, the north-east zone and the north-west zone. The planes upon the north and east zones, or those upon the other two zones, are either subordinate or predominant, according to the relative size of the two octahedrons, P X MT and P X M, P X T. The annexed diagram shows the relative positions of the equators of the octahedrons which belong to these two classes. The lines m a and t a represent the two equatorial axes, and the letters n e s w, the poles of these axes. Put the square marked No. 1 , equal to the base of an octa- hedron, whose axes p a is equal to the diagonal of the square, or to twice the semi-axis c to m a . The symbol for this octahedron will be PMT, equal to the regular octahedron. Let square 2 be an octahedron with the same axis p a . This will require the symbol PM, PT. The axes are again all equal, but the planes in this case belong to the north and east zones, whereas in the first case they belonged to the octahedral zones. With the same axis p a , take square 3 for the base of the octahedron. PRINCIPLES OF CRYSTALLOGRAPHY. 237 In this case, the equatorial axes are doubled in length, and the planes belong to the octahedral zones ; the symbol is consequently PM 2 T 2 , or better expressed P^MT. With the same axis p% take square 4 for the base of the octahedron. In this case, the equatorial axes are doubled in length as before, but the planes belong to the north zone and east zone ; consequently, the symbol becomes PM 2 , PT 2 ; or P^M, P|T. With the same axis p a , take square 5 for the base of an octahedron, having its planes on the octahedral zones. In this case the equatorial axes are trebled, and the form requires the symbol PM 3 T 3 , or PJMT. In this manner you may produce an unlimited series of octahedral forms, containing an octahedron of the first position and of the second position alternately : the octahedrons of the two kinds differing essen- tially in this, that with the same axes, those which have the symbol P X M, P X T, contain twice the equatorial base of those which have the symbol P X MT. Thus, the combination PM, PT, contains twice the base of the form PMT, although the axes of both of them are p a m a t% and the north and east meridians are precisely equal. Example of another series of Octahedrons. Let axis p a be equal to a side of square 4, which is twice the length of an axis of square 1. Then proceed as before, and with the given axis produce the following series of octahedrons : With square 1, you have P 2 MT 2, P 2 M, P 2 T 3, PMT 4, _ PM, PT 5, PfMT As in this "example the vertical axis p a , assumed to be common to all the bases, is made equal to the axes of squares 3 and 4-, the two smaller squares, 1 and 2, present examples of acute octahedrons. If we suppose the existence of a series of squares yet smaller than No. 1, we of course provide for so many octahedrons still more acute than P 2 MT. The next smaller square, for example, would produce the octahedron P 4 M, P 4 T ; the next P 4 MT, the next P 3 M, P 8 T, the next P 8 MT, and so on in regu- lar order, provided you take no intermediate squares, such as that marked No. 5, but only such as are each half the size of the next in order above it. It follows, from these considerations, that every octahedron of i\ie first position can be denoted by the symbol P X MT, and every octahedron of the second position by the symbol P X M, P X T ; and that in both of these symbols, the algebraic index x of any given form can be replaced by an arithmetical index, when the relation is known of axis p a to axis m a or t a . 476. Combinations of Octahedrons which replace each others edges. It is evident, on an examination of the diagram in 475, that the octahe- drons PMT and PM, PT, cannot appear on the same crystal, if the base 238 PRINCIPLES OF CRYSTALLOGRAPHY. of the latter is precisely twice the size of the base of the former, because the planes of PM, PT, lye exactly in the same position, and at the same distance from the axes, as the edges of PMT ; yet, we find by the com- bination of the regular octahedron PMT, with the rhombic dodecahedron MT. PM, PT, that the forms PMT and PM, PT, do combine with one another. The solution of this difficulty is found in the fact, that, while the axes of the two combining forms continue relatively the same, the forms differ absolutely in size. Hence the form PMT can have its oblique edges replaced by the combination PM, PT if the base of the latter is ever so little smaller than twice the size of the base of the former ; and the com- bination PM, PT, can have its oblique edges replaced by the form PMT, if the base of the latter is anything more than half, or anything less than twice the extent of the base of the former. Thus, we have Model 64, mt. pm, pt, PMT, in which the axes of mt. pm, pt are p a m a t a , and those of PMT are the same, but in which the base of the octahedron, pin, pt, is more than half, but less than twice, the extent of the base of the co-exist- ing octahedron PMT. A similar example is afforded by Model 77, con- taining the forms, p. pm, pt, PMT. 477. A word or two may be said respecting the indices of the symbols of the octahedrons that replace one another's edges. If the edges of the form P|MT are replaced by tangent planes, these planes must have the same relations to p a m a t a as the planes of the form P|MT. Thus, if square 3 is the base of P|MT, then square 4 will be the base of the re- placing octahedron, and will require the symbol P|M, P|T. But if square 2 is the base of the combination whose edges are replaced, and this combination bears the symbol P|M, P|T, then the form which re- places the edges of P|M, P|T, must have a base corresponding to square 3, the equatorial axes of which are twice the length of the axes of square 2, while the axes p a of both is the same. Hence the symbol of the re- placing octahedron having a base like square 3, must be PjMT. 478. Every Mineral of the Pyramidal System has belonging to it, a double series of Quadratic Octahedrons. As we cannot fix a limit to the number of forms which may occur in any given zone, we must admit that every different mineral may have belonging to it a series of octahedrons of each of the two positions, all the octahedrons of one position having the symbol P X M, P X T, and all those of the other position, the symbol P X MT. The indices of these symbols may be either _j_ or _. The value changes with every mineral, and with every different octahedron of the same mineral. Yet the indices of every octahedron of the same mineral have a very simple relation to one another. See 826. We cannot fix a theoretical limit to the possible variety of indices, yet practically, the variety is found to be very small, as I have shown at Part II. page 32. 479. Rule according to which the Octahedrons of a given Mineral arc ascribed to the north and east zones, or to the north-cast and north-west zones. PRINCIPLES OF CRYSTALLOGRAPHY. 239 a.) " In order to examine the relations to one another of the octahe- drons of a given mineral, they are all made to depend upon a chosen variety, which is called the Fundamental Form or Principal Octahedron. b.) " Which, among all the octahedrons that occur, is chosen for this purpose, is quite immaterial ; but we generally take that which occurs most frequently, or that whose planes commonly predominate in the combinations, or that to which all the other forms bear the simplest rela- tions. No more definite rule than this can be given for the choice of the fundamental form. c.) " The designation of the different quadratic octahedrons is then as follows : The fundamental form (a : a : c), Forms of the first position (a : a : mc\ Forms of the second position (a : oo a : me), in which signs, m is always a simple, whole, or fractional number, [and c the name of the vertical axis]." ROSE, Elemente der Krystallographie, p. 63. 480. It will be learnt from this quotation, that in the symbols given by ROSE For the octahedrons of this system, there is no measure of unity retained. Hence, the symbol (a : a : c) serves equally to designate the forms represented by Model 12. PfMT, the fundamental form of Zircon. Model 13. Pf M, PT, the fundamental form of Anatase. And the means by which the symbol (a : a : c) is made to discrimi- nate one fundamental form from another, is the accompanying register of the measurements of the angles of each fundamental form. This is a capital defect, and which I think exists in all the various editions of the German system of crystallography. There is no way of showing the relation of the octahedrons to one another so easy and distinct, as to mark those of different zones by different symbols, and to state the rela- tive lengths of their axes in figures. Next, observe, that while the fundamental form, once chosen, is placed in the first position, and all the other octahedrons are made to depend upon it, ROSE declares that no definite rule can be given for the choice of this fundamental form ; and that, in fact, the choice is of no moment. I point out this declaration for the purpose of remarking, that it is a mis- taken principle, and that on the contrary as much care must be taken in placing the forms of the pyramidal system as in placing those of the octa- hedral system ; and this I think a point of sufficient importance to merit a full explanation. If we take the combination P,M,T,mt. pm,pt, PMT, Model 64, and place the planes of PMT on the north and east zone, then, the planes of mt. pm, pt, fall of course into the octahedral zones, for the oblique planes of the octahedron and those of the dodecahedron, occupy respectively what are called i\\e first and second positions of the octahedrons of the pyramidal system. If, after thus changing the position of Model 64, we 240 PRINCIPLES OF CRYSTALLOGRAPHY. attempt to describe its forms in accurate" symbols, we obtain the follow- ing results : m,t PVJflfjpM, PVWT, pj-^mt. These are awkward indices, but if we shorten them to m,t.PVM, PVT,pio mt , the angle intimated across an edge of P ] 7 M will be 110 1' instead of 109 28', which includes an error of half a degree. This error is avoided by turning the crystal 45 horizontally, and throwing the planes into their usual positions which require the symbols, mt. pm, pt, PMT. Hence, the reason why the regular octahedron is called PMT, and not P X M, P X T, is simply, because the first is very convenient, and the last very inconvenient. The case is precisely the same with the two kinds of octahedrons of the pyramidal system. If you take all the octahedrons of any given mineral of this system, and give them the shortest and most convenient symbols that their measurements admit ; and if you then throw their planes from the first position into the second, you will find that their symbols cannot be completed by indices equally simple with those that were used before. Hence, the positions of the octahedrons of any given mineral are to be regulated by the indices demanded to express the relation of their axes. This is a case in which the principle of expediency is in fact the soundest philosophy. 2. THE HORIZONTAL PLANES. P. 481. It seems needless to say more of this form, than that it is the form P, consisting of two horizontal planes. Rose's symbol for it is (oo a : oo a : c). 3. THE QUADRATIC PRISMS. MT and M,T. 482. The quadratic prisms are four-sided vertical prisms, having a square base. There are two kinds, one of them containing the forms M and T, the other, the form MT. Rose designates the latter (a : a : oo c), or the prism of the first position ; the former, (a : GO a : oo c), or the prism of the second position. The prism of the first position = MT, is that which replaces the horizontal edges of the pyramids of the first posi- tion = P X MT. The prism of the second position = M,T, is that which replaces the horizontal edges of the pyramids of the second position = P X M, P X T. The first of these prisms is shown by Model 2, P_,MT. The second by Model 3, P_f.,M,T. The two prisms often occur together, as shown in Model 4, P+,M,T, mt. or P+,m,t, MT. 4. THE DIOCTAHEDRON, OR EIGHT- SIDED PYRAMID. P X M_T, P X M + T. 483. This combination would be represented by Model 22, P_MT, PM_T, PMT_, provided the four upper and four lower planes consti- PRINCIPLES OF CRYSTALLOGRAPHY. 241 luting the form P_MT were away, and the planes of the other two octahedrons, PM_T, PMT_, were continued both ways from the equa- tor till they met in eight-faced solid angles at the poles Z and N. The combination would then consist of sixteen scalene triangles. It is a com- bination that never produces a complete crystal, and only in one or two cases appears predominant ; but it occurs pretty frequently in combina- tion with the quadratic pyramids and quadratic prisms, exhibiting four pair of two planes on the zenith end of the crystal, and a similar number of planes on the nadir end. 484. The dioctahedron is a combination of two similar and equal octahedrons with rhombic bases, one of them having the equatorial rela- tion of m t a , and the other the relation of mj|- t a . The following diagram shows at once the equator of the dioctahedron, and of its two component pyramids. Assume, as an example, the existence of the dioctahedron, P 3 M,T 2 , PgMgT!. Then the rhombus n t a s t a , will be the base of the octahedron P 3 M,T 2 ; and the rhombus em a w m a will be the base of the octahedron PsMaTj. These two forms cut one another at the points aiEo, and the base or equator of the resulting combination is the octagon noeasiwE. The eight faces of the pyramid which rests upon this equator, incline upon the eight edges n o, o e, nE, Ew, &c. The solid angles of the dioctahedron are therefore of three different kinds, and are situated as follows: one kind at the poles Z, N; another kind at the poles n, e, w, s, and a third kind at the poles nw, ne, se, sw. 2i 242 PRINCIPLES OF CRYSTALLOGRAPHY. The edges of the dioctahedron are also of three different kinds: Eight edges of one kind occur round the equator; eight of a second kind on the north and east meridians ; and eight of a third kind on the north- east and north-west meridians. 5. THE EIGHT-SIDED PRISM. M_T, M + T. Varieties of this Combination, accordiny to MILLER : M|T,MfT. MiT, Mf-T. MJT, Mf T. M|T,M|T. , M T. 485. The equator of the eight-sided prism has the same characters as the equator of the right-sided pyramid or dioctahedron. Refer to the diagram, page 241. The rhombus nt a st a is the equator of the rhombic prism M|T. The rhombus m a em a w is the equator of the rhombic prism Mf T. The combination of these two rhombic prisms produces an equiaxed eight-sided prism, the equator of which is the octagon, shown by the thick lines noeasiwE in the diagram. As the axes m a and t a of the two separate rhombic prisms differ, so do the external angles of the combination ; but there are always four angles of one value, and four of another value : the angles at n e s w being all similar, and those at o a i E also similar. A single angle of one kind added to a single angle of the other kind, make together an angle of 270. See 399. Hence, if the angle across the north pole is called n ; and that across the north- west pole is called E ; then, E is 270 * n ; and n is 270 E. The eight-sided prisms of most frequent occurrence are MJT, Mf T. and MT, M^T. In general, the eight-sided prisms occur subordinately, replacing the edges of the prisms M,T, or MT, or M,T, mt. They very rarely occur predominant, or without the square prisms. 486. It is easy to see in the diagram, page 241, the positions which the planes of M_T, M + T, occupy in reference to the two square prisms. Let the square oai E be the equator of the prism M,T. Then if mjt, mft, replace the edges of M,T, the planes of mjt appear on the north and south sides of the combination, and near the bipolar normals ; while the planes of mft appear on the east and west sides of the combination, yet also near the bipolar normals. Again, let the line AB represent a side of the square prism MT. Then, the planes of mjt will appear attached to the north and south poles of the prism MT, and the planes of mf t to the east and west poles. Finally, let the square prism mt replace the corners of M,T, and produce the combination M,T, mt, as is shown at the corner marked v. Then, the planes of mjt replace the edges between mt and M, and those of mf t the edges between mt and T, as will be perceived if the octagon in the diagram be supposed to become small enough to touch the angles near v and r. PKINCirLES OF CRYSTALLOGRAPHY. 243 B. Hemihedral Forms of the Pyramidal System. 1. THE TETRAHEDRON. 487. This is a hemihedral form with inclined faces, bearing the same relation to the octahedron, P X MT, that the regular tetrahedron, iPMT, bears to the regular octahedron, PMT. It differs from the regular tetrahedron in these particulars : its axes are p*m a t a ; its edges at Z and N have not the same angle as its vertical edges; its planes are isosceles triangles; its north meridian and east meridian are rhombuses, while its equator is a square. The angle across the edges at Z and N is the complement of the angle across the equator of its corresponding homohedral octahedron. The angle across the lateral edges is the complement of the angle across the terminal edges of the corresponding homohedral octahedron. 2. THE HEMI-DIOCTAHEDRON. 488. The hemihedral variety of the dioctahedron is a parallel-faced hemihedral form, of the same character as the hemihedral forms described in 272. Refer to the diagram in page 241. Conceive the axis of a rhombic cutting prism to cross the axes m a t% in the direction of the line s t a . The four planes of such a prism would meet at the lines marked s i and o n. Conceive the axis of another rhombic cutting prism to cross the axes m a t a in the direction of the line m a w. The four planes of such a prism would meet at the lines ea and Ea. The eight planes thus pro- duced constitute the hemi-dioctahedron. This combination never con- stitutes a complete crystal, but only appears subordinately upon the angles of other combinations. There may be another hemi-dioctahedron in the alternate octants of the dioctahedron. The edges of the planes of this second variety would of course meet at the alternate equatorial lines iw, eo, as, ne. The different kinds of the hemi-dioctahedron are easily distinguished by means of the signs denoting the polaric positions of their planes. At present, they are forms of rare occurrence and little importance. 489. ZONES or THE PYRAMIDAL SYSTEM. The equatorial zone embraces the forms M, M_T, MT, M + T, T, and therefore includes all the different prisms of this system. The north zone embraces the form, P, P_M, PM, P + M, M. The east zone embraces the forms, P, P_T, PT, P + T, T. The two zones to- gether, therefore, include all the square-based octahedrons of the second position, the horizontal planes P, and the square prism of the second position. The north-east and north-west zones embrace the forms P, P_MT, PMT, P + MT, MT. They, therefore, include all the square-based octa- hedrons of the first position, the horizontal planes P, and the square prism of the first position. 244 PRINCIPLES OF CRYSTALLOGRAPHY. The onljrform of the pyramidal system that is not crossed by the above zones, is the dioctahedron, the planes of all the varieties of which fall in the open triangular spaces situated betwixt the equator and the intersections of the four meridians. Hence, a plane found in the division Z 2 n 2 w will belong to the form P X M_T, while a plane found in the adjoining division Z 2 nw 2 will belong to the form P X M + T. If a combination of the pyramidal system be divided into octants by the equator and the north and east meridians, and if the octants be four of one kind and four of another kind, the combination contains a tetra- hedron. 285. If a combination of this system be divided into vertical octants by the intersection of the four meridians, and if the octants be four of one kind and four of another kind, the combination contains a hemi-dioctahedron. Mathematical Properties of the Forms of the Pyramidal System. QUADRATIC OCTAHEDRONS OF THE FIRST POSITION. 490. Let the annexed figure represent the octant of a square-based octahedron of the first position, having the properties which are described in 300, &c., as being common to right-angled solid tri- angles so formed. N 491. PROBLEM. Given, Model 12, P_MT, with the angle across the equator, namely, the inclination of plane Znw on plane Nnw = 84 20' (Zircon, ROSE) ; required, the value of the index _ in the symbol. Take a right-angled solid triangle with pole n for its vertex. Then, half the angle across the equator = = 42 10', will be angle A; the plane angle cnw = 45, will be side b; and the angle across the edge M, will be angle C = 90. With these data, you can find side a of the solid triangle, which is equal to the plane angle Znc in the diagram, the tangent of which side is the required value of axis p a , when axis m a = axis t a , is unity. Given, A = 42 10'; b = 45; to find, a. Formula 7. log tan a = log tan A + log sin b 10. log tan A = 42 10' = 9-9570 + log sin b = 45 = 9.8495 log tan a = 32 S8J' = 9.8065 The natural tangent of this product is .6405, the nearest vulgar frac- tion to which is JJ. This is the value of the index _, and it produces the symbol Pf MT. I have, however, called the combination PMT, but this PRINCIPLES OF CRYSTALLOGRAPHY. 245 is perhaps taking too much liberty in the abridgment of the sign, since PfMT would indicate the cotangent .6667, which differs 4 per cent, from that found by calculation. HADY'S measurement across the equatorial edge of the octahedron is 83 38'. Let us see what symbol this requires. 83 38' -~ 2 = 41 49'. log tan 41 49' = 9.9516 + log sin 45 = 9.8495 log tan 32 19' = 9.8011 The cotangent of 32 19' is .6326 or nearly |f, which is nearer to than to . These statements show that we possess the power to describe these combinations either by^short indices, which indicate their angles approxi- mately, or else by larger indices, which show the precise angles quoted by any given authority. 492. PROBLEM. Given, Model 12, P_MT, with the angle across the equator = 84 20' ; required, the angle across a terminal edge. ROSE and PHILLIPS both call the angle across the equatorial edge of this crystal of Zircon, 84 20', while the angle across a terminal edge, is called by ROSE, 123 19', and by PHILLIPS, 123 15'. If the same triangle and the same quantities are taken as in the last problem, then half the angle across a terminal edge of the model, will be angle B of the given solid triangle. Given, A = 42 10'; b = 45; to find, B. Formula 8. log cos B = log cos b -j- log sin A 10. log cos b = 45 = 9.8495 + log sin A = 42 10' = 9.8269 log cos B = 61 39J' = 9.6764 Twice this product, or 61 39i' X 2 = 123 19', is the required angle across a terminal edge of the model, and this result agrees with the angle quoted by ROSE. 493. PROBLEM. Given, Model 12, P_MT, with the angle across a ter- minal edge = 123 19' ; required, the angle across the equatorial edge. Take a right-angled triangle, consisting of an octant of the model, with pole n for its vertex. Then, with angle C = 90 ; angle B = ^-^' == 61 39^' > and side b = 45, which is one-fourth of the square equa- tor, seek for angle A, which is half the inclination across the equator, from plane Znw to plane Nnw. Given, B = 61 39' ; b = 45 ; to find, A. Formula 22. log sin A = log cos B + 1 log cos b. 10 + log cos B = 61 39J' = 19.6764 log cos b = 45 = 9-8495 log sin A = 42 10' = 9.8269 246 PRINCIPLES OF CRYSTALLOGRAPHY. Twice this product, or 42 10' x 2 = 84 20', is the required angle across an equatorial edge of the model. 494. PROBLEM. Given, Model 12, with the symbol Pf MT ; required, the angle across a terminal edge of the model. Seek in the table of indices, page 139 for the cotangent equivalent to , which is .6667. D The angle corresponding to this cotangent is 56 18|'. This is the inclination of a terminal edge of the form to axis p a . Now, take an octant of the form as a right-angled solid triangle, with pole Z for its vertex. You have then given, side a = 56 18V> and side b = 56 1 8 J'. Both these angles are alike, because the inclination of all the four oblique edges of a quadratic octahedron to axis p a is the same. With these data, you have to find angle A or angle B, either of which is half the required inclination across a terminal edge of the form. Given, a = 56 181'; b = 56 18'; to find, A. Formula 13. log tan A = log tan a 4- 10 log sin b. 10 + log tan a = 56 18' = 20.1760 log sin b = 56 181' = 9.9201 Ipg tan A = 60 69' = 10.2559 Twice this product, or 60 59' X 2 = 121 58', is the required angle across a terminal edge of the form. This product, however, is 1 21' too little, because the index does not express the relations of the axes with sufficient exactness. 495. PROBLEM. Given, Model 12, with the symbol PJf- MT ; required, the angle across a terminal edge of the model. Proceed exactly as in the last problem, but change the value of the quantities. The equivalent of J J- given in the Table of Indices, page 1 39, is .640. The corresponding angle is 57 23'. Then the equation is Given, a = 57 23' ; b = 57 23' ; 'to find, A. 10 + log tan 57 23' = 20.1939 log sin 57 23' = 9-9255 t log tan 61 401' = 10.2684 Twice this product, or 61 401' X 2 = 123 21', is the required angle across a terminal edge of the model, and this product agrees with the measured angle within two minutes; consequently, the symbol P^fMT expresses very exactly the form described by the measurements of ROSE and PHILLIPS, while the symbol PfMT does so only approximately. Through the remainder of this section, I shall indicate Model 12 by the term P^f- MT, in order that the reader may see clearly the nature of the calculations, and the relation of their products. 496. PROBLEM. Given, Model 12, P_MT, with the inclination of a plane to axis p a = 47 50'; required, the angle across a terminal edge. PRINCIPLES OF CRYSTALLOGRAPHY. 247 Assume the zenith pyramid of Model 12 to be divided into eight por- tions by the intersection of the four meridians. Take one of those eighths as a solid triangle, with pole Z for its vertex. Then you have given, angle C = 90 = inclination of an external plane to the north- west meridian ; A = 45 = an interior angle formed by the meeting of two adjacent meridians at p a ; and b = 47 50' = the given inclination of a plane to p a . With these data you have to find B, which is half the angle across a terminal edge. Given, A = 45; b 47 50'; to find B. Formula 8. log cos B log cos b -f log sin A 10. log cos b =2 47 50' = 9.8269 + log sin A = 45 = 9.8495 log cos B = 61 391' = 9.6764 Twice this product, or 61 39 X 2 = 123 19', is the required angle across a terminal edge of Model 12. 497. PROBLEM. Given, Model 12, P_MT, with the inclination of a terminal edge to axis p a = 5721 / ; required, the angle across a terminal edge of the model. When you have the inclination of one terminal edge to the principal axis, you have the inclination of all four. Take the inclination of two edges to p a as sides a and b of a solid triangle, each equal to 57 21', and then find angle A, as directed in problem 495. The following short Formula is useful in examining quadratic octa- hedrons. Let x be the inclination of a terminal edge to axis p a , and 2y the angle across a terminal edge, then cot y = cos x. Example: cos x = 57 21' = 9,7320 cot y = 61 39i' = 9.7320 Twice 61 39J', or 123 19', is the required angle across a terminal edge of the model. 495, 499. 498. PROBLEM. Given, Model 12, P_MT, with the angle across a terminal edge = 123 19'j required, the inclination of a plane to axis p a and to the equator. Take a solid triangle, consisting of an eighth part of the zenith pyra- mid of the model, divided in the manner described in 496, and having pole Z for its vertex. Then, you have given, C = 90; B = i~l = 61 -391'; A = 45; to find b, which is the required inclination of a plane to axis p a . Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 61 39i' = 19.6764 - log sin A = 45 = 9-8495 log cos B = 47 50' = 9.8269 248 PRINCIPLES OF CRYSTALLOGRAPHY. This product, 47 50', is the inclination of a plane to axis p a , and its complement = 42 10', is the inclination of a plane to the equator. 499. PROBLEM. Given, Model 12, P_MT, with the angle across a terminal edge = 123 19'; required, the inclination of the terminal edge to axis p*. Take an octant as a solid triangle, with pole Z for its vertex, and the following known parts: C = 90; A = i^J = 61 39i'; B = 61 39J'; and with these data, find a or b, either of which is the required incli- nation of the terminal edge to axis p a . The solution of this problem requires Formula 4 modified by 104; namely, cos a cot A. nat cot A = 61 39 9.7319 nat cos a = 57 21' = 9.7320 This product, 57 21', is the inclination of a terminal edge of Model 12 to axis p a . 500. PROBLEM. Given, Model 12, P*fMT, with the angle across a terminal edge = 123 19'; required, the plane angles of the external faces. a.) With the same solid triangle and the same given quantities as in problem, 499, and with the help of Formula 6, find c, Given, A = 61 39|'; B = 61 39^; to find, c. Formula 6. log cos c = log cot A + log cot B 10. log cot A = 61 39|' = 9.7319 + log cot B == 61 391' = 9.7319 log cos c = 73 5' = 9.4638 This product, 73 5', is the plane angle of a face at pole Z. b.) Take the triangle described in 496, with the following given quantities: A = 61 39i'; B == 45; C = 90; to find b, which is half the obtuse angle of a face at pole Z. Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 45 = 19.8495 log sin A = 61 391' = 9-9446 log cos b = 36 32V = 9.9049 Twice this product, or 36 32 \' x 2 = 73 5', is the plane angle of the external faces at pole Z. c.) If the obtuse plane angle at pole Z is called z, then the acute plane angles at the base of the pyramid will each be \ (180 z). Hence, 180 73 5' = 106 55'. And 1 -^- / = 53 27J'. This last product, 53 27 i', is the value of each of the plane angles of Model 12 at the poles news. PRINCIPLES OF CRYSTALLOGRAPHY. 24-9 d.) Check on this Calculation. Take the solid triangle described in 491, and with the quantities there given, find side c, which is the plane angle at pole n. Given, A = 42 10'; b = 45; to find, c. Formula 9- log tan c = log tan b -|- 10 log cos A. 10 + log tan b = 45 = 20..0000 log cos A = 42 10' = 9.8699 log tan c = 53 27 J' = 10.1301 QUADRATIC OCTAHEDRONS OF THE SECOND POSITION. Example: Model 13. PfM, PjT. 501. PROBLEM. Given, Model 13, with the symbol PJM, P|T; re- quired, the inclination of the planes to the equator and to axis p a . a.) Look in the Table of Indices, page 139, for the decimal fraction equivalent to |. You find it to be 2.500. This number is the tangent of the inclination of the planes to the equator = 68 12', and the cotan- gent of their inclination to axis p a = 21 48'. According to Haiiy, the inclination of the planes to the equator of the crystal of Anatase, which this model represents, is 137 ^ I0 = 68 35': according to Mohs, it is 36 22 = 68 IT. The latter quotation agrees very nearly with the index . 502. PROBLEM. Given, Model 13, P X M, P X T, with the angle across the equator =r= 136 24' ; required, the value of the index x in the symbol. Rule. The tangent of half the angle across the equator is the value of the index. Example: ! -^' = 68 12'. tan 2.5002 or f. 503. PROBLEM. Given, Model 13, PfM, PgT; required, the angle across the terminal edge of the model. Find by problem 501, the inclination of the planes to the axis p a . Call this a = 21 48'. Then take a solid triangle, containing an eighth part of the zenith pyramid of Model 13, with pole Z for its vertex. In this you have given, angle C = 90 = inclination of a plane to the north meridian ; B = 45 = interior angle formed by the intersection of the north and north-west meridian; a = 21 48' = inclination of a plane to axis p a . With these data, you can find A, which is half the desired angle across a terminal edge. Given, a = 21 48'; B = 45; to find, A. Formula 10. log cos A = log cos a + log sin B 10. log cos a = 21 48' = 9-9678 + log sin B = 45 = 9-8495 log cos A = 48 571' _. 9.3173 2 K 250 PRINCIPLES OF CRYSTALLOGRAPHY. Twice this product, or 48 57V X 2 = 97 55', is the required angle across a terminal edge of Model 13. 504. PROBLEM. Given, Model 13, P+M, P + T, with the angle across a terminal edge = 97 55' ; required, the inclination of a plane to the equator. Take an octant of the model as a solid triangle, with pole nw for its ver- tex. Then you have given, A = ^^-' = 48 571' = half the angle across a terminal edge ; a 45 = one-fourth of the equator, or the inclination of the north edge of the equator to the nw normal. With these data, you can find B, which is the required inclination of a plane of Model 13 to the equator. Given, A = 48 57J'; a = 45; to find, B. Formula 2. log sin B = log cos A + 10 log cos a.' 10 + log cos A = 48 57V = 19.8173 log cos a = 45 = 9.8495 log sin B = 68 12' = 9-9678 This product, 68 12', is the required inclination of the planes to the equation. Twice 68 12' = 136 24', is the inclination of a plane of the upper on a plane of the lower pyramid of the model across the equator. 505. PROBLEM. To transpose a quadratic octahedron from the first position to the second position, or from the second position to the first position. a.) Before determining upon the index of a quadratic octahedron, it is proper to ascertain what index it gives, both when reckoned as P X MT and as P X M, P X T. When the positions of a series of octahedrons of a given mineral have been previously fixed, we then perceive which member of the series we have in hand ; and when such a series has not been determined, we can, after calculating two indices for a crystal, take that which consists of the simplest numbers. 5.) The resolution of the following problems gives the necessary infor- mation for this purpose : Given, A = a?; B = 45; C = 90; to find, a. Given, A = x; B = 45; C = 90; to find, c. To resolve these problems, you take a right-angled solid triangle con- sisting of an eighth part of the upper pyramid of a quadratic octahedron, formed by the intersection of the four meridians, and having pole Z for its vertex. 496. Then A is half the angle across a terminal edge, which quantity varies with every octahedron ; B is an interior angle formed by the meeting of the north with the north-west meridian at axis p a ; C is the angle where a meridian cuts a plane. Hence, a is the inclination of a plane to axis p a , the cotangent of which inclination is the value of the index x in the symbol P X M, P X T; and c is the inclination of PRINCIPLES OF CRYSTALLOGRAPHY. 251 a terminal edge to axis p a , the cotangent of which inclination is the value of the index x in the symbol P X MT. The Formulae to be employed in resolving these problems are as follow : Formula 4. log cos a = log cos A + 10 log sin 45. Formula 6. log cos c = log cot A + log cot 45 10. c.) When you know the relation of the axes of an octahedron without calculation, as when the symbol P|M, P|T is given, and you want to know what must be substituted in the symbol P X MT in place of J when the position of the octahedron is changed, you need only multiply the equatorial axes of P X M, P X T by the natural secant of 45 = 1.4142136, to obtain the value of the equatorial axes of P X MT. Thus, 1.4142 x 2 = 2.8284, shows that the symbol P|M, PfT, must be changed to On the other hand, if P|;^MT is given, and the symbol is to be changed to P X M, P X T, you find the value of the equatorial axes of the latter by dividing the equatorial axes of the former by 1.4142. Thus, f;|f|| = 2. Hence, PJigggJMT becomes P|M, PfT. Again, change Pf MT into P X M, P X T. 2 f.J$Jg = 17.607. Hence, Pj JMT becomes Pjf jfM, PjfgT. But jf -*. 4 = JJ = |$, which gives P-J-JM, P-ff T. This seems as good a symbol for Model 12, as PLf-MT. But it is less accurate, since |& is the synonyme, see page 139, of .9091, which is the tangent of 42 16^'. This gives the inclination of a Zenith to a Nadir plane = 84 33' instead of 84 20', 491. Here a difference of 13' in the equatorial angle arises, partly from the division of 25 by 1.4142 instead of 1.4142136, and partly from abridging the product 17.607 to 17.6. Whenever the indices of the two symbols P X M, P X T, and P X MT, appear to be equally commodious, it is impossible to determine which is the preferable symbol, without examining a series of octahedrons of both positions belonging to the same mineral. 506. COMBINATIONS OF QUADRATIC OCTAHEDRONS WITH ONE ANOTHER. a.) In the same zones The middle octahedron PMT, may combine with the form p_mt, or with the form p_j_mt. The first, p_mt, replaces the apex of PMT, and produces four planes which incline on the planes of PMT, Model 14. The second, p+mt, bevels the equatorial edges of PMT. In the same way, PM, PT, is replaced at the summits by p_m, p_t, Model 14, and bevelled at the equatorial edges by p+m, p + t. b.) The predominant octahedron, PMT, with a subordinate octahedron, p x m, p x t. i. If the terminal edges of PMT are replaced by tangent planes, then the axes of p x m, p x t are unity, and the symbol is pm, pt. 252 PRINCIPLES OF CRYSTALLOGRAPHY. ii. If the summits of PMT are replaced by four planes that incline on the terminal edges, these indicate the combination, p_m, p_t. iii. If the equatorial angles of PMT are replaced by planes inclining on the terminal edges, these indicate the combination, p+m, p+t. c.) The predominant octahedron, PM, PT, with a subordinate octahe- dron, p x mt. i. If the terminal edges of PM, PT, are replaced by tangent planes, then axis p a of p x mt, will be the same as axis p a of PM, PT, while the two equatorial axes of p x mt will be twice as great as the two equatorial axes of PM, PT. ii. If the summits of PM, PT, are replaced by four planes that incline on the terminal edges, then axis p a of the form p x mt being unity, axes m a and t a of p x mt will be more than twice as great as axes m a and t a of the combination PM, PT. iii. If the equatorial angles of PM, PT, are replaced by planes that incline on the terminal edges, these indicate that if axis p a of p x mt is unity, then axes m a and t a are less than twice as great as axes m a and t a of PM, PT. There are, consequently, eight varieties of combination between two quadratic octahedrons. It requires but one measurement either across an equatorial or a terminal edge of each octahedron, to afford information sufficient to lead to its symbol and index. 507. COMBINATION OF THE QUADRATIC OCTAHEDRONS WITH THE HORIZONTAL PLANES P. P_. PM, Pf 1\ Model 76. Molybdate of Lead. p. pm, pt, PMT. Model 77. Copper Pyrites. These are incomplete prisms with incomplete pyramids. Class 5, Order 1, Genus 2, page 115, Part II. Analysis. The inclination of P upon a plane of a quadratic octahe- dron, is 90 -f- x > in which formula, x signifies the inclination of the plane of the octahedron to axis p a . 508. COMBINATION OF THE QUADRATIC PRISMS WITH THE HORI- ZONTAL PLANES P. P_,MT: orP,MT. Model 2. Rutile. P+,M,T: or P|,M,T. Model 3. Apophyllite. P+,M,T, mt. Model 4. Egeran. These combinations are complete prisms with a square equator. Class 1, Order 1, Genus 2, page 98, Part II. Analysis. The inclination of P, to M,T or MT is 90. The inclina- tion of M,T to the equatorial axes is 90, to the equatorial bipolar normals, 45. The inclination of MT to the equatorial axes is 45, to the equatorial bipolar normals, 90. Hence, M,T incline upon MT at an angle of 90 + 45 = 135. PRINCIPLES OF CRYSTALLOGRAPHY. 253 Very short prisms of this class = Pl,M,T, were formerly termed tables, or tabular crystals. 509- COMBINATION OF THE QUADRATIC PRISMS WITH THE QUAD- RATIC OCTAHEDRONS. MT. PfMT. Model 6t. Zircon. (M,T. P|M, PfT) x 2. Model 62. Oxide of Tin. M,T, mt. PfM, PfT. Model 59. Wernerite. M,T, mt. PfMT. Model 60. Zircon. These combinations are incomplete prisms with complete pyramids. Class 4, Order 1, Genus 2, page 111, Part II. Analysis. The inclination of a quadratic prism to a quadratic octa- hedron is 90 -f x, in which formula, x signifies the inclination of a plane of the quadratic octahedron to the equator, or half the angle across a horizontal edge of the octahedron. The prisms M,T and MT are transposed from the first position into the second, or from the second into the first, to suit the index of the octahedron. When the octahedron is measured and the index fixed, then the vertical planes are named accordingly. Upon this principle, the ver- tical or prismatic planes of Models 59 and 60 may be called M,T, mt, or m,t, MT, just as is most convenient to suit the terminating octahe- drons. This power of changing the description of the prism by a mere change in the kind of letter employed to designate it, is extremely con- venient. 510. COMBINATION OF THE QUADRATIC PRISMS WITH QUADRATIC OCTAHEDRONS AND THE HORIZONTAL PLANES. P| , M, T. p Jmt. Model 4 1 . Apophyllite. p+,m,t, MT. P|M, PIT. Model 42. Idocrase. These combinations are complete prisms with incomplete pyramids. Class 3, Order 1, Genus 2, page 107, Part II. Analysis See 508, 509. THE DIOCTAHEDRON. 511. The symbol of the dioctahedron is P X M_T, P X M + T, the index x in both of which forms is alike, since the dioctahedron contains two rhombic octahedrons in inverse positions as regards the equatorial axes. The value of the indices _ and + depends not only upon the relative dimensions of the two component octahedrons, but also upon the posi- tion assigned to the dioctahedron as respects the axes of the combina- tion to which it may belong. Occurring only as a subordinate form, its position is generally determined by that of the quadratic octahedrons with which it is found in combination ; and as we have assumed the power to give to all quadratic octahedrons an azimuthal change of posi- tion to the extent of 45 whenever we think fit to do so, it follows that we, in like manner, assume the power to give an equal azimuthal remove 254 PRINCIPLES OF CRYSTALLOGRAPHY. of 45 to the dioctahedrons by which the quadratic octahedrons may be modified. The consequences of this are easy to be seen, on examining the diagram in 484. If the dioctahedron is placed as shown in the figure, the line en represents the length of axis m a ; whereas, if the equator is moved 45azimuthally, the line cE comes into the position of the line en and becomes axis m a ; and since the lines en and cE always differ in length, the index showing the relation of m a to p a or t a also necessarily differs according to this change of position. In order to be able to calculate the ratios of the axes of the diocta- hedron, and to find its indices, we require the angle across two of the three kinds of edges which it presents externally, or if the measurement of only one edge can be procured, some other quantity must be had equivalent to the measurement of another edge, or to the value of one of the three axes of the form under investigation. I shall proceed to show in what manner the axes of the component forms of the dioctahedron can be calculated from any two measurements across the external edges. The example which I shall take is the dioc- tahedron of the mineral Zircon, the measurements across the edges of which are given by ROSE as follows : Across the equator = 127 29'. I edge = e =63 44J'. Across the n meridian = 147 3'. | edge * *'= 73 31^'. Across the nw meridian = 132 43'. i edge = w = 66 2 1|'. In each calculation, I shall show what the indices are to be on the two assumptions, that either the edge n or the edge w is ascribed to the north meridian. The instructions contained in these calculations, added to those already given respecting the dioctahedron, in 409, 410, with other problems contained in the article on the hexakisoctahedron, will, I hope, sufficiently elucidate the nature of this combination. 512. PROBLEM. Given," a dioctahedron, P X M_T, P x M_f_T, with an edge, e 63 44 J', and an edge, n, 73 31^'; required, the inclination of these two edges to axis m a , and the value of the indices x , _, + , in the symbol. a.) Take a right-angled Isolid triangle, with pole n for its vertex, and in which you have given, angle A 63 44 J', and angle B 73 3H'. Then angle C = 90 will be the inclination of the north meridian to the equator; side a will be the inclination of the edge n to axis m a , the co- tangent of which angle will give the length of m a when p a is unity ; and side b will be the inclination of the edge e to axis m a , the cotangent of which will give the length of m a when t a is unity. b.) Given, A = 63 44i'; B = 73 31'; to find, a. Formula 4. log cos a = log cos A + 10 < log sin B. 10 + log cos A =. 63 44J' = 16.6458 log sin B = 73 311' 9.9818 log cos a = 62 3U' = 9.6640 PRINCIPLES OF CRYSTALLOGRAPHY. 255 This product, 62 2H', is the inclination of the edge n to axis m a . cot 62 31 1' = .5200 = |f. That is to say, axis m a is to axis p a as 13 is to 25. c.) Given, A = 63 441'; B 73 3U'; to find, b. Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 73 31 F == 19.4527 log sin A = 63 441' = 9-9527 log cos b = 71 34' = 9.5000 This product, 71 34', is the inclination of the edge e to axis m a . cot 71 34' = .3333 = J. That is to say, axis m a is to axis t a as 1 is to 3, or if we make m a = 13, to bring the products of b.) and c.) into unison, the relation m a to t a 'is 13 to 39. d.) Hence, the relations of the three axes are, p a 25; m a 13; t a = 39. This relation gives the symbol Pf^M^JT. If we divide these fractions by 13, we obtain the symbol Pi~- ! MJT, which is very nearly PfMJT. But this symbol would only express the external angles of the form approximately, exactly as the external angles of the quadratic octa- hedron, Model 12, are expressed approximately by the symbol PMT. See 491. If one form of the dioctahedron is P a MiT 3 , or Pf MJT, the co-existing form is P 8 M S T,, or Pf Mf T. 513. PROBLEM. Given, a dioctahedron, P X M_T, P X M + T, with an edge, e, = 63 44 J', and an edge, n, = 66 21|' ; required, the inclination of these two edges to axis m a , and the value of the indices x , _, _j., in the symbol. a.) Proceed exactly as directed in 512 a.) only changing the value of angle B from 73 3H' to 66 21|'. The other quantities remain as they were. b.) Given, A = 63 44*'; B = 66 2H'; to find, a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 63 44| ' = 19.6458 log sin B = 66 21*' = 9.9619 log cos a = 61 1' = 9.6839 i This product, 61 7', is the inclination of the edge n to axis m a . cot. 61 1, = .5517 = nearly JJ. That is to say, axis m a is to axis p a as 11 is to 20; or if we make m a 10, to bring it into unison with the pro- duct of the following equation, we have the relation m a to p a = 10 to 1.8128 = }g = f. c.) Given, A 63 441'; B = 66 21^; to find, b. Formula 5. log cos b = log cos B + 1 log sin A. 256 PRINCIPLES OF CRYSTALLOGRAPHY. 10 + log cos B = 66 2H' = 19.6032 log sin A = 63 44|' = 9.9527 log cos b = 63 26' = 9.6505 This product, 63 26', is the inclination of the edge e to axis m a . cot 63 26' = .5000 = J = Jg. That is to say, axis m a is to axis t a as 10 is to 20 or 5 to 10. d.) Hence, the relations of the three axes are, p a = 9 ; m a = 5 ; t a = 10, which relations afford the symbol P^yM-^T; or, since -f^ is less than the true relations of p a to t a , the real indices of the form are very nearly P-i-$M T 5 oT or PM|T; and those of the co-existing form of the diocta- hedron are P^M^T, or P 2 M 2 T, or PMTJ. 514. PROBLEM. Given, a Dioctahedron, P X M_T, P X M+T, with an edge, n = 73 31|', and an edge, w = 66 211'; required, the inclina- tion of the edge n, to axis p a , and the value of the indices, x , _, + , in the symbol. a.) To find the inclination of the edge n to axis p a . The solution of this problem depends upon the principle explained in 409, according to which the present problem is, Given, A = 66 211'; B = 73 31J'; C = 45; to find, a. Formula 37. sin \ a = . Model 10. These are complete prisms with a rhombo-rectangular equator. Minerals : Class 1, Order 5. Model 7 belongs to Genus 1; Model 10 to Genus 2. Part II. page 99. 33 minerals occur in the form of Model 7, and 1 1 minerals in the form of Model 10. Analysis. Inclination of P upon any vertical plane, 90. 534. Combinations containing six-sided prisms and six-sided pyramids. T, Mif T 2 . Pf T, PfMjf T 2 : or V. 2Rf Zw Ze. Model 73. T, Mjf T 2 . Pff T, Pjf M|f T 2 : or V. 2R-}f Zw Ze. Model 74. PRINCIPLES OF CRYSTALLOGRAPHY. 267 These combinations represent incomplete prisms with complete pyramids ; and rhombo-rectangular equators. Minerals : Part II., page 113, Class 5, Order 4, Genus 1. Analysis.' A plane of a regular six-sided prism inclines upon a plane of a regular six-sided pyramid at an angle of 90 -f- x, in which formula, x signifies the inclination of a plane of the pyramid to the equator. 535. Combinations containing regular six-sided prisms, six-sided pyramids, and horizontal planes. P,T,MifT 2 .pft,pJX-ft 2 : or P,V.2rfZwZe. Model 58. P,T,Mjf T 2 . pm, pm 2 t|f : or P,V. 2r, Zn Zs. Model 56. P,m,T, m 2 tff, M-ff T 2 . pm, pjft, pm 2 t|f, p^'ft,: or P, V, v. 2r t Zn Zs, 2rf Zw Ze. Model 52. These combinations represent complete prisms with incomplete pyramids ; and rhombo-rectangular equators. Minerals: Part II., page 109. The axes of the two first are p x mf 5 t* 3 . They belong to Class 3, Order 5, Genus 1. The axes of the last are p x m* 4 tj 3 . It belongs to Class 3, Order 5, Genus 2. The description of Model 52 affords an example of the benefit to be derived from the use of the abridged symbols. Analysis. See 532534. 4. THE TWELVE-SIDED PYRAMID. 536. The twelve- sided pyramid bears the same relation to the six-sided pyramid, that the dioctahedron bears to the quadratic pyramid. It occurs very seldom, and always subordinately. Generally, it produces twelve pair of small planes, replacing the solid angles between the planes of six-sided prisms and six-sided pyramids; as, for example, the acute solid angles of Model 58. The base of this combination is shown by the twelve-sided external figure in the diagram in page 265, where the twelve sides are numbered 1 to 12. The combination consists of three rhombic octahedrons: 1st, the form PJMlT, whose equatorial edges are marked 1,2, 3, 4, and the angle of whose base is shown by the line 4af ; 2dly, the form P X M + T, whose equatorial edges are marked 5, 6, 7, 8, and the angle of whose equator is shown by the line 7 d ; 3dly, the form P X M_T, whose equatorial edges are marked 9, 10, 11, 12, and the angle of whose equator is shown by the line s w. In many cases, this twenty-four-faced pyramid is suffi- ciently w r ell indicated by the short symbol 3p x m y t z . 5. THE TWELVE- SIDED PRISM. 537. The twelve-sided prism bears the same relation to the two six- sided prisms, that the twelve-sided pyramid bears to the two six-sided pyramids. The shape of the equator and the positions of the twelve sides of this combination, are shown by the lines marked 1 to 12 in the diagram in 531. The forms belonging to this combination are, there- fore, MzT, M_T, M + T. I believe that this twelve-sided prism never occurs but in combination with the two six-sided prisms, or with the combination V, v, the edges of which it replaces, and forms a combina- 268 PRINCIPLES OF CRYSTALLOGRAPHY. tion containing 24 vertical planes. I propose to distinguish this twelve- sided prism by the term 3m x t. There will then be three kinds of prisms peculiar to the rhombohedral system, namely : The 6-sided prism = V = T, M|f T 2 . The 12-sided prism = V, v m,T, m 2 t|f , M|f T 2 . The 24-sided prism = V, v, 3m x t = m,T, m 2 t|f , M-ff T 2 , mlt, m_t, m + t. Of these, the six-sided prism is by far the most important, the one of most frequent occurrence, and the only one that occurs predominant upon any combination. B. Hemihedral Forms of the Rhombohedral System. 1. THE RHOMBOHEDRON. P X T, iP x M|fT 2 : or R x . Examples : }PT, |PMifT 2 : orR, Model 26 a . |P|T,P|M}fT 2 : orR* 26 b . |Pf T, iPf M$ T 2 : or Rf 26 C . P 2 T, |P 2 M|f T 2 : orR 2 26". Rf Zw, R| Ze, rf Ze 26 e . Varieties of this Combination : See the List of Indices, Part II., page 43. The rhombohedron is a complete pyramid with a rhombo-rectangular equator, and falls, therefore, into Class 2, Order 5. See Part II., page 103, where the minerals which occur in this form are enumerated. 538. The rhombohedron is a solid bounded by six equal and similar rhombuses, arranged in three pair of parallel planes. It has twelve edges and eight solid angles. 539. The Solid Angles. Every rhombohedron has two similar solid angles, different from the other six. One of these is situated at pole Z and the other at pole N. The other six solid angles, though different from the first two, are similar to one another. They "are called^the lateral solid angles. All the eight solid angles are three-faced: the faces which meet at poles Z and N are all alike, but those which meet at the lateral angles are different. The six lateral solid angles are divisible into two sets : the upper lateral angles are those at the lower ends of the upper terminal edges ; the lower lateral angles are those at the upper ends of the lower terminal edges. 540. The Edges. There are two kinds of edges on the rhombohe- dron ; six terminal edges similar to one another, and six lateral edges, also similar to one another, but different from the terminal edges. The terminal edges meet three together at pole Z, forming a three-faced pyramid, and three at pole N, forming a second three-faced pyramid. Axis p a connects the summits of these two pyramids. The terminal edges of the zenith and nadir pyramids do not meet together, nor do they any where touch the equator of the combination. They terminate in the lateral solid angles, which are connected by the lateral edges in a PRINCIPLES OF CRYSTALLOGRAPHY. 269 zigzag line of six equal divisions. Hence, the lateral edges are not hori- zontal nor parallel to the equator. 541. Oblique Sections. A section through a rhombohedron, across two terminal and two lateral edges, and parallel to an external plane, is a rhombus, whose angles are supplementary of one another. This is true of all rhombohedrons. Hence, if x denotes the angle across a terminal edge of a rhombohedron, and y the angle across a lateral edge, then the angle across x is 180 y, and the angle across y is 180 x. When the angle across the terminal edges is more than 90, the rhombohedron is called obtuse; when the angle is less than 90, the rhombohedron is called acute. The cube represents the rhombohedron, the angle across whose edge is 90, 363, and forms, therefore, the point of separation between the two kinds of rhombohedrons ; but the cube is not considered to belong to the rhombohedrons, nor is the rhom- bohedron whose symbol has unity for index, the variety whose terminal edge measures 90. 542. Horizontal Sections. A horizontal section through the three upper or three lower lateral angles of a rhombohedron, produces an equi- lateral triangle. See the horizontal planes of Model 114, which repre- sents a combination containing a rhombohedron with the terminal planes P. A horizontal section through a rhombohedron, any where betwixt the lateral angles and pole Z, produces an equilateral triangle. See Model 114 a . A horizontal section exactly through the middle of the crystal, and which is consequently equal to the equator, is a regular hexagon, or figure of six equal sides and six angles of 120. See the brown lines drawn on Models 26% 26 b , 26% 26 d , 1 14, and 114 a . The angles of this equator, or of the horizontal hexagonal section, are at the middle points of the six lateral edges. A horizontal section through a rhombo- hedron, any where between the equator and the lateral angles, is a hexa- gon whose angles are 120, but whose sides are alternately long and short. 543. Vertical Sections. A vertical section through a terminal edge of a rhombohedron, and, therefore, through the oblique diagonal of one of its planes, is a rhomboid. The direction of such a section is marked on Models 26% 26 b , 26% 26 d , by the purple line which shows the east meridian. The long sides of this rhomboidal section correspond with the oblique diagonal of the planes of the crystal, and the short sides corres- pond with its terminal edges. No matter whether the rhombohedron is acute or obtuse : this relation invariably holds true. The following diagram serves to illustrate this point : N 270 PRINCIPLES OF CRYSTALLOGRAPHY. Let Z N be axis p a , and e w be axis t a of a series of rhombohedrons. Then, ZaNb will be a vertical section of a rhombohedron nearly similar to Model 26 C ; Z m N n will be a vertical section of a rhombohedron similar to Model 26 a ; and Z x N y, will be a vertical section of a rhom- bohedron similar to Model 26 b . These sections are all rhomboids, and in all of them the long sides are the oblique diagonals of the external planes of the crystals, and the short sides, the section through their ter- minal edges. Thus, Zb, Zn, and Zy are the oblique diagonals of the planes of the three Models, 26 C , 26% and 26 b , while Za, Zm, and Zx show the corresponding sections through their terminal edges. 544. The diagram shows another fact, which I shall presently demon- strate by a trigonometrical calculation, but which I may notice in the meantime, because it is obvious in the figure. This fact is, that the dis- tance from any point of axis p a between pole Z and pole N, to a terminal edge of a rhombohedron, is twice as far as the distance to a plane, measured on the same horizontal line. Thus, line c to = line c to i X 2 ; line c to 2 = line c to 1 X 2 ; and line c to 4 = line c to 2 X 2. This, as I have shown in 360 362, is an important and con- stant property of all rhombohedrons, and it enables us to establish the following very useful principles : a.) The cotangent of the inclination of the planes of a rhombohedron to axis p a , is the length of axis p a , when axis t a of the form P X T, to which the planes are assumed to belong, is unity. b.) The tangent of the inclination of a terminal edge of a rhombo- hedron to axis p a , is twice the tangent of the inclination of a plane to axis p a . 545. The Crystallographic Forms that constitute a Rhombohedron. Every rhombohedron contains a hemihedral biaxial form and a hemi- hedral triaxial form. Place the models of the four rhombohedrons, 26 a , 26 b , 26 C , 26 d , in upright position, according to the coloured lines drawn upon them. The brown lines, indicating the equator, are to be horizontal. The blue line, indicating the north meridian, is to be on the north zone. The purple line, indicating the east meridian, is to be on the east zone. The plane marked T in ink, is to be exposed to the west. The letters stamped upon the models are to be disregarded. .) The Hemihedral Biaxial Form, |P X T Zw. It will now be seen that each of the four models contains two planes on the east zone : one plane from Z to w, and another from e to N. These constitute the hemihedral biaxial form ^P X T Zw Ne. b.) The Hemihedral Triaxial Form, 1P X M||T 2 Zne Znw. It will also be seen, that each of the four models contains two zenith and two nadir octahedral planes, occupying the positions Zne Znw Nnw Nsw, and forming the hemioctahedron with parallel faces described in 272, d.) 546. The Rhombohedron is the Hemihedral Form of the Regular PRINCIPLES OP CRYSTALLOGRAPHY. 271 Six-sided Pyramid. If lines are drawn from the terminal solid angle of a rhombohedron to the middle of every lateral edge, in the manner shown by the blue dotted lines on the under side of Model 26 b , these lines indicate the six terminal edges of the six-sided pyramid represented by Model 26. And if the terminal edges and lateral angles of Model 26 b were replaced by sections made through the dotted lines drawn upon one side of the model, and the brown lines drawn upon the opposite side, the resulting solid would be a regular six-sided pyramid. The central por- tion of all the six planes of the original rhombohedron would be left upon the new form, but with the addition of six other planes precisely similar. If these six new planes were extended till they met one another and formed a complete figure by hiding the residual portions of the original form, the new figure would be a rhombohedron exactly similar to Model 26 b , but having a different position. The rhombohedron is conse- quently the hemihedral form of the regular six-sided pyramid. 547. The Equator of the Rhombohedron is a regular Hexagon. It is evident that the replacement of the terminal edges and lateral angles of the rhombohedron, described in 546, would have no effect on the shape of the equator of the combination, because all the sections are assumed simply to meet at and not to cross the equator. Hence, as a regular hexagon is the form of the equator of the six-sided pyramid, so is it also the form of the equator of the rhombohedron ; and since the equatorial axes of the six-sided pyramid are described by the term m* 5 t* 3 , and its octahedral form by the symbol P X MJ-JT 2 , 522, so may also the axes of the equator of the rhombohedron be described by the term m? 5 tt 3 , and its hemioctahedral form by the symbol iP x M}| T 2 . 548, Every Rhombohedron may assume four different positions on the equatorial Axes. 1st. The < /zrs position is already described. Its symbol is, 1P X T Z w Ne, lP x Mif T 2 Zne Znw Nnw Nsw. 2nd. Turn the Model horizontally 180, so as to place a zenith plane towards the east, but still keep the blue line on the north meridian. The symbol of the combination will then be |P X T Ze Nw, IP MJf T 2 Znw Zsw Nne Nse. The above are therefore the two rhombohedrons that produce the six- sided pyramid of i\\e first position, 522. 3rd. Turn the Model horizontally 90, so as to reverse the meridians, and place a zenith plane towards the north, and the blue line in the direc- tion of the east meridian. The symbol of the combination is now, 1P X M Zn Ns, iP x M 2 T|f Zse Zsw Nne Nnw. 4th. Turn the Model horizontally 180, so as to place a zenith plane towards the south, but keeping the blue line on the east meridian. The symbol of the combination will then be |P X M Zs Nn, iP x M 2 Tff Zne Znw Nse Nsw. 272 PRINCIPLES OF CRYSTALLOGRAPHY. The rhombohedrons of the third and fourth division, are those which constitute the regular six-sided pyramid of the second position, 523. These four positions of the rhombohedron, may be denoted briefly as follows, it being unnecessary to particularize the polaric position of every individual plane, because a knowledge of the position of a single zenith plane, leads to a knowledge of the whole: 1. |P x TZw, 2. iP x TZe, 3. JP x MZn, iP x M 2 TLf. 4. iPJMZs, 1P X M 2 TJ|. 549. Abridged Symbol for the Rhombohedron.' As the symbol for the rhombohedron is somewhat complex, and as all the terms of it are con- stant, except the index that relates to axis p a , I have proposed, Part II. page 45, to abridge it to the single letter R, the initial of the word Rhombohedron, so as to be enabled to say, R x Zw, instead of P X T Zw, |P x M-}fT 2 . This greatly shortens the descriptions of complex combinations. The same index is in every case to be appended to R, that would be placed after P in the full symbol. .' 550. The Index of the Rhombohedron. The biaxial form JP X T, and the triaxial form f P x M-}f T 2 , of the rhombohedron, both take the same index, exactly as do the biaxial and triaxial forms which constitute the six- sided pyramid, 522. If, therefore, you find the index for the form |P X T, you find also the index for the form |P x M{f-T 2 , for it is the same. But the index of the form P X T, is the ratio of axis p a to axis t a . This ratio is the cotangent of the inclination of a plane of the rhombohedron to axis p a , 544. Hence, the finding of the index of a rhombohedron, is merely the finding of the inclination of a plane of the rhombohedron to its per- pendicular axis. 551. PROBLEM. Given, Model 26 a , with the angle across a terminal edge = 104 28f; required, the value of the index x in the symbol JP X T, This is merely an example of the general problem described in 359. Take a sixth of Model 26 a , divided as described in 359, as a right- angled solid triangle, with pole Z for its vertex. The known parts are C = 90; A = 60; B = 104 28f -f- 2 = 52 14', or half the angle across a terminal edge. With these given quantities, find b, which is the inclination of a plane of the rhombohedron to axis p a ; and c, which is the inclination of a terminal edge to axis p a . a.} Given, A = 60; B = 52 14J'; to find, b. Formula 5. log cos b = log cos B + 10 log sin A. 10 + log cos B = 52 14 ' = 19.7870 log sin A = 60 == 9.9375 log cos b = 45 = 9.8495 PRINCIPLES OF CRYSTALLOGRAPHY. 273 This product, 45, is the inclination of a plane of Model 26 a to axis p a . The cotangent of 45 is 1.0000, or }-, so that Model 26 a requires the symbol JPT, |PMJ-|T 2 , or Rj. It is the simplest form of the rhornbo- hedron, and bears the same relation to the acute and obtuse rhombohe- drons, that the cube bears to the short and long square prisms. Com- pare Models 26 b , 26% 26 d ; and Models 2, 1, 3. b.) Given, A 60; B = 52 14|'; to find, c. Formula 6. log cos c = log cot A -f- log cot B 10. log cot A = 60 = 9-7614 + log cot B = 52 14' = 9.8891 log cos c = 63 26' = 9.6505 This product, 63 26', is the inclination of a terminal edge of the rhoia- bohedron to axis p a . The tangent of 45 is 1.0000 The tangent of 63 26' is 2.0000 Hence, the tangent of the inclination of a terminal edge of a rhombohe- dron to axis p a is twice the tangent of the inclination of a plane to the same axis, as was stated in 544 b.) This determination justifies our ascribing to axis t a of the triaxial form $P X MJJ-T 2 , twice the length that we ascribe to axis t a of the biaxial form JP X T. 552. PROBLEM. Given, Model 26% with the angle across a terminal edge = 104 28'; required, the plane angles of the faces. Take the same solid triangle and the same given quantities as in 551, and find side a. Given, A = 60; B = 52 14J'; to find, a. Formula 4. log cos a = log cos A -f 10 log sin B. 10 + log cos A = 60 = 19.6990 . log sin B = 52 14' = 9.8979 log cos a = 50 46' = 9.8011 Twice this product, or 50 46' X 2 = 101 32', is the obtuse plane angle at pole Z. Its supplement, 180 101 32' = 78 28', is the value of the acute plane angle at a lateral solid angle of the model. 553. PROBLEM. Given, Model 26% with the symbol, R! ; required, the angle across the terminal edges and across the lateral edges. a.) The index of a rhombohedron is the cotangent of the inclination of a plane to axis p a . The index 1 or 1.0000 is the cotangent of 45, which is therefore the inclination of a plane of Model 26 a to axis p a . Take a solid triangle, consisting of a sixth of the crystal, with pole Z for its vertex, and find B, equal to half the angle across a terminal edge of the crystal, with the help of the following known quantities: C = 90; 2N 274 PRINCIPLES OF CRYSTALLOGRAPHY. b = 45; A = 60. See problem 551, of which the present problem is the counterpart. Formula 8. log cos B = log cos b + log sin A 10. log cos b = 45 = 9.8495 + log sin A = 60 = 9.9375 log cos B = 52 14|' = 9.7870 Twice this product, or 52 14J' x 2 = 104 28f, is the angle across a terminal edge of the rhombohedron R t . b.) The angle across a lateral edge is the supplement of the angle across a terminal edge, or 180 104 28f ' == 75 3lK 554. General Formula for calculating the relation between the Index and the Angles of a Rhombohedron. Put B = half the angle across a terminal edge, and b = inclination of a plane to axis p a . Then, a.) log cos b = log cos B + 10 9.9375 (log sin 60). b.) log cos B = log cos b + 9.9375 (log sin 60) 10. See 551 and 553 for the derivation and details of these formula?. Examples : 1.) Given, Model 26 b , R x , B = 67 13'; required, b, and the value of the Index. Formula a.) 10 + log cos 67 13' = 19.5880 9.9375 log cos 63 26' = 9.6505 = b. cot 63 26' = .5000 = \. Therefore, Model 26 b = R|. Haiiy's Chaux carbonatee equiaxe. 2.) Given, Model 26 d , R x , B = 39 14'; required, b, and the value of the Index. Formula a.) 10 + log cos 39 14' = 19.8891 9.9375 log cos 26 33' = 9.9516 = b. cot 26 33' = 2.000 = f . Therefore, Model 26 d = Rf . Haiiy's Chaux carbonatee inverse. 3.) Given, Model 26 C , Rf ; required, b and B. By the Table of Indices, page 139, you find f = 2.667 = cot 20 33' = b. Then, Formula b.) log cos 20 33 r = 9.9714 + 9.9375 log cos 35 50 == 9.9089 = B. PRINCIPLES OF CRYSTALLOGRAPHY. 275 Twice this product, or 35 50' X 2 = 71 40', is the angle across a ter- minal edge of Model 26 C . Haiiy's Mercure sulfure primitif. He states the angle to be 71 48'. 555. Combination of Rhombohedrons with one another. Several rhombohedrons are often found in combination upon one crystal. Model 26 e represents a combination of three different rhombohedrons. It will be useful to examine what are the possible varieties of combination that can take place among the rhombohedrons. Rhombohedron of the First Position predominant R x Zw. 548, 1st. .) It can combine with a rhombohedron of the same position, whose index is _. The planes of r_ appear at pole Z, inclining on the planes of R x . It can also combine with a rhombohedron of the same position, whose index is + . The planes of r + appear at the lower lateral angles inclining on the planes of R x . &.) It can combine with a rhombohedron of the second position, R x Ze, having any index whatever. If the index of the second rhombohedron is equal to the index of the first rhombohedron, and if the two rhombo- hedrons are similar in size as well as equal in their axial relations = R x Zw, R x Ze, then the combination will be a regular six-sided pyra- mid; but if the rhombohedron of the second position is subordinate, = r x Ze, then its planes replace the upper lateral angles of R x Zw, and incline upon the terminal edges; and in this case, the edge of combination of the planes of the two rhombohedrons is parallel with the blue lines drawn on Models 26 a , 26 b , or with lines drawn from pole Z to the angles of the hexagonal equator. c.) If the planes of the rhombohedron r x Ze are tangent planes to the edges of the rhombohedron R x Zw, as they are represented by Model 26% then the index of r x Ze has a divisor twice as large as the index of R x Zw, the dividend remaining the same. Thus, if R x Zw is RJ, then r x Ze must be r|, because the equator of the second form has twice the diameter of the equator of the first form. This is evident from the cir- cumstance, that the inclination to axis p a of a plane of the second form is equal to the inclination to the same axis of an edge of the first form, while the inclination to that axis of an edge of the first form has a tangent of twice the length of the tangent of the inclination of the planes of the first form to the given axis. Of course, the edges of the second form bear the same relation to its planes, as the edges and planes of the first form bear to one another. The equatorial axes of the second form are therefore twice as long as the equatorial axes of the first form. d.) If the rhombohedron of the second position, r x Ze, has an index whose divisor is greater than twice the divisor of the index of the rhom- bohedron R x Zw, then the second rhombohedron is obtuse, and its planes appear on R x Zw at pole Z, replacing the upper part of the terminal edges of R x Zw. e.) If the rhombohedron of the second position, r x Ze, has an index 276 PRINCIPLES OF CRYSTALLOGRAPHY. whose divisor is less than the divisor of the index of the rhombohedron R x Zw, then the second rhombohedron is acute, and its planes replace the upper lateral angles of the rhombohedron R x Zw, and incline on its terminal edges ; but the edge of combination between the planes of R x Zw and r x Ze is not, as in case 6.), parallel with the blue lines drawn on Model 26 b , but has a more vertical position. f.) The rhombohedron of the first position does not combine with the rhombohedrons of the third and fourth positions. The rhombohedron of the second position is never considered to be predominant, merely because there can be only one predominant rhom- bohedron on any given combination, and it is most convenient to follow the rule, always to place this in the first position. 556. Rhombohedron of the Third Position predominant = R x Zn. 548, 3rd. The rhombohedron of the third position can combine with other rhom- bohedrons of the third position having the indices _ or +, and with rhom- bohedrons of the fourth position having any index whatever, exactly as rhombohedrons of the first position can combine with other rhombohedrons of the first position, and with all kinds of rhombohedrons of the second position. But the rhombohedrons of the third and fourth positions (Jo not combine with rhombohedrons of the first and second positions, except when they are in the condition of homohedral six-sided pyramids. 557. ANALYSIS OF COMBINATIONS CONTAINING SEVERAL RHOMBO- HEDRONS. Take, as an example, Model 26 e . R| Zw, Rj| Ze, r| Ze. Chabasite. a.) According to Phillips, the inclination of the large planes, or the planes of Rf , to one another, measured over the tangent planes that re- place the edges of R|, is 94 46'. With this information, you can find the index of the predominant rhombohedron, by the method given in 554. Formula a.) 10 + log cos 9 -^' = 47 23' = 19-8306 9.9375 log cos 38 34/ = 9.8931 cot 38 34' = 1.2542 = nearly (within 6'). This gives the symbol RfZw. b.) The planes which replace the edges of RJZw require the symbol R Ze, according to the principle explained in 555, c.) c.) The planes which replace the lateral solid angles, and which con- stitute the rhombohedron r + Ze, may be calculated from two different measurements: one upon the zenith planes of R|-, which Phillips quotes at 143 59', the other upon the nadir planes of R| which angle he quotes at 120 5'. d.) The angle of inclination R| upon r + contains the following angles : The inclination of R| to the equator = x -\- 90 + the inclination of PRINCIPLES OF CRYSTALLOGRAPHY. 277 r + to axis p a = y. These three quantities compose the edge of combi- nation between the planes of any two rhombohedrons in the same zone and on the same pyramid. Consequently, if X is the angle across the edge of combination, then y = X (x -f- 90) and x = X ~ (y + 90). Now angle x in the present case, or the inclination of a plane of Rf to the equator, is the complement of the inclination of the same plane to axis p a . By the Table of Indices, page 139, you find f = .6250, which is the cotangent of 58, or the tangent of 32. The former is the incli- nation of a plane of RJ| to axis p a , the latter, its inclination to the equator. Therefore, x = 32, and the equation y = X (# + 90), becomes y = 143 59' (32 + 90 = 122) = 21 59'. This product, 21 69', is therefore the inclination of r + to axis p*. cot 21 59' = 2.4772 = nearly f (within 10'). e.) The angle of inclination of a zenith plane of R J Zw upon a nadir plane of r_j. Ze contains two angles, which are, the inclination of a plane of R| and of a plane of r + to the equator. Call the first x and the sec- ond y. These two quantities always compose the edge of combination between the planes of two different rhombohedrons that meet at the equator. [Hence, if the edge of combination is called X, then x = X y> and y = X x. Now, angle #, or the inclination of a plane of R J to the equator, is the angle of which = 1.250 is the tangent. This angle is 51 20|'. Hence, as X is given at 120 5', the equation^ = X x, becomes y = 120 5> 51 20|' = 68 441'. This product, 68 44J', is the inclination of a plane of r_[. Ze to the equator, the tangent of which angle is the index of the rhombohedron. Hence, tan 68 44|', 2.570 = nearly f (within 32'.) /) The index of r + Ze is found, by c?.), to be 2.4772, and by e?.), it is found to be 2.570. The index chosen as the correct relation is 2.5000, which stands nearly mid-way between the two determinations. These dif- ferences afford an example of what very frequently occurs in crystallo- graphy. The index of a symbol is often greater or smaller, according to the measurement after which it is calculated. It seldom happens that the measurements of two different edges of the same natural crystal are found to be mathematically accurate, or that the measurements of the same edge by different persons agree to a nicety. Hence, the necessity of correcting such measurements, by calculating the indices of symbols from as many different original measurements as possible, and thus making one measurement serve to check another. It is as unsafe to fix the index of a symbol from a calculation grounded on the measurement of a single angle, as it would be to determine the atomic weight of a chemical element from the evidence afforded by a single analysis. From the result of the foregoing calculations, I conclude that the rhombohedrons contained on Model 26 e are R| Zw, Rf Ze, r| Ze. The minerals which occur in the shape of complex rhombohedrons, are enumerated in Part II. page 103, in Class 2, Order 5. 278 PRINCIPLES OF CRYSTALLOGRAPHY. 558. COMBINATIONS OF THE RHOMBOHEDRON WITH THE HORIZON- TAL PLANES P. P_.iPiT,PlMJ-fT 2 : orP_.Rf. Model 114. Corundum. p_.lPT,PM|fT 2 : OPP_.R!. Model 114 a . Calcareous Spar. These combinations are incomplete prisms with incomplete pyramids, having a rhombo-rectangular equator. Minerals: Class 5, Order 5, Genus 1, Part II., page 118. Analysis. The inclination of plane PZ to a plane of a rhombohedron is 90 -\- x, in which formula, x signifies the inclination of the plane of the rhombohedron to axis p a . If you have given, the combination P.R X , with the inclination of PZ upon P X T = X, then the index of R x is the cotangent of (X 90). 559. COMBINATIONS OF THE RHOMBOHEDRON WITH THE REGULAR SIX-SIDED PRISM. T, Mif T 2 . iPM Zn, |PM 2 T-ff : or V. R t Zn. Model 71. T, Mjf T 2 . |PT Zw, IPMjf T 2 : or V. R, Zw. Model 72. These combinations are incomplete prisms with complete pyramids, having a rhombo-rectangular equator. Minerals : Class 4, Order 5, Genus 1, Part II. page 113. Analysis. The inclination of plane Tw to plane P X T Zw is 90 + x, in which formula, x signifies the inclination of plane P X T Zw or of any plane of the given rhombohedron, to the equator. The inclination of plane M_T to plane P X M Zn, or of plane Tw to plane |P X M 2 T|| Zsw, is 90 -f y, in which formula, y signifies half the angle across the lateral edge of the given rhombohedron. The common practice of crystallographers is to put the terminal planes of Model 71 into the same positions as the terminal planes of Model 72. In that case, the rhombohedron is the same in both combinations, but then, the prism of Model 71 must be denoted as the prism of the second position. Now, I think it is much better to consider the positions of the prisms to be fixed, and the positions of the pyramids to be variable. This has the great advantage of diminishing the estimated number of the prisms without increasing the estimated number of the pyramids, or adding any thing to the difficulty of naming and discriminating them. This, of course, was difficult to be done without the use of the polaric signs, Ze, Zn, &c., but with these signs, this arrangement can be readily carried into effect. 560. COMBINATION OF THE RHOMBOHEDRON, THE Six- SIDED PRISM, AND THE HORIZONTAL PLANES. P,T,M|fT 2 . p|mZn,ipim 2 ti-|: orP.V.rfZn. Model 67. Corundum. This is a complete prism with an incomplete pyramid, and a rhombo- rectangular equator. Minerals : Part II. page 1 08, Class 3, Order 5, Genus 1. PR^CIPLES OF CRYSTALLOGRAPHY. 279 2. THE SCALENOHEDRON. Model 26 f . 561. The scalenohedron is a six-sided pyramid with scalene triangular faces and a twelve-sided equator, similar to the twelve exterior edges of the following diagram : There are three kinds of edges on the scalenohedron : six acute and short terminal edges, which have the positions of the terminal edges of a rhomb&hedron of the first position, or as the lines p, q, r, in the above diagram ; six obtuse and long terminal edges, situated like the terminal edges of a rhombohedron of the second position, or like the lines x, y, z, in the above diagram ; and six lateral edges which connect the terminal edges by a zigzag, situated like the zigzag lateral edges of a rhombohe- dron. Hence, the equator passes through the middle of every lateral edge, and divides three upper lateral angles from three lower lateral angles. The figure of the horizontal section made through this combina- tion any where betwixt the lateral angles and the summit is six-sided, with three angles of one kind and three of another, placed alternately, as shown in the diagram in 563. The vertical section through the ter- minal edges of the combination is a rhomboid, whose sides show the inclination of the two kinds of terminal edges to axis p a . 562. The scalenohedron is the parallel-faced hemihedral form of the didodecahedron or twelve-sided pyramid. It occurs much more fre- quently than the homohedral form, is often predominant, and sometimes produces complete isolated crystals, uncombined with other forms. The planes of the twelve-sided pyramid which combine to produce the scalenohedron, are enumerated in 274, and are marked S in the diagram in 561. They are as follow: |P X M_T, marked 9 and 11, and forming part of the octahedron, 9, 10, 11, 12. P x MzT, marked 1 and 3, and forming part of the octahedron, 1, 2, 3, 4. JP X M + T, marked 7 and 8, and forming part of the octahedron, 5, 6, 7, 8. 280 PRINCIPLES OF CRYSTALLOGRAPHY. The complementary planes, marked 10, 12, 2, 4, 6, and 8, combine to produce a second or inverse scalenohedron. In like manner, scalenohedrons are formed, which have the same rela- tion to axis m% that the above two varieties have to axis t a . There are a great many varieties of the scalenohedron ; probably as many or more varieties than of the rhombohedron. They are, however, all so very imperfectly described in the books on mineralogy, that I have found it impossible to give a clear account of them, and have, therefore, denoted them in the Tables of Minerals merely by a temporary proxi- mate sign. The scalenohedron which is best known, because most abundant in a separate state, is the metastatique scalenohedron of calcare- ous spar, which is represented by Model 26 f . The axes of the three hemioctahedrons of which this combination is composed, are to be found by the following processes : 563. PROBLEM. Given, Model 26 f , with the angle across an obtuse terminal edge = 144 20 J', the angle across an acute terminal edge = 104 28-f', and the angle across a lateral edge = 133 26'; required, a.) the indices of the hemioctahedron whose planes meet at the obtuse ter- minal edge, Zw; b.) the indices of the hemioctahedron whose planes meet at the acute terminal edge, Ze; and c.) the indices of the hemioctahedron whose planes lye on the north meridian of the scalenohedron. Put o = !!i!2i' __ 72 io' = half the angle across the obtuse edge. 10428!' 66 43' = half the angle across the lateral edge. a.) The first calculation to be made is on the model of that described in 403, relating to six-faced pyramids. Assume Model 26 f to be divided into six portions, by planes pass- ing through the terminal edges. Take one of these portions as an oblique-angled solid triangle, with pole Z for its vertex. Then the known parts are, C = 60 = interior angle formed by the intersection of two planes at axis p a ; A = 52 14^' = half the angle across an acute edge ; and B = 72 10|' = half the angle across an obtuse edge. With these data, find a = inclination of an obtuse edge of the scalenohedron to axis p a . Given, A = 52 14|'; B = 72 10i'; C = 60; to find, a. Formula 37. sin a = */ ~ ^ ^ B ( T A) , where S = i(A +- B+ C). Log sin i a || | {log cos S + log cos (S A) + 20 (log sin B -|- log sin C) }. A = 52 14F S = 92 12' 17^" B = 72 10' A = 52 14 r 20" C = 60 - A = 3957 / 57| // 2)184 24/ 35" />k = 180 Supplement of S = 87 47 7 42*" PRINCIPLES OF CRYSTALLOGRAPHY. 281 log cos S + log cos (S A) 87 47f = 8.5852 390 58' = 9.8844 + 20 log sin B = 72 101' = 9-9786\ + log sin C = 60 = 9.9375 f 38.4696 19.9161 2)18.5535 log sin a = 10 54' = 9.27675 Twice this product, or 10 54' x 2 = 21 48', is the inclination of the obtuse terminal edge of the scalenohedron to axis p a . As this edge is on the east meridian, and proceeds from pole Z to pole w, the cotangent of its inclination to p a is the index of the octahedron whose two planes meet at the Zw edge, in so far as respects axes p a and t a . cot 21 48' = 2.5000 = f = P|M X T. #.) You next proceed to investigate the value of the index x in the symbol Pf M X T just quoted. The hexagon, A,B> in the following dia- gram, is a cross section through the zenith pyramid of the scalenohe- M dron. Line ns is axis m a , and line ew is axis t a . Line cw is axis t 1 of the hemioctahedron, found by .) to be f of the length of axis p a . What you have now to determine, is the length of the line cs or en, the relation of which to the line cw is the required value of x in the symbol P|M X T; for the triangle swn is half the equator of the given octahe- 'dron, and cw, en, are its two equatorial axes. The lines BV and B 2 w represent the width of so much of the planes of P|M X T as appear upon 2o 282 PRINCIPLES OF CRYSTALLOGRAPHY. the scalenohedron. The portions s B 1 and nB 2 , and the entire other half of the octahedron, are replaced by the planes of the two co-existent hemioctahedrons, as is shown in the diagram. To obtain the relation of line en to line ew, you take an octant of the scalenohedron, Model 26 f , which is assumed to be divided for this purpose by the equator, the north meridian, and the east meridian. You employ this as a right-angled solid triangle with pole Z for its vertex. The base of this octant is shown by the lines ccB 2 w; the base of the given triangle is shown by the lines cnw. The known parts of this right-angled triangle are C = 90 = inclination of the east meridian on the north meridian, or of line cw on en ; A = 72 10j r = half the angle across an obtuse terminal edge of Model 26 f ; b = 21 48' = inclination of the same obtuse terminal edge to axis p a . With these given quanti- ties, you can find , which is the inclination of the north terminal edge of the given octahedron to axis p a , the tangent of which angle is the required length of line en in the diagram. Given, A = 72 10i'; b = 21 48'; to find, a. Formula 7. log tan a = log tan A 4- log sin b 10. log tan A = 72 10' = 10.4926 + log sin b = 21 48' = 9.5698 log tan a = 49 6' = 10.0624 The tangent of this product, 49 6', is 1.1544 = ||, which gives p^m^. This is a ratio not reducible to a convenient vulgar fraction fit to replace the index x in the symbol P|M X T, at least, so long as the index | fol- lows the sign P. I think it better, therefore, to take axis p a for unity, and to write the indices of this symbol as follows : PiM|f Tf . In writing the symbols of the associated hemioctahedrons of the scalenohedron, axis p a must then always be taken for unity. c.) You proceed now to investigate the hemioctahedron, whose two zenith planes meet at the acute terminal edge of the scalenohedron, which edge is marked B in the diagram, page 281. There are two cal- culations to make with this view. By the first you find the inclination of the acute terminal edge of the scalenohedron to axis p a , the tangent of which inclination is the length of the line Be; and then by employing this" product in an equation similar to that given in &.), you obtain an angle whose tangent is the length of the line co or ci. The ratio of axis m a = co, and of axis t a = cB, to axis p a , being thus determined, you are enabled to fix the indices of the symbol of the hemioctahedron, whose planes are marked A*B and A 2 B in the diagram. You may find the inclination of the obtuse terminal edge of the scalenohedron to axis p a by means of Formula 37, as shown in .), merely changing the designations of the given quantities to A = 72 10^'; B = 52 14J'; C = 60; with which to find a, which is the required angle. But the following method is shorter : d,) Take the same solid triangle as was employed in .), but use the PRINCIPLES OF CRYSTALLOGRAPHY. 283 following data : A = 52 14^' = half the angle across an acute edge ; a = 21 48' = inclination of an obtuse edge to axis p a ; B = 72 101' = half the angle across an obtuse edge. With these data, find b = in- clination of an acute edge of the scalenohedron to axis p a . Given, A = 52 141'; a = 21 48'; B = 72 10J'; to find, b. Formula 31. log sin b = log sin B + log sin a log sin A. log sin B P. 72 10J' 9-9786 + log sin a = 21 48' = 9.5698 19.5484 log sin A = 52 14i' = 9.8979 log sin b = 26 34' = 9-6505 This product, 26 34', is the inclination of the acute terminal edge of the scalenohedron to axis p a . Its cotangent is 2.0, its tangent .5, which gives the ratio of pat a , or the symbol P 2 M X T, or PM X TJ. e.) You next take a right-angled solid triangle, whose base is described by the triangle Bco in the diagram in page 281. Pole Z is its vertex, and you know the following parts, A = 52 141' = half the angle across an acute edge of the scalenohedron ; b = 26 34' = inclination of that edge to axis p a . With these you have to find, a = inclination to axis p a of a line or an edge from o to pole Z, the tangent of which angle gives the length of the line o c, or the ratio of axis m a of the hemioctahedron to axis p a . Given, A = 52 14 j'; b = 26 34'; to find, a. Formula 7- log tan a log tan A -f- log sin b 10. log tan A = 52 141' = 10.1109 + log sin b = 26 34' = 9.6505 log tan a = 30 = 9-7614 The cotangent of 30 is 1.7532, or fj ; its tangent is .5774 or |f, which gives the ratio of p a 6 m a 5 . If, therefore, we again call axis p a of the sym- bol P x M y T z = 1, then the indices derived from c?.) and e.) will be PiMj|T*. This is the symbol of the hemioctahedron whose two zenith planes meet at the acute terminal edge Ze. /) You have now to examine the axial relations of the hemioctahedron whose planes lye on the north zone of the scalenohedron, and meet at the lateral edges marked A 1 B 1 and A 2 B 2 in the diagram. The angle across the lateral edge, from a plane of the upper on a plane of the lower pyramid, is given at 133 26'. The supplement of this angle, or 1802 133 26' = 46 34', is the inclination of plane A 1 B 1 upon plane" A 2 B 2 measured across the upper pyramid. The inclined edge which this angle measures does not appear upon the scalenohedron, but its posi- tion is from pole Z to the extremity of the line c e, where the lines which 284 PRINCIPLES OP CRYSTALLOGRAPHY. pass from A 1 B 1 and A 2 B 2 , converge to a point at e. Let this inclined edge be called X, the equatorial axes of the given hemioctahedron, are the line c M c = m a , and the line c m e = t a . By means of an oblique- angled solid triangle, you first ascertain the inclination of the edge X to axis p% then the length of the line erne, which is the tangent of the inclination of the edge to axis p a ; and finally, with these data, and by means of a right-angled solid triangle, you determine the length of the line c M c. Let the triangle A 1 , c, e, be the base of the oblique-angled solid triangle, which has pole Z for its vertex. Then angle B = 72 10J', is half the angle across an obtuse edge of the scalenohedron ; angle A = ~ = 23 17', is half the inclination of plane A 1 B 1 on plane A 2 B 2 ; and side a = 21 48', is the inclination of an obtuse terminal edge to axis p% or that angle whose tangent is the line A 1 c in the diagram. With these data, you can find side b, which is the inclination of the edge X to axis p a . Given, A = 23 17'; B = 72 101'; a = 21 48'; to find, b. Formula 31. log sin b = log sin B + log sin a . log sin A. log sin B = 72 10i' = 9-9786 + log sin a = 21 48' = 9.5698 19-5484 - log sin A = 23 17' = 9.5969 log sin b = 63 26' = 9-9515 This product, 63 26', is the inclination of the edge X to axis p a . Its cotangent is .5000; its tangent, 2.0000, which gives the ratio of p a t|, or the symbol PJM X T, or PM X T 2 . g.) Now form a right-angled solid triangle, with pole Z for its vertex, and whose base is shown by the triangle e c M In the diagram. The given parts of the triangle are as follow : C = 90 inclination of side m upon side M; A = 23 17' = half the angle across the edge X, or half the inclination of plane A 1 B 1 on plane A 2 B 2 ; b = 63 26' = incli- nation of the edge X to axis p a . With these data, you have to find , the tangent of which angle is the length of the line c M c, compared with axis p a of the hemioctahedron under examination. Given, A 23 17'; b = 63 26'; to find, a. Formula 7. log tan a = log tan A + log sin b 10. log tan A = 23 17' = 9.6338 + log sin b = 63 26' = 9.9515 log tan a = 21 3' = 9.5853 The cotangent of 21 3' is 2.5983, or '/ Its tangent is .3849, or T %. This gives the ratio of pfg mf. If, therefore, we again put axis p a of the form P M y T z = 1, then the indices afforded by/) and g.) become P,M-/ 5 T 2 . This is the symbol of PRINCIPLES OP CRYSTALLOGRAPHY. 285 the hemioctahedron whose planes lye on the north zone of the scaleno- hedron. h.) The solution of the problem proposed at the beginning of this sec- tion, page 280, is as follows : a.) is found by a.) and b.) to be P|M|f Tf b.) is found by d.) and - - 180 (.w = 6850'J f.) To find the value of the two indices -f- and in the symbols of Model 21, P + M_T, p_f_mt a . The value of the sign -{-, or the ratio of axis p a to axis t a , is the tan- gent of the inclination of the acute terminal edge to axis t a . PRINCIPLES OF CRYSTALLOGRAPHY. 295 The value of the sign , or the ratio of axis m a to axis t a , is the tan- gent of the inclination of the equatorial edge to axis t a . The inclination of the acute terminal edge to axis t% see a.) is 62 18', tan = 1.9047. The inclination of the equatorial edge to axis t a , see d.) is 39 2'. tan = .8107. These relations give the indices P t 9047 M . 8107 T. PLOW nio.8io7 ti a . For convenience sake, I have abridged these in- dices to P}-gM T 8 oT, pf 9 m a ti a , although they do not then express the exact relations of the external angles. How far it may be prudent to abridge indices in this manner, I am not prepared to determine. It very seldom happens that the ratios of the axes of Forms belonging to Minerals of the Prismatic system can be indicated by indices so simple as those which serve to indicate the axes of Forms belonging to Mine- rals of the Octahedral and Pyramidal systems. See 294, r.) The next problem shows the amount of error introduced by this abridgement of the indices. 575. PROBLEM. Given, Model 21, with the symbol P|M T 8 n T, re- quired, the angle across its three external edges. a.) To find the angle across the equatorial edge. Take a right-angled solid triangle with pole w for its vertex. , The given quantities are : side a or the inclination of the acute terminal edge to axis t a , which is the angle whose tangent is |g. By the Table of Indices, page 139, |g = 1.90. tan 62 141', and side b, or the inclination of the equatorial edge to axis t a , or the angle whose tangent is f G = .8 = 38 40'. With these given quantities, you can find angle A, or the inclination of a plane to the equator, and angle B or the inclination of a plane to the east meridian. Given, a = 62 14J'; b = 38 40'; to find, A. Formula 13, log tan A = log tan a + 10 log sin b, 10 -I- log tan a =62 14|' = 20.2788 - log sin b = 38 40' = 9.7957 log tan A = 71 48' = 10.4831 Twice this product = 71 48' x 2 = 143 36', is the angle across the equatorial edge. This is 12' more than the angle found in problem, 574, c.) and quoted by Rose as the true angle, 143 24'. The reader will perhaps consider this difference too much to be neglected. But then, we find Phillips quoting this angle at 143 25', Mohs at 143 17', Haiiy at 143 2', and Kupffer at 143 26.8'. Between the last two quotations there is a difference of nearly 25'. Hence the difference be- tween the measurements of different authorities is greater than the differ- ence indicated by short approximate indices, and by long indices which describe exactly the measurements of one selected authority. This case is not peculiar to the given crystal of Sulphur, but applies to many of the forms of the Prismatic system. We may object to approximate in- dices, and resolve to give exact indices, even though they be long. But when we attempt to put this resolution into practice, we find the angles 296 PRINCIPLES OF CRYSTALLOGRAPHY. quoted by different authorities, nay, the angles across different edges of the same crystal, taken by the same person, to be frequently so irrecon- cilable with one another, that we are forced after all, to adopt approxi- mate indices as a pro tempore expedient. b.) To find the angle across the acute terminal edge of Model 21. Given, a = 62 14J'; b = 38 40'; to find, B. Formula 14. log tan B = log tan b + 10 log sin a. 10 + log tan b = 38 40' = 19.9032 log sin a = 62 141' = 9-9469 log tan B = 42 71' = 9.9563 Twice this product, or 42 7|' X 2 = 84 15', is the angle across the acute terminal edge of the form. This, however, is 53' too little, as Rose quotes this angle at 84 58'. This difference is very considerable, but then Haiiy quotes this angle at 84 24', and Phillips at 85 5', be- tween which angles there is nearly an equal difference. But the two calculations serve to show that in replacing the ratios of p a to m a , namely, 1.9047 : 0.8107 by 1.9 : 0.8, we reduce the value of m a too much. c.) To find the angle across the obtuse terminal edge of Model 21. Employ a solid triangle with pole n for its vertex. You have given, A = 71 48' =r inclination of a plane to the equator, see .); b = 51 20' = inclination of the equatorial edge to axis m a . This angle is the complement of the inclination of the same edge to axis t a , formed by a.) = 38 40'. With these data, find B = inclination of a plane to the north meridian. Given, A = 71 48,; b = 51 20'; to find, B. Formula 8. log cos B = log cos b + log sin A 10. log cos b = 51 20' = 9.7957 -f log sin A = 71 48' = 9.9777 log cos B = 53 36' = 9.7734 Twice this product, or 53 36' x 2 = 107 12', is the angle across the obtuse terminal edge of Model 21. This angle is quoted by Rose at 106 33', so that the calculation gives an excess of 39', arising from the substitution of -& for 0.8107 as the length of axis m a . The value of this angle as quoted by Haiiy, is 107 18f, by Phillips 106 30', by Kupffer 106 16.5', by Brooke 106 20', and by Mohs 106 38'. d.) The chief purpose of the present problem is to show the reader in what manner the angles of a rhombic octahedron may be calculated from the indices of its symbol, and that of the previous problem to show how the indices of the symbol can be found from the external angles of the crystal. The calculations are in both cases very easy, and it may seem strange that where this is the case, there should be more difficulty with the indices of the symbols than there is with the indices of other forms which require much more difficult calculations. PRINCIPLES OP CRYSTALLOGRAPHY. 297 576. INDICES OF THE RHOMBIC PRISMS, M X T, P X M, P X T. M X T. The index of the vertical prism M X T is the cotangent of the inclination of its planes to axis m% or the tangent of their inclination to axis t a . See 322327. P X M . The index of the form P X M, is the cotangent of the inclination of its planes to axis p% or the tangent of their inclination to axis m a . P X T. The index of the form P X T, is the cotangent of the inclination of its planes to axis p a , or the tangent of their inclination to axis t a . When the index of any one of these prisms is known, and the exter- nal angle is required, it may be found by the following rule : - The angle across the edge of a rhombic prism is tivice the angle whose cotangent is equal to the index of the prism. Example: P_, MJ T. Model 6. = T 8 n = .8. cot 51 20'. Twice 51 20' = 102 40'. This product is the inclination of plane MJT n w on plane MJT n e. 577. ANALYSIS OF COMBINATIONS OF THE PRISMATIC SYSTEM. a.) P, M, T, or P + , M_, T. Model 5. A rectangular prism. The angle across any edge is 90. It contains 3 pair of rectangular planes. The length of the edges of the Model is as the numbers 12, 9, 10, affording the symbol Pf^M^T. It is a com- plete prism with a rectangular equator. Minerals ; Class 1 , Order 2, Part II. page 98. b.) P_, Mf T. Model 6. A right rhombic prism. P on Mf T = 90. MJT n w on MJT n e is twice the angle of which is the cotangent. Class 1, Order 3, Part II. page 98. c.) p+. PJ-gM&T. Model 80. A rhombic octahedron with its summits replaced by horizontal planes. The inclination of P on Pj-gM T 8 0T, is the supplement of the inclination of Pj-jjM^jT to the equator. In the present case, this last angle is 71 48', see 575, c.) Then, 180 71 48' == 108 12'. This is the inclination of p + to P|gM T %T. An incomplete prism with an incom- plete pyramid, having a rhombic equator. Class 5, Order 3, Part II. page 115. d.) m . PfJM&T. Model 70. A rhombic octahedron with the obtuse lateral solid angles replaced by one vertical plane. The inclination of m to P-j^M^T is the supple- ment of the inclination of PJgM^T to the east meridian. 180 42 29' = 137 31'. In like manner, the inclination of T to PfgM^T, is the supplement of the inclination of P-j-JM-^T to the north meridian. Model 70 is a complete pyramid with an incomplete prism, having a rhombo- rectangular equator. Class 4, Order 5, Part II. page 114. e.) M T yr. PjgM^T. Model 66. A rhombic octahedron with its equatorial edge replaced by a rhombic prism with similar equatorial axes. The inclination of M T ^jT to 298 PRINCIPLES OF CRYSTALLOGRAPHY. is 90 + x > in which formula, x signifies the inclination of to the equator. A complete pyramid with an incomplete prism, and a rhombic equator. Class 4, Order 3, Part II. page 112. /) pjgt, Pl&M&T. Model 120. A rhombic octahedron with its acute terminal edges replaced by the form Py|)t, which has the same relation to axes p a and t% as the form P|gM T %T. Hence the inclination of p}gt to pigM T 8 ^T is 90 + ar, in which formula, x signifies the inclination of a plane of PyM T 8 n T to the east meridian. An incomplete pyramid, with a rhombo-rectangular equator. Class 6, Order 5, Part II. page 122. g) P_, MfT. pf m. Model 44. The inclination of P_ upon p-f m, is 90 -f- x, in which formula, x signifies the inclination of pf m to axis p a . A complete prism with an incomplete pyramid and a rhombic equator. Class 3, Order 3, Part II. page 107. h.) P_, m, t x , Mf T. pf m, pft. Model 50. The inclination of m upon Mf T is 90 -f x, in which formula, x is the inclination of MJ- T to axis m a . For pf m on P, see g.) The inclina- tion of P upon pft is 90 + x, in which formula, x is the inclination of pf t to axis p a . The inclination of pf m to m, or of pf t to t, is 90 + x, in which formula, x signifies the inclination of pf m or of pft to the equator. A complete prism with an incomplete pyramid and a rhombo- rectangular equator. Class 3, Order 5, Part II. page 109. t.) M_, MfT. Pf M. Model 100. Inclination of M on Mf T, see h.) Inclination of M on Pf M, see h.) If the angle is required across the edge that connects M with the solid angles on the east meridian, it may be found by the problem given in 331 ; but you must know the inclination of M|T and of PfM to the east meridian. This rule applies to all combinations containing M_T, P X M. If, also, you know the angle across the oblique edge arid one of the two other angles, you can with that information find the remaining angle, and determine the plane angles of the faces, by means of quadrantal solid triangles, because the east meridian of all combinations of M X T, with P X M is a side of 90. See 332. Model 100 is an incomplete prism with an incomplete pyramid, and a rhombo-rectangular equator. Class 5, Order 5, Part II. page 118. j.) Mf T. PJT. Model 82. When the angles across the zenith edge and north edge are known, the angle across the inclined edge can be found by means of a quadrantal solid triangle, as shown in 331. The inclination of Pf T to the west vertical edge is 90 -\- x, in which formula, x is the inclination of P|T to the equator. An incomplete prism with an incomplete pyramid and a rhombic equator. Class 5, Order 3, Part II. page 115. k.) M T 6 n T. P T 7 D T. Model 82 a . This combination was the subject of investigation in 328 332, and serves as a model for the investigation of combinations formed by two rhombic prisms of different zones, which combinations produce octane- PRINCIPLES OF CRYSTALLOGRAPHY. 299 drons with a rectangular base, arid afford quadrantal solid triangles when divided through the rectangular base. An incomplete prism with an in- complete pyramid and a rhombic equator. Class 5, Order 3, Part II. page 115. /.) T_, MftT. pjm, Py T. Model 1 10. The inclination of M T 6 D T to T is 90 + ar, in which formula, x signifies the inclination of M^T to axis t a . The inclination of Py>T to T is 90 4- x, in which formula, a? signifies the inclination of P l -f T to axis t a . To find the inclination o/"pjm Zn to axis p a . This is a very important calculation. Assume Model 1 10 to be divided into two portions by the north meri- dian. This section will divide the small rhombic plane pjm Zn, exactly through the middle, and its inclination to that plane will be 90. There are now two methods of proceeding ; one to be used when you know the inclination of P X T Zw to P X T Ze across the zenith edge; another to be used when you know the inclination of M X T n w to M X T n e across the north vertical edge. A right-angled solid triangle is used in either case, but with the former angle, the vertex of the triangle is that solid angle where p x m touches the zenith edge, and with the latter angle, the vertex of the triangle is that solid angle where p x m touches the north vertical edge. Put the inclination of M X T to axis m a = B ; the inclination of M X T to p x m = A, and the inclination of p x m to the north meridian = C = 90. Employing these data, and Formula 4, log cos a = log cos A + 10 log sin B, you obtain in side a of the triangle, the inclination of plane p x m Zn to the north vertical edge of the combination, the supplement of which angle is the required inclination of plane p x m Zn to axis p a . By the other of the two methods of proceeding, above alluded to, you obtain in side a, the inclination of p x m Zn to the zenith edge, which angle is 90o more than the inclination of the same plane to axis p a . The great use of this method of calculation is to find the inclination of an oblique plane to a vertical or horizontal edge upon which it rests, and when no other planes occur in the same zone. Model 110 is an incomplete prism with an incomplete pyramid and a rhombo-rectangular equator. Class 5, Order 5, Part II. page 119. m.) M^T. Pf T. Model 82 b . The account of Model 82% see .), applies equally to this combination. n.) M_, MfT. P|T. Model 104. Inclination of M to P|T = 90: of M to M|T, see h.) An incomplete prism with an incomplete pyramid and a rhombo-rectangular equator. Class 5, Order 5, Part II. page 119. o.) M, T. PM. Model 79 a . Inclination of T to all the other planes = 90, of M to Pf M, see h.) An incomplete prism with an incomplete pyramid and a rectangular equator. Class 5, Order 2, Part II. page 115. p.) T+, Mf T. P T 7 n T. Model 111. M, T+, Mf T. P T 7 ( ,T. Model 97. 300 PRINCIPLES OF CRYSTALLOGRAPHY. Inclination of T to Mf T, and T to P X T, see /.) The inclination of m to M X T, see h.) The inclination of M X T to P X T, see k.) Both com- binations are incomplete prisms with incomplete pyramids and rhombo- rectangular equators. Class 5, Order 5, Part II. page 119. q.) P+, T, M T 8 7 T. pijm. Model 55. Inclination of P to p x m, see g.) Inclination of T to M X T, see /.) In- clination of M X T to p x m, see t.) Inclination of p x m to the north vertical edge, see /.) A complete prism with an incomplete pyramid and a rhombo-rectangular equator. Class 3, Order 5, Part II. page 109. r.) (P + , T, M T 8 T T) x 2. Model 9. A twin crystal, each individual of which is a complete prism with a rhombo-rectangular equator. Class 1, Order 5, Part II. page 100. *.) p + , ML, T. Pf Mf T. Model 43. The value of the axes of the pyramid is determined from measurements across the Zn and Zw edges of the Form, by the method given in 574. Inclination of p + to Pf Mf T, see c.) Of m to PfMf T, see d.) When you know the inclination of M to P 7 Mf T, and of P 7 Mf T Znw to PMf T Zne, then you can determine the inclination of the Zn edge of the pyramid to axis p a by the process given in /.) In like manner, you can determine the inclination of the Zw edge of the pyramid to axis p a , when you know the inclination of P^Mf T Znw to Tw and to PfMf T Zsw. Model 43 is a complete prism with an incomplete pyramid and a rectangular equator. Class 3, Order 2, Part II. page 107. 0- P+> M_, T, MjT. P|M, Py>T, pgmjt Model 51. Inclination of p + to PM, see g.) to P^T, see h.) to pfmft, see c.) Inclination of M to M|T, see h.) to P|M, see h.) to pfmjt, see d.) Inclination of T to MT, see /.) to P^T, see h.) to pfmft, see d.) Inclination of M|T to pfmjt, see e.) Model 51 is a complete prism with an incomplete pyramid and a rhombo-rectangular equator. Class 3, Order 5, Part II. page 109. u.) MJgT, MffT. PI f T, Plfm^t. Model 90. The inclination of M^T nV to M-}T nw 2 , is composed of three quanti- ties, namely, x -f 90 + y, in which formula, x signifies the inclination of MJT to axis t a , and y the inclination of m||t to axis m a . In the present example, x is the angle of which ^| is the tangent, and y is the angle of which i is the cotangent. These two angles added to 90 produce the interfacial angle of M^T on mi|t. Model 90 is an incomplete prism with an incomplete pyramid and a rhombic equator. Class 5, Order 3, Part II. page 115. B. Hemihedral Forms of the Prismatic System. 578. The only Hemihedral Form of the Prismatic System which is noticed by ROSE, is the Rhombic Tetrahedron, the hemihedral form of the rhombic octahedron, a hemihedral form of rare occurrence and little importance. But the Biaxial Forms M X T. P X M, P X T, which are of so much im- portance in the Prismatic System, are all subject to become hemihedral, PRINCIPLES OF CRYSTALLOGRAPHY. 301 and to produce the Forms |M X T. |P X M, |P X T. How are these disposed of? They are classed together, to form the Fifth System of crystallisa- tion. That is the reason why no hemihedral forms, save the rhombic tetrahedron, occur in the Prismatic System ; while they are so very abun- dant in the Fifth or Oblique Prismatic System. 579. ASPECT OF COMPLEX CRYSTALS BELONGING TO THE PRISMATIC SYSTEM OF CRYSTALLISATION. To belong to the Prismatic System of Crystallisation, a crystal must have the following characters: 1.) Its axes must be p" x m y t a ; that is to say, three diameters, at right angles to one another, must be all different. 2.) When put into position, it must exhibit no hemihedral forms on the meridional zones. 3.) Its equator may be rectangular, rhombic, or rhombo-rectangular ; but cannot be quadratic, rhombo-quadratic, or hexagonal with angles of 120. All the combinations enumerated in 572, and every crystal of all the minerals enumerated in pages 61 to 77 of Part II. possess these three distinctive characters, by which, indeed, any crystal of the Prismatic System can be readily distinguished from one belonging to either of the three preceding systems. Among themselves, however, the crystals of the Prismatic System differ so considerably, that it is difficult to effect any definite subordinate classification. But in fact, the discrimination of the seven Forms of this system is so extremely easy, that a very elabo- rate classification of combinations is unnecessary. THE RHOMBIC OCTAHEDRON predominant. P_j_M_T. Model 21. Angle at pole Z replaced by: 1 horizontal plane = p. 2 planes, inclining on the east zone = p_t. 2 planes, inclining on the north zone = p_m. 4 planes, inclining on the planes, with edges of combination par- allel to the equator = p_m_t, in which the equatorial axes of both octahedrons are similar. ; 4 planes, inclining partly on the planes and partly on the north meridian = p_m_t + , in which axis t a of the upper pyramid is longer than axis t a of the predominant pyramid, axis m a remaining the same. 4 planes, inclining partly on the planes, and partly on the east meridian = p_m + t, in which axis m a of the upper pyramid is longer than axis m a of the predominant pyramid, while axis t a remains similar. Angle at pole n replaced by : 1 vertical plane = m. 2 planes, inclining on the north zone = p+m. 2 planes, inclining on the equator = m_t. 4 planes, inclining on the planes, with edges of combination par- allel to the east meridian = p+mit, in which axis m a of 302 PRINCIPLES OF CRYSTALLOGRAPHY. the subordinate pyramid is shorter than axis m a of the predominant pyramid, while axis p a and t a remain similar. 4 planes, inclining partly on the planes and partly on the north meridian = p + mzt + , in which axis t a of the subordinate pyramid is longer than axis t a of the predominant pyramid, axis p a remaining the same. 4 planes, inclining partly on the planes and partly on the equator = pjmzt, in which axis p a of the subordinate pyramid is longer, and axis m a shorter, than the same axes of the pre- dominant pyramid. Angle at pole w replaced by : 1 vertical plane =r t. 2 planes, inclining on the east zone = p+t. 2 planes, inclining on the equator = m + t. 4 planes, inclining on the planes, with edges of combination par- allel to the north meridian p_j_m_t_, in which axis t a of the subordinate pyramid is shorter than axis t a of the pre- dominant pyramid, while axes p a and m a remain the same. 4 planes, inclining partly on the planes and partly on the east meridian = p + m + t_, in which axis m a of the subordinate pyramid is longer than axis m a of the predominant pyra- mid, axis p a remaining the same. 4 planes, inclining partly on the planes and partly on the equator = pljlm_t_, in which axis p a of the subordinate pyramid is longer, and axis t a shorter than the same axes of the pre- dominant pyramid. Hence the solid angles of every predominant rhombic octahedron may be affected by subordinate octahe- drons in nine different ways. In all of these, however, the subordinate octahedrons shows two different edges, from measurements across which, the axial relations of the Forms can be calculated by the method given in 574. Equatorial edges replaced by : 1 tangent plane = m_t, where the index has the same value as the index of the equatorial axes of the predominant octa- hedron. 2 planes = pljlm_t, in which the subordinate octahedron has the same equatorial axes as the predominant form, but a longer vertical axis. Edges of the north meridian replaced by : 1 tangent plane = p + m, in which the index expresses the same value as the relation of p a to m a of the predominant octa- hedron. 2 planes = p_j_m_t_j_, in which the subordinate octahedron agrees PRINCIPLES OF CRYSTALLOGRAPHY. 303 with the predominant octahedron in the relation of p a to m a , but has t a longer. Edges of the east meridian replaced by : 1 tangent plane = p + t, in which the index represents the relation of p a to t a in the predominant octahedron. 2 planes = p_|_m + t, in which the subordinate octahedron agrees with the predominant octahedron in the relation of p a to t a , but has m a longer. Hence the edges of every predominant octahedron may be affected by subordinate octahedrons in three different ways. In all of these subordinate Forms, the relation of two axes is the same as that of the corresponding two axes of the predominant form, and the value of the third axis is found by taking the angle across the new bevelled edge, and using it as an equation with one of the known quantities proper to the predominant octahedron. THE RECTANGULAR PRISM predominant. P + , M_, T. Model 5. Solid angles replaced by 1 plane = p x m y t z . Vertical edges replaced by 1 plane = m x t. Edges across the north meridian replaced by 1 plane = p x m. Edges across the east meridian replaced by 1 plane = p x t. THE RHOMBIC PRISM predominant. P, M_T. Model 6. Obtuse solid angles replaced by : 1 plane = p x m. 2 planes = p x m y t z . Acute solid angles replaced by : 1 plane = p x t. 2 planes = p x m y t z . Terminal edges replaced by 1 plane = p x m y t,. Obtuse vertical edges replaced by : 1 plane = m. 2 planes = mlt. Acute vertical edges replaced by: 1 plane = t. 2 planes = m + t. THE COMBINATION M_T. P X T, (commonly called the RECTANGULAR OCTAHEDRON) predominant. Model 82 a . The position is supposed to be = M_T. P X T. Solid angles at e and w replaced by : 1 plane = t. 2 planes, in the equatorial zone = m x t. 2 planes, in the east zone = p x t. 4 planes, inclining on the edges = p x m,t,. 304 PRINCIPLES OF CRYSTALLOGRAPHY. In this case, the indices of the octahedron p x m y t z are found from measurements across the two new edges produced. Horizontal terminal edges replaced by: 1 plane = p. 2 planes = p_t. Obtuse vertical edges replaced by : 1 plane = m. 2 planes = m_t. Solid angles on the north meridian replaced by : 1 plane = p x m. 2 planes, inclining on the oblique edges = p x m y t z . In this case, only one edge of the resulting subordinate rhombic octahedron can be measured, but with that angle the inclination of that edge to axis p a can be calculated by means of an oblique-angled solid triangle, in which you have three given quantities: Angle B = half of M X T n e on n w; angle C = half of p x m y t z upon p x m y t z ; angle A = inclination of M X T upon p x m y t z . Taking these as the three angles of an oblique-angled solid triangle, then, side a of that triangle will be the supple- ment of the inclination of the edge between the two planes of p x m y t z to axis p a . See 328. When this is found, the rest of the calculation is made after the methods given in 574, 575. The formula to be used is No. 7. Given, A, b ; to find, a ; where A = half the inclination of plane p x m y t z Znw on plane p x m y t z Zne ; and b = inclination of the Zn edge between these two planes to axis p% and consequently the angle whose cotangent shows the relation of p a to m a of the given octahedron. Hence, side a of the triangle is the inclina- tion of the Zw edge of the same octahedron to axis p a , or that angle whose cotangent gives the relation of p a to t a , desired to complete our knowledge of the rela- tions of the three axes p a m a t a . V. THE OBLIQUE PRISMATIC SYSTEM OF CRYSTALLISATION. 580. The character of the Forms belonging to this system, as given by ROSE, is this: They have three axes, which are all unequal, and of which two cut one another obliquely, and are perpendicular to the third. The distinction between Homohedral and Hemihedral Forms is dropped in this system. See 578. ROSE'S enumeration of the Forms belonging to it is as follows : PRINCIPLES OF CRYSTALLOGRAPHY. 305 A, Rhombic Prisms: 1. Vertical Prisms = M X T. 2. Oblique Prisms: fZ 2 nwZ 2 ne. a. Basic =iP,M y T. either b. Front = iP + M,T. either c. Rear = P x M y T z either | Zne 2 3^ B. Single Planes : 1. Vertical Planes: a. Length Planes = T. b. Transverse Planes = M. 2. Oblique Planes: c iPJVl Z 2 n. a. Basic ^either | ^ ^ Zn 2 . b. Front = either (iPJMZs. c. Rear = either f^j ^ 581. The Models numbered 79, 84, and 87, exhibit three combinations of the oblique prismatic system of crystallisation. If, in these combin- ations, two of the three axes be considered to lye at right angles to one another, in a plane parallel to the larger terminal plane of each combin- ation, and if the third axis be considered to lye in the direction of the. four shortest edges of each combination, then the three axes will have the properties ascribed to them by ROSE; that is to say, two of the axes will cut one another obliquely, and be perpendicular to the third axis. All the descriptions which ROSE gives of the Forms and Combinations of the oblique prismatic system of crystallisation, are founded upon this hypothesis. But it appears to me, that the Forms and Combinations of this system can be much more conveniently described by reference to p a m a t a , the same three rectangular axes that are employed in all the other systems of crystallisation, than by reference to any system of oblique axes peculiar to itself. I do not intend, however, to institute here a com- parison between the two methods, but shall adopt the system of rec- tangular axes, and merely describe my own method of notation. 582, The peculiarities of the Forms and Combination of the Oblique Prismatic System of Crystallisation, considered in reference to the new method of notation, are pretty fully described in 340, 5), page 151 ; and in Part II. pages 77, 78. The chief peculiarities are as follow: a.) First, as to Axes: The axes are p x rap t*, or all three unequal. No single crystal of any mineral which belongs to this system ever possesses the axial relations of p 1 m a t a , or p* m a t a , or p x m^ t^. 2 it 306 PRINCIPLES OF CRYSTALLOGRAPHY. b.) Secondly, as to Forms: Except in one or two rare, perhaps doubtful, cases, the horizontal planes P are never present. All the vertical prismatic planes may be present, namely, M, M X T, T. The homohedral forms, P X M, P X T, and P x M y T z , are of rare, perhaps doubtful, occurrence. The biaxial hemihedral Forms iM x T, |P X M, iP x T, and the triaxial Form |P x M y T z , are abundant, and characteristic of this system. c.) Thirdly, as to Combinations: The minerals whose crystals belong to this system fall into two groups, whose combinations are as essentially different from one another, as the whole of them are collectively different from the crystals of the prisma- tic system. North Combinations: The crystals of one of these groups are characterised by hemihedral biaxial Forms belonging to the north zone, and hemihedral triaxial Forms, consisting of a single eidogen, whose axis lies in the plane of the north meridian. See 271, 272, a. b.), and 294 p.) East Combinations: The crystals of the other group are characterised by hemihedral biaxial Forms belonging to the east zone, and hemihedral triaxial Forms, con- sisting of a single eidogen, whose axis lies in the plane of the east meri- dian. See 271, 272, c. d.), and 294 p.) The minerals which belong to one of these groups never present crystals that belong to the other group. Hence, we may divide the Forms which ROSE ascribes to the oblique prismatic system of crystallisation, into two separate systems, as follows: Forms of JZast Combinations. M. T. MT. ZV. Forms of North Combinations. M. T. M X T. 1P X M Z ? n. 1P + M Zn 2 . 1P X M Zs. |P x M y T z Z 2 nw Z 3 ne. lP + M y T z Zn 2 wZri 2 e. 1P X M X T Z Zs 2 w Zs 2 e. 583. The prismatic portions of the crystals of these two groups have nothing peculiar, the equatorial planes of both being drawn from the series M, M_T, M + T, T. But this is not the case with the pyramidal planes. Here the two groups differ essentially, presenting two different systems of pyramidal terminations, totally irreconcileable with one another. Both of these combinations, as I have said at page 78, Part II., are commonly called Oblique Prisms; and it is said of the NORTH COMBINATIONS, that the terminal planes, namely, the Form ^P X M Zn, is set on the obtuse lateral Zw 2 . P X T Ze. |P X M T z Z ? nw Z 2 svr. |P+M y T z Znw 2 Zsw 2 . |P x M y T 2 Zne 2 Zse 2 . PRINCIPLES OF CRYSTALLOGRAPHY. 307 edge of the prism, while of the EAST COMBINATIONS it is said, that the terminal planes, namely, the Form JP X T Zw, is set on the acute lateral edge of the prism. The additional fact, that the hemioctahedrons of the North Combinations have always the positions ZnwZne NswNse, or ZseZsw Nne Nnw, and the hemioctahedrons of the East Combinations, always the positions Znw Zsw Nne Nse, or Zne Zse Nnw Nsw, and the farther peculiarity, that the cleavage of a mineral of either group is generally typical of the inclination of its terminal planes, are circum- stances which afford very decisive characteristics for the two groups of combinations, which collectively compose the crystals of this system. 584. To find whether a crystal belongs to the group entitled North Combinations, or to that entitled East Combinations. Put the prismatic planes into position, so as to suit the symbol M_T or M_, T. Then observe whether the terminal planes are biaxial or tri- axial ; that is to say, whether the terminal planes are crossed by the north or east meridian, or are merely separated from one another by these me- ridians. If the Forms are biaxial, and are on the north zone, the crystal belongs to the North Combinations. If the biaxial Forms are on the east zone, the crystal belongs to the East Combinations. If the Forms are triaxial, and the pair of zenith planes lye on the same side of the east meri- dian, the crystal is a North Combination. If the pair of zenith planes lye on the same side of the north meridian, the crystal is an East Com- bination. ROSE'S catalogue of the minerals that belong to the oblique prismatic system, is given in Part II. pages 5 13. A symbolic catalogue of the combinations presented by the crystals of each of these minerals, is given in Part II. pages 7791. 585. EXAMPLES OF NORTH COMBINATIONS. Model 79. M_, T. IPfMZn. Gypsum. 1 1 5. T_, M T %T. 1P T 5 3 M T %T Zne Znw. Gypsum. 75. T_, M&T. I P^M^T Zne Znw, JP& M i T Zse Zsw. Gypsum. 103. m, MfT. iP T \M Zn, JP+M_TZnV ZnV, |p x ra y t z Zne 9 Znw 2 . Azure Copper Ore. 84. M}T. |P^M Zn, Hornblende. 1 1 2. T, MjgT. 1 P T * y M Zn, P x M y T z Zse Zsw. Hornblende. 113. ( T, M|J T. 1 P T ^ M Zn, P x M y T z Zse Zsw) x 2. Hornblende. 67. MfgT. |P_ MJgT Zne Znw, ^P_ Mf$T Zse Zsw. Mesotypc. 79 b . M_, T. mM Zn. Epidote. 101. M_. JPiM Zn, pjjm Zs, ^P + M_T Zne ? Znw 2 . Epidote. 308 PRINCIPLES OF CRYSTALLOGRAPHY. 101 a . M_, t, mjf t |P|M Zn, Jpf m Zs, IP + M_T Zne 2 Znw 2 . Epidote. 81. MJT. iPiM Zn, HY^M Zs. ^fcpar. 109. T, MJJT. P|M Zn, JP&M Zs. Fefcpor. 105. T, |M|f T ne sw. iP^M Zn Ns. Felspar. 81 a . IMif T ne, |M| T nw. iplM Zn. 586. EXAMPLES OF EAST COMBINATIONS. Model 87. MfaT.|P~ 2 6 r TZw. Augite. 98. M, T, M|}T. iPjfj MiGT Znw Zsw. Augite. 99. (M, T, Mff T. IP^MJfT Znw Zsw) x 2. 53. p + , M, T, M|T. iP/j-MJJT Znw Zsw. ^w^'fc. 587. CLASSIFICATION OF CRYSTALS BELONGING TO THE OBLIQUE PRISMATIC SYSTEM. As respects the Classes. The occasional, perhaps doubtful, occurrence of the Form P, gives rise to a few complete prisms, Class 1. No homohedral octahedrons belong to this system. Therefore no complete pyramids occur. The presence of P gives rise to one or two complete prisms com- bined with incomplete pyramids, Class 3. The simultaneous presence of two hemioctahedrons, very much alike, in combination with a prism, presents occasional combinations resembling incomplete prisms combined with complete pyramids, Class 4. With these few exceptions, every combination of this system is an incomplete prism combined with an incomplete pyramid, Class 5. As respects the Orders. In consequence of the constant inequality of the equatorial axes m a and t% only three kinds of equators, and therefore only three Orders, can possibly occur among the crystals of this system, namely, the rectangu- lar, the rhombic, and the rhombo-rectangular. As respects the Genera. All the crystals of this system belong to the genus denominated p a m a t*. Hence the crystals of the oblique prismatic system, although very nu- merous, can only be referred to the following few Genera, and indeed are mostly comprised in the two last, namely: Class 5, Order 3, Genus 1, and Class 5, Order 5, Genus 2. Class 1. Order 3. Genus 1. Part II. page 98. _ 3. __ 5. 3. II. 109. _ 4. _ 3. 3. ~ II. 112. ^_ 5. 2. 1. II. 115. __ 5. 3. __ i* _ n. 115. _ 5. 5. 2f II. 118. PRINCIPLES OF CRYSTALLOGRAPHY. 309 * Groups c and g, North Combinations. * Groups e and h, East Combinations. f Groups /, m, n, o } q, s, u, and a:, North Combinations. Groups n,p, r, t, v, w, and y, East Combinations. MATHEMATICAL ANALYSIS OF THE COMBINATIONS OF THE OBLIQUE PRISMATIC SYSTEM. 588. PROBLEM. Given, Model 79 b , with the symbol M_, T. IPJM Zn; required, the value of x in the symbol. Figure s n n s in the margin represents the north meridian of Model 79 b , or rather of Model 79, in so far as regards the length of the prism, but of Model 79 b , in so far as regards the angle of in- clination of the pyramidal planes on the prismatic planes. The length of the prism is of no moment, as regards the following calculations. Line s n is the diagonal of plane |P X M Zn, and line s n, the diagonal of plane 1P X M Ns. If we take line s n for axis m a of the combination, and line Z N for axis p% then it is evident, that the value of the index x in the symbol, is the relation of line Z c to line c n, and it is also evident, that this relation is expressed by the tangent of the angle Z n c. But the angle which can be measured on the crystal is not the angle Z n c, but the angle Z n n, which is equal to the angle Z n c, added to the angle c n n, the prisma- tic angle of 90. Hence, if we put angle Z n n, or the inclination of ^P X M Z n on M n = X, then, tan (X 90) = value of x in JP X M Zn. Put X = 1 16 34'. Then, X 90 = 26 34'. tan = .5, or x = J. Hence the symbol is JP^M Zn. But if Model 79 b is taken to represent a crystal of Epidote, then, ac- cording to Phillips, X is 115 41'. Then X 90 = 25 41'. tan = .4809, or x = if. Here the symbol is iPf M Zn. According to Haiiy, X is 1 14 37'. Then X 90 = 24 37'. tan = .4582, or x = JJ. Here the symbol is iPJJM Zn. According to Mohs, X is 116 17'. Then X *- 90 = 26 17'. tan = .4939, or x = nearly i (within 17'). This gives the symbol |PJM Zn. These examples show that we have in the oblique prismatic, as in the prismatic, system, very different measurements given by different authori- ties, and which cannot be expressed by the same symbol. We have 310 PRINCIPLES OF CRYSTALLOGRAPHY. therefore either to choose a leader among these authorities, or else to adopt simple approximate numbers for indices, whenever we are uncertain which" authority it is most proper to follow. 589- The indices of the symbols of Forms belonging to the east zone, as |P x TZw, or |p x t Ze, are of course found in the same manner as the foregoing, from the inclination of |P X T Zw on T w, or of |p x t Ze on T e. 590. Sometimes several hemihedral biaxial forms occur on one crystal, as is shown by Models 101 and 101% the former of which exhibits the Forms iP|M Zn, Jpjjm Zs, and the latter the Forms iP|M Zn, |p|m Zs. The nature of these Forms is explained in the diagram in 588, where line i n represents the zenith plane of the Form ^PJMZn, and line i s the zenith plane of the Form ^pf m Zs. The value of the indices \ and \ is recognised at once on comparing the relation of the lines Z c to c n, and of p c to c s, since p c is four times as long as Z c. Upon the crystal, the angles that can be measured, are first, the angle Z n n already explained in 588, secondly, the angle i s w, or more frequently the angle of plane i s upon a vertical plane, as in Models 101 and 101 a , and thirdly, the angle s i n. When the angle measured is that of plane i s upon a vertical plane, the index of plane i s is found precisely in the same manner as the index of plane s n, 588 ; because the abstraction of 90 from the measured angle, leaves the residual angle i s o, the tangent of which is the relation of i o to o s, or of p c to c s. But when the angle measured is i s w, then it contains the angle i s o, and the angle o s w. The latter is equal to the angle Znc, the abstraction of which, when known, from i s w, leaves the angle i s o. When the measured angle is that at the summit, s i n, it contains two angles, namely, the angle n i o, which is the complement of Znc, sup- posed to be known, and the angle s i o, or the angle whose cotangent gives the index required for the plane s i. Take the case of the combination |P|MZn, Ipfm Zs, represented in the diagram, and examine the angles according to the above principles. Angle ino is 26 34', because i n n is 11 6 34'. Then, angle nio is 90 26 34' = 63 26'. Next, as s p c = s n s, it follows that s i o = 26 34'. Therefore, s i o + o i n = 26 34' + 63 26/ = 90. For the same reasons, angle i s n = 90, because i s o = 63 26' and o s n = 26 34'. 591. Sometimes the zenith pyramid of a combination of this kind, contains three, four, or more planes in one zone, the indices of all of which have to be separ- ately calculated. The figure in ,.-- the margin represents an exam- -*-- pie of this sort ; where s a, a b, and bn, represent the three zenith planes of the biaxial forms |p + m Zs 2 , |p x m Z ? s, iP x M Zn, Line s n is PRINCIPLES OF CRYSTALLOGRAPHY. 311 put = axis m a and line Z c = axis p a . A prismatic plane = M n is sup- posed to fall perpendicularly from n, and a similar plane = M s, from s. The mechanical measurements of the combination are assumed to be as follow : b n on M n 1 baonas b a on b n a s on M s The index of plane b n is derived from angle b n on M n, on the princi- ple explained in 588. The index of the plane a s is found from angle a s on M s in the same manner. When by these means, we know the angles n b y and s a x, or either of them, it is easy to find the index of the plane a b. If, for example, we know angles sab and sax, and wish to find q a b, then from angle sab, we first deduct sax, and then 90 x a q. The residue is = q a b, which is the angle whose tangent is the index of plane a b. If it is angle n b y that we know, then from a b n, we deduct n b y, and obtain as a residue a b q, the cotangent of which angle is the requir- ed index of plane a b. In analysing combinations of this kind, the only difficulty experienced is in finding the inclination of any one of the oblique planes to axis p a or to that equatorial axes to which the plane inclines. When this is done correctly, the indices of all the associated oblique planes of the same zone, however numerous, are found with ease. 592. The last diagram presents us with an explanation of the three different kinds of oblique planes which ROSE has termed Basic, Front, and Rear, 580. The plane marked a b is arbitrarily assumed to be the base of the prism peculiar to a given Mineral. Then the inclination of this plane to the vertical edge of the prism, or, what is equivalent, the inclination of the line b z to a line perpendicular to the line z n, is the fundamental or characteristic angle of the series of Forms belonging to that mineral. If now we assume the plane a b to be directed northwards, then, the plane s a, will be a front oblique plane of the series, and the plane bn will be a rear oblique plane of the series. There must be only one basic plane for a given mineral, but there may be any number of front or rear oblique planes. Which among all the oblique planes that occur in the series of forms belonging to a given mineral, is the one that is to be chosen for the basic plane, is a thing needful enough to be known, but which it seems cannot be taught. " The different front and rear terminal planes have the same properties as the basic planes, and we are at liberty to take whichever of these oblique planes we please for the basic planes, and to determine the fundamental Form accordingly. No other rule for choosing the basic planes can be laid down, than that already given for the choice of fundamental Forms in general, [ 479.] But the great variety of single planes and prisms that belong to the oblique prismatic system, often render the choice of the base of a mineral of this system much more difficult than is the choice of the fundamental Form of a mineral belong- 312 PRINCIPLES OF CRYSTALLOGRAPHY. ing to any other system." ROSE. [He then cites a combination which has three oblique terminal planes, any one of which may be denoted as the basic plane, or all of which may be denoted as front or rear oblique planes, occurring independently of basic planes.] It is evident upon a review of this matter, that by adhering to the system of rectangular axes, and considering all these oblique planes as hemihedral Forms of P X M or P X T, we can readily give every plane an ex- act symbol to indicate its obliquity to the equator, and by this means get rid of all doubt, difficulty, and confusion. 593. The explanation just given of the front, rear, and basic oblique planes, leads to the explanation of ROSE'S front, rear, and basic Oblique Prisms, 580. These oblique prisms are all, as already explained, hemi- octahedrons with parallel planes ; they differ among themselves in their axial relations, and consequently in their polaric positions. .) The Basic Oblique Prism is that which has the same relation to axes p a and m a , or to axes p a and t a , as the basic oblique planes. An example is shown in Model 103, where the large upper seven-sided plane is a basic plane, and the two scalene triangles near the poles Ze and Zw are the zenith planes of the basic oblique prism. Again, in Model 101, the four similar five-sided planes which are crossed by the east meridian, are the basic oblique prism, and the upper and lower planes where the north and east meridians cross, are the basic planes. Finally, the two terminal planes of Model 87 are the basic planes of the prism of Augite, while the four terminal planes of Model 98, are the basic oblique prism of the same mineral. The relation of the basic planes to the basic oblique prism is shown in the fact that the oblique diagonal of the basic plane has the same obliquity to the equator of the combination, as the oblique edge which separates the planes of the basic oblique prism. Hence, the edges of combination between the basic planes and the basic prism, are always parallel to one another. See Models 101, 10 l a , and 103. b.} The Front Oblique Prism is that which, on a north combination, occurs to the north of the basic oblique prism, and which, on an east combination, occurs to the west of the basic oblique prism. In other words, the front oblique prism is the hemihedral form of an octahedron that is more acute than the octahedron of the basic oblique prism, but which occurs in the same octants. Model 103 exhibits a front oblique prism in combination with a basic oblique prism, the former predominant. c.) The Rear Oblique Prism is merely the oblique prism of the in- verse octants, and^has, on a north combination, the positions Z s w Z s e, and on an east combination, the positions Z n e Z s e. Model 112 exhibits a basic plane and a rear oblique prism. Model 67, a front oblique prism, and a rear oblique prism. Model 75, the same. c?.) This classification and nomenclature of planes and prisms is of very little use, because the indices of the symbols of the different forms, and the notation descriptive of their polaric positions, give similar informa- tion in a more precise manner. PRINCIPLES OF CRYSTALLOGRAPHY. 313 594. PROBLEM. Given, Model 79> o, crystal of Gypsum, with the symbol M_, T. JPf-M Zn ; required, the inclination o/^Pf M Zn on Mn awe? on M s. .) The inclination of Pf M Zn on M n is that angle whose tangent is f , added to 90. By the Table of Indices, page 139, you find f .4286 = tan 23 12'. Then 23 12' + 90 = 113 12'. Upon applying the goniometer to Model 79> this will be found to be nearly correct. Ac- cording to Haiiy, the inclination in question is 113 8'. 5.) Upon referring to the figure in 588, it will be seen that the in- clination of plane JP X M Zn to plane M s is the supplement of its inclina- tion to plane M n, for as the four angles ,9 s n n are together equal to 360, so s and n, or s and n are together equal to 180. Hence, if the inclination of |Pf M Zn on M n is 113 12', then, |PfM Zn on Ms is 180 113 12' = 66 48', which agrees with measurement by the goniometer. 595. PROBLEM. Given, Model 84, a crystal of Hornblende, with the symbol M X T. |P X M Zn, and the angles M x Tnw on ne = 124 34', and |P X M Zn on M x Tnw = 103 13' ; required, the value of the Indices in the symbol. .) The index of M X T is the cotangent of 1 -^1 Cot 62 1 7' == .5254, (within 3' of .5265) = | g. This gives the symbol MjgT. .) Suppose Model 84 to be divided into two halves by a section through the north meridian. Take the west half as a right-angled solid triangle with pole Zn for its vertex. You have then the following given parts : C = 90 = inclination of P X M to the north meridian ; A = 76 47' = supplement of 103 13' the inclination of |P X M Zn on MfgTnw; and B = 62 IT = inclination of MlgTnw on the north meridian. With these data, you have to find a inclination of |P x MZn to the north vertical edge between planes M||j T nw and M-JT rie. This is the angle which is named X in 588, and the rule respecting which is again applicable here, namely : tan (X 90) = value of x in 1P X M Zn. Given, A, 76 47'; B = 62 17'; to find, a. Formula 4. log cos a = log cos A + 10 log sin B. 10 + log cos A = 76 47' = 19.3591 log sin B = 62 17 r = 9-9471 log cos a = 75 2' = 9.4120 In all calculations of this kind, one of the measured angles is always greater than 90, namely, the inclination of ^P X M or of |P X T on M X T. Hence, the supplement and not the measured angle, must be taken into the calculation, and hence also the product of the calculation is not the required angle but its supplement. See 330. In the present calculation, therefore, the negative product 75 2', must be changed for its supplement, 104 58', which is the required inclination of JP X M Zn on the north vertical edge. Deducting the constant quantity 2s 314 PRINCIPLES OF CRYSTALLOGRAPHY. 90, we have a residue of 14 58', the tangent of which is .2673, which differs only 2' from .2667 = -fj, which fraction I have therefore chosen to indicate the required value of x . Hence the symbol for Model 84, is 596. PROBLEM. Given, Model 87, a crystal of Augite, with the sym- bol Mf&T. iP^ 6 T T Zw; required, the inclination of MffT nw on ne, and the inclination of^P&T Zw on MfT nw. a.) The inclination of M JT nw on ne, is twice the angle whose co- tangent is . By the Table of Indices, page 139, the index |- = .952, and the corresponding angle is 46 24'. Then twice 46 24' = 92 48', which is the required inclination of plane ne on plane nw. b.) To find the inclination of plane IP^-T Zw on plane MfJTnw, across the Znw edge, proceed as follows : Assume Model 87 to be divided into two halves by the east meridian, and take the north half as a right- angled solid triangle with the solid angle near pole Zw for its vertex. Then, angle C of this solid triangle, = 90, will be the inclination of plane IP^yT Zw to the east meridian. Angle A will be the inclination of plane M|JTnw to the east meridian. This is the complement of the inclination of the same plane to the north meridian, found by a.) to be 46 24', and the complement of which is therefore 43 36'. Side b of the triangle will be the inclination of plane P^ 6 T T Zw to the west vertical edge between the prismatic planes nw and sw. This angle is equal to the prismatic angle of 90 added to that angle whose tangent is / T . By the Table of Indices, page 139, you find ^ T = .2857, which is the tangent of 15 57'. Then, 90 + 15 57' = 105 57' is the value of b. But since this angle is greater than 90', you cannot take it into the calculation, but must employ its supplement, which is 180 105 57' = 74 3'. With these given quantities, you have to find angle B, which is the required angle across the Znw edge of Model 87. Given, A = 43 36'; b = 74 3'; to find, B. Formula 8. log cos B = log cos b + log sin A 10. log cos b = 74 3' = 9.4390 + log sin A = 43 36' = 9.8386 log cos B = 79 4i' = 9.2776 Then, 180 79 4J-' = 100 55J'. This product, 100 55*', is the required inclination of IP^TZw on Mf^Tnw. c.) Investigation of the agreement of these calculated angles with the measured angles of natural crystals of Augite. According to Hatty, the inclination of MfJT nw on ne, is 92 18', which is 30' different from that expressed by MfQ-T, and would require the symbol Mf^T. According to Phillips, the same angle is 92 55', which, although differing but 7' from 92 48', could be better expressed by the symbol MJ-T. Hence the approximate symbol which I have adopted, M|T, indicates an angle betwixt the two quotations of Haiiy and Phillips. PRINCIPLES OP CRYSTALLOGRAPHY. 315 The inclination of the terminal to the prismatic plane of Model 87, calculated from the symbols, Mf$T. fcP&T Zw, to be = 100 55|', is according to Haiiy, 100 5', and according to Phillips, 100 10'. These quotations are rather wide of the mark, but I have retained the index ^y, because it appeared to agree better than any other, with the measure- ments and calculations of other forms of this mineral. 597. PROBLEM. Given, Model 112, a crystal of Hornblende, with the symbol T, M X T. iP x M Zn, |P x M y T z Zse Zsw ; with the angles nw on ne = 124 34'; P X M Zn on M X T nw = 103 13'; lP x M y T z Zse on Zsw = 149 38'; and |P x M y T z Zsw on M X T sw = 110 2'; required, the value of the indices in the symbol. a.) M X T is M$T. See 595, a.) b.). JP X M Zn is JP-^M Zn. See 595, b.) c.) To find the Indices of the Hemioctahedron |P x M y T z Zse Zsw. Assume Model 112 to be divided into two halves by the north meridian, and take the west half as an oblique-angled solid triangle, having for its vertex the solid angle near the pole Zs, where the two zenith planes of the hemioctahedron meet the two south planes of the prism. The given parts of this oblique angled solid triangle are as fol- low: Angle A = supplement of the inclination of P x M y T z Zsw on M X T sw, or 180 1 10 2' = 69 58^. Angle B = half the inclination of plane Zse on plane Zsw, or 1^ = 74 49'. Angle C = half the inclination of plane sw on plane se, or i~2? = 62 17'. With these data, you have to find side a, which is the inclination of the Zs oblique edge between the octahedral planes to the s vertical edge between the prismatic planes. The use to be made of the auxiliary angle denoted by a will be explained afterwards. Given, A = 69 58'; B = 74 49' ; C = 62 17'; to find, a. Formula 37. sin * a = V ~ ^ B^C"^ where S = \ (A+B+C). log sin \ a = i {log cos S + log cos (S A) + 20 (log sin B + log sin C)} A= 69 58' B = 74 49' S = 103 32' .C= 62 17' A= 69 58' 2)207 4' S A = 33 34' S = 103 32' Supplement of S = 76 28' log cos S = 76 28' = 9.3692 + log cos (S - A) = 33 34' = 9.9208 + 20 = 39.2900 _ / log sin B = 74 49' = 9.9846 ) \ + log sin C = 62 17' = 9.9471 / 2)19.3583 sin | a = 28 32' = 9.6791 316 PRINCIPLES OF CRYSTALLOGRAPHY. Twice this product, or 28 32' x 2 = 57 4', is the supplement of side a of the given oblique-angled solid triangle. See 330. The auxiliary angle is therefore 180 57 4' = 122 56'. This is the Zs 2 angle of the north meridian of Model 112. Hence, its supplement = 57 4' is the inclination of the edge betwixt the two octahedral planes to axis p a , the cotangent of which angle shows the relation of the given hemiocta- hedron to axes p a and m a . Cot of 57 4' == .6478. tan 57 4' = 1.5438. d.) To find now the relation of axes p a to t a of the same two octahedral planes, you need to employ a right-angled solid triangle with pole Z for its vertex, and the given quantities in which are as follow : angle A = inclination of plane P x M y T z Zsw to the north meridian = 74 49', and side b = 57 4' = inclination of the oblique edge between the octahedral planes to axis p a . With these given quantities, you can find side a of the right-angled solid triangle, which angle is the inclination to axis p a of an edge of the same octahedron assumed to rise from axis t a . This edge is not visible on the com- bination, but the cotangent of the angle thus found is the required rela- tion of axis p a to axis t a of the given hemioctahedron. Given, A = 74 49'; b = 57 4'; to find, a. Formula 7. log tan a = log tan A -f log sin b 10. log tan A = 74 49' = 10.5664 + log sin b = 57 4' = 9-9239 log tan a = 72 5' = 10.4903 Cot 72 5' = .3233 Tan 72 5' = 3.0930. e.) By the first operation, you find the relation of p a to m a to be as 1.0000 to 1.5438. By the second operation, the relation of p a to t a to be as 1.0000 to 3.0930. This gives the symbol Pi.ooroMi.wa8Ts.o93o. Here it is easy to see that axis m a is exactly half of axi-s t a . But the length of t a has been taken at 19 for the associated prism M}J T, and it is convenient to denote the transverse axis of the octahedron in re- ference to that standard. The relation of 1.5438 to 3.0930 is that of 91 to 19 or of 19 to 38. The equivalent number for p a is found by the proportion 3093 : 1000 :: 38 : 12 Hence the indices for |P x M y T 2 Zse Zsw, are %$ and $, affording the symbol iPJf MT Zse Zsw, and the complete symbol for Model 112 is T, M|$T. JP&M Zn, JPJf M$JT Zse Zsw. It is given erroneously in page 136, Part II. PRINCIPLES OF CRYSTALLOGRAPHY. 317 598. Miscellaneous Remarks on Calculations peculiar to the Forms of the Oblique Prismatic System. a.} The symbols of all the hemioctahedrons of the oblique prismatic system may be found by the method described in 597. The calculations are tedious, but unavoidable. There must always be two equations re- solved; one to determine the inclination of the single oblique edge be- tween the two zenith planes of the hemioctahedron to axis p a , and the other to determine the inclination of the absent oblique edge belonging to the transverse meridian. b.) Calculation of JP x M y T z Znw Zsw. When the oblique terminal edge of a hemioctahedron falls upon a vertical PLANE instead of a vertical EDGE, as it does on Model 98, then the calculations are much easier. In this case, we measure the inclination of |P X M V T Z Znw, first on Zsw and then on T w, and call the first A and the second B. We divide the Model into two halves by the east meridian, and take the north half as a right-angled solid triangle, with the solid angle at pole Zw for its vertex. Then, with the above given quantities, we find side b of the solid tri- angle, which is the inclination of the oblique edge to plane T w. The supplement of this angle is the inclination of the oblique edge to axis p a . When this is known, a second calculation is made on the model of that just given in 597, e?), in which the quantities used are the inclination of the oblique edge to axis p a , and half the angle across that edge, and by which we find the relation of axis p a to the third axis of the octahedron. This completes the calculation. c.) Model 101. To find the indices of JP x M y T z Zne 2 Znw 2 . The in- clination of one of these planes to plane |P^M Zn, is 90 more than its inclination to the north meridian. According to Haiiy, the given inclin- ation is 124 57' = 90 + 34 57'. The inclination of plane ^PIM Zn to axis p a (= 63 26', see 588), is equal to the inclination to that axis of the oblique edge between the planes Zne 3 and Znw 2 of the hemi- octahedron, which edge is replaced by the plane ^PJM Zn. Take an octant of a rhombic octahedron as a right-angled solid tri- angle with pole Z for its vertex, and the following given quantities: Given, A = 34 57', b = 63 26'; to find, a. Formula 7. log tan a = log tan A + log sin b 10. log tan A = 34 51' = 9-8444 + log sin b = 63 26' = 9.9515 log tan a = 32 Oi' = 9.7959 This product 32 0|' corresponds to cot 1. 6 = f . The relation of the three axes of the Form under investigation are therefore p a to t a = 8 to 5, and p a to m a ' 1 to 2 = 8 to 16. Hence the symbol for the hemioctahedron of Model 101 is JPf M'/T Zne Znw. d.) Model 109. T, Mf T. JPM Zn, iP/jM Zs. Felspar. The in- dex of PM Zn, is found from a right-angled solid triangle, consisting 318 PRINCIPLES OF CRYSTALLOGRAPHY. of the west half of the model, with the solid angle at pole Zn for its vertex, and with these given quantities: A = inclination of ^P|M Zn on M^f T nw. B = inclination of MJT nw on the north meridian. Then, a = supplement of the inclination of -JPiM Zn to axes p a . See 595, ). Next, the index of JP r 7 jM Zs is found, either by a similar process, or else by the method described in 590. ft) Model 103. m, Mf-T. 1P&M Zn, iP + M_T ZnV ZnV, ip x m y t z Zne 2 Znw 2 . Blue Carbonate of Copper. To find the indices of P + M_T ZnVZn 2 w 2 . Take the angle across the edge between ZnV and Zn 2 w 2 . Call the half of this B. Take the inclination of plane Zn 2 w 2 on ^P^M Zn. Call the supplement of this angle, A. Assume the model to be divided into two halves by the north meridian, and take the west half as a right-angled solid tri- angle, with the solid angle at pole Zn for its vertex. Then, with the aforesaid given quantities, A, B, and Formula 4, find side a, which is the inclination of plane JP/^M Zn on the oblique edge between planes Zn 2 e 2 and Zn 2 w 2 . From this auxiliary angle, first deduct the angle whose tan- gent is -f^, and then deduct 90. The residue is the inclination of the oblique edge to axis p a . When this is known, a second equation, on the model of that given in 597, d), gives the rest of the information re- quired for completing our knowledge of the length of the three axes of P + M_T ZnV Zn 2 w 2 . According to Hatty, the angle of plane ZnV on plane Zn 2 w s is 107 34'. The half of it = B = 53 47'. And the angle of plane Zn on plane Zn 2 w 2 is 116 36'. Its supplement = A = 63 24'. Formula 4. log cos a = log cos A -f 10 log sin B. 10 + log cos A = 63 24' = 19.6510 log sin B = 53 4V = 9.9068 log cos a = 56 18' = 9.7442 This being a negative cosine, we have to take its supplement = 123 42', as the value of the inclination of plane Zn on the front oblique edge. This will be found to agree with the measurement of the Model. The angle whose tangent is & = .2143, is 12 6'. This added to 90 is 102 6'. Deducting this joint sum from 123 42', we have 21 36' for the inclination of the oblique edge at pole Zn to axis p a . The tan- gent of 21 36' is .3959, which is the value of m a when p a is 1.0000. We next use the Formula given in 597, d) with these given quanti- ties : A = 53 47' ; b = 21 36'. Formula 7. log tan a = log tan A -f- log sin b 10. log tan A = 53 47' = 10.1353 + log sin b = 21 36' = 9.5660 log tan a = 26 41' = 9.7013 The tangent of 26 41' is .5026, which is the value of t a when p a is 1.0000. PRINCIPLES OF CRYSTALLOGRAPHY. 319 If we take p a = 10, then m a = .3959, is nearly 4, and t a = .5026, is nearly 5. This gives the symbol JPy>Mf T ZnV Zn 2 w 2 . The indices of the hemioctahedron |p x m y t z Zne 2 Znw 2 , also contained on Model 103, are found as follows. The inclination of the plane Znw 2 on the plane ^P_M Zn is 90 more than its inclination to the north meri- dian. The inclination to the equator of the edge across which this angle is taken, is equal to the inclination to the equator of the plane |P_M Zn. From these quantities we deduce the inclination of p a to m a of the axes of the form |p x m y t z . Then by a second calculation on the model of that given in 597, d), tne operation is completed. /) Model 75. T_, M T yJT. iP T 5 ? M T %T Zne Znw, JP&M^T Zse Zsw, Gypsum. Model 67. MfT. |P_MfgT Zne Znw, iPJMJgT Zse Zsw. Mesotype. In combinations of this sort, the indices of the hemioctahedron that produces the two planes Zne Znw, and of that which produces the two planes Zse Zsw, are found by separate calculations, both when the dif- ferent hemioctahedrons are quite unlike, and when they are so similar and equal as almost to constitute a homohedral octahedron. The angles required in all cases are six, namely, 1.) Prismatic plane nw on ne. 2.) Octahedral plane Znw on Zne. 3.) Octahedral plane Znw on prismatic plane nw. These are for the front hemioctahedron. The rear hemioc- tahedron also requires three angles: 1.) sw on se. 2.) Zse on Zsw. 3.) Zsw on sw. The six angles being found by measurement, the cal- culations are made according to the examples given in 597 c and d). The ratios of the axes being thus determined, the indices are easily found for the symbols. 599- The combinations of this system are so exceedingly numerous and diversified, that it is impossible to reduce the instructions for calcu- lating the indices of their symbols to so orderly an arrangement as was contrived for the instructions given in the foregoing systems. I have endeavoured, however, to give such a collection of examples, as will serve to convey a variety of information relative to the calculations best suited to particular cases. I have also endeavoured, by grouping the Forms and Combinations of the system in several different modes, to show their mutual relations and dependencies in as striking a manner as possible. See 580, 582, 587; and Part II. pages 1 15 and 118. The calculations might have been carried into much greater detail, but not without frequent repetitions of calculations given in preceding sec- tions, which did not appear to me to be expedient. I hope the principles upon which the different calculations are to be made, are explained so fully as to enable the reader to supply the details easily. 320 PRINCIPLES OF CRYSTALLOGRAPHY. VI. THE DOUBLY OBLIQUE PRISMATIC SYSTEM OF CRYSTALLISATION. 600. The character of the Forms belonging to this system, as given by ROSE, is this : They have three Axes, which are all unequal, and which cut one another obliquely. ROSE'S enumeration of the Forms belonging to this system of Crystal- lisation, is as follows : The planes that belong to the Forms of this system all occur in sets of two, and they are of three kinds : 1. Planes that cut three Axes = JP x M y T z . 2. Planes that cut two Axes = M X T. 3. Planes that cut one Axis = M or T. In referring crystals of this class to a system of three rectangular axes, we have merely to give each pair of planes a specific denomination. But according to both systems of notation, every pair of planes must have a different name, and as it is as easy to name them in reference to rectangular axes as to oblique axes, it does not appear that any advantage is gained by the adoption of the doubly oblique system of axes. The properties of the Forms and Combinations belonging to this sys- tem, are described in 340, 6), and in Part II. page 91. The only homo- hedral forms which commonly occur are M and T. In Part II. I have admitted M X T to be an occurring homohedral form, especially on Axin- ite ; but I have had no opportunity of examining good minerals of this class, and having founded this opinion merely on the figures given in books, I am doubtful of its correctness. The forms of most frequent occurrence on the minerals of this system are M X T and |P x M y T z . These forms consist invariably of a pair of parallel planes. Every doubly ob- lique combination must contain at least three of such pairs of planes, of which two pair must belong to the prismatic zone, and one pair to the octahedral zone, and none of which must meet at a right angle. A single combination sometimes contains as many as 12 pair of planes, all belong- ing to the series M, T, |M X T. iP x M y T z . There are never present any planes of the Forms P, P X M, P X T. The Axes of all the Combinations belonging to this system are p x m y t*. Rose's Catalogue of the minerals that belong to the doubly oblique prismatic system, is given in Part II. pages 7 13. A symbolic catalogue of the Forms and Combinations presented by the crystals of each of these minerals, is given in Part II. pages 91 94. Every crystal of this system is an incomplete prism, combined with an incomplete pyramid. They all fall therefore into Class 5. The equator of every combination is either rhombic, or, when M or T is present, rhombo-rectangular. Hence all the minerals embraced by this system, belong either to Class 5, Order 3, Genus 1, Group , Part II. page 116, or to Class 5, Order 5, Genus 2, Group z, Part II. page 119- - PRINCIPLES OP CRYSTALLOGRAPHY. 321 601. MATHEMATICAL ANALYSIS or THE COMBINATIONS OF THIS SYSTEM. Model 81 b . Mf T. iP x M y T z Z 2 nw, Jp x M y T z Zn 2 e, ip x m y t z Z 2 ne 2 . a.) The index j of Mf T is determined from the obtuse angle of the equator at the north pole. When a combination contains several pair of prismatic planes, you begin by determining the obliquity of any one pair to axis m a or axis t a , and then estimate the obliquity of the rest accord- ing to their inclination to the known pair of planes. b.) The Model presents an oblique-angled solid triangle at pole Zw, formed by the meeting of Mf T with P x M y T 2 Z 2 nw. You take the three angles across the edges which meet here, and by means of an equa- tion with Formula 37, you determine the plane angle at pole Zw of the plane Mf T nw. Secondly, you assume the Model to be divided by the east meridian into two halves, and you take the north half as an oblique- angled solid triangle, with pole Zw for its vertex. The known quantities are now : A = inclination of P x M y T z Z 2 nw on plane Mf T nw ; B = inclination of Mf T nw on the east meridian ; c = plane angle of M f T nw at pole Zw. With these data, you can, by two separate processes, determine the inclination of plane |P x M y T z Z 2 nw to the east meridian, and the inclination of the Z 2 w oblique edge of the east meridian to the west vertical edge. Thirdly, by means of the last two products, em- ployed in an equation with a right-angled solid triangle, you can deter- mine the inclination of axis p a to the edge of intersection between plane P x M y T z Z 2 nw and the north meridian. By these calculations you find the inclination of axis p a to the purple line drawn on the Model from Z towards w, and to the blue line drawn from Z towards n. The tangents of these two angles are the indices of the Form |P x M y T z Z 2 nw. The calculations are made exactly in the manner of the calculations described in the last section. See 597, 598. But as the determination of the indices of one pair of planes requires three measurements to begin with, and these to be employed in three oblique-angled and one right-angled solid triangle, it follows that the investigation of Forms belonging to this system is particularly tedious. Hence it follows that they have been very little attended to by Crystallographers, and that the descriptions we meet with of crystallised Minerals belonging to the doubly oblique pris- matic system, are not much to be relied on for correctness. or their Hemihedral or Tetartohedral Varieties. 602. I have stated, in 244, that Mr. BROOKE'S symbols for Combin- ations are of the same character as HAUY'S. This, however, alludes to MR. BROOKE'S symbols for expressing "the character of the modifying planes of crystals, and their geometrical relations to the primary Form, as connected with the theory of decrements ;" according to which theory, " the secondary Forms of Minerals consist of Modifications of the pri- mary, occasioned by decrements on some of their edges or angles ;" and the symbols alluded to constitute "a system of notation connected with the same theory, and capable of expressing the figure of any secondary Form." H. J. BROOKE'S Familiar Introduction to Crystallography, 8vo, London, 1823. The book just quoted, contains, however, another and different system of Crystallographic notation, which, being independent of mathematical calculations, and comprehensible without much thought, has been oft- ener adopted and referred to by English writers, than any of the more scientific systems of Crystallography. This method consists in giving figures of a certain number of Forms, under the name of "primary Forms;" and then representing all the different angles and edges of these primary Forms as being replaced by one or more modifying planes, which are labelled a, b, c, and so on to x, y, z. Mr. Brooke gives 1 76 figures and descriptions of these primary Forms, their Modifications, and com- plete secondary Forms which they produce. The number of particular modifications thus classified, is one hundred and fifty. The number of primary Forms is fifteen. The number of complete secondary Forms is twenty-three. Now it will be easy to show that the whole of these Forms, simple and compound, consist merely of varieties of the seven Forms, P x , M x , T x , M X T. P X M, P X T, P x M y T z , or of their Hemihedral or Tetar- tohedral Varieties ; and that consequently the symbols of these Forms can be used to replace the unsystematic notation employed by Mr. Brooke. PRINCIPLES OF CRYSTALLOGRAPHY. 323 Catalogue of MR. BROOKE'S Primary Forms and their Modifications. Every symbol expresses the primary Form as well as the Modification. In some cases, the same symbol expresses several of Mr. Brooke's Modi- fications. See the Rectangular Octahedron, h, i> k, I, and the Rhombic Octahedron, d, e,f. But when the algebraic indices x,y, z, are replaced by numerical radices, 1, 2, 3, &c., every symbol acquires a specific meaning. THE CUBE, P, M, T. a. P, M, T. pmt. b. P, M, T. 3 p_mt. c. P, M, T. 3 p + mt. d. P, M, T. 6p_mt + . e. P, M, T, mt. pm, pt. / P, M, T, m_t, m+t. p_m, p+m, P-t, p+t. g. P, M, T. |pmt. h. P, M, T. l (3 p_mt). i. P, M, T. 3 p_mt+. h. p, m, t, M_T. P_M, P+T. THE TETRAHEDRON, |PMT. a. IPMT, Jpmt. rmt. pm, pt, IPMT. \|PMT, | (3 p+mt) Znw. c. IPMT Znw, l(3p_mt) Zne. IPMT Znw, | (6 p_mt+) Zne. m_t, m+t. p_m, p+m, p_t, p+t, IPMT. m_t. p_m, p+t, |PMT. e. p, m, t. JPMT. THE OCTAHEDRON, PMT. a. p, m, t. PMT. b. PMT, 3p_mt; [PMT. c. m_t, m+t. p_m, p+m, p_t, p+t, d. PMT, 6 p_mt+. e. mt. pm, pt, PMT. / PMT, 3 p+mt. THE RIGHT RECTANGULAR PRISM. [Mr. Brooke places this Form so as to make it = Px, M_, T+. J. J. Gr.] a. P+,M_,T. Px m y t z . b. P+,M_,T, m_t. c. P+,M_,T. Px t. d. P+,M_,T. p x m. b. d. THE OCTAHEDRON WITH A RHOMBIC BASE. P+M_T: orP x M y T z . a. p + . P + M_T. b. p_m, P + M_T. c. p_t, P + M_T. d. P + M_T, p x m y t z Z 2 nw. e. P+M_T, p x m y t z Z 2 nw. \ f. P + M_T, p x m y t z Z 2 nw. g. m. P + MT. h. p + m, P + M_T. t. m_t. P + M_T. h. P+M_T, p x m y t z Zn 2 w. /. P + M_T, p x m y t z Zn 2 w. m. P + M_T, p x m y t z ZnV. n. t. P + M_T. o. p + t, P + M_T. p. m + t. P + M_T. q. P + M_T, p x m y t z Znw 2 . r. P + M_T, p x m y t z Znw 2 . s. P + M_T, p x m y t z Znw 2 . t. p + m, P + M_T. u. P+M_T, p x m y t z Z 2 n 2 w. v. p+t, P + M_T. x. P + M_T, p x m y t z Z 2 riw 2 . y. m_t. P+M_T. z. P + M_T, p x m y t z Zn 2 w 2 . THE RIGHT RHOMBIC PRISM. P+,M_T. a. P X ,M_T. p x m. b. P X ,M_T. p x m y t z Zn 2 w Zn 2 e. c. P X ,M_T. p x t. d. P X ,M_T. p x m y t z Zne 2 Znw 2 . e. P X ,M_T. p x m y t z ZnV Zn 2 w 2 . / P x ,m,M_T. g. P x ,mrt,M_T. h. P x ,t,M_T. . P X ,M__T, m + t. 324 PRINCIPLES OF CRYST ALLOGRAPH Y. THE OCTAHEDRON WITH A SQUARE BASE. P X M, P X T : or PJMT. [Either of these symbols may be taken to represent a square-based octahedron. The choice is regulated by the value of the index x- See 480. Mr. Brooke's Modifications, however, refer to the latter symbol. J. J. G.] a. p. P X MT. b. P X MT, p_rat. c. p_m, p_t, P X MT. d. P X MT, p x ra y t z , p x m z t y . e. m, t. P X MT. / p+m, P+ t, P,MT. g. m_t, m + t. P X MT. h. P X MT, Px m y t z ,p x m 2 t y . i. P X MT, p + m y t z , p + m 2 t y . k. P X MT, p+m y t z , p+m 2 t y . /. p x m, p x t, P X MT. m. P X MT, p x m y t z , p x m.t y . n. int. P X MT. o. P X MT, p + mt. THE OCTAHEDRON WITH A RECTANGULAR BASE. P X M, P X T. [I consider these Combinations to be Mo- difications of the Rhombic Prism, and alter the positions' accordingly. J. J. G.] M_T. P X T. a. t, M_T. P X T. b. M_T, ra+t. P X T. c. M_T. P X T, p+t. d. M_T. P X T, P+ m y t z . e. M_T. p x m, P X T. / M_T. P+ m,P x T. g. M_T. p_m, P X T. h. M_T. P X T, p x m y t z . t. M_T. P X T, p x m y t z . k. M__T. P X T, Px m y t z . /. M_T. P X T, Px m y t z . m. m, M_T. P X T. n. nO, M_T. P X T. o. p, M_T. P X T. p. M_T. p_t, P X T. THE RIGHT-SQUARE PRISM, P X ,M,T: orP x ,MT. a. P X ,M,T. p x mt. b. P X ,M,T. p x m y t z , p x m z t y . c. P X ,M,T. p x m, p x t. d. P X ,M,T, mt. e. P X ,M,T, m_t, m+t. THE OBLIQUE RHOMBIC PRISM. Either M_T. JPJVIZn: or M_T. JP X T Zw. a. M_T. JP X M Z 2 n, |p + m Zri 2 . b. M_T. |P X M Zn, ip x m y t z Zne Znw. c. M_T. JP X M Zn, |p x m Zs. d. M_T. ^P X M Zn, lp x m y t z Zse Zsw. e. M_T. 1P X M Zn, ip x m y t 2 Zne 2 Znw 2 . M^T. 1P X M Zn, ip x ra y t z ZnV Zn 2 w 2 . g. MJJT. iP x M Zn, ^p x m y t z ZsV Zs 2 w 2 . h. m, M_T. |P X M Zn. i. net, M_T. |P X M Zn. k. t, M_T. !P x MZn. /. M_T,m + t.iP x MZn. THE RHOMBIC DODECAHEDRON. MT. PM, PT. . p, m, t, MT. PM, PT. b. MT, m_t, m + t. PM, p_m, p + m, PT, p_t, p+t. c. MT. PM, PT, 3 p_mt. d. MT. PM, PT, 6 p_mt + . e. MT. PM, PT, pmt. / MT. PM, PT, 3 p + mt. g. MT. PM, PT, 3 p_mt. h. MT. PM, PT, 6 p_mt + . t. MT. PM, PT, 3 P_MT. k. MT. PM, PT, 6 P_MT + . THE RIGHT OBLIQUE-ANGLED PRISM. Either M_, T. JP X T Zw: or M x , T. iP x M Zn. [Placed by Mr. Brooke in the position of P_, iMxTnw, ^MxT ne. J. J. G.] a. M_,T. P X T Zw, |p x m y t z ' Znw Zsw. b. M_,T.iP x TZw,ip x m y t z ZneZse. c. M_, T, m_t. ^P X T Zw. d. M_, T.1P X T Zw, ip x m y t z Zn 2 w 2 Zs 2 w 2 . e. M_, T. |P X T Zw, ip+t Zw 2 . / M_, T. |P,T Zw, Jp+t Ze. PRINCIPLES OF CRYSTALLOGRAPHY. 325 THE REGULAR HEXAGONAL PRISM. P X ,T, M-JfT 2 :orP x , V. a. P X ,V. 2r x Zn Zs. b. P X ,V. 3p x m y t z . c. P X ,V. 2 Bx Zw Ze. A P X ,V, v. e. P x , 3m x t. THE RHOMBOID. iP.T, iP.Mif T 2 , or R x . [Rx is assumed to be Ri. J. J. Gr.] a. p. RJ. 6. R, Zw, r_ Zw. c. R! Zw, r_ Ze. rf. R! Zw, s_ Zw. Tetrahedron, e. Octahedron, . Dodecahedron, a. The Right- Square Prism == P X ,MT, or P X ,M,T. Right- Square Prism, d. Quadratic Octahedron, a with e, or a with n. The Right Rectangular Prism. = P X ,M_,T. Rectangular Octahedron, a, m, ando. Rhombic Octahedron, , g, and n. Right Rhombic Prism,/and h, with two primary planes 326 PRINCIPLES OF CRYSTALLOGRAPHY. The Right Rhombic Prism. = P X ,M_T. "c and m, p and m, Rectangular Octahedron,- a and e, a and /, Rhombic Octahedron, a and g, b and o, ^n and o. 'a and t, 05 and /?, a and y, 6 and w, A and w, t and TZ, c and g, o and ^, ^v and g. -\ with one pair d I of planes of the > /> primary at right angles to the J modification. Right Rhombic Prism, g, or t. Right Rectangul Prism ght f ngular -s ism, l 7%e Ttig^f Oblique-angled Prism. or M X ,T.|P|M Zn. Oblique Rhombic Prism, h arid , with two primary planes remaining. The Rhomboid. = R x . Rhomboid, modifications b, c, #, , or m. 3. CONTAINED WITHIN EIGHT PLANES : The Regular Octahedron. = PMT. Cube, . Tetrahedron, with a. R. Dodecahedron, e. The Quadratic Octahedron. = P X MT: or P X M, P X T. Quadratic Octahedron, by c, f, /, or o. Right square Prism, a or c. The Octahedron with a Rectan- gular base: M_T. P X T: or M_T. P X M. Right Rectangular Prism, b and c. Right Rhombic Prism, a or c, with four primary planes added. The Octahedron with a Rhombic base. = P x M y T z . Rectangular Octahedron, h, i, k, or /. Rhombic Octahedron, d, e, f, k, I, m, q, r, s, u, x, or z. Right Rectangular Prism, a. Right Rhombic Prism, b, d, or e. The Hexagonal Prism = P x ,T,Mjf T 2 : or P X ,V. Rhomboid, a and e, or a and o. Hexagonal Prism, d. Right Rhombic Prism of 120, h. 4. CONTAINED WITHIN TWELVE PLANES : a. The planes being isosceles trian- gles. (Figure without name). =. |(3P_MT) Zne. The left Hemiicositessarahedron. Tetrahedron, c. Cube, h. (Figure without name). = i(3P_MT) Znw. The right Hemiicositessarahedron. Tetrahedron, f. (Figure without name). = 2R+, orP + T,P + MjfT 2 . An Acute six-sided Pyramid. Hexagonal Prism, a or c. Rhomboid, particular planes of d, h, i, I) or n. [This appears to be a mistake. I do not think it possible for particular planes of the Scalenohedron to produce regular isosceles six-sided pyramids, with lateral edges situ- ated in the plane of the equator. J. J. G.] b. The planes being scalene triangles. (Figure without name). S x . The Scalenohedron. Rhomboid, d } h, ?', /, , or p. PRINCIPLES OF CRYSTALLOGRAPHY. 327 c. The Planes being Rhombs. The Rhombic Dodecahedron. = MT. PM. PT. Cube, e. Tetrahedron, b. Octahedron, e. d. The planes being trapezoids. (Figure without name). = i(3P+MT). The Hemitriakisoctahedron. Tetrahedron, b. e. The planes being pentagons. (Figure without name). =. M_T. P_M, P+T. The Pentagonal Dodecahedron. Cube, h. Tetrahedron, d? 5. CONTAINED WITHIN SIXTEEN TRIANGULAR PLANES. (Figure without name). =. P x M y T x , P x M z T y . An eight-sided Pyramid of the Pyramidal System Octahedron with square base J, h, i, k, or m. Right square Prism, b. 6. CONTAINED WITHIN TWENTY- FOUR PLANES. a. The planes being isosceles trian- gles. (Figure without name). = M_T, M+T. P_M, P+M, P_T, P+T. The Tetrahishexahedron. Cube,/ Tetrahedron, d? Octahedron, c. R. Dodecahedron, b. (Figure without name). = 3P+MT. The Triahisoctahedron. Cube, c. Octahedron,/ R. Dodecahedron,/ b. The planes being equal trapezoids. (Figure without name). = 3P_MT. The Icositessarahedron. Cube, b. Octahedron, b. R. Dodecahedron, c, g, or t. 7. CONTAINED WITHIN FORTY- EIGHT TRIANGULAR PLANES. (Figure without name). = 6P_MT + . A Hexahisoctahedron. Cube, d. Octahedron, d. R. Dodecahedron, d, h, or k. " On the Application of the Tables of Modifications; The preceding Tables of Modifications are adapted principally to two purposes. The first is, by the remarks they contain upon the comparative characters of the secondary Forms belonging to the different classes of the primary, to assist the mineralogist in determining the primary Form of any Mineral ' from an examination of its secondary Form. And the second is to enable him to describe any secondary crystal, whose primary Form is known. An attempt is thus made to supply a language, by means of which the second- ary Forms of crystals may be described independently of the theory of decrements, and without the assistance of mathematical calculation." BROOKE, Introduction, p. 223. The " remarks" alluded to by MR. BROOKE, describe the Forms and Combinations which I have represented in symbols. In MR. BROOKE'S book, these remarks, and the figures to which they relate, occupy 129 328 PRINCIPLES OF CRYSTALLOGRAPHY. octavo pages, and yet convey little more information than is given by the present brief abstract of six pages. I mention this circumstance as a proof that the new symbols possess great descriptive powers, and may therefore well replace the notation of MR. BROOKE, which is so inartifici- ally constructed that no memory can retain it. The system presented by Mr. Brooke is, indeed, one of apparent ease but real difficulty, as a single investigation must satisfy every mineralogist- Suppose the student to have for examination the crystal represented by Model 32, containing the Forms, P, M, T, mt. pm, pt, PMT, and exhibiting a cube truncated on the edges and angles, or a combination of the cube, the rhombic dodecahedron, and the regular octahedron. Well, this is a secondary Form, and the student wants to know the primary, and the reason of his wanting to know the primary is, that in MR. BROOKE'S Cata- logue of Minerals, the primary Form of each mineral alone is named, so that the most perfect knowledge of the secondary Form, or natural crystal, is not sufficient to give the name of the Mineral, even if only a single Min- eral should be known to exist in the shape of that particular secondary Form, and the same Mineral be totally unknown in the shape of its pre- sumed primary Form. The student therefore looks in the Table of second- ary Forms for P, M, T, the Cube, where he finds that this secondary Form indicates as its primary, The Tetrahedron, Modification e. The Regular Octahedron, Modification a. The Rhombic Dodecahedron, Modification a. He next looks in the Table of secondary Forms for MT. PM, PT, the Rhombic Dodecahedron, which Form is also upon his crystal. This indi- cates as its primary, The Cube, Modification e. The Tetrahedron, Modification b. The Octahedron, Modification e. He finally looks in the same Table for the Form PMT, the Octahedron, which also forms part of his secondary crystal, and finds that this indi- cates as its primary, The Cube, Modification a. The Tetrahedron, with Modification a. The Rhombic Dodecahedron, Modification e. He has therefore A CHOICE of no less than four primary Forms, namely, The Cube, The Rhombic Dodecahedron, The Octahedron, The Tetrahedron. And here ends the power of the Table to guide him in finding the re- quired primary Form. What is the remedy for this difficulty ? First, he may destroy the crystal to find the cleavage. But this is a method always expensive, and often impossible. Secondly, he may guess at the primary from the predominant Form, which in this case is the cube. But suppose that Model 33, P, M, T, mt. pm, pt, PMT, had been the secondary crystal under examination, the Tables would then lead to the same result as in the above case, while the predominant Form would induce the student to PRINCIPLES OF CRYSTALLOGRAPHY. 329 choose the octahedron as the required primary. And if Model 34, p, m, t, MT. PM, PT, pint, had been the given secondary Form, the student would be induced by the same routine of examination, to choose the Rhombic Dodecahedron as its primary Form. And after all this examination, the chances are exactly two to one that the examiner comes, to a wrong conclusion. For, suppose him to have a Mineral in the Form of Model 32, P, M, T. mt. pm, pt, PMT, and that he chooses the predominant Form, or the cube, to be the primary, and con- sistently with this choice names his crystal, the Cube with Modifications a and e. Then, if his Mineral happens to be Grey Copper, he will be wrong, because Mr. Brooke says the primary Form of that Mineral is the tetrahedron, in which case Model 32 must be described as the Tetrahe- dron with, Modifications , e, and b. If the given Mineral was Fluorspar, the student would be equally in the wrong, because the primary Form of that Mineral is assumed to be the regular octahedron, in which case, Model 32 is described as the Octahedron with Modifications a and e. Finally, if the Mineral was Galena, the student would be right, because the primary Form of this Mineral is the cube, and Model 32 has then to be labelled the Cube with Modifications a and e. The conclusion which I draw from this investigation is, that the doc- trine of primary or primitive Forms, is worthless and mischievous, when made the basis of a system of Crystallographic notation. The search for '' primary Forms'* with the assistance of MR. BROOKE'S Tables, or with any other help, is a search for a nonentity a mere waste of time. WEISS'S discovery of the relations of the axes of crystals to their planes, has abolished primary Forms for ever. SECTION XV. ON THE UTMOST POSSIBLE ABRIDGMENT OF EXACT CRYSTALLOGRAPHIC NOTATION. 603. Brevity is a desirable qualification in Crystallographic Notation, but it is of much less importance than accuracy and intelligibleness. We must never therefore displace a symbol which is accurate and intelligible, by a sign, however short, which gives an ambiguous indication, or which is liable to be forgotten or to be mistaken for something different. In constructing the notation employed in this work, I have therefore not struggled to attain the utmost possible limit of shortness. I think indeed that brevity may in this matter be carried to an absurd and injuri- ous extent, and it has been a rule with me to carry out my principle of notation, consistently and faithfully, to its full extent, rather than to take alarm at the occasional occurrence of a long symbol, and for the sake of avoiding that long symbol, to cut and carve at the notation, and break down the unity of the system, and produce, not a useful working nota- tion, but a collection of enigmas. 2 u 330 PRINCIPLES OF CRYSTALLOGRAPHY. The notation which I have adopted conveys information on four prin- cipal points, namely : * 1. The names of the axes that are cut by each Form or set of planes. 2. The lengths measured from the centre of the crystal, at which these axes are cut. 3. The comparative magnitude of each set of planes on a combination. 4. The polaric positions of the planes of hemihedral Forms. None of this information can be prudently suppressed. Therefore, no- tation is inadmissible which is unable to convey all these particulars. On the other hand, that system of notation which indicates all these rela- tions of planes most completely and most conveniently, is the best. By these principles the value of any notation may be estimated. To name the~axes that are cut by each Form, I denote the three axes by the letters P, M, T. To denote the lengths of the axes, I employ the vulgar fractions J, J, &c., as Indices. When a plane cuts one axis, the index is written after the letter, as PJ. When it cuts two axes, the index is written between the two letters, as Mf T, where it indicates the rela- tion of the first-named axis to the second. When it cuts three axes, the relation of the first to the third is expressed after P, and the relation of the second to the third is expressed after M, as P|M|T. There must always be two indices to the symbol of a triaxial form, and one index to the symbol of a biaxial form, but the indices of axes that are unity may be suppressed. Thus, M|T is written MT, and P-{M}T is PMT. Professor MILLER has given the following method of shortening the symbols of the biaxial and triaxial forms. Instead of Mf T, he writes {023J, and instead of P|M|T he writes {236}. In the first example, 2 means M 2 and 3 means T 3 . In the second example, 2 means P = i of T, 3 means M = \ of T, and 6 means T = unity. But, in reality, very little advantage is gained by this abridgment, for {023} and J2 3 6}, brackets included, are as long as Mf T and PiMJT, while they are less explicit, and have the great defect of being unable to denote the comparative magnitude of different Forms in the same combination, which information is of con- siderable utility, but can never be given by symbols that consist solely of figures. For this reason, I think that letters ought not to be dispensed with, even in a single symbol. It is still more evident that it is impossible to remove the figures which denote the lengths of the axes of the different Forms, for these, being the only marks of the individuality of the Forms, are an indispensable portion of their names. Hence, the liberty of abridging crystallographic nota- tion is placed under considerable restrictions. I would almost say, that the restrictions are such as render abridgment inexpedient. They do not, however, render it impossible, and as I know that many persons are pecu- liarly alive to the merits of a short notation, I shall proceed to show what can be done with safety in the way of abridgment. The best plan, then, of constructing short symbols, is to take the " Forms" and fundamental " Combinations" of each of the six systems of crystal- PRINCIPLES OF CRYSTALLOGRAPHY. 331 lisation, and ARBITRARILY replace ALL the letters of each symbol by ONE letter, preserving the characteristic indices and" signs of position. Thus, C may indicate the cube = P, M, T ; O|, the icositessarahedron P|MT, PM|T, PMT ; K|, the tetrakishexahedron = M|T, Mf T. PJM, P^M, P|T, PfT; C, pi, pf , o, oj, of, ^hl, h| = Haiiy's Parallelique or maximum crys- tal of Iron Pyrites, which contains 134 faces. See his symbol at page 22, Part II. And so on. I shall give a Table of such abridged signs, and add a few examples and observations. 604. TABLE OF ABRIDGED NOTATION. 1. OCTAHEDRAL SYSTEM OF CRYSTALLISATION. Axes: p a m a t*. O = PMT = octahedron. C = P,M,T = cube. D = MT. PM, PT = rhombic dodecahedron. O__ = 3PJMT = icositessarahedron. O + = 3P+MT == triakisoctahedron. x Oy = 6P_MT + = hexakisoctahedron. K x = M_T, M + T. P_M, P+M, P_T, P+T = tetrakishexahedron. P x = M_T. P_M, P + T = pentagonal dodecahedron. x H y = 3P_M T + = hemihexakisoctahedron with parallel faces. T = IPMT = tetrahedron. I_ = ^(3P_MT) = hemiicositessarahedron. S_|_ = |(3P.|.MT) = hemitriakisoctahedron. x X y = I(6P_MT + ) = hemihexakisoctahedron with inclined faces. 2. PYRAMIDAL SYSTEM. Axes : p! m a t a . O x = P X MT = quadratic octahedron of the ne and nw zones. N x = P X M, P X T = quadratic octahedron of the n and e zones. P = P = horizontal planes. Q = M,T = quadratic prism of the n and e zones. Q = MT =' quadratic prism of the ne and nw zones. *D y = P x M y T z . P x M z T y = dioctahedron, or eight-sided pyramid. x H y =;j(P^MyT z , PJVI Z T) = hemidioctahedron. V x = M_T, M + T = Eight-sided prism. 3. RHOMBOHEDRAL SYSTEM. Axes: p.m^tia, or p* m^ t,V 2R X = P X T, P x Mff T 2 = six-sided pyramid. P = P = horizontal planes. V T, M}f T 2 = six-sided prism, first position. V =. M, M S T{ = six-sided prism, second position. % : y : z = or simply J = 3m x t = subordinate twelve- sided prism, with alternate similar angles. o x : 7 : 2 = or simply jj = 3p x m y t z = subordinate twelve-sided pyramid. Rx = |P X T, lP x M}f T 2 = rhombohedron. S x . i(3P x MT+) == scalenohedron. V, v = six- sided prism with vertical edges replaced. V, v, f = twenty-four-sided prism. 332 PRINCIPLES OF CRYSTALLOGRAPHY. 4. PRISMATIC SYSTEM. Axes : p* m* t*. x O y = P x M y T; =" rhombic octahedron. P = P = horizontal planes. M = M north and south vertical planes. T = T = east and west vertical planes. V x = M X T = vertical rhombic prism. N x = P X M = rhombic Form of the north zone. E x = P X T = rhombic Form of the east zone. 5. OBLIQUE PRISMATIC SYSTEM. Axes: p*m y t*. M = M = north and south vertical planes. T = T = east and west vertical planes, V x = M X T = vertical rhombic prism. N x n = |P X M ZnS N x * ~~ ! p* M Zg f > = hemihedral biaxial Forms of the north zone. N] I = fp*MZ i sJ E x ~ = |P X T Zw 2 >| u x i = = il^^T >>= hemihedral biaxial Forms of the east zone. k x = iP x A ^e | E x | = iP x TZ 2 eJ X y = |P x M y T z Z 2 ne Z 2 nw -N x O y n = |P x M y T z Zn 2 e Zn 2 w [ = hemioctahedrons belonging to X O T = iP x M y T z Zne 2 Znw 2 f north combinations. X = ^PMT z Zs 2 e 2 Zs 2 w 2 J ^ x O, " = |P x MyT z Zn 2 w Zs 2 w [ = hemioctahedrons belonging to x O y w = JP x M y T z Znw 2 Zsw 2 f east combinations. x O y e = 1P X M 7 T Z Zne 2 Zse 2 J 6. DOUBLY OBLIQUE PRISMATIC SYSTEM. Axes: p*m y t*. M = M == north and south vertical planes. T = T = east and west vertical planes. V x w=iM x Tnw=\ . y __ iiyr T _ f hemihedral prismatic rorms. x OyZnw = iP x M y T z Znw = tetarto-octahedron. Examples from the Octahedral System of Crystallisation : Model 36. P, M, T, mt. pm, pt, ipmt = C, d, t. Model 37. p, m, t, mt. pm, pt, |PMT = c, d, T. Model 32. P, M, T, mt. pm, pt, PMT = C, d, o. Model 33. p, M, T, mt. pm, pt, PMT = c, d, O. Model 34. p, m, t, MT. PM, PT, pint . = c, D, o. Model 45. P, M, T,m|t,mft.p|m,pfm,p|t 5 pft= C, k. Model 46. p,m,t. 3P^M|T = c, iHJ. Model 48. p, m, t, MJT. PM, Pf T, PMT = c, P, O. PRINCIPLES OF CRYSTALLOGRAPHY. 333 Examples from the Pyramidal System : Model 4. P + , M, T, mt = P + , Q, q. Model 42. p +5 M, T, MT. PfM, PJT = p + , Q, Q. Nf. Model 59. M, T, mt. PfM, Pf T = Q, g. Nf . Model 77. p.pm, pt, PMT = p. n, O. Examples from the Rhombohedral System : See Part II. pages 4560. Examples from the Prismatic System : Model 80. P+ . PjgM T * -T = p + . tfOft. Model 50. P_, m, t + , Mf T. pf m, pf t = P_, m, t + , Vf . nf, e j. Model 100. M_, Mf T. PJM = M_,Vf . Nf. Model 51. p+, M_,T,M|T. PJM, P\T,pfraft = p+,M_,T,v|.N|,EV , Model 97. M, T, Mf T. P^ T = M, T, Vf . E-&. Model 79 a . M, T. PfM = M, T. Nf Model 90. Mif T, M| T. Pf | T, P ||m^t. = VJ, vif . E|g Examples from the Oblique Prismatic System : Model 84. M}JT. |P r 4 5 -M Zn Model 87. Mf $T. JP^-T Zw Model 79 b . M_, T. 1P|M Zn = M_, T. Model 112. T,MHT.iPAMZn, ) iP^fMigT Zse Zsw } *^ Model 105. T, JMJ|T ne. iP|M Zn = T, Model 115. T_,M T 9 3 T.P T 5 3M T %T Z 2 neZ 2 nw == T Model 98. M, T,M|a T .|P 2 6 T M|5-TZnwZsw = M,T, Examples from the Doubly-oblique Prismatic System: Model 8P.MfT.iP x M y T z ZW, ip x M y Zn 2 e, ip x m y t z Z 2 ne 2 Model 107. M, Jm/st nV, ^MJ Tnw 2 , ^mjt ] _ M 7 y7 7 Q ne.iP_M + TZ 2 ne /' 605. The reader will perceive from the foregoing Table and Examples, that it is by no means a difficult task to provide a short notation, when we possess a complete systematic notation upon which to found it. A short notation is only difficult of construction when we are without a complete systematic notation. The reason of this is, that when the arbitrary signs which constitute the short notation are merely syrionymes of another more complete and fully-described notation, we are freed from the neces- sity of giving those exact definitions and detailed descriptions of the short notation, which would in other circumstances be indispensable. I believe that the notation just described, is shorter, and yet as exact as any that has hitherto been proposed. It has one advantage over any that I am acquainted with, inasmuch as the single letters belonging to each symbol can be varied in size to indicate the comparative magnitude of zne 334 PRINCIPLES OP CRYSTALLOGRAPHY. different Forms present on any one combination. But the notation, never- theless, possesses several defects, from which no notation of this kind can be free. In the first place, we burden the memory with a long catalogue of synoymes ; a'nd in the second place, we lose the opportunity of looking over the zones of complex combinations, which are all represented in an orderly manner by the notation employed in Part II. of this work, but which are put out of sight when the short notation is employed. Thus, when a combination of the cube and the rhombic dodecahedron is repre- sented by P,M,T, MT. PM, PT, we see at a glance the number of planes in the three zones, but when the same combination is represented by C, D, that convenience is lost, and in very complex combinations this is a con- siderable disadvantage. I attach no particular value to the letters which I have chosen for the short synonymes of the regular symbols of the Crystallographic Forms. Any other set of letters would answer as well as those which are given in the Table. I would lay it down as a rule, however, that no letters should be taken for a purpose of this kind except such as can be printed with the types to be found in a common printing office ; because the adoption of strange types produces a sort of unknown tongue, which no ordinary printer can print. Thus, Dalton invented a set of symbols for chemical notation, which could not be printed; and more recently, Berzelius spoiled his chemical notation, by inventing crossed letters^ which no printer could print without founding new types expressly for that purpose. The consequence of the introduction of these crossed letters is, that Berzel- ius's notation will never come into common use. Even in Germany, it is being displaced by the modified notation proposed by Liebig and Poggen- dorff, in which the crossed letters are omitted. A second rule to be observed, in constructing a useful notation, is to have no characters that occupy more than one line of print, because cha- racters that run up and down into two or three lines are perplexing to the eye, troublesome and expensive to print, and, while they pretend to be short, are in reality long, since they occupy a great deal of space in a book. Of this character are the symbols employed by HAUY, see page 81, Part I. and page 22, Part II. of this work; and so also are the fol- lowing symbols, taken from MOHS : 2 /2 L. ("IT" P- 2) 3 T " 2 The form represented by the above symbol is a hemihedral form of a dioctahedron of the pyramidal system = |(P x M y T z , P x M 2 T y ) or x H y , p* m a t a . It is clear that Mohs's symbol must occupy two or three lines of print in a book, so that it is really not a short symbol. 2 2 The first of these symbols represents a hemihedral biaxial form of the east zone, belonging to the oblique prismatic system = z^xT Zw or E x w . PRINCIPLES OF CRYSTALLOGRAPHY. 335 The second represents a similar form belonging to the north zone = |P X M Zn or N x n . Both of Mohs's symbols appear short, but as each occupies two or three lines of print, they are in reality not short. A third rule to follow is, to avoid too many brackets, braces, and simi- lar arbitrary signs, such as are contained in the following symbols: It is perhaps a matter of taste, but it appears to me that the longest symbol contained in the second Part of this work is much more conveni- ent, and much easier to follow, than a symbol which presents so many abstractions as the shortest of those which are here quoted. The symbol of a Form or Combination expressed in this short nota- tion can never be printed alone, but must always be accompanied by the characteristic of the system to which it belongs; because the same letter is used in different systems to indicate different things, and this cannot be otherwise, so long as the letters of our alphabet are fewer in number than the Forms and fundamental Combinations of all the six systems of Crystallisation collectively. Thus S x indicates a Hemitriakisoctahedron of the first system, and a Scalenohedron of the third system. Hence, to be perspicuous, we must say, S x , p a m a t a , and S x , p x m t, a 3 . The symbols which refer directly to the seven fundamental Forms, P, M, T, MT, PM, PT, PMT, are free from this serious defect, and therefore, though longer, are in most cases preferable. SECTION XVI. TABLE OF SINES AND TANGENTS. 606. In 49, I promised to give a short Table of Sines and Tan- gents, which is accordingly here appended. It is not adapted to replace the common Tables of logarithms, which are more extensive, and there- fore more useful; but it will nevertheless often save the reader of this work the trouble of seeking his book of logarithms when he wishes to find either the angle indicated by a particular index, or the index required to denote a particular angle. During the composition of the present treatise, I have been in the habit of marking in my logarithmic tables the angles that I had occasion to refer to, and as the whole of the angles thus marked are printed in the present Table, the reader will find it more frequently useful than from its limited extent he might imagine to be possible. Another motive which prompted me to give this Table, was the wish to present the reader with a system of tangents and cotangents in vulgar fractions, adapted for use as indices of the seven crystallographic Forms. This is accordingly done, and serving as a counterpart to the Table of Indices given in page 139, will save the reader much calculation, and consequently much time. 336 PRINCIPLES OF CRYSTALLOGRAPHY. The Table contains natural tangents and cotangents, and logarithmic tangents, cotangents, sines and cosines. I have not introduced any nat- ural sines and cosines, because the system of calculation which I have adopted does not require them. The words sine and cosine appear, in- deed, in the formulae which relate to solid triangles, but the calculations are always made by means of the logarithmic equations. On the other hand, the natural tangents and cotangents are constantly employed in the construction and interpretation of the indices of the symbols. The indices in vulgar fractions are placed against those decimal tan- gents and cotangents, which are their nearest synonymes. Any term not contained in the Table may be found by taking the pro- portion of difference between the two terms nearest related to it in the Table. EXAMPLE: To find the tangent of 26 24'. The Table contains tan 26 34' = .5000 tan 26 11' = .4917 difference = 23' = .0083 23rd part of .0083 = .00036 This last product, .00036, is the tangent of V or the 23rd part of the difference between 26 34' and 26 11'. The number of minutes in 26 24' more than in 26 11' is 13'. Therefore, the tangent of 26 24' is to be found by adding 13 times the tangent of 1' to the tangent of 26 11'. Hence, tan 26 11' = .4917 .00036 X 13 = .00468 tan 26 24' == .49638 The large Table gives these tangents as follow: 26 11' 26 24' 26 34' 4916997 4964043 5000352 This shows how far this method of finding intermediate terms is to be depended upon. The approximation of the numbers is however often closer, though sometimes less so, than in this example. For calculations of importance, the larger table of sines and tangents should be invari- ably referred to, but for the calculations which occur in the course of studying the principles of the science, the numbers given in the present Table will generally suffice. TABLE OF SINES AND TANGENTS. Angles. N. tan. T. c. N. cot. L. sin. L. cos. L. tan. L. cot. Angles. 00.00 0000 infinite. inf. neg. 10.000 inf. neg. infinite. 90.00 00.01 0003 3437.7 6.4637 10.000 6.4637 13.5363 89.59 00.30 0087 114.59 7.9408 10.000 7.9409 12.0591 89.30 00.38 0111-90 90 90.463 8.0435 10.000 8.0435 11.9565 89.22 00.57 0166 LL 6_0 60.306 8.2196 9.9999 8.2196 11.7804 89.03 1.00 0175 57.290 8.2419 9.9999 8.2419 11.7581 89.00 1.09 0201 L& 5_0 49.816 8.3025 9.9999 8.3026 11.6974 88.51 1.14 0215 46.449 8.3329 9.9999 8.3330 11.6670 88.46 1.26 0250 U- 39.965 8.3982 9.9999 8.3983 11.6017 88.34 1.30 0262 X 38.188 8.4179 9.9999 8.4181 11.5819 88.30 1.47 0311 , 3 l > 32.118 8.4930 9.9998 8.4933 11.5067 88.13 1.48 0314 ** T* 31.821 8.4971 9.9998 8.4973 11.5027 88.12 1.55 0335 J* 5.0 29.882 8.5243 9.9998 8.5246 11.4754 88.05 2.00 0349 ^ 28.636 8.5428 9.9997 8.5431 11.4569 88.00 2.07 0370 Ly 27.057 8.5674 9.9997 8.5677 11.4323 87.53 2.17 0399 ' 25.080 8.6003 9.9997 8.6007 11.3993 87.43 2.23 0416 | A 24.026 8.6189 9.9996 8.6193 11.3807 87.37 2.52 0501 i 20 2_0 19.970 8.6991 9.9995 8.6996 11.3004 87.08 3.00 0524 19.081 8.7188 9.9994 8.7194 11.2806 87.00 3.01 0527 n A! 18.976 8.7212 9.9994 8.7218 11.2782 86.59 3.11 0556 * 17.980 8.7445 9.9993 8.7452 11.2548 86.49 3.22 0588 & 16.999 8.7688 9.9992 8.7696 11.2304 86.38 3.35 0626 I 15.969 8.7959 9.9992 8.7967 11.2033 86.25 3.49 0667 j ij 14.990 8.8232 9.9990 8.8242 11.1758 86.11 3.57 0690 & <29 14.482 8.8381 9.9990 8.8392 11.1608 86.03 4.00 0699 & y 14.301 8.8436 9.9989 8.8446 11.1554 86.00 4.05 0714 Tl 1_4 14.008 8.8525 9.9989 8.8536 11.1464 85.55 4.14 0740 /Y 2 J 13.510 8.8682 9.9988 8.8694 11.1306 85.46 4.24 0769 T3 13 1 12.996 8.8849 9.9987 8.8862 11.1138 85.36 4.34 0799 8.** Zinc Blende 9. Sulphuret of Molyb- 8 9. denum '5 L Galena j 2. Seleniuret of Lead cc 3. Do. of Lead and 'S Cobalt 4. Do. of Lead and 1 Mercury g 5. Do. of Lead and % Silver *3 1O. Sulphuret of Silver 11. Sulphuret of Cobalt 12.* Iron Pyrites 1 3. Muriate of Ammonia 1. Chloride of 1O. Fluorcerium 1 14. Chloride of Sodium Mercury 15. Chloride of Silver s 16. E 1. Fluorspar 1 2. Yttrocerite on PMT. 1. 1. P,M,T, Cornwall Model 1. J 199 . L657 3 . H106 19 *. P210 4 . 1. 1. P_,M,T. Axes: pAm a t a Model 2. 1. 1. P + ,M,T. Axes: p4_m a t a Models. 1. 2. P+,M_,T. Axes: p4_mlt a Similar to Model 5. 3. 1. PMT, Persberg... Model 15. J 203 . L657 5 . P210 1 . H106 195 D 4 . 3. 1. 3 P 2 MT x 2, Schoharie, New York S i7 . D406 2 . 3. 3. 3 P^MT, ...Corsica... Model 22. S 266 . J295 12 . HIOS 197 . D 16 . L658 15 . 2. 3. SP^M^T, Model 35. R 15 . 3. 4. PMT, 3pjmit, J 20 *. D 48 . 3. 1. P,M,T.pmt, ... Similar Model 29. L657 7 . J 202 . D 2 . P210 3 . H106 00 . 3. 1. P,M,T.3pimt,...Md.39. R I9 .D U . S 266 . Jiii293 11 . H107 205 . L657 11 . 3. 1. P,M,T.pmt,3p^mit,... Monroe, Connecticut. Facebay^.R 533 . Mr 33 . H107 210 . L658 13 . S 264 . 3. 1. p,m,t.PMT, Model 30. D 3 . P210 2 . 3. 4. P,M,T,mt 2 .pm 2 ,p,t, Elba R 53 . D 42 . 3. 4. P,M,T,MT 2 .PM 2 ,P 2 T, Cornwall, Elba Sim. Md. 47. J . H106 201 . L657 2 . P210 8 . S 382 . D 43 . 3. 4. P,M,T,MT 2 .PM 2; P 2 T,PMT, ...Sim. Md. 48. L657 6 . H107 109 . S 261 . 3. 4. P,M,T,mt 2 .pm.,p,t, 3p^mit, Rossie, New York D407. 3. 4. PjM^jmt^mlt.pm^plm^tjptljpmt, 3pjmt,3p 2 mt,3p^mlt,3pimit, Peru H108 216 .* 3. 4. p,m,t,PMT, Spjmjt, J\ 3. 4. p,m,t.PMT, Spjmt, 3r 2 MT, Gpimjt, H108" 15 . 3. 4. p,m,t.3PlMT, D 15 . 3. 4. p,m,t.3P^MiT, Brosso Thai, Md.46. R 47 .H106 -. J' 09 . S 267 . L657 9 . 3. 4. p,m,t,MT 2 .PM 2 ,p 2 T, 3P^M1T, 3 p Jmjt, Piedmont R 47a . 4. 1. mt.pm,pt,PMT, Model 64. H106 204 . 4. 3. MT 2 .PM 2 ,p 2 T, SP^MT, 3plmt,3p 2 mt, H108 214 . 4. 4. mt 2) m 2 t.pm 2 ,p 2 m ? pt 2 ,p 2 t,PMT, 3pjmt, H107 211 . 5. 4. MT 2 .PM 2 ,P 2 T, Elba...Md. 91. R 49 . D 44 . H106 198 . S 263 . P210 7 . J 201 . L657 t . 5. 4. (MT 2 .PM 2 ,P 2 T) X 2, L658 17 . 5. 4. MT 2 . PM 2 ,P 2 T, pmt, Similar Model 92. D 47 . 5. 4. MT 2 .PM 25 P 2 T,PMT, Elba.. .Md. 92. R 52 . J 207 . H107 206 . L657 4 . R210 6 . D 46 . 22 MINERALS OF THE OCTAHEDRAL SYSTEM. 5. 4. MT 2 .PM 8 ,P 2 T, Spjmjt, ......... Elba ...... R 51 . H107- 08 . S* 8 . L657 8 . 5. 4. MT 2 .PM 2 ,P 2 T,pmt, Spjmjt, .................. Elba ...... R 51a . H107 212 . 5. 4. mt,MT 2 .pni,PM 2 ,pt,P 2 T,PMT, Spjmit, ............... L658 14 . H107 213 . 5. 4. mt 2 .pm 2 ,p 2 t 5 PMT, ......... Model 93. J 206 . D 45 . P210 5 . H107 207 . S 260 . * " This variety of iron pyrites the ParalUlique offers at present the maximum rela- tive to the number of faces observed upon different bodies produced by crystallisation. The number of faces upon this crystal is 134. The symbol for it is: ?. J I M.Pde"e' e y" y 1 y /" f f s" *' s ( A^G 1 B* )( AB > C'X^AC 1 G 3 ) " HAUY, Traite de Mineralogie. Paris, n" ri n IW2,t.w.p.B?. In W. PHILLIPS'S Mineralogy., London, 1837, page 210, there is a figure nearly resem- bling Hatty's parallelique crystal, and said to be "given on the authority of Hatty," which contains 194 planes, or 60 above Hatty's maximum quantity. The combination exhibited by Phillips, contains the following planes: P,M,T, mt,rat 2J m|t. pm,pm 2 ,p|m, pt,p 2 t,ptf , prat, 3pjmt, Spjmt, 3p 2 mt, Spjmjt, 6pjmlt. Now, the Forms of Iron Pyrites are collectively as follows: P,M,T, MT,MT 2 ,M 2 T,Mf T. PM,PM 2 ,P 2 M,P|M, PT,PT 2 ,P 2 T,PTf . PMT, 3PJMT,3PiMT, 3P 2 MT, SP^M^T, SP^MTJ, 3PJM1T. These forms contain 206 planes in all, but Hatty's maximum crystal is without the forms MT.PM,PT; M 2 T,P 2 M,P 2 T; 3P^MT; and 3 P^MTi; which contain together 72 planes. Consequently, Phillips's crystal is a diagram intended merely to represent all the forms of Iron Pyrites, and is not a drawing of a real crystal. The Eidogens whose intersections produce the forms of Iron Pyrites, have the following dimensions: 90 00^= P,M,T,MT,PM,PT. 109 28'%= PMT,PiMT,PMT,PMT|. 129 3V = PMT,PMT,PMTi. 112 37 X = M|T,P|M,PT|. 126 52' = MT 2 ,M 2 T.PM 2 ,P 2 M ? PT 2? P 2 T. 141 3' = P 2 MT,PM 2 T,PMT 2 . 149 00' = 154 47' = Here are no less than eight Eidogens or primary forms of crystallised Iron Pyrites, that is to say, evidence of the operation of the power of crystallisation, within the limits of eight forms of different dimensions. Yet crystallographers have chosen to fix the term primary form exclusively upon one combination produced by the action of one Eidogen, differing, however, among themselves as to which one of the many they should elevate to the post of honour, thus, Hatty chose the Cube=P,M,T; Leonhard, the Pentagonal Dodecahedron = MT 2 ,PM 2 ,P 2 T; and Phillips, the Octahedron = PMT. But it does not appear to me, that any one of these forms or combinations, can be said to have exercised any more control over, or to have been in any more peculiar manner affected by the power of crystallisation, than any one or all of the other forms; so that there is no evidence of the existence of any single primary form adapted to originate all the other forms, and there is no benefit de- rived from the assumption of such a primary form. MINERALS OF THE OCTAHEDRAL SYSTEM. 23 13. MURIATE OF AMMONIA, Salmiak. Ammoniac muriatee. Cleavage = PMT. 1. 1. P,M,T, ......... St, Etienne by Lyons ...... Model 1. L588 2 . D'. S 169 . 1. 1. P+,M,T, .............................................................. L588 2 . 2. 1. PMT,... Model 15. Jiii 12 1 . H ii 223 2 . L588 1 . P201. H55 187 . D175. . 3. 3PMT, Glan in Bavaria. Sim. Md.22. Jiiil2 5 .L588 3 .H55 188 . D 16 . 3. 1. P,M,T,mt.pm,pt, .................................... Model 27. Jiiil2 3 . 4. 1. MT.PxM,PT, ................................. Model 63. L588 4 . J iii 12 4 . 4. 1. M,T.pm,pt, ................. : ....................................... Jiii 12 2 . 14. CHLORIDE OF SODIUM. Muriate of Soda. Steinsalz. Rock Salt. Soude muriatee. Cleavage = P,M,T, mt.pm,pt. 1. 1. P,M,T, ............... Tyrol ...... Model 1. Jiii3. H53 171 , R 13 . L584 1 . 1. 1. P_,M,T, .......................................... Model 2. Jiii3. L584 1 . 1. 1. P + ,M,T, .......................................... Model3. JiiiS. L584 1 . 3. 1. P,M,T,mt.pm,pt, ................................. Model 27. L584 3 . D 5 . 3. 1. P,M ? T.pmt, ...Salzberg...Sim. Md. 29- P200. L5S4 2 . H53 173 . S 145 . 3. 4. P,M,T,mt,,m 2 t.pm 2 ,p 2 m,pt 2 ,p 2 t, ......... Similar Model 45. S 146 . D 10 . 4. 1. MT.PM,PT, ............... Berchtesgaden ...... Model 63. L584 4 . D 7 . 15. CHLORIDE OF SILVER. Hornerz. Argent muriatee. Cleavage = 0. 1. 1. P,M,T,...Peru, Cornwall... Md. 1. H iii294 1 . D 1 . L581 1 . Jii351 la . 1. 1. P + ,M,T, ..................... Poland ...... Models. Hiii 294 1 . L581 1 . 2. 1. PMT, ............ Siberia ...... Model 15. D 4 . S268 2 . L58I 5 . Jii 351 2 . 3. 1. P,M,T,mt.pm,pt } .......... Job. Georgenstadt...Md. 27. D 5 . L581 2 . Jii351 lb . S 253 . 3. 1. P,M,T.pmt, ...Cornwall... Sim, Md. 29. S' 52 . L581 4 . D 2 . Jii351 lc . 3. 1. p,m,t,MT.PM,PT, ........................................ Model 28. D 6 . 3. 1. p,m,t.PMT, ...................... Cornwall ...... Model 30. D 3 . L581 . 4. 1. MT.PM,PT, ...... Siberia ...... Md. 63. D 7 . L581 3 . J H351 3 . S268 4 . 1C. An Isomorphous Group, 1 and 2 : 1. FLUORSPAR. Flusspath. Chaux fluatee. Cleavage = p,m,t,mt.pm,pt,PMT. P,M,T, Kongsberg.Derbyshire. Md.l. L574 5 . P173 3 . R 13 . H27 2 . J 155 . PMT, Derbyshire. Puy-de-D6me. Md.15. P173 1 .H27 1 . L574 1 . J 159 . 3P 3 MT, ...................................... Similar Model 17. L575 12 . PMT,3p 3 mt, .................................. Kongsberg ...... R 23 . Mr 26 . P,M,T,mt.pm,pt, ...... Derbyshire, Drammen..,Md. 27. H28 14 . R 17 . L575 9 . J 156 . P,M,T,mt.pm,pt,pmt, ................................. Model 31. P173 4 . P,M,T,mt.pm,pt, 3plmt, .................................... Pl73 10 . R 2. 3. 3. 3. 3. 3. 3. 1. P,M,T.3p|mt, ...... Saxony ...... Model 39- P173 9 . H28 12 . JH591 7 . 10a P,M,T.PMT,... Derbyshire, Zinnwald...Md. 29. H28 10 . L574 4 . J 158 24 MINERALS OF THE OCTAHEDRAL SYSTEM. 3. 1. P,M,T.6pimit,...Munstherthal Baden, Derbyshire... Sim. Md. 40. P173 11 . H28 15 . R'. Mr- 7 . JH591 8 . L575". 3. 1. P,M,T,MT.PM,PT, 3pmt, 6pimit, GpJj-mit^ England... R' 96 . 3. 1. p,ra,t.PMT, Model 30. P173 2 . 3. 1. p,M,T,mt.pm,pt,PMT, Cornwall, Zinnwald. Md. 33. H28 16 - L575 8 . 3. 1. p,m,t,MT.PM,PT,PMT, 3pimt, D185 1 . 3. 4. P,M,T,mt 3 ,m3t.pm3,p3m,pt3,p 3 t,... Alston Moor. Zinnwald. Durham. Md.45. P173 7 . R 21 . H28 13 . L575 7 . J 161 . 3. 4. P,M,T,mt,mt3,m 3 t.pm,pm 3 ,p 3 m,pt,pt 3 ,p 3 t, H29 19 . 3. 4. P,M,T,mt 3 ,m 3 t.pm 3 ,p 3 m,pt 3 ,p 3 t, 6pmt> Miinsterthal...L575 10 . 3. 4. P,M,T,mt,mt3,m 3 t.pm,pm 3 ,p 3 rn,pt,pt 3 ,p 3 t, 3p|mt, H29 20 . 3. 4. P,M,T,MT,mtJ,m|t.PM,pmf,pfm,PT,pt|,pJt,PMT,... England... R 21a . 3. 4. p f m,t. 6P*MjT, P173 12 . 3. 4. P,M,T,mt J mt| 5 mt 10 ,m|t,m 10 t.pm,pnTj,pmj 0r p|m,p I0 m,pt,pt|,pt IO ,p|t, Pi t,pmt, 3pmt, 5 (6 p x m y t 7 ).* 3. 4. p,mt,MT 3 ,M 3 T.PM 3 ,P 3 M,PT 3 ,P 3 T,... Zinnwald.. .Sim. Md.45. R 32 . 4. 1. MT.PM,PT, Chalons... Model 63. P173 6 . H28 9 . L574 3 . J 157 . 4. 1. mt.pm,pt,PMT, ....Saxony... Model 64. P173 5 . H28 11 . L574 2 . J 160 . 4. 3. MT 3> M 3 T.PM 3 ,P 3 M,PT 3 ,P 3 T, Derbyshire Sim. Model 68. P173 8 . L574 6 . H27 4 . J 16 '. 4, 4. mt.pm,pt,PMT,3pJmt, H28 18 . L575 14 . 4. 4. mt,mt 3 ,m3t.pm,pm3,p 3 m,pt,pt 3 ,p 3 t,PMT, H28 17 . L575 :3 . * A crystal from Devonshire, in the possession of W. Phillips, exhibited all these faces, in number 338. The figure shown in his Mineralogy, page 174, is an imaginary crystal, containing 434 planes, as follow: P,M,T,MT, 3mt + , 3m + t. PM, 3pm + , 3p + m,PT, 3pt + , 3p + t, PMT, 4 (3p_mt), 5 (6p x m,t z ). The measurements given are not sufficient to supply a more significant symbol for either of these complex combinations. 2. YTTROCERITE. Yttrocerit. 3. 1. PMT? Model 15. R162' 6 . IS*. RED OXIDE OF COPPER. Rothkupfererz. Cuivre oxidule. Cleavage = p,m,t. PMT. 1. 1. P,M,T, Moldava Model 1. J 176 . P317 4 . H99 124 . L567 4 . 2. 1. PMT, Siberia, Cornwall, Chessy Model 15. J 171 . P317 1 . H99 U3 . L566 1 . 2. 1. PMT, 3p 3 mt, P317 9 . J 181 . L567 10 . 2. 1. 3P 3 MT, Gumeschewskoi Sim. Model 17. J 82 . L567 11 . 2. 4. PMT,3pimt, P317 10 . J 83 . L567 5 . 2. 4. PMT,6 Px m y t z , J 184 . L567 7 . 3. 1. p,m,t,MT.PM,PT, Model 28. H100 158 . 3. 1. P,M,T,rat.pm,pt.pmt Model 31. P317 5 . 3. 1. p,m,t,mt.pm,pt,PMT, Corn wall... Model 33. H100 129 . L567 12 . 3. 1. p,m,t>MT.pM,PT,PMT, 3p|mt, Gumeschewskoi R lob . 3. 1. p,m,t.PMT, Cornwall... Model 30. J 175 . P317 3 . H99 126 . L567 3 . MINERALS OF THE OCTAHEDRAL SYSTEM. 25 3. 4. p,M,T.PMT,3pimt, ......................... Guraeschewskoi ...... R 16a . 3. 4. p,m,t,mt,mt 2 ,m 2 t.pm,pm 2 ,p 2 m,pt,pt 2 ,p 2 t,PMT, ............ J 180 . L567 13 . 3. 4. P,M>T,MT,mt 5 ,m 5 t.PM,pm 5 ,p 5 m,pT,pt 5 ,p 5 t,PMT, ..................... R 33 . 3. 4. p,m,t,MT,mt 2 ,m t.PM,pm 2 ,p 2 m,pT,pt 2 ,p 2 t, PMT,3p|mt,3p 3 mt,6p x m y t z . P317 11 . L567 14 . 4. I. MT.PM,PT, ...Chessy ...... Model 63. J 178 . P317 8 . H99 125 . L567 9 . 4. ]. MT.PM,PT, pint, .................................... Model 65. P317 7 . 4. 1. mt.pm,pt,PMT,... Chessy.. .Model 64. J 177 . P317 6 . H100 127 . L567 8 . 4. 4. MT.PM,PT,PMT, 3p 3 mt, .................. Gumeschweskoi ...... R 33 . 4. 4. mt^mat.pm^para^p^PMT, ............................... J 179 . L567 6 . . 3. M/^T.P^T. Axes:p4.mt% ............. : ................ J 174 . L566 1 . 5. 3. M-j-^T. l P T 7 Q T. Axes: p4.mt a , .................. P317 2 . J 173 . L566 2 . . 2. iPMT Znw,Nse, f pmt, ................................................ J 172 . 1.8. OXIDE OF ARSENIC. Arsenikbliithe. Arsenic oxide. Cleavage = PMT. 3. 1. PMT, ................................................... Model 15. P281 . 3. My^T.P^T. Axes: m^tt a , .................................... L334. 19.* An Isomorphous Group, 1, 2. 3 : 1. BRIGHT WHITE COBALT. * Kobaltglanz. Cobalt gris. Cobaltine. Silver White Cobalt. Cleavage = P,M,T,mt 2 .pm 2 ,p 2 t. 1. 1. P,M,T, ...Hokanbo, Tunaberg...Md. 1. P285 1 . L654 3 . H1V228 1 . . 1. PMT, ......... Tunaberg... Model 15. L654 5 . J 280 . P285 4 . H iv 228 2 . 3. 1. P,M,T.pmt, ...Tunaberg... Sim. Mod. 29. P285 2 . L655 7 . Mil 456. 3. 1. P,M,T.PMT, ........................ Tunaberg ...... Model 29. L655 7 . 3. 1. p,m,t.PMT, ......... Tunaberg... Model 30. P285 3 . L655 7 , M ii 456. 3. 4. P,M,T,mt 2 .pm 2 ,p 2 t, ......... Tunaberg ...... Sim. Md. 47. R 53 . L654 2 . 3. 4. P,M,T,mt 2 .pm 2 ,p 2 t,pmt,... Tunaberg, Sim. Md. 48. L655 6 . R 51 . Mr 31 . 3. 4. P,M,T,MT 2 .PM 23 P 2 T, ...... Tunaberg... Model 47. P285 5 . L654 2 . 3. 4. p,M,T,MT 2 ,mt 4 .PM 2 ,pm 4 ,P 2 T,p 4 t,PMT,3pfmlt ......... Mr 32 . P285 9 . 5. 4. MT 2 .PM 2 ,P 2 T, ...Hokanbo, Tunaberg... Model 91. P285 6 . L654 1 . 5. 4. MT 2 .PM 2 ,P 2 T,PMT, ......... Tunaberg... Model 92. L654 4 . P285 7 . 5. 4. mt 2 .pm 2 ,p 2 t, PMT, ............ Tunaberg ...... Model 93. P285 8 . R 48 . 2. NICKEL GLANCE. Nickelglanz. Sulpho- Arsenide of Nickel. Cleavage = P,M,T. 1. 1. P,M,T, .............................. Sweden, Hartz ...... Model 1. D 400 . 3. 1. P,M,T.pmt, ........................ Hartz ...... Similar Model 29. S81. 3. SULPHO- ANTIMONITE OF NICKEL. Nickeliferous Grey Antimony. Nickelantimonglanz. Cleavage = P,M,T. 1. 1. P,M,T, ........................... Hartzgerode ...... Model 1. TH531. . 1, PMT, ................................................ Model 15. Tii531. 3. 1. P,M,T.pmt, ................................. Similar Model 29. Tii531. d 26 MINERALS OF THE OCTAHEDRAL SYSTEM. 3O. PURPLE COPPER. Buntkupfererz. Variegated Copper. Cleavage = pmt. 1. 1. P,M,T, Model 1. D 1 . 3. 1. PMT, Cornwall Model 15. S254 2 . 3. 1. PMT X 2, Model 16. L643 2 . D 129 . 3. 1. P,M,T.pmt, Cornwall Similar Model 29. P310. D 2 . 3. 1. p,m,t.PMT, Model 30. L643 1 . D 3 . S254 3 . 31. An Isomorphous Group, 1, 2, 3, 4, 5, 6 : 1. SPINEL. Spinell. Alumine magnesiee. Cleavage = pmt.' 3. 1. PMT, Ceylon Model 15. J 13 . L542 1 . H51 151 . P81 1 . R 1 . 3. 1. PMT x 2, Model 16. L542 6 . J 16 . P81 2 . H51 151 . 3. 1. PMT, 3p 3 mt, S202 8 . D 21 . 4. 1. MT.PM,PT, Model 63. L542 3 . H52 156 . P81 5 . J 18 . 4. 1. mt.pm,pt,PMT, ... Ceylon ... Md. 64. L542 2 . H52 157 . P81 3 . R 2 . J 17 . 4. 1. MT.pM,PT,PMT,3pJmt, P82 1 . L542 4 . 4. 1. MT.pm,pt. Axes: p4.m a t% J 19 . 4. 4. MT.PM,PT,PMT,3pJmt, H52 15 '. 5. 3. M^T.P^T. Axes: p4.mt% J . Hiil68 a . L542 1 . . 1. 1PMT, * Model 117. J 15 . 6. 1. 1PMT, ipmt, ModelllS. J 14 . 6. 1. (lPMT,iLpmt) x*2>* Ji44 7 . . 2. iPMTZnw, Nse,|pmt,* Ji45 12 . * Many of the crystals of spinel are segments of the combinations PMT, iPMT, and ^PMT,^pmt, the sections of these forms being always parallel to a plane, and the segments occurring either singly or combined, two, three, or more together, in variously inverted positions. See P8U. Jip. 44, and Hiip. 168. It would be useless to give symbols for such irregular combinations. Their plane angles show the character of the crystallisation, and their interfacial and re-entering angles distinguish the octahedral from the tetrahedral forms. 2. PLEONASTE. Zeilanit. Black Spinel. Spinelle noir. Ceylanite. Cleavage = PMT. 3. 1. PMT, ...Amity, N.Y. Ceylon... Md.l 5. J 9 . Hii 170 1 . P83 1 . L543 1 . 3, 4. PMT,3jmt, J 11 . 3. 1. p,m,t,MT.PM,PT,PMT,3p^mt Hamburgh, New Jersey... S 416 . 4. 1. MT.PM,PT, Model 63. J 12 . H ii 1 70 2 . P83 3 . 4. 1. MT.PM,PT,pmt, Model 65. P83 2 . 4. 1. mt.pm,pt,PMT, Model 64. S202 1 . J 10 . L543 2 . H ii 170 3 . 4. 4. MT.PM,PT,PMT, 3pjmt, HiH70 4 . pi. 52 158 . 4. 4. mtpm,pt,PMT, 3pjmt, Vesuvius.. "....R 10 . L543 4 . 4. 4. MT.PM,PT,PMT, Sp^mt, 3p 3 mt, P83 4 . L543 5 . 3. AUTOMALITE. Gahnite. Zinciferous Spinel. Cleavage = PMT. 3. 1. PMT, Model 15. L544. Hii 171. J89 1 . 3. 1. PMT x 2, Model 16, Hii 171. D 129 . MINERALS OF THE OCTAHEDRAL SYSTEM. 27 6. 1. iPMT,lpmt, ....................................... ModelllS. Ji39 2 . 6. 1. IPMT x 2, ......................................................... Ji39 5 . 4. MAGNETIC IRON ORE. Oxydulated Iron. Magneteisenerz. Fer oxidule. Cleavage = pmt. 1. 1. P,M,T, ............ Arendal. Steyerraark ...... Model 1. S20 4 . L553 8 . 2. 1. PMT,... Fahlun. Piedmont... Md.l5.L553 l .H103 165 . P215 1 .R 1 . J 193 . 2. 1. PMT x 2, .................. Model 16. Jiii 190 7 . L553 12 . Hiii 562 C . 2. 1. 3 Pf MT, ............ Zillerthal ...... Similar Model 17. S282. L553 6 . 2. 1. PMT, 3p|rat, .................................. . ................ J 195 . L553 5 . 2. 4. PMT,3pJmt, ............... Traversella ...... L553 9 . R IOa . H iii 562 4 . 3. 1. P,M,T.pmt, .................................... Similar to Model 29. J 198 . 3. . p,m,t,MT.PM,PT, ........................ Model 28. Mii400. L553 4 . 3. . p,m,t,MT.PM,PT,pmt, Spjmt, ................................. L553 11 . 3. . p,m,t,MT.PM,PT,PMT, 3pjmt, .................................... L553 11 . 3. . p,m,t.PMT, ............. ..Zillerthal ...... Model 30. Jiii 189 C . L553 7 . 4. . MT.PM,PT,... Piedmont... Model 63. H103 167 . L553 3 . P215 3 . R 4 . 4. . MT.PM,PT,pmt, ..................... Normarken ...... Model 65. R 3 . 4. . mt.pm,pt,PMT,... Traversella.. .Md.64. H103 166 . L553 2 . P215 2 . J 19 *. 4. 1. MT.pm,pt. Axes:p^m a t a , ............................................. J 197 . 4. 4. MT.PM,PT, 3pimt, ............... Zillerthal ...... Model 69. L553 10 . 4. 4. MT.PM,PT 5 PMT, 3pjmt, ..................... Piedmont ...... R 9 . Mr 29 . 4. 4. mt.pm,pt,PMT, 3pjmt, ........................ Piedmont ...... L553 10 . 5. 3. M/oT.P^yT. Axes: p4.m_lt a , ................................. L553 1 . 5. 3. M T 7 oT.JP T 7 oT, Zw,Ne. Axes: p.|.mlt a ...Bournon...L553note. 6. 1. iPMTjipmt, .................................... ModelllS. Jiiil90 5 . 5. FRANKLINITE. Franklinit. Zinc oxide ferrifere. Cleavage = pmt. 3. 1. p,m,t,mt.pm,pt,PMT, ................................. Model 33. L551 2 . 4. 1. mtpm,pt,PMT, .................. Model 64. MH403 1 . L551 1 . P219. 4. 1. mt.pm,pt,PMT,3p + mt, .......................................... Mii403 2 . 6. CHROMATE OF IRON. Chromeisenerz. Fer chromate. Cleavage = pmt. 2. 1. PMT, ............ Baltimore ...... Model 15. L558. P275. MH396. 4. 1. mt.pm,pt,PMT, ...... Hoboken, New Jersey... Model 16. S 131 . D 9 . 22.** An Isomorphous Group, 1, 2: 1. BORACITE. Borazit. Borate of Magnesia. Magnesie boratee. Cleavage = pmt. 1. 1. P,M,T, ......... Segeberg, Holstein ...... Model 1. H46 101 . Mii348. 3. 1. P,M,T,MT.PM,PT4PMT4pmt, ...... Luneberg ...... Model 35. R 39 . 3. 1. P,M,T,mt.pm,pt,!pmt, ...Luneberg.. . Md. 36. R 40 . H46 103 . D347 1 . 3. 1. P^T^T.rM^pmt, l (3pJmt)Z 2 nw, ..... ........... P187. H46 104 . 28 MINERALS OF THE OCTAHEDRAL SYSTEM. 3. 1. P,M,T,MT.PM,PT4PMT Znw, -Jpmt Zne, J (Spjmt) Z 2 ne, ' ZVw, Luneberg R 39a b . 3. 1. p,m,t,MT.PM,PT,ipmt, Luneberg R 42 . D347 1 . 3. 1. p,M,T,MT.PM,PT, 1PMT Znw, IPMT Zne, 1 (Spimt) Z 2 nw, Luneberg R 41 . 3. 1. p,m,t,MT.PM,PT, ipmt Znw, | (3pjmt) Z 2 ne, H46 105 . 3. 1. P,M,T,MT.PM,PT,|PMT Znw, (6pjmit) Zne, H46 106 . 3. 1. p,m,t.lPMT, Luneberg R 27 . 3. 1. p,m,t,mt.pm,pt,PMT, ... Luneberg... Model 37. R 33 . L288 1 . Mr 28 . 3. 1. p,m.t,mt.pm,pt, PMT, pmt, L288 2 . 4. 1. MT,PM,PT,pmt...... H46 103 . 2. RHODIZITE. Rhodizit. G. Rose, Pogg. Ann. xxxiii. 253. 4. 1. MT.PM,PT, Siberia Model 63. D 7 . R164. 4. 1. MT.PM,PT,pmt, Siberia Model 65. D 8 . R164. 4. 1. MT.PM,PT, ^pmt, Siberia G. R. in Pogg. 4. 1. mt.pm,pt,PMT, Siberia Model 64. D 9 . R164. S3.** FAHLERZ: An Isomorphous Group, 1, 2, 3: 1. ARSENICAL GREY COPPER. Arsenikfahlerz. Tennantite. Cleavage = mt.pm,pt. 3. 1. P,M,T,mt.pm,pt,pmt, Model 31. P313 3 . S 431 . p,m,t,MT.PM,PT,pmt, Model 34. P313 2 . S 432 . p,M,T,MT.PM,PT,PMT, 3p|mt, 3pjmt, P313 & . 3. . p,m,t.PMT, Model 30. P313 1 . L604 1 . 3. . p,m,t,mt.pm,pt,PMT, Model 33. L604 2 . . MT.PM,PT, Model 63. P313 4 . L604 5 . . mt.pm,pt,PMT, Model 64. L604 a . 4. 4. MT.PM,PT,3P^MT, Model69. P313 5 . 2. MIXED GREY COPPER. Vermischtes Fahlerz. Cuivre gris. Panabase. Cleavage = J pmt. 3. 1. p,m,t. |PMT, H97 103 . L648 3 . 3. 1. p,m,t,mt.pm,pt, PMT, Jpmt, H98 109 . L648 11 . 3. 4. p,m,t. |PMT, |(3p|mt) Anhalt L648 10 . 3. 4. p,m,t,mt.pm,pt, iPMT, (3pmt) L648 6 . 3. 4. p,m,t,mt.pm,pt, iPMT, | pmt, i (3p|mt) Z 2 nw, H98 m . L648 13 . 3, 4. p,m,t,mt.pm,pt, |PMT, 1 (SP|MT) Z 2 nw, 1 (3p^rat) Z 2 ne,...L648 15 . 3. 4. p,m,t,mtpm,pt ; i(3P|MT) Z 2 iiw,i(3p^mt)Z 2 ne,^(3piMT)Z 3 nw, L648 16 . 3. 4. P,M,T, mt.pm,pt, |PMT, |pmt, ^ (SP^MT) Z 2 nw, ^ (3plmt) Z 2 ne, ... P312. L648 17 . 4. 1. MT.PM,PT, Model 63. J 219 . 5. 1. mt.pm,pt,iPMT,...Kapnik. Dillenberg. Md.78. R 32 . H97 104 . L684 4 . 5. 4. mt.pm 5 pt,PMT, |(3p^mt) Z 2 nw, Felsobanya, Clausthal, Schemnitz Model 94. R 33 . H9S 107 . L648 5 . 5. 4. mt.pm,pt, |PMT, |PMT, i(3p|MT)Z 2 nw, i(3p|mt) Z 2 ne,...H98 in . MINERALS OF THE OCTAHEDRAL SYSTEM. 29 5. 4. mt.pm,pt, JPMT, ipmt, (3pmt) Z 2 nw, ............ H98 108 . L648 12 . 5. 4. MT.PM ? PT, PMT, (SP^MT) Z 2 nw, (3p|mt) Z-ne,...Dillenberg... R 33a . H98 110 . L648 14 . 5. 4. mt.pm 5 ptaPMT,^(3p^MT)Z 2 nw,^(6pim^t)Z 3 n 2 w 5 ...Ilanz...R 83b . 5. 4. MT.PM,PT, (3P|MT) Z 2 nw, * (Splmt) ZVw, ... Dillenberg...R 34 . 6. 1. APMT, ......... Heidelberg ...... Model 117. R 25 . H97 100 . J 213 . L648 1 . 6. 1. lPMT,lpmt, ...... Kapnik ...... Model 118. H97 102 . R 31 . J 2U - L648 2 . 6. 1. (JPMT, Jpmt) x 2, ............................................. L648 18 . 6. 1. IPMT Znw, i (Splmt) Z 2 ne, ........................... H97 105 . L648 7 . 6. 3. !(3PMT) ...... Clausthal...Sim.Md. 119. H97 101 . R 29 . J 217 . L648 9 . 0. 4. IPMT, 1 (3pjmt), l (3pimt) ..................... Clausthal ...... R45. 6. 4. |PMT Znw, K3P^MT) Z 2 nw, ................ R 28 . J 216 . H97 106 . L648 8 . 3. ANTIMONIAL GREY COPPER. Antimonfahlerz. Schwartzerz, Black Copper. Cuivre gris arsenifere. Cleavage = 0. 3. 1. p,m,t,mt.pm,pt,lPMT, ........................... Model 37. Jiii321 b . 3. 1. p,m,t,mt.pm,pt,lPMT,lpmt, .................................... Jiii321 c . 4. 1. MT.PM,PT, ....................................... Model 63. Jiii321 2 . 4. 1. MT.PM,PT,lpmt, .................................................. P313 7 . 6. 1. IPMT, ............................................. Model 117. Jiii321 a . SULPHURET OF TIN. Zmnkies. Etain sulfure. Tin Pyrites. Cleavage = p,m,t,mt.pm,pt. 1. 1. P,M,T? ..................... Cornwall ...... Model 1. Miii 163. L624. 25. GARNET. Granat. Grenat: comprehending the following species: 1. Almandine. Precious Garnet. Edler Granat. 2. Cinnamon Stone. Kaneelstein. 3. Grossular. Green Garnet. Aplome. 4. Common Garnet. Gemeiner Granat, 5. Melanite. Black Garnet. Pyreneite. 6. Manganesian Garnet. Mangangranat. 7. Rothoffite. Colophonite. Brown Garnet. Cleavage = MT.PM,PT. . 3. 3PJMT, ...... Arendal ...... Model 22, H61 39 . L488 4 . J 57 . P14 4 . R 6 . 3. 3. 6 PfMJT, ......... Mussain Piedmont ...... Similar Model 23. P18. 3. 1. p,m,t,MT.PM,PT, ... Czilklowa, Siberia... Model 28. L488 2 . P16. 4, 1. MT.PM,PT, ...... Arendal ...... Model 63. H60 36 . P14 1 . R 4 . L488 1 . 4. 1. MT.pm,pt. Axes: pm a t% ................................. L488 1 note. -4. 4. MT.PM,PT, 3pjmt 5 ...... Melanite from Frascati ...... Sim. Md. 69. J 58 . H61 40 . P14 2 . R 5 . L488 3 . 4. 4. MT.PM,PT, l(3plmt) ....................................... L488 3 note. 4. 4. MT.PM,PT, 3pjMT, 3p|mt, ........................ Brossothal ...... R te . 4. 4. MT,mt|,m|t.PM,pm|,p|m,PT 5 pt|,p|t, 3pimt,...Friedberg...R31. 4. 4. mt.pm,pt, 3 P^MT,... Grossular from Siberia... Md. 69. P14 3 . R24. 4. 4. MT.PM,PT,3PJMT,6pimJt, ......... Arendal... R 11 . L488 5 . Mr 30 . 30 MINERALS OF THE OCTAHEDRAL SYSTEM. 4. 4. MT.rM,PT } SPiMT, Gpjmjt, Cziklowa R35. 4. 4. MT,mt 2 ,nL,t. pM,pm 2 ,p;jn, PT,pt 2 ,p 2 t, 3P,}MT, Dognatzka L488 6 . R 5b . J 60 . H61 42 . 36. LEUCITE. Leucit. Amphigene. Cleavage = p,m,t,mt.pm,pt. 2. 1. PMT, Model 15. L435 2 . 2. 3. 3PMT, Model 22. R 6 . L435 3 . P105. H78 218 . 3. 1. P,M,T.pmt, Similar Model 29. L435 1 . 25 1 . SODALITE. Sodalit. Cleavage = MT.PM,PT. 3. 1. p,m,t,MT.PM,PT, 3plmt, Vesuvius L46P. 4. 1. MT.PM,PT, Vesuvius. Greenland... Model 63. L461 1 . P134. 28.** BISMUTH BLENDE. Wismuthkieselerz. Arsenical Bismuth. Cleavage = mt.pm.pt. 6. 1. iPMT, Model 117. D 30 . 6. 3. i(3PiMT) Similar Model 119. P279. D 34 . 6. 4. IPMT Znw, i (3PJMT) Z 2 nw, D 35 . 29. ANALCIME. Cubicite. Analcim. Kubizit. Sarcolite. Cleavage = p,m,t. 1. 1. P,M,T, Model 1. L202 1 . P138 1 . J 78 . H85 287 . 2. 3. 3PJMT, Kilpatrick Hills.. .Md. 22. H85 288 . L202 3 . D 16 . J 80 . R 6 . P138 3 . 2. 4. 3PzMT,3P|MT, Catania Levy 45 2 . 3. 1. P,M,T.3pfmt, Model 39. H85* 89 . L202 2 . D 14 . J 79 . P138 2 . R 19 . 3. 1. P } M,T.3plMT, Similar Model 39. L202 2 . P139 1 . 3. 1. P,M 5 T,mt.pm,pt,PMT, (Sarcolite) Model 31. P139 2 . 3. 4. p,m,t.3PiMT, D 15 . 30. ARSENIATE OF IRON. Wurfelerz. Cube Ore. Per Arseniate. Cleavage = p,m ; t. 1. . P,M,T, Model 1. Jii 341 1 . L165 1 . P235 1 . 3. . P,M,T,mt.pm,pt, Model 27. Jii 342 3 . Ll65 2 . 3. . P,M,T,mt.pm,pt,pmt, Model 31. Jii342 4 . L165 5 . P235 4 . 3. . P,M,T.pmt, Similar Model 29. L165 3 . 3. . P,M,T. ipmt, Cornwall Model 38. R 37 . P235 2 . J ii 342 2 . 3. . PjMjTjiut.pnijpt, ipmt, Cornwall Levy 70 4 . 3. . P,M,T.pmt,3p,mt, L165 4 . 3. . P,M,T.ipmt 5 K 3 P:: mt ) P235 3 . 6. 3. l(3P VMT), Cornwall Levy 70 2 . P235 5 . Rose has not marked this mineral with **, as having hemihedral forms with inclined faces, although he has given an example of such a form. MINERALS OF THE OCTAHEDRAL SYSTEM. 31 31. ALUM. Alaun. Alun. Alumine sulfatee. Comprehends the following varieties : 1. Potash Alum. 2. Ammonia Alum. 3. Soda Alum. Cleavage = pint. 1. 1. P,M,T, Model 1. H98 124 . 3. 1. PMT, , ...Model 15. H48 125 . 3. 1. PMT x 2, Model 16. HH117 5 . 3. 1. P,M,T.pmt, Similar Model 29. H48 1SG . 3. 1. P,M,T, mt. pm,pt,PMT, Model 33. H48 126 . The forms of factitious crystals of alum are often very irregular, presenting any number and combination of the planes of the cube and rhombic dodecahedron, subordinate to the planes of the octahedron. The following symbols represent some of these combinations: Jmtnw,ne. PMT. r. ipmZn, PMT. p, Jmn, te, Jmt ne, se. PMT. p,m,t. I PM Zn, i pm Nn, { PT Nw, \ pt Zvv, PMT. These combinations prove that, although there is no substantive crystal, which can be called a hemi-rhombic-dodecahedron, yet the planes of the three zones of that combina- tion, namely, the forms MT, PM, and PT, are all susceptible of producing either hemihe- dral or tetartohedral forms. In the same manner, the crystals of commercial alum present all kinds of odd numbers of the planes of the cube. 33.** HELVINE. Helvin. Cleavage =r -Jpmt, or 0. 5. 1. mt.pm,pt,PMT, Model 78. L463 3 . 6. 1. JPMT, Model 117. R 25 . L463 1 . 6. 1. PMT, |pmt, Schwartzenberg Model 118. L463 2 . P243. 33. An Isomorphous Group, comprehending : 1. LAPIS LAZULI. Lazurstein. Azurestone. 2. HAUYNE. Haiiyn. 3. NOSIAN. Nosin. Spinellane. Cleavage =. mt.pm,pt. 3. 1. PMT, Model 15. L457 4 . 3. 1. p,m,t,MT.PM,PT> Model 28. L457 5 . 3. 1. p,m,t,MT.PM,PT, 3p^mt, L457 6 4. 1. MT.PM,PT, Model 63. P131. Hiii56'. R 4 . Pill. L457 1 . 4. 1. MT.PM 5 PT,pmt ? Model 65. L457 3 . 4. 1. MT.PM,PT, Spjmt, Model 64. Hiii56 2 . L457 2 . 34. PYROCHLORE. Octahedral Titanium Ore. Cleavage = 0. 3. 1. PMT, Norway Model 15. P260. 32 MINERALS OF THE PYRAMIDAL SYSTEM. 35. PYROPE. Red Garnet. Cleavage = MT,PM,PT. 4. 1. MT.PM,PT, .: Model 63. R174. 36. CANCRINITE, a variety of Sodalite. Cleavage = mt.pm.pt. 4. 1. MT.PMPT, Siberia Model 63. P134. R174. 39*. UWAROWITE. 4t. 1. MT.PM,PT, Model 63. P408. R174. The reader is requested to make the following addition to the list of authorities given at page 14. The work in question came into my hands only a few days before the printing of this page, and, therefore, is not quoted in the previous section: Ly = Levy, A. Description d'une Collection de Mineraux, formee par M. Henri Heuland, et appartenant a M. Ch, Hampden Turner, de Rooksnest dans le Comte de Surrey. London, 1837, 3 vols. 8vo, with a folio Atlas. The large figures refer to the plates, and the small figures to the subjects. The letter v = variety, and refers to descrip- tions given in the text. This work contains about twelve hundred figures of crystals, apparently drawn with great accuracy, but it is published without a single measure- ment of their angles, which greatly detracts from its utility. There is a Table of Angles spoken of in the preface, but none printed in the book. CLASS II. MINERALS BELONGING TO THE PYRAMIDAL SYSTEM OF CRYST ALLIS A TION. The AXES of all Combinations belonging to this Class are = p| m a t a . Different Combinations of the same Mineral have Axes which are sometimes = p+ m a t% and sometimes = pi m a t a , but never =. p a m a t a . The FORMS are either equiaxed or unequiaxed. The following are those of most frequent occurrence : Forms of the North Zone. P PJM. PfM. A P^M. I PiM. 1 P|M. 1 . PfM. cS P|M. 9 P|M. J PJM. PfM. PfM. a.) Homohedral Forms. Forms of the East Zone. P. PfT. PIT. P|T. PfT. PfT. P|T. PfT. PIT. Forms of the Forms of the Octahedral Zones. Prismatic Zones. P. M. PiMT. T. P^MT. MT. PfMT. M_T,M + T. PfMT. MfT,M 3 T. P T 5 2 MT. MJT,M 8 T. P^MT. MfT, M|T. PfMT. PfMT. PfMT. PfMT. PfMT. PfMT. MINERALS OF THE PYRAMIDAL SYSTEM. 33 Forms of the Forms of the Forms of the Dioctahedrons North Zone. East Zone. Octahedral Zones. which always occur subordinately. PM. PT. PMT. r P.y>M. P^T. pjm^t, pjmtj. P|M. P|T. PfMT. plm^t, p^mt^. PfM. PfT. PfMT. pm_t + , pm_|_t_. P 2 M. P 2 T. P|MT. p 2 mt 3 , p.m 3 t. P|M. P|T. PfMT. p 3 mt 2 , p 3 m 2 t. . PfM. P|T. P 2 MT. p 4 mt 2 , p 4 m 2 t. PfM. PfT. P|MT. p 4 mt 3 , P4,m 3 t. P 5 M. P 3 T. P 3 MT. p 5 mt 3 , p 6 m 3 t. P 4 M. P 4 T. p+mt_, p + m_t. M. T. MT. p x m y t z , p x m z t y . 2.) Hemihedral Forms. i(M_T,M+T). iP|M. iPMT. i (3 P MT). PJMT. PiMT. PfMT. (p x m y t z , p> z t y .) (pim|t, pimt J.) The Table shows the order in which these Forms are arranged, when several of them occur together upon one Combination. The angle of P upon PM, or of PM upon M, is 135, and all the forms quoted betwixt P and PM make with P a more obtuse angle than 1 35, and appear upon a combination exactly in the order in which they are placed in the Table. On the other hand, the planes betwixt PM and M make with P a more acute angle than 135, and correspondingly a more obtuse angle with M. In short, the series from P to M, from P to T, or from P to MT, repre- sents the gradual passage from a horizontal to a vertical plane, the forms PM, PT, and PMT, which are equiaxed, constituting the middle points of each oblique series. 1. CHLORIDE or MERCURY. Quecksilberhornerz. Mercure muriate. Axes : p^ m a t a . Cleavage = m,t. PfM Zn on Nn : 136 Brooke. Cot. 68 = 0.404 = PJ-.J5M = P|M. PfMZn on Mn : 129 32' = PfM on m a : 39 32 X Cot. 1.21166 = P|gM = P|M. PfMT on MT = 119 30 r . 119 30 r 90 = 29 30 r = cot. 1.7675. 1.7675 x sec 45 = 2.4996 = PJJMT = PfMT. 3. . P|M,P|T, Jii357 3 . L580 2 . 3. . P + ,MT.pfm,p|t,p|mt. Factitious. Brooke, Annals Phil. Oct. 1823. 4. . M,T. P|M, PfT, J H356 1 . L580 3 . 4. . MT.P|M, P|T, JH357 2 . L580 1 . A 65 . MH156 97 . 4. . M,T.p|m,p|t,pfmt, JL580 4 . 31 MINERALS OP THE PYRAMIDAL SYSTEM. 4. 1. M,T,mt.p3m,p|t,pfmt, ..................... ...... L580 5 . M ii 157 102 . 4. 1. M^mt.pfmjpfmjpft, pt,pfmt, ...................... D249. P380. 2. BRAUNITE. Brachytypous Manganese Ore, HAIDINGER. Edinburgh Journal of Science, January, 1826. Axes: p*m a t a . Cleavage = PMT. PMT Znw on Nnw == 108 39'. Haidinger. Cot. 54 19|' = 0.7179. 0.7179 X 1-4142 = 1.015 = PffiMT = PMT. P 2 MT Znw. on Nnw = 140 30'. Cot. 70 15' = 0.359- 0.359 X 1.4142 = 0.5077 = PiSgMT = P^MT = P 2 MT. 2. 1. PMT,P 2 MT, ....................................... Haidinger 2 18 . L759 2 . 3. 3. PMT,P 2 M 3 T,P 2 MT 3 , ................................. Haid. 2 19 L759 4 . 5. 1. p. PMT, ................................................ Haid. 2 17 . L759 1 . 5. 1. p.pmt,P 2 MT, .......................................... Haid. 2 20 . L759 3 . 3. An Isomorphous Group, 1,2: 1. OXIDE OF TIN. Zinnstein. Etain Oxide. Zinnerz. Axes: p*m a t a . Cleavage = m,t,mt.p|M,pT. Pf M Zn on Nn : 67 52' P. Cot. 33 56' : 1.4863 == P|M = P}-M. PfMT Znw on Nnw = 86 58' Haiiy. Cot. 43 29' == 1.0544. 1.0544 X 1.4142 = 1.493 = Piooo MT _ jno MT == p|MT. 3. . Pf M, Pf T, (Axes: pJmStS) ............ L355 1 . H112 253 . P250 1 . J 186 . 3. . P^,M,T,HT.p|ni,pft,pfmt, .................................... HI12 262 . 4. . MT. PfM,P|-T, ........................... L355 2 . H112 954 . P250 2 . J 190 . 4. . MT.PfMT, ............................... L355 6 . H112 ?55 . P250 4 . J 187 . 4. . (MT.PfMT) X 2, ......... Similar Model 62. H113 569 . J 193 . S 438 . 4. . (M,T.PfM,PfT) x 2, ................................. Model 62. S 438 . 4. . MT. PfM^fT^m^psmta, .............................. J 191 . H112 257 . 4. . MT. Pf MT ; p 3 Tn 2 t,p 3 mt 2 , ................ . ...................... P250 5 . J 193 . 4. . M,T,mt.PfMT, .............................. Cornwall ...... D363 1 . S 435 . 4. 1. M,T,mt. pf m>p ft,PfMT, ............ Goshen, Massachusets, ...... S 436 . 4. 1. m,t,MT.PMT, ................................... . ...... L355 7 . H112 256 . 4. 1. MT.pf M,pf T, pf mt, ................................................ P250 3 . 4. 1. m,t,MT.pfm,pft, pfmt, .............................. H112 260 . R 63 . J 183 . 4. 1. MT.p|mt,p 3 M 2 T,p 3 MT 2 , ........................... J 192 . P250 6 . H112 268 . 4. 1. MT.pf m,pft 5 pfmt,p 3 M 2 T,p 3 MT 2 , .............................. H112 259 . 4. 4. m,t,MT,mt 3 ,m 3 t.p|m,pft,PfMT, ........................ H112 261 . J 189 . 4. 4. MT,m|t J mft.pfm,pft J pfmt,p 3 m 2 t,p 3 mt 2 , .................. D363 2 . S 437 . 2. RUTILE. Rutil. Titane oxide. Axes: p^m a t a . Cleavage = m,t,MT. PfM on M : 122 51'. 122 51' 90 = 32 51'. Cot. 1.5487 = Pioo M = po M = P | M> PfMT on MT : 132 20'. 132 20' 90 = 42 20'. Cot. 1.0977. 1.0977 X 1-4142 == 1.5521 = P|JMT = Pj^MT = PfMT. 1. 1. P/ T ,M,T. (Hatty's primitive form) Md. 2. L360 1 .P255 1 . Hiv333. 1. 1. Pi,MT,,.. (Ditto, position reversed) ...... Md. 2. H iv 234. L360 1 . MINERALS OF THE PYRAMIDAL SYSTEM. 35 2. 1. PfMT, ................................................ Hiv235'. Hll7 310r . 3. 1. P,m,t,MT .......................................................... L360 3 . 3. 1. P^m^MT.pfmt, ................. .......... Aschaffenberg ...... L360 4 . 3. 1. P^,MT. pf mt, ...................................................... L360 7 . 3. 4, PJ, mlt,m 2 t,MT, ................................................... L360 3 . 4. 1. MT.PfM,P|T, .............................. Aschaffenberg ...... L360 2 . 4. 1. MT.PfMT, ......................................................... L360 8 . 4. 1. m,t,MT.PfMT, ............... Aschaffenberg ...... L360( 4 ). Mii377 2 . 4. 3. MT 2 ,M 2 T.PiMT, ....................................... H117 309 . L360 6 . 4. 4. M,T 5 mt,mt 2 ,m 2 t.pfm,pft,PfMT, ................................. HH7 310 . 4. 4. m,t,MT 5 mt 3 ,m 3 t.pfm,pft,PfMT, .................. S 385 . P255 2 . D359 2 . 5. 1. (m,t,MT.JPM) x 2, .......................................... H117 311 . 5. 3. (MT 2 . JPfM) X 2, ....................................... H117 312 . J 170 . 5. 4. (MT,MT 2 . JPf M) X 2, .......................................... H117 313 . 5. 4. (M, T ,MT,mt,m 3 t.iPfM) x 2, .............................. P255 3 . S 386 . 4. ANATASE. Anatas. Titane Anatase. Ootahedrite. Axes: p*m a t a . Cleavage = P. P|M, P|T. P|M Zn on Nn : 137 10'. H. tan 68 35' : 2,5495 pf-iM = P|M; P|M Zn on Nn : 136 22' M. tan, 68 11' = 2.498 = PM. n on Nn : 53 6' M. cot. 26 33' = 2.0013 = P'M = PM 8 . Z. I. P5-M,PfT, ................................. Model 13. L358 1 . H117 314 . 3. 1. (PfM,PiT) X 2, ................................................... L358 20 . S. 1. PjM,pim,P!T,p!t, ......... Model 14. Mii379 M05 . L358 3 . Hll7 3!<3 . . 1. P!M,pJm, P !m,PJT,pAt,pJt? .................................... L358 6 . 3. 1. P5-M,P!T,pimt,plmt? ............................................. L358 5 . 3. 1. PfM^WI-T^plmt, .......................... . .................. -R 57 - 2. 1. P!M,pm,PfT,p^,p5mt, ........................................ L358 14 . 3. 1. P|M 5 PiT ; pim^pimti, ....................................... H117 317 . 3. 1. p 5 mt.P|M,PfT, ...................................................... L358 8 . 3. 1. p,mt.P|M,P|T,pimt, ............................................. L368 9 . 3. 1. p,m^PfM,P|T,pfmt, ............................................. L358 15 . 3. 1. p,m,t, mt.PfM,P|T J pmt, ....................................... L358 16 . 4. 1. M,T 3 mt.PM,P|T, ................................................ L358 13 . 5. 1. p.P5M 5 P|T, ................................. Mii379 l . L358 2 . H117 315 . 5. 1. (p.PiM,P5T) X 2, .............................. Dauphiny ...... L358 20 . 5. 1. p.P!M,pim,PT>pit, ............................................. L358 4 . 5. 1. p.PM,P5T,pJmt,plmt? .......................................... L358 7 . 5. 1, p.P|M,p|T,p|mt,p|int, ....................................... MH379 3 . 5. 1. p.PM,PT,p5mt, .. ........................................ L358 12 . R 58 . 5. 1. pi P|M,P5T,p3mt, ...................... ! ......................... L358 10 . 5. 1. p. P5M,PJT,p5mt 5 p 3 mt, ............................... , .......... L358 11 . 5. 1. P|M,P5T,pimt,pimt 5 p 3 mt, ....................................... L358' 7 . . 1 . p+. P^M, pjm, PJ T,plt,pS mt,pjma,pjm Jt, ............ M ii 379 flfJ< 10 36 MINERALS OF THE PYRAMIDAL SYSTEM. &.** COPPER PYRITES. Kupferkies. Cuivre Pyriteux. Cu Fe S. Axes : p*m a t a (sometimes p a m a t a ?) Cleavage = p. P 2 M,P 2 T. The following measurements and computations tend to prove the ex- istence of equiaxed pyramidal forms upon combinations of the two-and- one-axed Class : Forms of the Octahedral Zones. PMT = P = 109 53'. 108 40'. Mohs. 108 40' -j- 2 = 54 20'. log cot 54 20' = 9.8559376 log sec 45 = 10.1505150 log cot 44 34' = 10.0064526 cot 44 34|' = 1.0149465 = PMT. = P 4 (d) = 38 15' Mohs. 38 25' -v- 2 = 19 121'. log cot 19 12^ = 10.4583286 log sec 45 = 10.1505150 log cot 13 49 1' = 10.6088436 cot 13 49^= 4.0636171 =PiMT. P_3(e)=4950' u Mohs. 49 50' -f- 2 = 24 55'. log cot 24 55' == 10.3329786 log sec 45 = 10.1505150 log cot 18 11'= 10.4834936 cot 18 11' = 3.0445018 = PJMT. PiMT = P 2 = 69 44'. .Mohs. 69 44' -*. 2 = 34 52'. log cot 34 52' = 10.1569261 log sec 45 = 10.1505150 Forms of the Octahedral Zones. PMT = P + 2 = 140 31 r . Mohs. 140 31 r ~ 2 = 70 I5y. log cot 70 15J 7 = 9-5553374 log sec 45 = 10.1505150 log cot 63 4' = 9-7058524 cot 63 4' = 0.5080607 = P 8 MT. Forms of the N. and E. Zones. PM, PT = P 1 (b) = 120 30', 89 9'. Mohs. 899' -f- 2 = 44 341'. Cot 1.01 49465 = PiM = PM. P - 2 (g) = P|M,P|T = 66 36'. Mohs. 66 36' ~- 2 = 33 1 8'. cot 1.5223545. 56'. Mohs. ^-2 = 555S'.cot.0.6753553 = PfM. P,M, P 2 T = P + 1 (c) = 126 11'. Mohs. 35J'. cot 0.5075119 = PVM = P 2 M. log cot 26 14' = 10.3074411 cot 26 14' = 2.0292873 = PJMT. The axes m ft t a of these forms are all, according to the results of the above calculations, invariably 1 J per cent, greater than they should be, to agree with the short symbols that I have given to express their rela- MINERALS OF THE PYRAMIDAL SYSTEM. 37 tion to the axis p a . To make the symbols agree with the reckoning, I should say : Pi:8888?M, PfcgfS wr, Pi:8iSS8 MT > instead of PM,PT,PMT ; also, PmtyfflM* Pi:j}MttT, instead of P 2 M,P 2 T; and Pi:gS$$$& MT instead of P-JMT ; and I admit that these proportions may possibly represent the combina- tions which occur in nature, although I may at the same time be per- mitted to inquire whether or not the natural combinations have in this case been correctly measured. It will be observed that, according to Mohs, the form PMT measures across the equator 108 40', which measurement gives for the axes p a and m a the relation of 1 to 1.0149465. But Phillips (Introduction to M. p. 315) states this angle to be 180 71 10' = 108 50', which gives for the axes p a m a the relation of 1 to 1.0118215. But, again, he states that a plane of the form PMT makes with the horizontal plane P Z an angle of 126. Now, 126 90 = 36 = PMT on p a . This gives 54 as the value of the -J angle at the equator, and the relation of the axes p a m a comes thence to be that of 1 to 1.0274253. Finally, we find Haiiy stating the relation of the axes to be that of 1 to 1. Hence we have: p a to m a or t a . Haiiy, 1 : 1.0000000. Phillips, 1 : 1.0118213. Mohs, 1 : 1.0149465. Phillips, 1 : 1.0274253. Both of Phillips's measurements cannot be correct, because they are not consistent ; and we have no means of calculating whether either of them is correct. Mohs has taken the mean of these as the true measure, from which it may be concluded, that Mohs's angles are those of arithmetic and not of nature ; and that, consequently, all the above calculations give only approximate values for the lengths of the axes, and are insufficient to prove the impropriety of denoting the forms by symbols with short fractions, such as those which I have given in the Table. 9. 1. PMT, (Mohs's assumed primitive) L645 1 . Jiii311 la . 3. 1. P 2 M,P 2 T, (Phillips's assumed primitive) P315 1 . 2. 1. P 2 M,P 2 T,pmt, Freiberg...... L645 5 . P315 2 . Mii470 2 . 3. 1. p,m,t.PMT, JiU311 lc . 3. 1. p,m,t.PMT, H97 103 . 5. 1. p. PMT, L645 2 . 5. 1. P_. iPMT,ipMT, Cornwall M 11470'. 5. 1. p.pm,pt,PMT, Model 77. L645 3 . 5. 1. p.p 2 m,p,t,^PMT,|rMT, D408 1 . A 153 . L645 4 . 5. 1. p.pm,p,m ? pt ? p,t,PMT,ipMT, Freiberg D408 3 . M ii 470 3 . 38 MINERALS OF THE PYRAMIDAL SYSTEM. 5. 1. p.pmjplmjplm^mjp^pftjplt^tjlPMT^PMTjipJmt Z 2 ne,Jpimt Z 2 ne, i(p Jm Jt,p Jmt J) Z 2 ne, ............ M ii 470 4 fig - 178 6. 1. JPMT, ......... (HaUy's assumed primitive) ...... H97 100 . Jm311 2b . 6. 1. iPMT,ipmt, ................................. P315 3 . Jiii311 2a . H97 102 . 6. 1. p 2 m,p 2 UPMT,iPMT, ................................................ D408 3 . 6. 3. i(3PiMT), ......................................................... H97 10 '. 6. CRYOLITE. Kryolith. Alumine fluatee alkaline. Axes: p x m a t a . Cleavage = p,m,t. 1. 1. P X ,M,T? ............................................. P204. L570. R164. f. HAUSMANNITE. Black Manganese. Manganese oxide hydrate. Axes: p x m a t a . Cleavage = pfm,pft. PfM Zn on Nn : 117 54' Haid. cot. 58 57' = 0.602 = P\PM = PfM. 3. 1. PfM,PfT, ....................................... L760 1 . Haidinger 2 11 .*. . 1. (PfM,PfT)x2, ..................... Mii 106 - 107 . L760*. Haidinger 2 13 . 3. 1. PfM^lm^fT^ft, ..................... Mii 105 . L760 2 . Haidinger 2 12 . 2. 1. PfM,pfm,PfM,p!t 5 p!mt, .................. L760 3 . Haid. p. 46. com - 2 . * Edinburgh Journal of Science, Jan. 1826. Mohs ii 416. 8. PHOSPHATE OF YTTRIA. Phosphorsaure Yttererde. Axes: p^m a t a . Cleavage = M,T. 4. 1. m,t.PM,PT ................... Lindenaesin Norway ...... P194. L276. 9.* FERGUSONITE. Fergusonit. Axes : p4-m a t a . Cleavage = p 2 m,p 2 t. 1. 3. P, |MT + ,iM + T, ................................................ Miii98'. 3. 3. p, IMT+^M+T. P 2 M,P 2 T, ............... M iii 98 3 . fig - no . P274. A 139 . 3. 3. p, |mt + ,|m + t. p z m,p a t, |(p x m y t z , p x m t y ) ............... M Hi 98 4 . %< i93 . 1O.* An IsomorpJious Gi'oup, 1, 2, 3 : 1. TUNGSTATE OF LIME. Tungstein. Scheelin Caleaire. Axes: p4.m a t a . Cleavage = P|M } P|T,P|MT. P|M Zn on Nn : 1 13 36 r . Zn on Ze : 107 27 r = P|MT Znw on Zne 100 8 r , Znw on Nnw : 130 20' = P \8PMT = P|MT. 9. 1. P1M,P1T 3 .............................. Ly. v. 1 H119 329 . P182 3 . L347 3 . . 1. P|MT, .......................................... H119 328 . P182 1 . L347 1 . . 1. P|M,P|T,p|mt, ........................ Ly v. 2 . H119 330 . P182 2 . L347 2 . . 1. P|M,pfm,P|T 5 pft 5 ............................................. M ii 114 2 . . 1. PlM^nvPIT^t, .......................................... MH114 3 . 3. 4. P|M,P|T,p|mt,|pm_t +J ipm + t_, ........................... MH114 4 . 2. 4. PIMjPITjplmtap+mt^^jnj;, ....... . .......... Miill4 5 . Ly79 3 . 3. 4. P|M,P|T 5 plmt,ipm_t + ^pm + t_,|p + mt_,ip + m_t,...Miill4 6 . fig - 108 . 5. 1. .pMmp^tjPMT, ....................................... Ly79 2 . MINERALS OF THE PYRAMIDAL SYSTEM. 39 5. 1. P.PfM,PfT, MH114 1 . 5. 4, p.PJM,p^ m,p 4 ra,P|T ? pV t 5 p 4 t,pmt,PJMT,pfmt,p 4 mt 2) p 4 m 2 t, PI 83. 2. TUNGSTATE OF LEAD. Scheelbleierz. Plomb Tungstate. Axes: p4.m a t a . Cleavage = P. P|MT. 1. 1. P + ,M,T, L345 1 . 1. 1. P_,M,T, L345C). 3. 1. P 3 MT, Levy, Annals Phil, xxviii. 364. L345 4 . 3. 1. PfM,pfm,PiT,pft,plmt, Ly59 2 . 3. 1. P + ,M,T.p|mt, L345 2 . 3. 1. P + ,M,T.pim,plt, P370, 4. 1. MT.pfm,pft,P|MT, Levy Annals Phil, xxviii. 364 1 . 4. 1. MT.pfm,pft,P|MT,P 3 MT, idem. 5. 1. P + . P|M,P|T, L345 5 . 5. 1. P.P|MT, L345 3 . 3. MOLYBDATE OF LEAD. Gelbbleierz. Plomb molybdate. Axes: pm a t a . Cleavage = pjm,pt,pfmt. Pf M Zn on Zw 128. Zn on Nn : 76 40'. H = P|;ooo M = PfMT Znw on Zne : 116 22'. Znw on Nnw : 96 22' H. = = PfMT. 1. 1. P_,M,T, P367 3 . H95 52 . L341 9 . 1. 1. P^m,t,mt P367 5 . H95 86 . L341 10 . I. 4. P_,mt,mft,m|t, H95 87 . L341 n . 3. 1. PfM,PfT, Bleiberg Ly v. 1 . P367 1 . H94 77 . L341 1 . 9. 1. p|m,pft,P^MT, H95 81 . L341 4 . 3. 1. P_,M,T.pjm,p|t, P367 2 . H95 83 . L341 8 . 3. . p,mt.PfM,Pfr, H95 84 . L341 3 . p,MT.PfM,Pf T,pfmt, P367*. H95 88 . L341 6 . MT.PIMT, P367 6 . Ly58 7 . 5. . P_.pfM,pfT,pfmt, H95 85 . L341 5 . 5. . P_.PfM,PfT, Model 76. Ly58 9 . H95 83 . L34P. P.P|MT, Ly58 8 . Levy quotes many other combinations of this mineral, but his descriptions are not ac- companied by measurements of angles, so that they cannot be accurately expressed in symbols. II. ZIRCON. Zirkon. Hyacinth. Axes: p^m a t a . Cleavage = m,t. PfMT. PfMT Znw on Nnw : 84 2(X. Znw on Zne : 123 19' = P}?MT = PfMT. P 2 MT on MT : 159 35 r P. 159 35 r 90 = 69 35' = Py$$>MT = PVMT = P 2 MT. , 1. PfMT, Model 12 pm a t% L388 1 . R 55 . H58 19 . P95 1 . 4. 1. M,T.PfMT, p+m't% S 490 . J 6 . L388 5 . R 62 . H58 20 . P95 3 . 4. 1. M,T.PfMT, Ceylon plm a t a , H58 20a . Mii 368 2 . 40 MINERALS OF THE PYRAMIDAL SYSTEM. 4. 1. MT.PfMT,...p4.m a t%...St. Gotthardt...Model 61. L388 2 .Mii 368 1 . R 61 . J 3 . H59 21 . S 491 . 4. 1. m,t.P|MT, pm a t% P95 2 . 4. . mt.PiMT, pm a t% L388 3 . H59 22 . 4. . M,T,mt.PfMT, p^m a t%...Md.60. H59 24 . J 7 . S 492 .L388 6 . 4. . m,t,MT.P|MT, p4.m a t% ." " P95 4 . 4. . m,t,MT.pfm,pft,PfMT, ... pm a t% R 63 . 4. . MT.PfMT,p 2 mt, p.m a t a , H59 26 . S 493 . L388 4 . 4. . M,T.PMT,p 2 mt 3 ,p 2 m 3 t, . ...p4_m a t a , R 64 . H59 23 . L388 9 . 4. . MT.Pf MT,p 2 mt 3 ,p 2 m 3 t, p4.m a t% . . . J 4 . S 495 . H59 25 . P95 5 . L388 7 . 4. . MT.PMT,p 2 mt,p 2 mt 35 p 2 m 3 t,p4.m a t a , H59 28 . P95 6 . J 5 . L388 8 . 4. . M,T,mt.PfMT,p 2 mt 3 ,p 2 m 3 t, p4_m a t a , ...Mii 368 3 . H59 27 . J 8 . L388 10 . 4. . m,t,MT.pfm,pft,P|MT,p 2 mt, 3 ,p 2 m 3 t, p4_m a t a , . . . H60 29 . S 496 . L388 11 . 194 4. . m ; t,MT.PfMT,p 2 mt, p4.m a t a , S 4. . m,t,MT.p 2 mt,P-|MT,p 2 mt3,p 2 m 3 t, Norway Mii 368 , fi ^- 68 . 4. . m,t,MT.pf m,p|t,PMT,p 2 mt 3 ,p 2 m 3 t,p 4 mt3,p 4 m 3 t, Carinthia M ii 369 5 . 4. 1 . nijtjMT. p^Mjp^T,p^mt ; p 2 MT3jP2M 3 Tjp4mt 3 jp4m 3 t,p5nit 3 ,p 5 Tn 3 t ? Sanalpe M ii 369 6 . fig " ". 12. MURIO-CARBONATE OF LEAD. Hornbleierz. Axes: p^m a t a . Cleavage = p,M,T. P|M on P : 123 6'. 123 6' 90 = 33 6 = PJigJgM = P|M. 1. 1. 3. 3. 3. 3. 3. P+,M,T, L295 1 . P362. P + ,M,T,mt ? L295 2 . P362. P + ,M,T.pJm,pjjt, Ly 56 2 . L295 4 . P + ,M,T.pmt, L295 3 . P + ,M,T,mtpfm,pft, L295 5 . P362. D226. P + ,M,T,mt.pmt, L295 8 . D226. P + ,M,T.p|m,pft, L295 4 . 3. 4. P + ,M,T,mlt,m 3 t.p|m,p|t, L295 6 . 3. 4. P +) M,T,mt,mJt,m 3 t.p|m 5 p|t 5 L295 7 . 3. 4. P^Mj r r ) MT 9 m^t ) m s t.p^m ) p^t ) p 3 mt 2) p 3 m 2 t, Brooke, Phil. Mag. Series III. xi. 175. 4. 1. MT.PMT, L295 10 . 4. 1. M,T,mt.PMT, L295 9 . 13. MELLITE. Honigstein. Mellate of Alumina. Axes: pm a t a . Cleavage = pjmt. P|M Zn on Nn : 73 44' M. = P|;-JJM = P|M. P|MT Znw on Nnw : 93 22' H. = P| ; oooo MT = p|MT. 2. 1. PjMT, Thuringia L790 1 . H120 347 . 3. 1. p,m,t.P|MT, L790 2 . H120 349 . P395. R 56 . 3. 1. p,M,T.pjm,p|t,P|MT, Miii56 4 . fig - 104 . 4. 1. M,T.P|MT, L790 3 . H120 318 . Miii56 2 . 4. 4. m_t,m + t. PfMT, ' L790*. 5. 1. P_.p|mt, Miii56 1 . 8im - fi f- 92 . MINERALS OF THE PYRAMIDAL SYSTEM. 41 14. IDOCRASE. Vesuvian. Egeran. Pyramidal Garnet. Axes : p^.m a t a . Cleavage = p, M,T, mt. P|M Zn on Nn = 74 14' Mohs. = P|;O>O M = pj M . PJM Zn on Nn = 28 19' M. = P^go M _ P i M . Pf MT Znw on Nnw = 56 8' M. = P^MT = Pf MT. 1. 1. P+,MT, H72 156 . P20 1 . L483 1 . 1. 1. P + ,M,T,MT, (Egeran) Model 4. H72 15? . P21 2 . L483 2 . 1. 4. P + ,M,T,MT,mit,m 3 t, L483*. 3. 1. P + ,MT.pfmt, L483 2 . 3. 1. p +J M,T,MT.Pf M,PfT,... Siberia... Md. 42. M ii 354 1 . H72 158 . J 53 . P20 2 . L483 5 . 3. 1. p + ,M,T,MT.pfmt, P80 3 . L483 6 . 3. 1. P+ ,M,T,MT.PjM,PfT,pimit,plmti, H73 159 . L483 9 . 3. 4. p + ,m,t,MT,mit,m 3 t.PfM,PfT, Mii354 2 . H73 ieo . J 54 . L483 7 . 3. 4. p + ,M,T,MT,mit,m 3 t.PfM,PfT.pimJt,pimti, H73 161 . L483 10 . 3. 4. P + ,M > T,MT,mJt,m 3 t.p|m 5 pjt ) pJmit,plmtJ, R 65 . 3. 4. p^m^M^m^niat.PIMjPjT^fmt, J J5 . H73 16 '. M ii 354 3 . L483 8 . 3. 4. p + , M,T, MT, mit, m 3 t, pjm, P|M, p 3 m, pjt, P|T 3 p 3 t, pfmt, pjmjt, pjmti H73 163 . L483 11 . 3. 4. p^MjTjMTjmJtjingt.plmjPaM.pj^Ps^pfmt, 3 (p x m y t z> p x m z t y ) H73 161 . L483 12 . 3. 4. P + ,M,T, MT, m^t,m 3 t.pjm, P|M, p|m,p|ra,p 3 ra, pjt, p|T,p|t,pt,p 3 t, pfmt, 4(p x m y t z ,p x m z t y ) P21. fig. 3. 4. P + ,M,T, MT,mlt,m 2 t,mit,m 3 t.P|M,plM,p 3 m,P|T ) p|t > p 3 t,p|mt, 5(p x m y t z , p x m z t y ) Vesuvius M ii 355 4 . fig. ii % . 15. GEHLENITE. Axes: p x m a t a . Cleavage = p,m,t. 1. 1. P X ,M,T, Fassain Tyrol P22. L212. Miii 103. 16. WERNERITE. Pyramidal Felspar. Paranthine. Scapolite. Meionite. Axes : p x m a t a . Cleavage = p,m,t,mt. Pf M Zn on Nn = 62 56'. tan 31 28' = 6120 = Py^M = PfMT Znw on Nnw = 60. tan 30 X sec 45 = 2.449 = = PfMT. 1. 3. P + ,M,T,mt, (Paranthine) L473 1 . H75 184 . 1. 1. P + ,M,T,imtnw, J. J. G. 3. 4. rjM^MT^TjMaT.PfMjPfTjpfmt^niyt,, p x m z t y ,...( Meionite)... Vesuvius P150. 4. 1. M,T,MT.PfM,PfT, ...... (Wernerite)... Model 59. L473 2 . H75 182 . 4. 1 . M,T,MT.pf mt, (Paranthine) L473 3 . H75 185 . 4. 1. m,t,MT.PJM,PfT, Akudlek, Greenland... M ii 265 1 . Hii 585. 4. 1. M,T,MT.PJM,PfT,pfmt, (Scapolite) MH265 2 . P143. 4. 1 . M,T,mt.Pf M,Pf T,pf mt, p x m y t z , p x m z t y , Vesuvius. . .M ii 265 3 . 4. 4. M,T>mt,mit,m 3 t.PM,PfT, p x m y t 2 , p x m z t y , ... Vesuvius... Mii 265*. 42 MINERALS OF THE PYRAMIDAL, SYSTEM. IS?.* HUMBOLDTILITE. Axes : P4_m a t a . Cleavage = P. 1. 1. P + ,M,T,mt, .......................................... Vesuvius ...... D447. 1. 1. PMjTjm^mt, ................................. Vesuvius ...... D447- 18. URANITE. Uranglimmer. Two varieties : 1. Copper Uranite. Kupferuranit. Chalkolite. 2. Lime Uranite. Uran Mica. Kalkuranit. Axes: pm a t a . Cleavage = P,m,t. 1. 1. P_,MT, ................................. G230 1 . L141 1 . H116' 05 . P269 1 . 1. 1. Pz,mt, ............................................................... P269 2 . 1. 1. P^MT, ............................................................ L141 6 . 1. 4. Pz,m,mt, ............................................................ P269 3 . 3. 1. PJMT, ..... ........................................... L141 4 . Hiv321 2 . 3. 1. P_,m,t,mt, ................................................... ......... G230 3 . 3. 1. P,m,t.P_MT, ...................................................... G230 5 . 3. 1. P,MT.p_mt, ......................................................... L14P. 3. 1. P_,MT.p_m,p_t, ................................................... P269 5 . 3. 1. P_,MT.p_mt, .................................... G230 3 . L141 2 . P269 4 . 3. 1. P_,m,t,MT.p_m,p_t,p_mt, .............................. L141 7 . P269 6 . 3. 4. P,m,t,MT.m_t,m + t, ................................................ L141 9 . 3. 4. P_,m,t,MT,m_t,m + t. pm,p_m,pt,p_t,plmt,pf int,pf mt,pf mt, ......... P269 7 . L141 10 . 5. 1. P.P_MT, ................................................... G230 4 . L141 3 . 19. APOPHYLLITE. Fischaugenstein. Fish-eye-stone. Axes : p*mltl. Cleavage = P,m,t. PfMT Znw on Zne : 104 2'. Znw on Nnw 121 = P^^MT = PJMT. 1. 1. PJ,M,T, .............................. Models. L214 1 . H85 293 . P109 1 . 3. 1. PfMT, ............................................................... L214 7 . 3. 1. P,M,T.p|mt, ................................. Model 41. L214 2 . H85 295 . 3. 1. Pf,M,T,mt.pmt, ................................................... L214 9 . 3. 1. p + ,M,T.PfMT, ............................................. P109 3 . L214 3 . 3. 1. P_,m,t.p|mt, ....................................... Mii 11 . P109 2 . L214 4 . 3. 1. Pf,M,T.pm,pt 5 p|mt, ................................................ L214 8 . 3. 4. p + ,M,T,mlt,m 2 t.PfMT, ................................. H85 297 . L214 10 . 3. 4. P,M,T,mlt,m 2 t.pim,p|m,pit J p|t,PJMT,pimt J p 1 5 2 mt,... Mii fi ^ S8 . 4. 1. M,T.P|MT, P 4.m a t% .............................. Mii 97 . H85 294 . L214 5 . 4t. 4. M,T,mlt,ra 2 t.PJMT, .................. R 66 . H85 296 . L214 11 . Mii 244 3 . 5. 1. p.P|MT, ............................................................ L214 6 . SO. BLACK TELLURIUM. Blattererz. Nagyagererz. Axes : pm a t a . Cleavage = P. 1. 1. P_,M,T, ..................................................... .L689 1 . P343 1 . 1. 1. P_,M,T,mt, ......................................................... L689 2 . 3. 1. P_,M,T.p|m,pft, ........................... . ....................... L689 3 . 3. 1. P_,M,T.pfmt, ...................................................... L689 5 . 3. 3. 3. 5. 5. MINERALS OF THE RHOMBOHEDRAL SYSTEM. 43 . P_,M,T.pfm,pft,pfmt, L689 6 . . P_,M,T,mt.pfm,pft, L689 7 . . P_,M,T,mt.pfm,pft,pfmt, L689 9 . . P_.PfM,PfT, L689 4 . . P_.pfm,p|t,pjmt, A 160 . L689 8 . Ly80 2 P343 2 . D424. 21. MELLILITE. Mellilith. Axes: p_|.m a t a . Cleavage = ? 1. 1. P +5 M,T 5 mt P48. Levy 46 2 . 22. OERSTEDTJTE. Axes: p.|.m a t a . Cleavage = ? 3. 1. P,M,T,mt.pm,pt,pfmt, Dana 368. 4L 1. M/r,mt.PJMT,pfmt,p:j:mt,p x m y t z ,p x m z t y ? . . . Forchhammer, Pogg. Ann. xxxv. 630. 23. SOMERVILLITE. Axes: pm a t a . Cleavage = P. 3. 4. P,M,T,MT,mt > m s t.pJm,pft, P406. 24:.** EDINGTONITE. Axes: plm a t a . Cleavage = M,T. 5. 1. MT. JPJMT Znw, Zse, Nne, Nsw, JPf MT Zne, Zsw, Nnw, Nse, P150. CLASS III MINERALS BELONGING TO THE RHOMBOHEDRAL SYSTEM OF CRYSTALLISATION. The AXES of most Combinations belonging to this Class are = Pxrn^t^; but in Combinations which include the twelve-sided prism, = m,T,m 2 tif, M}f T 2 , the length of m a fluctuates between |J and if, without reaching either of those limits ; while the Axes of the Combina- tions of Tourmaline are out of all rule. The following are the chief FORMS and COMBINATIONS of the Rhom- bohedral System: 1.) The horizontal planes = P. 2.) The six-sided Prism = T,M}f T 2 . The twelve-sided Prism = m,T,m 2 t}J,M|f T 2 . The twenty-four-sided Prism = m,T,m 2 tL|-,Mif T 2 , 3m x t. The positions of these prisms, and the number of their planes, are im- mutable. Therefore, m, m 2 1}| cannot occur without T, My T 2 , nor can 3m x t occur without both the other combinations. 3.) A series of Rhombohedrons, whose general symbol is this : and whose characteristics (i. e. the equivalents of x in this symbol, or, in other words, the length of p a when t a is considered unity,) are these : Y. V. > > f, f. V. V 6 . f f, f, f, i, f.ff, f. J. ^ i. V. t, f- ti, f, i- I?' I. if. f. ^ It. I- f. f, f. f TT. , = P,m,T,m 2 t}-|,Mff T 2 == the 12 sided prism. P,V,, 3m x t = P,m/I\m,tif,Mif T 2 ,3ni x t = the 24 sided prism. The six-sided pyramid = PT,PM|f T 2 , which, as shown in 5) above, contains two equal and similar Rhbmbohedrons in opposite positions, may be expressed in abridged signs as follows : R! Zw + R! Ze, or 2R, Zw Ze. Also, R 1 Zn + R! Zs, or 2^ Zn Zs. The abridged signs, representative of the principal combinations, are given in the following catalogue within brackets, after the full symbol corresponding to each of them. About fifty of the most complex com- binations of calcareous spar, placed at the end of the list relating to that mineral, are particularised irr abridged signs alone. The expansion of the abridged signs into full symbols can be readily effected by substi- tuting the equivalent symbols in the place of each R, S, and V. Thus, fig. 153, plate 21, Haiiy = V. Rl Zw, R 4 Zw, r^Ze, rj Ze, r 2 Ze, S { , Becomes, T,M|f T 2 . iriZw, P 4 TZw, Ip^tZe, |pf tZe, |p 2 tZe, *PMff T 2 > T 2 , frimj-jt* ipjmift 2 , ^mift>, i(PM}-f Tf, PMJf T|, 1. An Isomorphous Group, 1, 2, 3: 1, NATIVE ANTIMONY. Antimon. Antimoine natif. Cleavage = P.^PJT, iPjMjf T 2 = (P.Rf). 5. 5. P4 p |TZw,iPJMf|T 2 = (P.RJ) Similar Model 11 4. P343. 46 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 2. NATIVE ARSENIC. Arsenik. Arsenic natif. Cleavage = p. 5. 5. p.iPJT Zw, iPfMffTo = (p.RJO-.. Sim. Mod. 86. P280. A226. 3. NATIVE TELLURIUM. Tellur. Cleavage = pft,p|mift 2 = (2R|). 3. 5. P + ,T,Mjf T 2 .plt,pfmift 2 = (P + ,V.2r|) ...... Sim. Mod. 58. P340. 3. GRAPHITE. Plumbago. Black Lead. Cleavage P. 1. 5. P_,T,M|f T 2 = (P_Z,V) ..................... Model 7. L674 1 . P385. 3. 5. P_,T,Mjf T 2 .p x t,p x mift 2 = (P_V. 2r x ZwZe) Sim. Md.58. L674 2 . 3. 5. P_,T,M[f T 2 .p x m,p x m 2 t-;f = (P_,V. 2r x ZnZs) Sim. Md.56. L674 3 . 3. OSMIUM-!RIDIUM. Iridium osmie. Cleavage = P. 1. 5. P_,T,MffT 2 = (P_,V) .............................. Model 7. Ly47'. 3. 5. P_,T > M 1 t|T 2 . P Jt ; p|t J pirnift 25 pfmfft 2 = (P_,V.2rl2rf) P340. Ly47 2 . 5. 5. P_.PiT,P|T,PiMifT 2 ,P|MjfT 2 = (P_.2R1, 2Rf) ... Ly v.2. -4. ANTIMONIAL NICKEL. Antimonnickel. 1. 5. P x ,T,MifT 2 = (P XJ V) ........................... Model 7. Geiger 63. 5. COPPER NICKEL. Kupfernickel. Arsenical Nickel. 1. 5. P x) T,M}fT 2 ?= (P X ,V) ....................................... Rose 162. 6. TELLURIC SILVER. Tellursilber. Weiss- Tellur. 1. 5. P X ,T,M}|T 2 ? ................................................... Rose 162. U?. SULPHURET OF NICKEL, Haarkies. Nickel sulfure. 1. 5. P XJ T,MifTo, ....................................... Model 7. Rose 162. 8*. CINNABAR. Sulphuret of Mercury. Zinnober. Cleavage = T,Mff T 2 = (V). 1. 5. P +) T,MjfT 2 = (P+,V.) .............................. Model 7. H89 32 . . 5. iP|TiPMifT 2 = (Rf) .................. Model 88. H89 31 . Ly50 1 . 3. 5. P^^MilT^lplt^PT^plmift^iPMifT^ ............... H89 34 . 3. 5. P-,T,MiT 2 4 P t 5 iP|T,i P mfft 2 , ^MftT* ............ H89 35 . 3. 5. P-,T,Mi|T 2 ,iptt 5 iP|T,ipimi|t 2 ,iPfMifT 2 , ............ H89 36 . 5. 5. P+^Pi-T^Plt^pltaPIM-JfT^ipgmilt^ipfmi^ = (P + .R|,r-e 5 rf) ...... H89 33 . Ly50 6 . 0. SULPHURET or MOLYBDENUM. Molybdanglanz. Cleavage = P. 1. 5. P_,T,M|f T 2 , .................. Model 7. J 231 . H116 306 . P249. L667 1 . 4. 5. T 5 Mi|T 2 .P x T,P x M-JfT 2 ==(V. 2R X ) ...... Sim. Md. 47. J 232 . P667 2 . H116 307 . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 47 10. FLUORIDE OF CERIUM. Fluorcerium. Cerium fluate. Cleavage = P. 1. 5. P_,T,M;-f T 2 . = (P_,V) Fahlun Model 7. P267. 1. 5. P_,m,T,na}f,M|fT 2 = (P_V,iO ... Model 10. L571. Hiv399- 11. ICE. Eis. 1. 5. P x ,T,MifT 2 ? = (P X ,V) Model 7. Rose 162. 12.* An Isomorphous Group, 1, 2, 3: 1. CORUNDUM. Korund. Corindon. Sapphire. Oriental Ruby. Cleavage = PJT,P}Mj$T, = (2RJ). 1. 5. P + ,T,MjfT,, Bengal.. .Sim. Md. 7. J 23 . P66 3 . L536 4 . H47 113 . 2. 5. JPfT,iPfM|fT 2 = (Rf)... Bengal ... Sim. Md. 89. P66 1 . L536 1 . H47 107 . 2. 5. PfT,PMjf T 2 = (2Rf ZwZe)...Sim. Md. 26, but more acute... P66 5 . L536 7 . H47 108 . 2. 5. Pf T,Pf Mjf T 2 = (2R| Zw Ze) ...Sim. Md. 26, but more acute... P66 4 . L536 7 . J 25 . H47 109 . 2. 5. Py^PfTjP^M^TftPJMjfT, = (2RV 6 ZwZe, 2Rf ZwZe) H47 114 . 3. 5. P+,T,M|fT 2 .Pf T,PfM}fT 2 , ... Sim. Md. 58. J 27 . L536 6 . H47 116 . 3. 5. P,T,M|JT 2 .lpimZn 5 ip|m 2 tif, Model 57. L536 3 . H48 118 . 3. 5. P+T,M|JT s ."}p5m,pJt,ip2m ! ,t|f,pJmUt ...P65 4 . L536 5 . H48 119 . 3. 5. P + ,t,m||t 2 .PT,P|MjfT 2 =(P + ,v.2Rf ZwZe) H48 10 . 3. 5. P,T,M}fT 2 .p|m,p?,mot{f = (P,V.2r|Zn Zs) Sim. Md, 56. P65 1 . 4. 5. T,MifT,.P|M,P^M 2 ff| = (V.2R|ZnZs) P65 2 . 4. 5. T,M-Jf T 2 4PlM,iP|M 2 Tif = (V. R| Zn) P66 2 . 5. 5. P.PfT,PfMifT 2 = (P.2Rj)Sim.Md.96.J 6 .P65 3 .L536 8 .H47 U2 . 5. 5. P^.PV 6 T,P^M||T 2 = (P^.2RV 6 ) H47 115 . 5. 5. P + .lplm ) PfT,lplm 2 ti|,P|M-i|T 2 = (P + .RiZn J 2R|Zw)H48 117 . 5. 5. P_.iP|T 5 iPplifT 2 = (P_.R|)...Sim. Md. 114. Ly v. 1. H47 111 . L536 2 . 2. SPECULAR IRON. Eisenglanz. Fer oligiste. Cleavage = ipit4p|m|ft 2 = (r|). 2. 5. iP|T,iP|M|^T, == (R|)...Sim. Md. 83. L545 1 . P217 1 . H104 170 . 2. 5. iP|T,JPf M|f T, = (R|) Sim. Md. 85. L546 4 . H104 171 . 2. 5. iP!T, P ft,iPfMifT 2 ,ipfm}ft 2 == (R|,rf) L545 3 . H104 173 . 2.5. iP|T,| P ft,iPlMifT 2 ,^ P fm-{ft 2 ,K3p x m+t) = (R|,r|,s x ) ... P217 6 . H104 181 . 2. 5. iPfTZw, JpfTZw, JpfttZe, ^P|M|fT 2 , ip|MJ|T ip T Vnfft 2 , ^(3p x M+T+) = (R}Zw,B|Zw,p^ r Ze,s ) H105 184 . 2. 5, 2 Pf T, Jpft, JP}M|J T 2) 2 pm}f t 2 [i(3p x m+t+)] = (R|,rf ,2s,) . . . H105 185 . 2. 5. P,t,mj|t 2 .P7M,P7M 2 Tl| = (P,v.2RJZnZs) H105 182 . 5. 5. P4PiTaP|MifT 2 = (P.R|) Sim. Md. 114. L545 2 P217 2 , H104 172 . 5. 5. P.PJTjPJMif T 2 = (P.2RJ) Sim. Md. 96. H104 174 . 48 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 5. 5. P.P 5 T,P 5 M|fT 2 = (P.2R 5 ) Sim. Md. 96. P217 5 . H104 1?5 . 5. 5. P_.PTipft,PJMffT 2 .|p|m|ft 2 = (P_.2RJ,rf) H104 176 . 3. 5. p.P|T,P|M|fT 2 ,l(3p x m+t) = (p.2R|,s x ) H104 177 . 5. 5. P_4P|T,ip|t^P|Mf|T 2 ,i P |mlft 2 = (P_.R| 5 r|)L546 5 .H104 179 . 5. 5. P_.pit,p|m{ft 2 = (P_.2r|) P217*. H104 180 . 6. 5. P_,t 5 m^|t 2 .p|m,p|m,tl| = (P_,v. 2rf ZnZs) H105 183 . 3. TITANITIC IRON ORE. Titaneisenerz. Cleavage = P. 5. 5. P_.^P|T, 2 P|M4|T 2 ,i(3p x m+t+) Mii"* 141 - 1 ". 5. 5. P_^P|T, 2 P|M^|t 2 , 2 (3p x m+t) ; Mii f -' 5. 5. P_* |P|T Zw, 2 p_t Ze, 2 p_j_t Ze, 2 p| M ffT 2 , p_m-^t 3 , 2 p + m^t 2 , J(3P X M+T+) = (P_. R| Zw, r_ Ze 3 r + Ze, 2 S X ) M ii f - "*. 13. SULPHATO-TRICARBONATE OF LEAD. Schwerbleicrz. Cleavage = P. 3. 5. R|, Brooke, Edin. Phil. Jour. iii. 119- 2. 5. R|,3r_, idem. 4. 5. T,M|JT,.R|, idem. 14. QUARTZ. Quarz. Rock Crystal. Cleavage = PfT,PJMIT 2 = (2R|). 3. 5. PfT^IM-^T, = (2R|) Sim. Mod. 26.. .J 67 . P2 3 . R 67 , H55 2 . 2. 5. 2 PJT,IP|M^T 2 = (R|) Sim. Mod. 83 P2 1 . R 71 . H55 1 . 3. 5. ^P|TZw^p|tZe, 2 P4M!|T 2 ^p|miIt 2) = (Rf Zw, rf Ze) P2 2 . 3. 5. P + ,T,Mi|T 2 .PfT,P|M^T 2 , H56 9 . 4. 5. T,M^T 2 .P|T,PfM^T 2 = (V.2R| ZwZe) Model 73... J 66 . P2 6 . R 68 . H55 3 . 4. 5. T 5 M{|T 2 . 2 PfT Zw, ipJtZe, ^PIM1T 2 , ipjm^t, = ( V. Rf Zw,r| Ze) H56 4 . 4. 5. t ) ml|t2^PiT^ P |t, 2 P|M^T 2 , 2 p|ml|t 2 , P2 4 . H56 5 . 4. 5. T_,MJ-fT 2 .P|T 5 pfMA|T 2 , H56 6 . 4. 5. T,M1|T 2 . 2 Pf T Zw, 2 p|t Ze, pfm-i|t 2 = (V. Rf Zw, rJZe) H56 8 . 4. 5. T,MA|T 2 . 2 p 5 m,P|T, 2 p 5 m 2 t^,P|Mf|T 2 . = (V.r 5 Zn, 2Rf ZwZe) J 68 . H57 10 . 4. 5. T,MjlT s . P fm,PfT,pim^{,P|M^T a , H57 12 . 4. 5. T,M^T 2 .p|t,P 39 T, P Jml|t 2 ,P 39 MUT 2 , H57?. 4. 5. T,M{ 5 5 T 2 .p 5 t,PfT,p 5 mi|t 2 ,P|Mf|T 2 , P2 11 . H57 U . 4. 5. T,Mi^T 2 . P|T, P|M1|T 2 , \ (3 P+ mt+)Ze, i (3p + mt+) Zw P2 8 . H57 15 . J 69 . 4. 5. T,Mi 5 3T 2 .PfT,PfMi|T 2 ,2[ 2 (3p + mt+)] = (V.2RZwZe, 2s+)... H58 16 . 4. 5. T,M^T 2 .rjT,P 4 T,pjMilT ,P 4 M 1 1 lT 2 , 2 (3 P4 .mt+) H58 17 . 4. 5. T,Ml|T 2 .P4T,PfM^T 2 ,3[ 2 (3 Px mt+)] = (V.2R| ZwZe, 3s + ) H58 18 . 4. 5. t,m{&.P4T,P!M}rr 2 = (v.2RfZwZe) P2 9 . 4. 5. t,mBt 2 . 6 m,PT, 6 mXSPMST 2 = v.2r 6 ZnZ8,2RZwZe) P2 7 . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 49 15.* TETRAD YMITE. Cleavage = P. . 5. Fi G r T 'FAMMT 2 = (RJ?) .................................... S230. 16*. POLYBASITE. 1. 5. P_,T,MjfT 2 = (P_,V) ..................... Mexico ...... D417. P300. 3. 5. P,T,M T fT 2 .3p x t,3p x m|ft 2 = (P, V. 3r x Ze, 3r x Zw) ......... S114. 3. 5. P,T,MifT 2 .6p x t,6p x mift 2 = (P,V.6r x Ze,6r x Zw) ......... SI 14. Iff.* RED SILVER. Rothgiiltigerz. Ruby Silver. Argent rouge. Argent antimonie sulfure. Two varieties : 1. LIGHT RED SILVER; Sulphuret of Silver and Arsenic. 2. DARK RED SILVER ; Sulphuret of Silver and Antimony. Cleavage = PT,PM|f T 2 . (The forms PT,P1T,P 2 T should perhaps be Pf T,P/ T T,Pf$T.) 1. 5. P,T,M}fT 2 = (P,V.) .............................. Model 7 ...... H87 11 . 3- 5. ipJt>Mnitf^K3P+M+T+) = (r i ; S+) ..................... H87 i 2< 2. 5. i(3P+M+T), i(3p-m+t) = (S+,s_) ........................ H87 13 . 3. 5. | P+ t,i p+ mift,,i(3P x M:;T) = (r + , S x ) ..................... H87 14 . 3. 5. P ,T,M|fT 2 .iPiM,iPiM 2 Tif = ( P ,V.riZn) ............... H87 16 . 4. 5. T,MifT 2 .iPMiPM 2 Tif. = (V. R, Zn) ........................ H87 10 . 4. 5. ^MifT^JPMZn^pimZsaPM^jf^pim.tJl ............ H87 15 . 4. 5. T,M]fT 2 4PMZn,i P imZs,|p 2 mZs J iPM 2 Tif } i p im 2 tif, |p 2 m 2 ti|, |(3p+m+t+) - (V.R 1 Zn,r^Zs,r 2 Zs,s + ) ...... H88 53 . 4. 5. T,M-}fT 2 .i(3p_M+T+), i(3 P+ m+t) = (V.s_,s + ) ......... HS8 20 . 4. 5. T, M|f T 2 . IPM Zn, ipirn Zs, \ PM 2 T|f , iplm.tif , i(3p_m+t) = (V. R.Zn.r^Z^sJ) ......... H88 21 . 4. 5. m 5 T,m 2 t[f,Mif T 2 4P|M, JPiM 2 T|| = (V, v. RJ Zn) ...... H88 19 . 4. 5. m^^.tj^MlfT^-iPiMZn^pimZs^PM.^!,^^^! = (V,.R l Zn,rJZs) ......... H88 22 . 4. 5. m,T 5 m 2 tif,MifT 2 .iPMZn,|pJmZsaPM 2 Tif 5 |p|m 2 tJ-f, i (3p + mt+) = (V, v. R! Zn, r J Zs, a+) ......... H88 24 . 5. 5. P.PT,P 2 T,PMifT 2 ,P 2 MifT 2 .= (P. 2^,211,) ............... H88 17 . 18. MAGNETIC IRON PYRITES. Magnetkies. Fer sulfure magnetique. Cleavage = P,T>Mif T 2 = (P,V). 1. 5. P_,T,Mif T 2 = (P_,V) .............................. Model 7. L665 1 . 1. 5. P 5 m,T 5 m 2 t|| 3 M|f T 2 = (P,V, v) ............ ". ..... Model 10. L665 2 . 3. 5. P,T,M}-f T 2 .pm,pm 2 t|f = (P,V.2r,ZnZs).. ................... L665 3 . 3. 5. P_,T,MifT 2 .pit,pimjft 2; .............................. D404. L665 4 . 3. 5. P,m,T,m 2 t}|, M}|T 2 .pm, pm 2 tj| = (P,V,v.2r 1 ZnZs) ...... L665 7 . 3. 5. P,i%lVm,1#,M#^^^ = (P, V, v. 2r, ZnZs, 2r ZwZe, 2rf ZwZe) ...... L665 8 . 9 50 MINERALS OP THE RHOMBOHEDRAL, SYSTEM. 3. 5. P,m,T,m 2 tif,MifT 2 .pm jP |t,pm 2 t|f,pfmfft 2 = (P,V,v. 2 ri ZnZs,2rf ZeZw) ...... P213. 4. 5. T,M}fT 2 . PJT, PMjf T 2 , .................. Sim. Model 74. L665 6 . 5. 5. P.PJT,PiMJ-fT 2 ==(P.2RiZwZe) ...... Sim. Model 96. L665 5 . 19.* An Isomorphous Group of Carbonates, which crystallise or cleave in nearly the same form as the cleavage rhombohedron of cal- careous spar. I have written the symbols as if the forms were all precisely alike, but the following table corrects the error of this uniformity by giving the numbers derivable from the most recent measurements of each form. [The angle quoted is that across the obtuse edge of the Rhombohedron.] 1. Calcareous Spar: 105o 5' Mohs = Do. 104o 28'Hauy= 2. Dolomite : 106o 15' Mohs = 3. Brown Spar: 107o 30' Levy = 4. Carbonate of Magnesia : 107o 22' Levy = iP|:g{f^?JT, iP}:8tt???8 M M T 2 . 5. Mesitin Spar: ? Rose 6. Carbonate of Iron : 107 0' Mohs = 7. Carbonate of Manganese : 106 51' Mohs = 8. Carbonate of Zinc : 107 40' Mohs = I have adopted Haiiy's measurement, and written the symbol JPT, IPMif T 2 , as equivalent to the whole of the above cited forms. It may be that this is inaccurate, but, at any rate, the differences of the forms are not such as to serve as a useful means of discriminating the minerals from one another, while, at the same time, chemistry affords us methods of discrimination which are both simple and certain. The Cleavage of all the varieties is = PT, iPMf T 2 . 1. CALCAREOUS SPAR. Calc Spar. Carbonate of Lime. Chaux Car- bonate. Kalkspath. Kohlensaurer Kalk. Prisms. 1. 5. P+, T,M|fT 2 . = (P+,V) ...(Hauy's Prismatique)...Md. 7. H6 22 . 1. 5. P+, 2 T, 2 t,iMifT 2>2 mift 2 , ............ (alternante) ...... H6 23 i.315 a . 1. 5. P + ,T,mift 2 , ............... Model 8...(comprimee) ...... H6 22 i.315 b . 1. 5. P + ,t,MffT 2 , ................................. (evasee) ...... H6 22 i.315 c . MINERALS OF THE RHOMBOHEDRAL, SYSTEM. 51 1. 5. P_,T,MffT 2 = (P_,v) .................. (raccourcie) ...... H6 22 i.315 d . 1. 5. Pz,t,mJft 2 = (P=,v) .................. (lamelliforme) ...... H6 22 i. 315 e . 1. 5. P + ,M,T,M 2 T&Mirr 2 ==(P + ,V,v) Cumberland... Md. 10. HI I 73 . Rhombohedrons. 3. 5. iPT^PMifT 2 = (R,)... (Hauy's Primitive)... Md. 83. H4 1 . R 71 . 2. 5. |PiT,PiMiJT 2 = (R|) ...... (Hauy's Equiaxe)...Md. 85. H4 2 . 3. 5. iP 2 T,P 2 MifT 2 = (R 2 ) ......... (Hauy's In verse)... Md. 89. H4 3 , 3. 5. 2 P 4 T, 2 P 4 M|fT 2 = (R 4 ) ............ (Hauy's Contrastante)...H4 7 . 2. 5. J P 5 T, lP 5 Mif T 2 = (R 5 ) ... (Hauy's Mixte) ...Derby shire... H4 8 . 3. 5. P 5 T } P 5 M}JT 2 = (2R 5 ZwZe)... (Hauy's Leptomorphique)...H4 4 . 3. 5. 2 PfT J2 P|MifT 2 ==(Rf) ......... (Hauy's Cuboide) ............ H4 9 . 2. 5. 2 PTZvv,|p^tZe, 2 PMf|T 2 ^pXft 2 = (R 1 Zw,r 2 Ze) (Hauy's Semi-emargine). . . Isere ...... H5 11 . 3. 5. I pt Zw, 2 P 2 T Ze, 2 pmlf t 2 , 2 P 2 M jf T 2 = (r t Zw, R 2 Ze) ...... (Hauy's Unitaire) ...... Lyons. Ireland ...... H5 12 . 2. 5. 2 PT )2 P 4 T 52 PM}-|T 2)2 P 4 Mi|T 3 = (R,,R,) ...... Derbyshire ...... H5 17 . . 5. ^ptZw, 2 P 5 TZe 5 Jpmift 2 , 2 P 5 MifT 2 = (p, Zw, R 5 Ze) ......... H5 18 . . 5. 2 p^ 2 P 2 T, 2 pimi|t 2 , 2 P 2 MifT 2 = (r 2 ,R 2 )...Clermont-Farrand, H6 25 . 3. 5. 2 ^T, 2 P 5 T J ^MJfT 2 , 2 P 5 M}|T 2 . = (r 2 ,R 5 ) ...... Derby shire... H7 33 . 3. 5. 2 P 13 T Zw, JpjT Ze, 2 P 13 M{f T 2 , 2 p 2 Mj-|T 2 . = (R., Zw, r^ Ze) ...H? 33 . 2. 5. 2 P 14 T, 2 p 2 T, 2 P 14 Mf|T 2 , 2 pXliT 2 . = (R M ,Bi) ..................... H7* 4 . 3. 5. 2 P 2 T Zw, 2 p 4 t Ze, iP,MgT 2 p 4 m!it 2 = (R, Zw, r 4 Ze) ......... H8 42 . 3. 5. 2 Prr Zw, 2 p 4 t Ze, 2 P^MJIT 2 , 2 p 4 m^t 2 . = (R 5 4 Zw, r 4 Ze) ......... H9 53 . 2. 5. 2 ptZw, 2 r 4 TZw, 2 P 5 TZe, 2 pml 5 3 t,, 2 p 4 M{|T 2 , 2 P 5 M[rr 2 . = (P! Zw, R 4 Zw, R 5 Ze) ...... Jura ...... H10 65 . 3. 5. 2 p 2 T, Jpjt, 2 P 14 T, >p 2 Mi 5 3 T 2 , 2 pX 5 at 2 , iP u MST 8 = (Rj, rj, R I4 ) ...H13 91 . 3. 5. ^ZwR.Zw^sZ^r^Ze .......................................... H16 111 . Scalenoh edrons. 3. 5. i(3 1 PMT+) = (S 1 )...(Hauy'sMetastatique)...Md. ! ll6. H4 5 . R 79 . 3. 5. i(3P:j:MT+) = (Sjl) ............... (Hauy's Axigraphe) ...... H4 6 . 3. 5. 2 (3P_MT+), 2 (3P+MT+) = (S_S+) ........................ H7 36 . 3. 5. 2 (3p,mt+), 2 (3P_MT+) = (s 1? S_) ............ Simplon ...... H7 37 . Rhombohedrons with Scalenohedrons. 3. 5. ipt,ipmJ$t 8 ,J(3P I MT+) = (r,,^) ...... Derbyshire ...... R 81 . H5 U . 3. 5. lpt, 2 pmfft 2 , 2 (3P + MT+) = (r,,S+) ......... Oisans ...... R 81 . H5 16 . 3. 5. 2 P 2 T, 2 P 2 MifT 2 , 2 (3 P+ m+t) = (R 2 ,s+) ........................ H6 26 . 3. 5. 2 P 2 T, 2 P 2 MifT 2)2 (3p^m+t) = (R 2 ,slj:) ........................ H6 28 . 3. 5. 2 P 2 T, 2 p 2 M}fT 2 , 2 (3P+M+T+) = (rJ,Sj) ........................ H6 29 . 3. 5. 2 p 2 T, 2 p 2 Mi^T 2 ,l(3P + MT+)=(R 2 ,S + ) ........................... H7 39 . 52 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 3. 6. 2 p 2 Up 2 mllt 2 , 2 (3P:t:MT+) = (r 25 S+) ........................... H8 4 . 3. 5. !P 4 T,P 4 Mirr 2 .^(3p 1 mt+) = (R 48l ) ...... Derbyshire ...... H8 45 . 3. 5. 2 P 4 5 T,^PX|T 2 , i(3P!MT+) = O 4 ,, SO ...Bex, Switzerland... H8 46 . 3. 5. KT, iPJMiST,, J (3P,MT+) = (R$,S,) ........................ H8 47 . 3. 5. Jp 4 t, JpXfe. ^(3P^MT+) = (r 4 , Sj) ............... Hartz ...... H8 48 . 3. 5. Jp5tipWltl(3PtMT+) =(48^) ........................... H9 49 . 3. 5. {pt Zw, ^p 2 t Ze, Jprngt,, ^p m}^, 2 (3PJM+T+) = (r L Zw, r 2 Ze, Sj) H9 57 . 2. 5. 2 PT Zw, 2 p 4 t Ze, ipMiiTa JpJmStg, 2 (SP^fTqi) = ( Rl Zw, r 4 . Ze, S 1 ) Mexico ............ H10 62 . . 5. ^PT,^PMi 5 3 T 2 , 2 (3P + MT+), 2 (3P_MT+) = (R 1 ,S + ,S_) H10 63 . 3. 5. fet, 2 p 2 m!it 2 . 2 (3P + MT+), 2 (3 Pl mt+) = (r 2 ,S + , Sl ) ......... H14 9 *. S. 5. 2 P 2 TZw, 2 p 14 tZe, 2 P 2 M^T 2 , 2 p 14 m!|t 2 , 2 (3 P+ m+t+) = (R 2 Zw,r 14 Ze, s + ) ...... Lyons... H14 95 . 3. 5. 2 P 4 TZw, 2 p 2 tZe, ^P 4 M{^T 2 , 2 p 2 m^t 2 , 2 (3p mt+) = (R 4 Zw,r 2 Ze, s,) Hl4 i S. 5. R I ,S 1 ,SJ, ............................................................ H15 103 . 3. 5. ^Zw, r 2 Ze, s,, S+, ................................................... H15 106 . . 5. p 1 Zw,B 4 Zw,iiZe, S,, ............................................. H15 107 . 2. 5. ^Zw^Zw^s., ................................................... H15 109 . . 5. R,Zw, R 4 Zw, R 14 Ze, s,, ............................................. H16 109 . 3. 5. R,,r 4 ,s+ s_, ......................................................... H16 110 . . 5. R 4 Zw, r 2 Ze, r 2 Ze, s w ................................................ H17 121 . 3. 5. r 4 Zw, r 8 Ze, S,, SJ, ................................................... H18 131 . 3. 5. r^r 2 , Sl ,S+, ......................................................... Hl8 ia2 . 3. 5. r l Zw,r{Ze,itZn,rJZ8,S 4 ,8j, .................................... H19 135 . 3. 5. r,Zw, n 4 Ze, r^Ze, R 2 Ze, S 15 .......................................... H19 137 . 3. 5. R 4 Zw, R 14 Ze, r 2 Ze, r 4 Ze, i*Zn, r^Zs, S 13 ........................ H20 150 . Complete Prisms with Incomplete Pyramids. 3. 5. p_|_,T,M}sT 2 . 2 r 2 T, ^p 2 Ml|T 2 , ...... Kongsberg. Andreasberg . . . H10 66 . 3. 5. P + ,T,M^T 2 . 2 p 2 T, 2 p 2 M}lT 2 , ......... ............ Derbyshire ...... HI I 69 . 3. 5. P + ,T,M{|T 2 .P M,P 5 M 2 Tl| = (P + ,T,M!|T 2 .2R 3 ZnZs) ...... Hll 72 . 3. 5. P,t,m!|t 2 . 2 (3P+MT+) = (P.t 5 mfeS+) = (P,v.S+) ...... Hll 74 . 3. 5. P +J T,Mi|T 2 . 2 plt, 2p Xit 2 = (P+,V,ri) ...... Derbyshire ...... Hll 75 . 3. 5. P+,T,MliT 2 . 2 p 5 t, 2 pX3t 2 = (P + ,V,r 5 4 ) ........................... H12 76 . 3. 5. p + ,tX 5 3 t 2 ^P 4 M^P 4 M 2 T|i == (p + ,v.R 4 Zn) ..................... H12 77 . 3. 5. p + , V.RtZwjpiZe, ................................................ H15 105 . 3. 5. P+ ,V.i4Ze,s_, ...................................................... H16 112 . 3. 5, P,V.r^Ze,r 2 Ze, ...................................................... H16 113 . 3. 5. P+, V.r 2 Ze } S,, ............................... ....................... H16 114 . 3. 5. P+, V,v.r,Zw, ...................................................... H16 116 . 3. 5. p + , V.BiZe,r 2 Ze, ................................................... H16 115 . 3. 5. P+ 5 V. r 4 Zw, R 5 Ze, ................................................ Hl6 nz . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 53 3. 5. p + ,v. R 2 Zn, rf-Zn, s+, H19 1 ' 8 . 3. 5, P + , v, 3m x t. R 2 , 8$, H19 139 . 3. 5. P+, V, v. R 4 Zn, r 2 Zs, H19 140 . 3. 5. P + , V, v. S+, s+, H19 141 . 3. 5. p,v, v. r 2 , R 2 , s+, H20 K3 . 3. 5. P+, v. r,Zw, r 2 Ze, R 14 Ze, sljl, H20 149 . Incomplete Prisms with Complete Pyramids, a.) Without Scalenohedrons. 4. 5. T,M}-fT 2 .PfM, PfM 2 Tff, Derbyshire. Cumberland... H7 35 . 4. 5. T,MffT 2 .PfM,PfM 2 T}f, H7 38 . 4. 5. T, Mi|T 2 . PT, PM}nV == (V. 2R,) Sim. Md. 73. HlO 64 . 4. 5. T,MifT 2 . 2 PMZn 52 PM 2 Tf| = (V.^ Zn) Md.7l. H5 13 . R 78 . 4. 5. T,M{|T 2 . 2 PT Zw, 2 PM}f T 2 = (V. R^ Zw) ...Md. 72. H5 16 . R 77 . 4. 5. T,M|fT s .ipjT,bX4 T Sim. Model 72. H7 30 . 4. 5. T,MifT 2 . 2 p 2 m, 2 p 2 m 2 ti| } Cumberland... Sim. Model 71. H6 27 . 4. 5. T,M^T 2 , 2 P 2 T, 2 P 2 M!|T 2 .= (V. R 2 ) Lyons H8 41 . 4. 5. T,M}rr 2 . 2 p 4 T, 2 pX3T 2 , Sim. Model 72. H9 50 . 4. 5. T,Mj|T a . JPJT, JPJMJSTs, Sim. Model 72. H9 51 . 4. 5. T,Mi|T 2 .>PiT,^M^T 2 , ...Castelnaudary... Sim. Model 72. H9 53 . 4. 5. T, Mi^T 2 . 2 pm Zn, JP 2 M Zs, Jpm,t8, JP^TIS, Lyons H9 58 . 4. 5. T, MgTg. 2 PT Zw, |p 2 tZe, ^PM}|T 2 , Ip 2 m}|t 2 , ...Derbyshire... HlO 69 . 4. 5. T,MBT a .ipJtZw,iP 1 TZw,JpX5t{P.MST 1> ... Derbyshire. H12 79 . 4. 5. T,MiiT,.JPJT,ipSt,iPiMgT}pX5t a = (V.Ri,r H13 88 . 4. 5. T,M^T 2 .^t,^PiT,.ipXit 2 ^PiMl|T 2 , Hartz H13 89 . 4. 5. T,MiIT 2 . 2 ^T,^P H T,^X|T 2 ^P 14 M!|T 2 , Freyberg H13 M . 4. 5. t,mif t 2 .^P 2 T, JP5M-14 T 2 , Derbyshire. Norway H7 31 . 4. 5. t, mlgtfiPT Zw, 2 p 2 t Ze, ^PMiiT 2 , JrimX Isere. H9 56 . 4. 5. m^^t&MST^iPJT.iPlMUT,. = (V,f>.RJ) H12 80 . ^.) P^'^A Scalenohedrons. 4, 5. T,MBT 2 .i(3P 1 MT+) = (V. SJ Derbyshire H8 41 . 4. 5. T,MliT 2 .^PM,^PM 2 Tli^(3p+mt+) =(V. R t Zn, sj) ... HlO 60 . 4. 5. T,MHT 2 .| P t^pmi|t 2 ^(3p 1 MT+) = (V.p 1|8l ) HlO 61 . 4. 5. T,MBT,.}PiT,JP{MST5(3PiMT+)== (V. R 1 , S,) H12 81 . 4. 5. T,M!^T 2 . frit, JpXstg, K3PiMTi{L) H12 82 . 4. 5. [T J M} 5 3 T 2 .^t, 1 pX3t 2 ,K3P 1 MT+)] X 2, H12 83 . 4. 5. T^ilT^KSpim+t+^KSP-M+T^z^CV.s^s.) H14 93 . 4. 5. V.riZe, r^Zn, r 5 4 Zs, S lf H17 118 . 4. 5. V.r'Ze^s., H17 119 . 4. 5. V.R|Ze,r 2 Ze, s 1} H17 120 . 4. 5. V. RJZe, B!, s+, H17 122 . 4. 5. V. B, 3 Zw, R^Ze, s + , HIS 126 . 54 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 4. 5. V. R 13 Zw, S+, s_, HIS 127 . 4. 5. V.RwZw.sJZejS!, HIS 129 . 4. 5. V. R 2 Ze, s,, s_, HIS 130 . 4. 5. V. ri Zw, R^Ze,r 2 Ze,s,, H19 ia4 . 4. 5. V.r,Zw,r*Ze, S,,s_, H19 136 . 4. 5. V. r 2 Ze, r 2 Ze, rJZn, r!>Zs, S 1? H19 142 . 4. 5. V. R 2 Ze, r^Ze, 2s + , H20 144 . 4. 5. V. R|Ze, r 6 Ze, B 4 Zw, s,, H20 145 . 4. 5. V. ri Ze, r 4 5 Zn, r^Zs, s 15 s_, H20 146 . 4. 5. v,t;.r4Zw,r5Zii,r5Zs,8:k H20 H7 . 4. 5. V. ^Zw, R 4 Zw, rlZe, r 4 Ze, r 2 Ze, S 15 H21 153 . 4. 5. T,Ml 3 'T 2 .i(3P 1 MT+) = (v. SO Derbyshire H8 43 . 4. 5. T,M} 5 3 T 2 . pX pXt!8. (3PjMT+) = (v. 2r ZnZs, S,) HIS 92 . R 82 . 4. 5. TX 5 3T 2 4p 2 tZe,^pX^,K3PiMT+) H14 9 l 4. 5. v. S 1>S _, HI5 101 . 4. 5. v.Sj,s + , H15 102 . 4. 5. v. R 4 , s_, H15 104 . 4. 5. v. RtZw,TlZe,8i, H17 123 . 4. 5. v. R 4 Zw,r^Ze,s_, HIT 125 . 4. 5. t,mj&. JPJT, iPJMST,, i(3p+mt+) (v. RJ, s + ) H13 8G . 4. 5. tX&.^t,Mm& 2 ,K3PM+T+) = (v.rj, Sj)... Norway. HIS 87 . 4. 5. tXit 2 .^P 2 T, P 2 M}j>T 2 , K3pimt+) = (v. R 2 , s,) India... H14 96 . 4. 5. t,m!&.^P 2 M, ^P 2 M 2 T{i, J(3 P ^mtqi) = (v. R 2 Zn, sj) H14 97 . 4. 5. tX3fc-2P a M^P 2 M 2 T^^(3p^mtq:) H14". 4. 5. v. p Ze, R 5 Ze, s,, HIT 124 . 4. 5. v. r!>Zn, r^Zs, S x , 2s+ H18 m . 4. 5. v.R,Zw, S M S+, H19 133 . 4. 5. v. R 4 Zn, r 4 Zs, r 2 Zs, s 15 s_, H21 151 . 4. 5. v. r 4 Zw, r 5 Ze, S,, 2sJ, H21 152 . 4. 5. V, v. r^Ze, s + , s + , H20 143 . 4. 5. y,.R 1 Zw,r 18 Zw,8^,-a^8^ H21 154 . 4. 5. V,v. R 13 Zw } R^Ze,R 2 Ze, s_,s+ H21 155 . Pyramids with the apex truncated. 5. 5. P_^PT^PMjfT 2 ==(P_.R 1 )...(Hauy'sBas6) Md. 86. R 73 . H5 10 . 5. 5. p- JPiT, iPJM|T 2 . = (p_.Ry... Mexico... Sim. Model 86. H5 19 . 5. 5. P 4P 2 T^P 2 M;-fT 2 . = (p.R 2 ) ; Offenbanya H6 20 . 5. 5. p+. iP 4 T, JP 4 MjfT 8 == (p+.R 4 ) Derbyshire H6 53 . 5. 5. p.^P|T^P|M||T 2 . = (p.R|) Hartz H6 24 . 5. 5. P. Jpt Zw, ^P 2 T Ze, ipm{t ^P 2 Mi 5 3 T 2 = (P. r t Zw, R 2 Ze) H9 54 . 5. 5. P.^^T^pmfc^Ml^^CP.r^R,) H9 55 . 5. 5. p+. Jpjt, ^P 5 T, iplmStg, lP 5 MiiT 2 . = (p + . ri, R 5 ) Hartz. . .Hi I 67 . 5. 5. p + ^P 2 TZw,^ 13 tZe^P 2 M^T 2 ^p 13 mj|t 2 = ( P+ .R 2 Zw,r 13 Ze)Hll 70 . 5. 5. p + 4p2t^PuT^pXit 2 ^P,4M 1 JT 2 .= ( P+ .r 8f R 14 ) ... Hartz... Hi I 71 . 5. 5. p. iPJT Zw, ^p 5 tZe, iM!|T^ ip 6 miSt > =- (p. RfZw, r 5 Ze) ... H12 :8 . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 55 5. 5. P.>p 2 tZe^p 2 mJ3t 2 ,K3PiMT+) = (P.r,Ze, SO... Mexico.. .Hll*. 5. 5. P + .K3P^MT+) = (P+.SJ) ........................ Hartz ...... H6 21 . 2. DOLOMITE. Bitter Spar. Carbonate of Lime and Magnesia. Rhomb Spar. Rauterispath. 2. 5. 2 PT, 2 PMIiT 2 = (R,) .................................... P168. Lyl2'. 2. 5. r,, R 2 , ............................................................... Lyv. 5. 3. 5. R 4 ,R,, .................................................................. Lyl2 3 . 3. 5. p + , v.r,Zn, R 4 Zn, r 2 Zs, sj, ....................................... Lyl2 6 . 4. 5. v. RjZn^Zn, ......................................................... Lyl2 4 . 4. 5. v. ri Zn, R 4 Zn, r 2 Zs, ................................................ Lyl2 5 . 5. 5. P.^PT,|PMi^T 2 = (P.R 1 ) ....................................... Lyl2 2 . 5. 5. p + .R 4 , ......................................................... '. ..... Lyv. 4. 5. 5. P.r,,R 2 , ............................................................ Ly v.6. 3. BROWN SPAR. Braunspath. Carbonate of Magnesia and Iron. Breunnerite. Hallite. Pearl Spar. 2. 5. JPT, JPM1JT, = (RO .................................... Lyv.l. P168. 4. CARBONATE OF MAGNESIA. Talkspath. 2. 5. JPT,1PMST, = (R,) ............................................. Lyv.l. 5. MESITIN SPAR. Mesitinspath. . 5. 2 PT,^PM1|T 2 ? 6. CARBONATE OF IRON. Eisenspath. Spathose Iron. 1. 5. P,t,mifo = (P, v) ................................................... Ly69 2 . . 5. ^PT, 2 PMi|T 2 = (R t ) ............................................. Lyv.l. 8. 5. ^P 2 T,^P 2 M{rr 2 =:(R 2 ) ............................................. Lyv. 2. . 5, 2 P 5 T, 2 P 5 Mi^T 2 =(R 5 ) .......................................... Lyv.3. 3. 5. ^PT, 2 p 5 t, 2 PMiiT 2 , 2 p 5 m{it 2 = (R 1 ,r 5 ) ........................... Lyv. 4. 4. 5. V.RjZn,^, ......................................................... Ly70 5 . 4. 5. P_T,M{ST sl .}ptZw,ipitZe > ipmiStJpim 1 1 t > = (P_,V. ri Zw 5 r 2 Ze) Ly70 6 . 5. 5. P. 2 PT, 2 PM ! , 5 3 T 2 = (P. RO ....................................... Lyv.5. 5. 5. P+. 2 P 5 T 52 P 5 Mi|T 2 = (P + .R 5 ) ................................. Ly70 3 . 5. 5. P_. 2 PTZw, 2 PrrZe, 2 PM!iT 2 , 2 P 2 Mirr,-(P_.RiZw,R 2 Ze)Ly70 4 . 7. CARBONATE OF MANGANESE. Manganspath. Red Manganese. 3. 5. 2 P 2 T, 2 P 2 M} 5 3 T 2 = (R 2 ) ............................................. Lyv.l. 2. 5. S 1? ..................................................................... Lyv.2. 5, 5. P. 2 PT, iPMUT, = (P.Ri) .......................................... P246. 8. GALMEI. Carbonate of Zinc. Calamine. Zinc spath. 1. 5. P x ,T,Mirr 2 = (P x , V) .......................................... Hivl84. 3. 5. 8 PT, JPMBT, = (RO ........................ Ly73 1 . P375. HivlSl. . 5. 2 P 2 T, 2 P 2 MJ 5 3 T 2 = (R 2 ) ................................. P375. H iv 184. 56 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 2. 5. JPiT, JPMT, = (RJ) .................................... Lyv.2. P375. 2. 5. ^P 2 T, SpitjiPsMHTa.JpXStg = (R rj) ........................ Ly73 2 . 9 ? PLUMBO-CALCITE, Carbonate of Lime and Lead. 2. 5. RI, ............................................................... Johnstone. 2O*. NITRATE or SODA. Saltpetersaures Natron. Cubic Nitre. Cleavage = JPT, iPMjJT 8 = (R^. 2. 5. iPT, iPMliT 2 = (Rj) .................. Model 83. P198. Hii214. 21. TALC. Talk. Hexagonal Talc. Cleavage = P. 1. 5. P_T,Mi|T, = (P_,V)... Model 7. Ti 357. Ly v. 1. H71 139 . P120. 22.* PHENAKITE. Cleavage = T,MiT 2 = (V). 4. 5. tXJt,. JPJM, iPJMjiTS, ............ Beirich, Pogg. Ann. Bd 34 fig 11 . 4. 5. T t M8TrFgT,FWiKTft .................. Sim. Model 73. idem fig 13 . 4. 5. T,M}iT 2 . ip{m, P^T, ^m 2 t|^, PJMilTg = (V. r,p^ P Vm 2 t!I,pX 5 3t 25 .............................. H26 8 - 3. 5. P,M,T, M.T1S, M{|T 8 . P 5 2 T, P 2 M1 5 3 T 2 = (P,V,t;. 2R^ZwZe) J 10 . H26 9 . 3. 5. P,m,T,m 2 t!LMiiT 2 .pV t ; pVmilt 2 ,2s, .............................. H26 10 . 3. 5. P,m,T, m 2 t!|, M^T 2 . 2r^ ZnZs, 2r^ZeZw, ...... Sim. Md. 52. H26". 3. 5. P,m,T, m,tS, M1|T,. 2ryZnZs, 2r 4 ZeZw, 2rJZeZw, ............ H27 12 . 3. 5. P,m,T, m,tS, Mj|T 2 . Sr^ZnZs, 2r 4 ZeZw, 2riZeZw, ............ H27 13 . 3, 5. P,m,T, m 2 t{l,Mi|T 2 .2rVZnZs,2rlZeZw, 2iiZeZw,2rSZeZw, 2s, H27 14 . 4. 5. T,MBT,.P|T, P^Mi|T 2 =(V.2R^ZwZe) .................. J 153 . H26 2 . 4. 5. m,T,m 2 tii, M!|T 2 . P 4 T, P 4 M1|T 2 , ..................... P171 3 . H26 6 . J 154 . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 57 3. PHOSPHATE OF LEAD. Braunbleierz von Paoullaouen. Plomb phosphate. Cleavage = pgt, pJXst. = Oi 3 .). 1. 5. P + ,T,M{j;T 2 == (P + ,V)...Hofsgrund. Clausthal. Beresof...Md. 7. J 131 . L273 5 . H93 70 . 1. 5. P+,M,T, M 2 Tlj|, M!|T 2 = (P + ,V, a) ......... Huelgoet ...... Model 10. J 132 . L273 6 . H93 71 . 2. 5. PgT, PgMgT,, (Primitive} = (2Rg ZwZe)...Md. 26. Leonhard. 3. 5. P + ,T,MijjT 2 .pgt,plX3t2v. ....Job. Georgenstadt ...... Sim. Md. 58. L273 3 . J ;3t . H94 75 . 3. 5. P + ,m,T,m 2 t^M!|T 2 .p! 5 3 t,plX 5 3t 2 , ............... L273 4 . P363. H94 76 . 4. 5. T,M{*T 2 . P! 3 T, PjJMT,, ...... Beresof. Corn wall... Sim. Model 74, but the prism longer, like Model 73 ...... J 133 . L273 3 . M 117 . H93 72 . 4. 5. T 5 M!|T 2 .P^T,P^M1 5 3 T 2 , ............... Sim. Model 74. L273 2 . H94 73 . 5. 5. p_. Pi 3 ,!, PgMST ... Joh. Georgenstadt... Md. 96. L273 1 . H94 7 *. 4. ARSENIATE OF LEAD. Griinbleierz von Johann- Georgenstadt. Plomb phosphate- arseniate. Cleavage = t, ml^. 1. 5. P,T,MiJT .................................... Model 7. Ly 52 l . P364. 3. 5. P,T,MgT*pfcpGiD8b ...... Sim. Model 58. Ly52 4 . Ti574. P364. 3. 5. P_, T, MiiT 2 .pit,pX3t 2 , ..................... Sim. Model 58. Ly52 5 . 3. 5. P,T,M^T 2 .p^p^ P X 5 3 t 2 ,pX 5 3t 2 , .............................. Ly52 7 . 4. 5. T,Mi|T 2 . P^T, P^M1|T 2 , ........................ Sim. Model 74. Ly52 2 . 4. 5. T,Ml|T 2 .P^T,p|t,P^M^T 25 pX3t 2 , .............................. Ly52 6 . 5. 5. P_. p|t, p|mj|t 2 , ..................... Sim. Model 76, but flatter. Ly 52 3 . 25.* COPPER MICA. Kupferglimmer. Rhomboidal Arseniate of Copper. Cleavage = P. Jp 3 t, |p 3 m!|t 2 = (P.r 3 ). 2. 5. iM^foT,ip i mfciP^M}tT 1 ,= (r 3 ,R 2 -io) ............ P330 3 . 5. 5. P-. ^p 3 t Zw, \plt Ze, ^p 3 m{ 5 3 t 2 ; JpXSt .................. Ly 65 3 . P330 2 . 5. 5. P=.fet,p^^ 3 m^ 2 ,pX 5 3 t 2 , ....................................... Ly65i. 5. 5. Pl.}p&tPmBt Tingtang, Cornwall. H102 149 . Ly65 2 . P330.Mii 119 . 36.* DIOPTASE. Emerald Copper. Kupfer-Smaragd. Cleavage = ^PfT, JPlMgT, = (R 2 ). 4. 5. T,Mi 5 3 T 2 .>P|M,^M 2 Tl^...Si.n.Md.71. M 118 . H100 135 . P323 2 . R 78 . 21?. COQUIMBITE. Persulphate of Iron from Chili. Cleavage = t, m^t 2 . p 9 t, pXStt = (v. 2i$). 3. 5. p + ,T,M!|T 2 .P 9 T, P 9 M^T 2 = (p + ,V.2R^ZeZw)...Dl78. T1450. 28, VANADIATE OF LEAD. Vanadinbleierz. 1. 5. P,T,M!^T 2 = (P,V), ................................. Model 7. Ti573. h 58 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 29. ONE- AXED MICA. Einaxiger Glimmer. Mica with one axis of double refraction. Cleavage == P. 1. 5. P_,T,M{iT 2 , ...... Siberia. Vesuvius. Model 7. J 105 . H82 260 . P102. 1. 5. P=,tXstrp6t,pX!t .......................................... J 107 . H82 262 . 3O. NEPHELINE. Sommite. Cleavage = p,tXits = (p,v). 1. 5. P,T,M1|T 2 = (P,V)... Vesuvius. Md.7. L469 1 . J 101 . Mii250 ! . H62". 1. 5. P,m,T, m,tS, MSTjp ........................... Sim. Model 10. Ly28 2 . 3. 5. P,T,M}5T,. plt,plmgt Sim - Md.58. L469 2 . J 102 . M 111 . H62 80 . Ly28 3 . 3. 5. P,T,M^T 2 .p^pt,pX|t 2 ,pmllt 2 , ........................ L469 3 . PI 32. 3. 5. P,m,T,m 2 t!i, MgT,. pjt, pirnS** ................................. Ly28 4 . 3. 5. P,m,T,m,tJS,MBT r pJt,pt,pXXpmX ..................... Ly28 5 . 31, BERYL, Emerald. Smaragd. Emeraude. Aquamarine. Cleavage = P,T,M^T 2 . 1. 5. P,T,Mi!>T 2 , ............... Peru ...... Model 7. Ly33 T . L392 1 . H7l 110 1. 5. P,m,T,m,t& Mj'T,, ... Sim. Model 10. Ly v.2. L392 2 . J 35 . H71 141 . 3. 5. P,T,MlJT f .pra,pm 8 tlg, ...... Model 56. J 36 . Lyv.3. L392 3 . H71 142 . 3. 5. P^T,MgT r j^t,|gm8t ......... Sim. Model 58. J 58 . L392 4 . H71 148 . 3. 5. P/^M^T^pn^pl^pm^plXIt^ ...... J 37 . Ly v. 6. L392 7 . H71 145 . 3. 5. P,T,M! 5 3 T 2 .pm,p^t,pm 2 t!i,p>lit 2 , Md. 52. Lyv.5. L392 5 .H71 116 . 3. 5. ^^M^T^pm^l^p^pmApiX^pM^, ...... Lyv.7. L392 8 . H71 M7 . 3. 5. P,T,Mi 5 3T 2 ,3m x t. P m,p^ P m 2 t!i,piX|t 2 , ..................... Ly33. 3. 5. P^T,in^MJJT 1 .pin f j^pm^pgm^ ............... Model 52. Ly v. 8. L392 6 . H72 149 . 3. 5. P,m,T, in, tS, M1 5 3 T 2 . pm^yfet, pgt, pgt, pm a tB, p^o m J^P>^t 2 , plXit 2 , ............ P99. 3. 5. P,m,T, mrfj, M{ 5 3 T 2 . pm, plgt, pgt, pm.tB, pgrngt,, pJSmJSt, = (P,V,w.2riZnZs, 2r\$ ZvvZe, 2rgZwZe) ......... Ly33 3 . 4. 5. T,M^T 2 .pm,p^T,pmAplX|T 2 , .............................. Ly33 2 . 32. PYROSMALITE. Fer muriate. Cleavage = P. 1. 5. P,T,M{ 5 3 T 2 , ...... Model 7. L772 1 . J iii M1 . P227. Miii 143. Ly v. 1. 3. 5. P,T,M^T 2 .p^ P tt,pX3t 2 ,pX 5 3 t 2 =(P,V.ii,rO ...... P227. L772 2 . 33. EUDIALYTE. Cleavage = P-sptm, iplmgtla = (P. r|). 3. 5. P,T,MBT a . ^P^M, iplm, illMsTg, ^m 2 tj 5 3 = (P,v. RiZn, r^Zn) M 127 . P153. 3. 5. PjV^.RjJZ^rlZn^Zs, ................................. Ly27 2 . L391 2 . 34. CH ABA SITE. Chabasie. Cleavage = R| = Rhombohedron of 94 46'. 9. 5. R2, .................................. . .......... J 81 . Ly44 1 . L199 1 . H84 284 . . 5. R 5 4 X 2, ...................................................... Ly44 2 . L199 7 . MINERALS OF THE RHOMBOHEDRAL SYSTEM. 59 2. 5. RJZw,r|Ze, ......................................................... L199 2 . 2. 5. RiiZw^Ze, ......................................................... L199 3 . 2. 5. R2Zw,nSZe,rSZe, ............ Model 102. L199 4 . J 82 . M 10 . H84* 5 . 3. 5. r^Zw,r*Ze, S_, ............................................. L199 6 . H84 286 . 3. 5. rSZw,r|Ze,rSZe,S^ .......................... . ..................... PH5 2 . 2. 5. (RJZw,rSZe) x 2, .......................................... L44 3 . L199 7 . 2. 5. (R*Zw,R*Ze,iiZe) x 2, .......................................... L199 7 . 3. 5. (R},S_) X 2, ...................................................... Ly44*. 3. 5. (Ri,R$,Su) X 2, ................................................... Ly44 5 . 2. 5, (RJ, Ri, Ri S_; X 2, ............................................. Ly44 6 . 4. 5. v.RJZn,BtZs,rgZs, ................................................ L199 5 . 4L 5. (v. RJZn,BZ8,i$Zs) X 2, ................................. M 173 . L199 7 . 4. 5. (V. R5, Rl, R, S_) X 2, .......................................... Ly44 7 - 35. LEVYNE. Cleavage = R 2 == OP 2 T^P 2 Mi|T2). 5. 5. (P.pn^P^pmAP.Mjrr,) X 2, ..................... Pi 46. Ly45 3 . 36. ALUNITE. Alum-stone. Alaunstein. Cleavage = P. 2. 5. R| .................................................... Ti307. LlSl 5. 5. P. RJ, .................. Sim. Model 86. T1307. M ll! . L131 2 . Lyl9 2 . 5. 5. p.R|,r_,rZ, ........................ ......... . .............. L131 3 . P203. 3. TOURMALINE. Turmalin. Axes varying from p+mM to P+ m *A in which respect this mineral differs from all others belonging to the rhombohedral system. The prism T,M}f T 2 never occurs in a complete state, but is always irregularly modified by hemihedral varieties of the prism MjM^T-jf. The obtuse rhombohedral terminations of the prisms are extremely irregular, dissimilar combinations of rhombohedrons occurring at the two ends of almost every crystal. When the crystals are heated over a spirit lamp, and then allowed to cool gradually, one end exhibits positive electricity and the other end negative electricity, during the time of cooling. It is the zenith, or upper end of the combinations described below, which exhibit NEGATIVE electricity. When the symbol of a rhombohedron is embraced between the signs 1 and N, (for example, J^Nn,) it signifies that only three planes of the rhombohedron are present on the combination, and that these are all situated on the nadir end of the crystal. But if Z instead of N is used in the symbol, it signifies that the three rhombohedral planes are all on the zenith end of the crystal. The references are to figures which accompany an article on the connection between the form and the electrical properties of Tourmaline, written by G. Rose, and printed in Poggendorff's Annalen for Oct. 1836. Black Tourmalines : 3. 5. PZ, Mn, t, iM/rijj se sw, m^U R^Zn, M Zs, ir,Nn, Bavaria, Rose 1 1 . 60 MINERALS OF THE RHOMBOHEDRAL SYSTEM. 4. 5. Mn,M 8 Ti|sesw. R^Zn, ........................... Ceylon ...... Rose 1. 4. 5. Mn, M 8 T{j; se sw. RJZn^r.Nn, .................. Arendal ...... Rose 2. 4. 5. Mn, t, M 2 T}i se sw, mltt a . R 2 Zn, Jr.Nn, ...... Siberia. St. Gotthardt. Zillerthal. Schneeberg ...... Rose 3 4. 5. Mn,t,M 8 Tlj;sesw,mltt 8 . R^Zn,XZs,^Nn, ...... Sweden. ..Rose 4. 4. 5. Mn, t, M 8 T{| se sw, ml|t 2 . 2 R 2 Zn, 8 r a Ns, &Nn, 2 s 5 N, . . . Arendal. R.5. 4. 5. mn, T, m 2 t{jj se sw, MJ|T 2 . R\ Zn, ............ Greenland ...... Rose 6. 4. 5._mn, T, 2 m 2 tlt se sw, M}j|T 2 . \r\ Zn, 2 r 2 Zn, 8 RiZs, 2 R 2 Ns, Andreasberg, Rose 7. 4. 5. Mn, ms, T, Xtg ne nw, ^M 2 T}| se sw, M1|T 3 . R 2 Zn, IrJZs, ^Nn, Silesia ...... Rose 9. 4. 5. Mn, ms, T, Xtjl ne nw, JM a Tg se sw, MiiT 2 , Xt n 2 e n s w. R^ Zn, riZs.irJZs, ............... Rose 10. Green Tourmalines: 3. 5. PZ, mn, T, Xtls se sw, M!|T 2 . ^r^Zn, X Zs, \r\ Ns, ^Nn, St. Gotthardt ...... Rose 14. 3. 5. PZ, p N, Mn, t, ^M 2 T{i se sw, m{lt 2 . Jr^Zn, XZs, ^RiNs, ^s 8 Nn, ^s 3 Ns, Saxony ...... Rose 15. 4. 5. Mn, t, M 2 T}j; se sw, m',it 2 . ^Zn, ^RiNs, ^Nn, 2 R 2 Nn, Brasils,R.12. Brown Tourmalines: 4. 5. Mn,t,^M 2 Tl|sesw,ml|t 2 .R^ZnNs^r,Nn,...St. Gotthardt,... R.I 6. Red Tourmalines: 3. 5. PZ, mn, T, Jm 2 tS se sw, M^T 2 . |r 8 Ns, ^s 3 Ns, ......... Katharinenburg, Rose 17. 3. 5. P, Mn,t,iM,T8sesw,mXWZs,{rlNs, ......... Elba ...... Rose 18. 3. 5. pZ,mn, T, Xtll se sw, M}|T 8 . ^R^Zs^RoNs, ... Saxony.. .Rose 19- 4. 5. mn, T, Jm 2 tJl se sw, M|^T 2 . R 2 Zs Nn,^Zn, ...... Saxony... Rose 20. The contrast betwixt the prisms of Tourmaline and the regular hexagonal prisms may be thus shown: The figure e o w s, page 17, Part I., is a rhombus having angles of 1 20 at o and s, and of 60 at e and w. When the acute angles are replaced by the lines ez and wx, the resulting figure has six equal sides and six angles of 120 each. This is the prism = T,M|JT 8 of the rhombohedral system. But when the rhombus e o w s, is divided by the single line o s, the two products, e o s and sow, are three-sided forms, having three angles of 60 each, and, therefore, are equilateral triangles. This is the form of the equator of the Tourmaline crystal, and in the example marked " Rose 1," it occurs without modification. 38. PALLADIUM from Tilkerode. Seleniet of Palladium. 1. 5. P_,T,MjjjT 8 ,.-.. Cleavage == P, ............... Model 7. D388. T1656. 30.* to CBICHTOMTE. Fer oxidule titane. Cleavage = P. 2, 5. R 8 , ......................................................... Ly69 ! . Hiv99'. MINERALS OF THE PRISMATIC SYSTEM. 61 2. 5. R 8 Zw,r,.Ze, Ly69 3 . H iv 99 3 . 5. 5. P. R 8 Ly69 2 . Ti467. P257. H iv99 2 . 5. 5. P.R, Hiv99*. 40. CHLORITE. Talc? 1. 5. P,T,M^T 2 , Model?. R174. 41. CRONSTEDTITE. Hydrous Silicate of Iron, Cleavage = P } t,m{^t 2 . 1. 5. P,T,M} 5 3 T 2 , Model?. T461. P223. L211 1 . 1. 5. P,m,T, m 2 t! 5 3 , M!|T 2 , Model 10. L211 2 . 42.* SlDEROSCHISOLlTE. 1. 5. P,T,M}$T SI ... Cleavage == P Model 7. P225. 43. FINITE. Cleavage == t, mi^. 1. 4. P,M,T, m 2 ti|, miltj, Axes: p a x m a t a , H62 54 . 1. 5. P,T,M1|T 2 ,... Greenland. Salzbourg. Md. 7. Ly v. 1 L464 1 . H62 51 . 1. 5. P,m,T,nU!|,Mij;T 2 , Puy-de-D6me, Md. 10, Lyv2. L464 2 . H62 53 . 3. 5. P,m,T,m 2 ti*,M!iT 2 .p?t,pX 5 3t 2 , Ly29 2 . L464 3 . H62 53 . 3. 5. P,m,T, m 8 til, MJ5T,. pjm, p}t, p?t, pXC pXtt pX&, L464 4 . PI 14. 44.* DREELITE. Dreelith. 2. 5. R!i, (Cleavage = ?) D203. CLASS IV. MINERALS BELONGING TO THE PRISMATIC SYSTEM OF CRYSTALLISATION. The AXES of all Combinations belonging to this Class are = p^niyt^. In many cases the Axes are = pj}_mt a , as in the examples P_j.,M_,T and p 1 9 M 8 T r \ o 1V1 1 o L The constituent FORMS of the Combinations of this Class are as fol- low : Number of varieties of the Zones ' Forms ' unequiaxed Forms.. Prismatic, M, M_T, M + T, T 5 About 70. North, P,P_M, P + M 5 M, 20. East, P 3 P_T,P + T,T, 50. Octahedral, PJM y T z , 60. No other Forms than these occur upon crystals of this Class. There are two or three hemihedral octahedrons, and many twin crystals. There are no Hemihedral Forms in the Prismatic, North, or East Zones. 1. SULPHUR. Schwefel. Soufre. Cleavage = mftt. p}gmftt. 9. 3. PjgM-ftT, Model 21. L596 1 . Ly2. J 213 . R 85 . P383 1 . H331, 62 MINERALS OF THE PRISMATIC SYSTEM. 2. 3. PjJM&T.p&mftt, ........................ L596 3 . J 248 . P383 6 . H337. 9. 5. p+t^J-gM^-T, ...................................................... L596' 4 . V. 5. pfgt,P#!M&T,p T Vnftt f ........................ L596 12 . J 251 . H339. 3. 5. P+ ,m T y.pigt, P|m T y, ............................................. Ly6. 3. 5. p^m&t.piSt,PfgM&T, P Amftt, ............... M 18 . L596 6 . Ly8. 4. 5. t,mftt.P|8MftT, Pl Vii;ftt, .................................... L596' 3 . 4. 5. m_. PjgM^T, ............... L596 7 ... Model 70. J 246 . P383 4 . H334. 4. 3. M^T.PfgM^T,..." ......... L596 9 ... Model 66. J 247 . P383 5 . H335. 5. 3. p+.P}gM T %T, ......... L596 2 . Ly3... Model 80. J 245 . P383 3 . H333. .5. 3. P+ .P|gMJLT,p T %m r V, ..................... L596 4 . Ly5. J 249 . H338. 5. 5. P+ .p[gt,PigM&T, Pl Vn T y, ............... L596 5 . Lf. R 86 . H340. 5. 5. P+ ,m_.PfgM T %T, ............................................ . ...... L596 8 . 5.5. p + .pigt,PigM&T, ................................................ L596". 6. 5. iPjgMftT, ipigm T V, .............................. J 244 - p 3^3 2 . H332. 6. 5. pfgt, PfgM^T, ............... L596 10 . Ly4... Model 120. J 25 . . H336. 58. ANTIMONIAL SILVER. Antimonsilber. Argent antimonial. Cleavage = p^m/gt.pjt. 1. 2. P X ,M_,T, ............................................................ L685 6 . 1. 5. P,T,M/ ff T, .............................. Sim. Model 7. Ly v 1. L685 1 . 3. 3. PJM^T, ............................... . ............................ L685 9 . . 5. Pf T, P|M T 7 2 -T, ...................................................... L685 8 . 3. 5. P,T,M T ^T.FjM T Vr, ................................................ Ly47 2 . 3. 5. P^M^T.pJ^pjM-^T, ................................. Ly47 3 . L685 3 . 3. 5. P,m,t,M T 7 2T, m 1 /t.pJt,pJm T 7 2t, ................................. L685*. 4. 5. T,M/^T.P|T,P|M r 7 2T, .......................... ................ L685 3 . 4. 5. M^-T.PJM/^T, ................................................... L685 5 . 5. 5. P.P|T,PJM T 7 2 -T, ................................................... L685 7 . 3. ARSENICAL IRON. Arsenikeisen. Cleavage = P,ra^t. 5. 3. M^T.P 2 M, ..................... Stiria ...... Sim. Model 82. Mohsii 1 . 4. An Isomorphous Group, 1, 2: 1. VITREOUS COPPER. Kupferglanz. Cuivre sulfure. Sulphuret of Copper. Copper Glance. 2. SULPHURET OF SILVER AND COPPER. Silberkupferglanz. Cleavage = p^t, p 2 m}f t 2 . 1. 5. P, T,Mf|T ? ... Cornwall... Sim. Md. 7. L640 1 . P308 1 . LyGO 1 . H98 1U . 1. 5. P,m,T, m,tif , M|J T 2 , ............... Mexico ...... Model 10. Ly60 3 . 2. 5. P 2 T, P 2 M{T 2 , ...... Sim. Md. 26, but more acute. P308 5 . H98" 5 2. 5. P^T, PlMf|T 2 , ........................ Similar to Model 26. P308 3 . 3. 5. P 2 T,PiT 5 P 2 M;fT 2 ,PiM}fT 2 = (2R 2 ,2Ri) 3. 5. P,T,MjfT 2 .p.t ; p::m I 1 -ft 2 , ...... Sim. Md. 5S. P308 4 . Ly6l 6 . H99 118 . 3. 5. P } T,M!fT 2 .pit,pimift 2 , ...... Sim.Md.58. P308 . Ly6l 4 . H99 119 . 3. 5. P,T,Mj-fT 2 .pit ; p,t,pimyt, 5 p 2 m}ft 2 , ............... Ly6l 8 . H99 120 . 3. 5. P,m,T, m,t;|, MJfT,.pit,i4m!5t, = (P,V,t>.2rl) ......... H99 121 . MINERALS OF THE PRISMATIC SYSTEM. 63 3. 5. P 5 m,T, nU] 4, M}f T 2 . pit, pt, p a t, = (P,V,tf. 24,2^,2^) ......... H99 122 . 3. 5. (P,T,MffT 2 .pit,pim}ft 2 ) X 2, ................................. Ly6l 5 . 4. 5. m,T,m 2 t||, M||T 2 . PJT, PiM{|T 2 ir (V,t;. 2RJ) ......... Ly6l 7 . 5. 5. P_.P 2 T, P 2 M[fT 25 .................. Sim. Md. 96, but flatter. H99" 6 . 5. 5. Pz. pt, pm}t 2 , ........................... Sim. Md. 96, but flat. H99" 7 . 5. SULPHURET OF BISMUTH. Wismuthglanz. Bismuth sulfure. Cleavage = p,t,m T 7 jt. 1. 3. P,M T 7 f T, .......................................... Sim. Model 6. L616 1 . 1. 5. P^M/yT, ............................................................ L616 2 . 3. 5. P,t,M T VT.p x m,p x t,p y t, .................................... P277. L616 3 . . An IsomorpJious Group, \, 2 : 1. SULPHURET OF ANTIMONY. Antimonglanz. Antimoine sulfure. Grey Antimony. Grau Spiesglanzerz. Cleavage = p,m,t. p^fom^t. 1. 5. P,T,M r 9 Q 6 oT, ........................... Sim. Models. L605 7 . H116 331 . 1. 5. P,M,T,m^yt, ................................................... H116 80 *. 4. 3. M^T.P-^M^-T, .................. Sim.Md. 67. L605 1 . H116 299 . 4. 3. M T ^T.p^_ M _2.6_ Tjp _ mT 9_6 o . t) ................................. L605 .. 4. 3. MflfcT.PJ&t, ................................................ L605 3 . 4. 5. ^M^T.P^-M-^oT, ........................ Mii 6 . L605 6 . H116 300 . 4. 5. m^M^T.P^M^T, .............................. L605 5 . H116. 4. 5. m,t,M T af D T.P_M_T, ..................... .'. ................ Hiv291 5 . 2. ORPIMENT. Auripigment. Arsenic sulfure jaune. Cleavage =. m. 5. 5. M,T, Mf T, mft, m|t. PT, p x m y t z , ..................... D434. P283. 5. 5. M,T,M|T,m|t.PgT, .......................................... Levy 74 2 . f. WHITE IRON PYRITES. Speerkies. Fer sulfure blanc. Cleavage = p,MJT. 1. 3. P,M|T, ...Cornwall. Derbyshire... Sim. Model 6. L661 1 . H109 223 . 3. 3. P,M}T.pJt, ............... Sim. Model 44. L661 2 . Ly v. 2. H109 22G . 3. 3. p,mftp|m,pjt, PfMfT, .............................. L661 7 . H109 229 . 3. 5. p,m|t. PfM, Pf-T, .................. Joachimstal ...... L661 5 . H109 2 - 8 . 4. 5. mjt. P|M, P|f, ........................ Freiberg ...... L661 6 . H109 227 . 5. 3. M|T. PJT, .................. Derbyshire ...... L661 3 . Ly v. 1. H109- 24 . 5. 5. mjt.P^T, .............................. Similar to Model 82. H109 2 ' 5 . 8. RED OXIDE OF ZINC. Zinkoxyd. 1. 3. P,MJT, ...... (Cleavage = mjt) ............ Sim. Md. 6. L563. P373. 9. WHITE ANTIMONY. Weissantimonerz. Antimoine oxide. 5. 5. T_,Mf-T.PVT,p x m y t 7 ,... (Cleavage = T) ......... P348. Mohs ii ' 4 . 64 MINERALS OF THE PRISMATIC SYSTEM. 10. PYROLUSITE. Grey Ore of Manganese. Cleavage = m,t, m-^t. 1. 3. P,Mj^T, m_t, Elgersburg Levy 75 3 . 1. 5. P,m, M^T, Thuringia Levy 75 3 . 3. 5. P,M 5 t, M T 9 n %T. p_t, P238. D376. 11. ARSENICAL PYRITES. Arsenikkies. Mispickel. Fer arsenical. 1. Common Arsenical Pyrites. 2. Cobaltic Arsenical Pyrites. Cleavage = p,Mf T. 1. 3. P,M|T,... Freiberg. ..Sim. Md. 6. L663 1 . Ly v.l. P214 1 . H105' 88 . 3. 3. P,MfT.pfm, Sim. Model 44. T497 2 . L663 7 . P214 3 . 3. 5. P,M|T.pa Mj L663 8 . P214 4 . 3. 3. ^MfT.p^T, Sim. Model 104, P214 2 . 5. 5. mftPfM, Sim. Model 82. T497 3 . P214 6 . 5. 3. MfT.pfm,P T 3 -T,... (Cobaltic) Scheerer. 5. 3. M|T. PfT,. . . Cornwall. Tunaberg. Freiberg. L663 2 . Ly v.4. Hi 05 18 '. 5. 3. MfT.P T 3 -T, ... Freiberg. Bohemia... M 2 . L663 3 . Lyv.2. H105 193 . 5. 3. M|T.P T 6 oT,pf-t, Tunaberg L663 4 . Ly v.5. H105 191 . 5. 3. MJT. P^yT, pjt, R 89 . L663 5 . H105 193 . 5. 5. T,MfT.pfT,P?MfT, L663 6 . H105 193 . . 3. (MfT. P T %T) X 2, Freiberg L663 9 . Ly67 2 . 12. BRITTLE SULPHURET OF SILVER. Sprodglaserz. Cleavage = mf k p x m y t z . 1. 5. P_,T,MfT, ......... . .......................... Schemnitz ...... Levy 50 2 . 3. 5. P_,T,M|T.p|t, p x m y t z , ..... . ......... Mexico ...... P298. Levy 50 3 . 3. 5. P_,T,Mf T. 2p x t, 2p x m y t z , ..................... Freiberg ...... Levy 50*. 3. 5. P_,T,Mf T. 3p x t, P X M X T Z , ..................... Freiberg ...... Levy 50 r> . 3. 5 P_,T,MfT. 3p x t, 2p x m y t z , ..................... Freiberg ...... Levy 5 O 6 . 13. BERTHIERITE. 1. 3. P,M X T? ...... (Cleavage = m x t ?) ............... Auvergne ...... P344. 14. JAMESONITE. Cleavage = P. 1. 3/P,MfT,... Cornwall. Siberia. Hungary... Sim. Md. 6. P346. D420. 15. ZINKENITE. Cleavage = 0. 4. 5. T,M}-f T 2 . P|T, P|MifT 2 = (V.2R|) ...... Sim. Md. 74. P347. 16. ANTIMONIAL COPPER. Kupferantimonglanz. Cu 2 S + Sb 2 S 3 . Cleavage = p,T. 1. 5, P,T_, mft, iDy^t, .................. G. Rose, Pogg. Ann. xxxv. 360. ff. STERNBERGITE. Cleavage r= P. . 5. P_,T. 2p x m t z , .......................... ............................... P297. MINERALS OF THE PRISMATIC SYSTEM. ( 65 IS. MENDIPITE. Muriate of Lead. Berzelite. 1. 3. RMfT, ...... (Cleavage = MfT) ................................. P361. 19. An Isomorpkous Group, 1, 2: 1. MANGANITE. Grey Oxide of Manganese. Hydrated Deutoxide of Manganese. Cleavage = p,M 2 T, MT, m-^t. 4. 3. MT, M 2 T, m/jjt. PJyMJT, pf mf t, pmf t, Jp ro s t Znw Zse, Haidinger, Edin. Jour, of Science, Jan. 1826, fig. 2, 3, 4. 5. 3. MfT,M 2 T,MfT,m T 5 2t.pfm,p|t,PiMfT,pfmft,p|mft,idem%5. 2. PRISMATIC IRON ORE. Nadeleisenerz. Brown Iron Ore. Fer hydro-oxide. Cleavage = T. 4. 5. T^M^T.P.M^T, .......................................... Levy69 2 . 4. 5. T^M^T,m^t,p,m$ ff T ....... .......................... Levy 69 3 . 4. 5. M, T_,M T 9 T f IJ T,M T %%T.p + m ) p_ra x t,2p + m x t, .................. P220 2 . 5. 5. T_, MI %%T, m^t. P_T, ....................................... Levy 69 4 . 2O. TANTALITE. ) 1. COLUMBITE. } Cleavage =M,T. 3. 5. P + ,M,T,mft,mft, mft.pjm, P_M X T, p + m x t,...D37l 3 . P272. T485. 3. 5. P +5 M,T,MfT, mft, mft.pjm, piT,p_M x T, 2p + m x t, ......... D37l 3 . 3. 5. M,T,mft, m|t, mf t. P_M X T, .................................... D371 1 . 22. AESCHYNITE. Cleavage = p. 4. 3. MJT.P x M y T z , ............................................. P261. Brooke. 23. An Isomorphous Group of Carbonates, 1, 2, 3, 4, 5 : 1. WITHERITE. Carbonate of Barytes. Baryte Carbonatee. Cleavage = t. p|t, pjmf-t. 2. 5. P|T,P|MfT, ....... .' ........................................... Levy 15 2 . 2. 5. PiT, P |t,PiMf-T,pim|t, .......................................... H43 77 . 3. 5. P + ,T,MfT.p|t,pfmft, .................................... L330 1 . H43 76 . 3. 5. P+ ,T,MfT.pfT,p|t,plt,p|MfT,pfmft,pimft,...H43 78 .Levyl5 3 . 4. 5. T,Mf T. PfT, P^Mf-T. .................................... L330 2 . H42 75 . 5. 5. T, Mf- T. pft, pf t, pjt, .................................... L330 3 . Mii 23 . 5. 5. T,MfT.PfT, ......................................................... Mii 9 . 2. STRONTIANITE. Strontian carbonatee. Strontites. Cleavage = MfT.pft. 1. 5. P_,T,MfT, ........................... Sim. Model 7. Levy 18 1 . H45 93 . 3. 5. P,T,Mf-T.pft,pfmft, ...... Sim. Model 58. L328 1 . Lyl8 2 . H45 94 . 3. 5. P,T,MfT.pft,pft,pJmft,p*/mft, ..................... L328 3 . H45 95 . 3. 5. P,T,MfT.pfT,pfmft,p 2 /raft, ............ D200. Mii 28 . Levy 18 3 . 4. 5. T,Mf-T.pfT,pfMfT, ..... .' .......................................... L328 2 . t 66 MINERALS OF THE PRISM A. TIC SYSTEM. 3. ARRAGONITE. Carbonate of Lime (and Strontian ?). Cleavage = t,Mf T. p T ^t. 1. 3. P,Mf T, .............................. Spain ...... Sim. Model 6. H23*. 3. 3. p,M|T.PyT, .......................................... Spain ...... H23 5 . 4. 5. T_,M|T.P T 7 oT,p 5 t, ................................................ Lyll 9 . 5. 3. MfT.P^T, ........................ Spain ...... Sim. Model 20. H23 1 . 5. 3. MfT.Py>T, ........................ Spain ...... Sim. Model 19- H23 2 . 5. 5. M,T, M| T. P T 7 oT, ............ Piedmont ...... Sim. Model 97. H23 6 . . 5. M,T,Mf T. P T 7 -T, ....................................... Model 97. H23. 5. 5. T,MfT.P T 7 oT, ...... Piedmont ...... Sim. Model 111. Lyll 7 . H23*. 5. 5. T,MfT.p^t,P/ n T, P yt.3p x M y T z , ................... . .......... Mr6 84 . [Many twin crystals. See Levy and Haiiy.] 4. JUNKERITE. Cleavage = t,Mf T. 5. 3. MfT. P X T, ......................................................... T1449. 5. WHITE LEAD ORE. Weissbleierz. Plomb carbonate. Carbonate of Lead. Kohlensaures Blei. Cleavage = P^^t. 2. 5. PyT,PJLM T 6 -T, .................................... L290 5 . J 138 . H91 5i . 3. 3. PvMjVT. P/ n T, .......... . ........................................ L290 1 . 3. 5. ^^M^-T.py^p/o-m-^t, ................................. J 136 . H92 59 . 3. 5. P,T 5 M T %T.pV 4 t,p 2 t 5 pV t,p 1 7 om 1 V, ........ .......... L290 9 . H92 63 . 3. 5. P_ 5 M,T,m 1 %t ) MiT.p>,p^t, P / n t J .................. L290 10 . H92 6 *. 4t. 5. T^M&T.PVTjP&M&T, .................. L290 6 . J 137 . R 96 . H92 56 . 5. 3. Mf T.P^T. Axes: p a m^ a , ..................... Model 82 a . H91 51 . The letters stamped on Model 82 a in agreement with the system of Hauy, who considers this combination to be a rectangular octahedron, give rise to the symbol P^M, P J 7 T. To make the Model agree with the symbol M^T. P-j^T, it must be turned' on the axis m a from east to west, 90, so as to make p a and t a change places. 5. 5. T_,M T 6 n T.pJm,PVT, ............... Model 110. Rf. Sim. H92 57 . 5. 5. T_,M T 6 oT.p^m,P r %T, ...... Sim. Model 110. J 139 . H92 57 . L290 7 , 5. 5. M^T.PaT, .................................... Mii 2 .L290 2 . J 135 . H91 53 . 5. 5. T,M T 6 T.PVT, .................................... Mil 9 . L290 3 . H91 55 . 5. 5. T,m 1 %t.pfm,P T 7 T) T,P 1 ^M T 6 -T, ................................. H92 58 . 5. 5. T,M T 6 T.pm,py t,p T 55t ......... ......................... L290 8 . H92 60 . 5. 5. M^m^mLgt.P/o^P^M^-T, .............................. H92 61 . 5. 5. T^T.pfm^T.pp^pMT, ..................... H92 62 . 24. PHOSPHATE OF MANGANESE. Triplit. Manganese phosphate. 1. 1. P X ,M_,T,...( Cleavage = P,m,t) ...... Limoges... P248, Lyiii304. 35. NITRE. Salpeter. Potasse nitrate. Cleavage = T,m}Jt. [The following descriptions apply to manufactured Crystals.} . 5. P?|T, Pff Mff T 5 ......... Axes: p^m^... Sim. Model 26. H52 161 . 3. 5. p + ,m,T_,miJt.pi|t, .............................................. H53 164 MINERALS OF THE PRISMATIC SYSTEM. 67 4. 5. T,MlT.r'fT,rffMf-T, Axes: p_jLm a t... Sim. Model 73. Aikin. H52 163 . 4. 5. T, M} J T. pf t, pi |t, pjf t, pff mf ft, pigmfgt, pf fm^t, . . .H53 166 . 5. 3. M}T. PJ^T,...Axes: p^mAtfg... Similar to Md. 20. Aikin. H52 159 . 5. 5. Tw,te,M}jT.P}|T, J. J. G. 5. 5. T_,MjjT.p}jT, Axes: p4.m a tl M ii 9. Levy. H52 162 . 5. 5. T_,m}Jt.pf^t, Aikin, Chemical Dictionary. 5. 5. T_, M !AT.p?t,p}2t,pMt, ..Mii 23 . H53 165 . 26. STAUROLITE. Staurolith. Staurotide. Cleavage = T t mfft. 1. 3. P_,M/yT,...Morbihan. Sim.Md.6. MH366 1 . L409 1 . P75 1 . H61 44 . 1. 5. P+jTjMiyjT, ......... Cayenne. St. Gotthardt...J 61 . Ly v. 1 . L409 2 . Mii366 l . P75 2 . H61 45 . 1. 5. (P+jTjM/^T) x 2, ... Two individuals crossing at a right angle. Model 9. J 63 . Ly v. s . L409 4 . P75 4 . H62* 7 . 1. 5. (P + ,T,M! 8 7 T) x 2, ......... Two individuals crossing at alternate angles of 60 and 120, ...... J 64 . Ly v. 4 . L409 4 . P75 5 . H62 48 . 3. 5. P + ,T,M^T.pjjM, ...... Aschaffenberg... Model 55. P75 3 . H6l 46 . M 12 . J 62 . Lyv.2. L409 3 . 27. ANDALUSITE. Feldspath apyre. Andalousite. Cleavage = m^yt. 1. 3. ^M^fiyT, .................. Lisenz, Tyrol ...... Ly42 ! . Mv. 1 . L405 1 . 3. 3. ^M^yT.p^yt, ............ Lisenz, Tyrol ...... Ly42 2 . Mii 3 . L405 2 . 3. 3. P,M^yT.p5Jm, ................................................... L405 3 . 3. 3. ^M^T.pJ^p^Lt, ................................. Ly42 3 . L405 4 . 3. 3. ^M^^M^-T.p^^p.m^, ................. . ............... P107. 3. 3. P,M r 9&T.p T yot,r x M y T z , .......................................... Ly42 4 . 3. 3. P,MA% T, m^t. pjjm, p^jt, p x m y t z , ........................ Ly42 6 . 3. 5. P^M^T.p^t, ................................................ L405 5 . 3. 5. P 5 m,M T _T.pJim,p T V G t, ....................................... L405 6 . 3. 5. P,m,M T 9 %T.p T V -t,p x M y T z , ........................................ Ly42 5 . 5. 5. M^T.P/^T, ............................................. Mv 2 . L405 7 . 5. 5. mMT.mPr .......................................... L405 8 . 2S. OLIVINE. Chrysolite. Peridot. Krysolith. Cleavage = p,ra,T. 1. 2. P|,M|,T, .......................................... Eisenach ...... L531 1 . 3. 2. p+,M_, T.p|m,p^T,pft,py>m|t,pJmjt, .................. Ly32 5 . 3. 5. p + ,M_,T,M|T.P|M,pfM|T, ........................... L531 5 . H70 130 . 3. 5. P+ ,M_, T,M-iT.PfM, P^T,pfm|t,... Model 51. L531 3 . R 93 . Ly32 3 . H70 132 . 3. 5. P+ , M_, T, MJT. P|M, PLOT, p|t, p^m|t, ... L531 4 . Ly32 4 . H70 133 . 3. 5. p + , M_, T, mjt, m|t, rn^ft. PJM, P^>T, pjt, pf mjt, ptfmgt, found in meteoric Iron, ............ L531 11 . P85. 3. 5. P +,M_, T,mjt,mgt.PiM,p\ft,pjt,pgmjt, .................. Ly32 7 . 68 MINERALS OF THE PRISMATIC SYSTEM. 3. 5. p+, M_, T, M| T, mft. Pf M, P y>t, pjt, pfm|t, pV>m?t, ...... Ly32 9 . 3. 5. p4.,ltt,MjT,in|t.P|M,p^T,pjt,pJmjt, ......... L531 6 . H70 135 . 3. 5. P+5 M_, T, M|T, mft, mV 6 t. P|M, P^T, pf MJT, p^mft, ...Ly32 8 . 3. 5. p^i^T,mft,MyT.p|m,Py>T;PfM|T, ......... L531 8 . H70 134 . 3. 5. P + ,T,MfT,m\ft.P\pT, ....................................... Ly32 2 . 3. 5. p^t,MfT,mft.P|M,Py>T, ........................... L531 7 . H70 131 . 5. 2. M_,T.PL g T, .......................................... Baden ...... L531 2 . 5. 5. M_,T, M|T,mft.PM,pVT,pfmft,p\fmft } ............... Ly32 6 . 29. SULPHATE OF POTASH. Schwefelsaures Kali. Potasse sulfatee. Cleavage = t,m^t. 1. 5. P x ,T,MfT. Axes: p a m^t a , ....................................... H53 169 . 3. 5. Pf T,PfMf T, ................ , ...................... PI 96. R123. H53 188 . 30. THENARDITE. Sulphate of Soda. Cleavage = P,mit. 2. 3. P+M1T, .................. Aranjuez ...... Sim. Model 21. D 76 . P407. 3. 3. P+,MJT. p+mlt, .................................... Aranjuez ...... D75. 31. An Isomorphous Group of Sulphates, 1, 2, 3: 1. SULPHATE OF BARYTES. Schwerspath. Baryte sulfatee. Heavy Spar. Cleavage = P,m,t,MjT. 1. 3. P,MfT. Axes: p|m a 4 t^, ......... Model 6. R 92 . LylS 1 . L256 1 . H33 1 . 1. 5. P-, m, mft, ...... pm a t?_, .................................... L256 2 . H33 8 . 1. 5. Pz,t,mft, ...... pmt a , .................... .J 142 . Lyv 3 . L256 5 . H33 9 . 1. 5. PZ, m, Mf T, m^t, ...... plm a t4., ........................... Lyl5 5 . H34 16 . 3. 3. P_,M|T.pJm, ...pm|t a , ...... Model 44. Lyv 4 . L256 7 . H33\ S 228 . 3. 3. P_,MfT.p|mft, ...... pm^t a , ....................................... H33 7 . 3. 3. P_, MJT,m T 8 3 -t.p|t J p|mft, ...... plnrM^, ........................ H36 32 . 3. 3. p +) M|T.pfM,pft,pfm|t, ...... p4.mt% ........................ H36 33 . 3. 3. p + ,M}T.pfT,p|m|t,pfm|t, ...... p4.m a t a , ..................... H37 36 . 3. 3. p + ,MfT.p|m,r|T,pfm|t } pfM|T, ...... p4.m a t a , ............... H38 48 . 3. 5. P_,m,t + ,MfT.p|m,p|t, pm a t4.,...Md. 50. Lyv 28 . L256 20 . H38 45 . 3. 5. p + ,M,t,MfT.p|T,pJm|t, ...... p4.mt% ........................ H38 46 . 3. 5. p+,M, T,M}T.^m,pjt,p|^p|t,p^mJ^ ...... p4.m a tl ....... H41 64 . 3. 5. p + ,M,T,MjT.pfM,p T 8 J m,P4T,p|m|t,pfm|t,p4.m a t,L257 30 .H42 70 . 3. 5. p + ,M_,T,MfT,mft.P|M,P|T,p|mJt,...p^mlt a , ...... S 237 . H40 61 . 3. 5. P+ ,M_,T, MJ T,mf t. P|M, P|T, pf mft, pj m|t, P 4.mt a , S 238 . H41 67 . 3. 5. p+,m, T,M|T.p|m,p| T , p fm|t, P 4:m a tl 5 ...Ly v 43 . L257 29 . H40 57 . 3. 5. p+,m,T,MjT.PfM,pft,pft,pJt, ...... p4_m a t, ............... H40 58 . 3. 5. P_, m, Mf T. pft, ...... pm a t4., .................................... H34 15 . 3..5. Pz,m_, Mf^m^tpfm, ...... plmt^ ........................... H36 30 . 3. 5. p + ,M,MfT.p|M,pft,p|m|t, ...... p4_mlt a , ............ Lyv 33 . H38 43 . 3. 5, P + ,M_,MfT, MjT.pft, ...... p4.mit% .................. L256 24 . H35 25 . MINERALS OF THE PRISMATIC SYSTEM. , 69 SULPHATE OF BARYTES Continued. 3. 5. P_,t, MfT.pfm,p|t, p^mHj?., S 235 . H36 29 . 3. 5. P_, t + , MJ- T. P|M, pf m, p|t, plnrt.?., Lyv 32 . H38 41 . 3. 5. p + ,t, MfT.pfM,pfT, pfmft, p4_m.lt a , Lyv 30 . H38 42 . 3. 5. P_,MfT.pft, pm a t|., Lyv 6 . R 94 . S 230 . L256 10 . H33 6 . 3. 5. P_, MfT,mj9 5 -t.pfm,pft, plm a t.?_, H36 S8 . 4. 5. T=,MfT.pfM,p 3 t, pfmft, p4.m a t, H36 31 . 4. 5. T_,MJT.p|-M,pjT,p|t,pfmJt, p4.m a t, Lyv 37 . H37 40 . 4. 3. M|-T.pJm,pJ T , p4-m|t a , L256 12 . H34 10a . 4. 3. MfT.pfm,pJt, p^mjti J 146 . L256 12 . H34 10b . 5. 3. Mf-T.PfT, p4-m a t a , Model 82. J 14 \ S 229 . L56 8 . H233 2 . 5. 3. MfT.PJT, p4-mt a , Lyv 2 . H33*. 5. 3. MJT. P|T, pfmft, p4-m a t a H34 12 . 5. 3. M}T. pf M, P|T, pf mft, p4Lm a t a , H34 17 . 5. 3. MfT,mft.P|T,pft, p4_mt a , H35 22 . 5. 3. MJT, mf t. p|t, pf t, pf m|t, p.?_inlt a , H37 35 . 5. 3. M|T, m|t. pf M, pf T, pfmf t, pf m|t, p4.mt% H38 47 . 5. 3. MfT.pfm, p4-m a t a , L256 11 . H33 3 . 5. 5. M_,MfT.PfM, p a mlt4., Model 100. R 91a . J 140 . L256 18 . 5. 5. M=, Mf T. P|M, p4.rnlt% Sim.Md. 100.R 91a .Lyv7. J 140 .L256 18 .H34 n . 5. 5. M_, M^T, mf t. P|T, p4.mt% Lyv 15 . L256 22 . H34 13 . 5. 5. M=, M fT.pJM,p|t, p4-mlt a , Lyv 11 . J 141 . L256 17 . H35 14 . 5. 5. M_, mft, m|t. P|M, p a mtJ|L, Lyv 14 . H35 18 . 5 5. M_, mf t, Mf T. P|M p.|_mt a , H35 19 . 5. 5. M_ mft, mft. PJM, P|T, p^Lmlt a ,...Ly v 18 . S 232 . D204.H35 23 . 5. 5. M_ 3 mft > MfT.pJM,p|t, p4.mlt a , Lyv 19 . S 233 . H35 24 . 5/5. M_, Mf T, mft, mft. P JM, P|T, p|m|t, p4_mt a , H37 39 . 5. 5. M=,t,mft.p|M,p|t, p4_mt a , Lyv 17 . L256 27 . H36 27 . 5. 5. M_,t,MfT.pjM,p|t,pV 5 t, p4.mlt a , :..L257 28 . H38 44 . S. 5. M_,t,MfT,MfT.p|M,p|T,p|m|t, p^.mt a , H40 65 . 5. 5. M_, t, MfT, mft.pJM, P|T, p^ 5 t, p|mft, p4.mt% H40 59 . 5. 5. M_, t, Mf T. Pf M, P|T, ...p4_mlt a , A 52 . 5. 5. m, T,M|T.pfm,p|T,pft,pfm|t, p.|.m a t, H39 53 . 5. 5. m, T_, MjT.pfm,p|T,pft,pfm|t,...p4.m a t a ,...Ly v* 7 . S 236 . H39 54 . 5. 5. m, T_,MJT.pfM,p|t,p4t,pft, p4.m a tl, Lyv 46 . H39 52 . 5. 5. m, T_, MjT. pf m, Pf T, p|m|t, pf mft, p4_m a t, H40 56 . 5. 5. t, MfT.pfM, p4.mt a , J' 45 . S. 5. t, MjT. pjm, p|t, p4-mlt% J l . 5. 5. t,M|T.p|t,pft,p|m|t,ip + M_TZneZnwNseNsw, p4.mt% H39 49 . 5. 5. T,M|T,mft.p|M,p|t,p|t,pft, p4.m a ti H39 50 . 6. 5. T,M|T.pfm,p|t,pft,pgt,pfmlt 5 P 4.m a t, Lyv 48 . H39 51 . 70 MINERALS OF THE PRISMATIC SYSTEM. SULPHATE OF BARYTES Continued. 5. 5. t,MfT.PJT, ...... p4mlt% ................................. R 91 . S 231 . J 144 . 5. 5. t + , MJT, m|t. P|T, ...... p4.mt a , ................................. H35 20 . 5. 5. t+, Mf T. PfT, pf raft, ...... p4.mlt a , .............................. H35 21 . 5. 5. t, MJT.p^M,p|T,p|m|t, ... P 4LniJLtV..S 231 . L256 16 . Ly v 23 . H36 26 . 5. 5. T_, MJT. pf t, pf t, pf t, ...... pm a t, .................. Ly v 26 . H37 31 . 5. 5. T_ M|T.pfM,p|t,pft,pft, ...... p-Jmti, ............ Lyv 35 . H37 37 . 5. 5. T_, MfT.pfM,pfT,pft,pf-m|t ; ...... p4_m a t^ ......... Lyv 36 . H37 38 . 1. SULPHATE or STRONTIAN. Strontspath. Strontiane sulfatee. Celestine. Cleavage = p,m,T, PJM. 1. 5. P_,t,mft, ..................... plm.|_t a , .................... L263 3 . H43 83 . 3. 3. p + , MT. pf MT, ............ p-t-m a t a , ...... Meudon, Paris ... Lyl8 2 . 3. 3. p + , T, MfT.pigm, pft, ...... p^Lm a t, La Catholica, Sicily... Lyl8 3 . 3. 5. p +) t,MfT.pfT,pfmft, ...... p4.rat a , .................. L263 8 . H54 89 . 3. 5. p +5 m,T,MfT,m 10 t.pfm,p|t,p|t,p|t,pfmft....p4.mt a , ...P193 2 . 3. 5. p +J T,Mfr.p|M,pfT,pfmft,pfmft,...p4.mt a , ... Sicily... Lyl8 5 . 4. 3. MfT.PfMfT, ............... p+m a t^ .............................. H43 81 . 5. 3. MfT.P|M, .................. p4-m;tS, ............... S 124 . L263 2 . H43 80 . 5. 3. Mf T.p}f M, P|T, ............... p+m a t a , Ly v } . A 53 . S 125 . L263 4 . H44 84 . 5. 3. MfT.pJt,P|M|T, ............ p.f_m a t a , .............................. H44 85 . 5. 3. Mf T. pim,pf t,pf Mf T, ...... p4.m a t, ...... Fassa, Tyrol ...... Lyl8 4 . 5. 5. M, Mf T, mft. PfJT, ......... p4.mlt a , ......... Lake Erie ...... D201 2 . 5. 5. T_,MfT.p;m,p|t,p|T, ...... p a m4_t, ......... Bristol... Lyv 5 . H44 87 . 5. 5. T_, MfT.piM,p|t,p|T,pfmJt J ...p a m4.t, ......... Verona... Lyl8 6 . 5. 5. t,MfT.P|M, ............. ..p4.mt a , ...... ; ....... S 123 . L263 1 . H43 82 . 5. 5. t, M|T.p}fM,p|t,.. .Etna... p_ a _m.lt a ,...D20] 1 . L263 5 . Lyv 2 . H44 86 . 5. 5. t,MfT.pijM,p|t,pJmft,...p4.mt a , ............... S 126 .L263 6 . H44 88 . 3. SULPHATE OF LEAD. Bleivitriol. Ploinb sulfate. Anglesite. Cleavage = pjt. 3. 5. p^M^fT.plM^fT^lMfT, ........................... L249>. H97 98 . 3. 5. p + ,M 5 MJT.p|M,p>,pfT,p4MfT, .................. L249 8 . H97". 5. 3. MfT. P|T, ...... Anglesea... Model 82 b , with p a and t a reversed, ... L249 1 . H96 89 . 5. 3. MfT.pfT, .......... ...................................... J 124 . S 21 . H96 92 . 5. 5. M^MfT.P|T, ............... Model 104. A 53 . J 125 . S 22 . L249 2 . H96 93 . 5. 5. M,MfT.pfM,PfT, .................. Anglesea ...... S 23 . L249 3 . H96 94 . 5. 5. M,M|T.p|T,p|MfT 5 .. .............. . ................... L249 4 . H96 95 . 5. 5. M, MfT. P|M, P|T, P|M| T, ................................. S 24 . H96 86 . 5. 5. ^MfT.pl^PfMJT^ImJt, ....... ............................. H96 97 . MINERALS OF THE PRISMATIC SYSTEM. 71 33. ANHYDRITE. Anhydrous Gypsum. Chaux sulfatee anhydre. Cube spar. Cleavage = p, 25 M 9 , T 10 , pt. 1. 2. Pfg,M l a J ,T,...Mont Blanc... Md. 5. MH63 1 . LyH 1 . L267 1 . H32 22 . 3. 2. P|,M T 9 n ,T.riT,...Bex...S 26 . Mii63 3 . L267 2 . Ly v >. P176. H32 23 . 3. 2. Pj&M^T.pjK/r,, ....................................... Lyl4 2 . L267 3 . 3. 2. Pif,M^,T.p x M y T z ,p + m y t z , ....................................... Lyl4 3 . 3. 2. P|,M, 9 n ,T.3p x m y t z , ...... Steyermark...Mii 26 . S 27 . L267*. H32 24 . 5. 5. M,T;m,tPiT, ................................................... L267 5 . 33. MURIATE OF COPPER. Atakamit. Salzsaures Kupfer. Cleavage = T,pjm. 5. 3. MfT.PjT, Chili JU344 1 . Ti621. L242 2 . Ly62 2 . 5. 5. T,MfT.PfT, Chili J ii 344 8 . L242 1 . Ly62 3 . 5. 5. T,Mf T, mft, mf t. P jT, p" x m y t z , Chili Ly62 4 . 5. 5. m,T,Mf T, m|t. P jT, p_m y t z , P X M } .T Z , p + m y t z , P326. 34. WAVELLITE. Alumine hydro-phosphatee. Devonite. Cleavage = T, Mf T. 5. 3. MgT.PfM, Ji389'. 5. 5. T,M|T.PfM, J1389 2 . L134 1 . D188. 5. 5. t,Mf T, mft, m&. PfM, L134 2 . P157 2 . 5. 5. t,MgT,mft.PjM, Ly24 2 . - 35. An Isomorphous Group of Copper Ores, 1, 2: 1. OLIVENITE. Right Prismatic Arseniate of Copper. Olivenerz. Cuivre arseniate octaedre aigu. Cleavage = P|T. 5. 3. M-^T.jPfT, plm a t4., Sim. Model 82 a . Hl02 150 .iii 510. 5. 3. M^T.pfT, p4.mlt a , ...Sim. Md. 82. Ly65 2 . Hl02 15l .iii510. 5. 5. M_,t,M T 9 T.p x m, PfT, Cornwall P332 2 . 2. LIBETHENITE. Cuivre phosphate. Oktaedrisches phosphorsaures Kupfer. Cleavage = P|M. 3. 3. p,M T VT.PfT, L143 4 . 5. 3. M^T.PfT, Ly62 2 . Mii 2 . L143 1 . 5. 3. Mj^T. Pf T, p x m y t z , Mii 5 . D24 2 . L143 2 . 5. 5. t,M^T.P|T,p x m y t z , Libethen. Cornwall Ly62 3 . P327 2 . 5. 5. m,M^T.PfT,p x m y t z , L143 3 . 36. EUCHROITE. Cleavage = M^T. 3. 3. P f MJT,mJt.P||T, Libethen L174 2 . Mm95 l . 3. 5. P,t,M5t,mAt,M?T.P|4T, ... Libethen... S 174 . L174 1 . Mii 192 . P333. 38 1 . HAIDINGERITE. Cleavage = T. 5. 5. M,T,MT. P + M, PJT, .................................... D190. P181. 72 MINERALS OF THE PRISMATIC SYSTEM. 38. SILICEOUS OXIDE OF ZINC. Kieselzinkerz. Electric Calamine. Zinc oxide silicifere. Galmei. Cleavage = MJT. 3. 5. p,T_, M|T. P X M, p x m, 3p x t, Aix-la-Chapelle Ly73 6 . 5. 5. T_,MfT.PfM, Siberia Ly73 2 . H113 270 . 5. 5. T_,Mf T .piT, Jii438 2 . H113 271 . 5. 5. T_, M fT.pfM,plT, S 165 . P374. 5. 5. T_,M|T. P X M, 2p x t, Bleiberg, Carinthia Ly73 3 . 5. 5. T_,M|T. 2p x m,p x t, Carinthia Ly73*. 5. 5. T_,M|T.P X M, p x m, P X T, p x t, Matlock. Bleiberg Ly73 5 . 39. PICROSMINE. Pikrosmin. Silicate of Magnesia. Cleavage = M,T, mjt. pfm. 5. 5. M,T>iT.PfM, Til72. P94. 40. MASCAGNINE. Ammoniaque sulfatee. 5. 5. T_,M x T.p x m,p x t? 41. BROCHANTITE. Cleavage = pft. 5. 5. M,t,M|T.P T 4 5 M,p|T, P324. 5. 5. M,MfT.P T * 5 M,pft, L724. D245. 43. An Isomorphous Group of Sulphates, 1, 2: 1. SULPHATE OF MAGNESIA. Bittersalz. Magnesie sulfatee. Cleavage = mi 9 ^)*' [The following symbols relate to artificial crystals.] 4. 3. M$&T. P^&M^&T, H45 97 4. 5. M T M-^-T. P-3-Q-M-2-9-T H45 98 . 4. 5. M^'m-^tmirmftP^-M^T, H45". 4. 5. m,t,M^T. pjm, p^ P/^M^LT, H45 1(W . 2. SULPHATE OF ZINC. Zincvitriol. Zinc sulfate. White vitriol. Cleavage = T, m$&t. 4. 5. T.M^jjyT.P^M^T, D179.L110 2 . P376. 4. 3. HMrT.P^MAVT. . ...T1547. Jiii21 ! . L110 1 . 43. NEEDLE ORE. Nadelerz. Acicular Bismuth -Glance. 5. 3. MiT.p x m,p x t? 5. 5. t,MJT.p x m,p x t? 44. BOURNONITE. Triple Sulphuret. Endellione. Cleavage = M, T, mJ-|t. 1. 5. P_, Mjf,T, m||t, L613 2 . P353 1 . 1. 5. (P x , M-f|,T, mfjt) X 2, P353 3 . 3. 2. P_,Mif,T.pf|m,p||t, Kapnik L6l3 4 .Ly51 2 . 3. 2. P_,Ml|,T.p|im 5 p||t, Ly51 3 . 3. 2. P, M^,T.p||m, L613 3 . P353 2 . MINERALS OP THE PRISMATIC SYSTEM. / 3 3, 2. P_, M ft , T. p||m, pfft, p x mJ-|t, ......... Kapnik ...... L613. Ly51 7 . 3. 2. (P,M^,T.p|m,pfft,p x mlt) X 2, ........................ Ly51 8 . 3. 5. P-, IVll|,T,Ml|T.p||m )P |lt 5 ........................... L613 5 . Ly51 6 . 3. 5. P-,Mlt,T,Ml| T .p||m,p|4t, Px m^t,p + m^t, ... L613 9 . Ly51 9 . 3. 5. P_, M}|,T, rajft, m_t. p||m, pf ft, p mjft, p+mj^t, ......... L51 10 . 3. 5. P_,M{|,T, mjft, m_t, M=T. P||M, p| j t,p x m jf t, p + mft...Ly51 u . 3. 5. P_, MfJ,T, mjft, m_t, MZT. p||m, p_m, pfft, pjn^t, p+m^ft, Ly51 12 . 3. 5. P_, Mlf, T, MfJT, m_t, mzt, m+t. p + m, P||M, 3p_m, P|f T, p_t, 8 p x m y t E , ...... (Imaginary Combination) ...... P35 3 5 . 5. 2.'P_.p|jM,pjT, .............................. Kapnik ............... Ly51 4 . 5. 5. P_.P||M,P||T J P X MJ|T, .................. Hartz ...... L613 8 . Ly51 5 . 45. TOPAZ. Topas. Topaze. Cleavage = P,mift,mift.p|M,p||T. 3. 3 p + , MJfT, MifT. pif t, pf f m Jft, pf f rajf *> ............ S 237 . H50. 3. 3. p + ,Mi-|T,M;-|T.p|fT,plfmift, ........................... J". H50 13i . 3. 3. p + ,Mi|T,nif|t,mi|t.p|-8T 5 pffm^|t, ............... P77 3 . H50 143 . 3. 3. p 3. 3. p 3. 3. P 3. 3. P+, Mi JT, Mi| T. P j J T, pff rni&t, pHmift, .................. H50 1 ". 3. 3. P + ,Mf|T J M||T,mi9t.p|8 T ,p|5T,pf|m^t,pffmf|t 5 piimf|t, S 411 . H50 150 . Mr6 87 . 3. 5. P + ,T,Mi|T.pM,pffm-Jft 5 ........................... L398 7 . H50 137 . 3. 5. p^^Mi^MigT.Pie^plfm^t, ........................ H50' 39 . 3. 5. P+,M,T, mif t, mif t. P||T, ....................................... H50 MI . 4.3. M^T/MifT.pliT.pifM^p^migt, ........................ H49 134 . 4. 3. MiT,Mf|T.P|fMfT, ........................... J 50 . L698 3 . H49 135 . 4. 3. Mf|T,M;-|T.Pf|Mi|T, ....................................... A 120 . R 87 . 4. 3. Mfl^MitT.PH^pllm^pf-gm^plJmift, ......... H50- 3 . 4. 3. MiT,Mj|T.pgm,p||T,pJfmJft,pJilmJ|t,p^mJgt,J H51 147 . 4. 3. M^T,M-JT.pIft,pffMi|T, pjjniift, ........................ Mii. 5. 3. M^fT, M!T. P|T, p||m^-t, ... Md. 90. J 53 . S 439 . L398 2 . H49 136 . 5. 3. MiT 5 MiT. P |m,PifT 5 pl|m^t 5 ........................... H49 133 . S. 5. M_, Mi9T.p^t,p|gT, .......................................... H49 132 . 5. 5. T The Atlas to Levy's Catalogue contains 82 figures of Topaz Crystals, all different from the above, but most of them apparently consisting of combinations of the same forms. 46. AMBLYGONITE. Cleavage = M|T. 1. 3. P,M|T, ...................................................... P158. L283. k 74 MINERALS OF THE PRISMATIC SYSTEM. 4?. CHIASTOLITE. Chiastolith. Made. Hohlspath. Cleavage p,m,t. 1. 3. P.MT, ...................................................... J" 7 - H64 6 '. 48. CHRYSOBERYL,. Cymophane. Prismatic Corundum. Cleavage = M,T. 4. 5. M,T, M/o T. P x M y T z , .................................... L540 7 . H60 3: 4. 5. M,T_,m_t. p x M y T z , ................................................... Ly27 3 5. 2. M,T.PM, ...... Sim. Model 8. Haddam...L540 l . J 28 . S 132 . H60 31 5. 2. M,T_.p x M, p x M/r z , p + m y t z , ....................................... Ly28 5 5. 5. M,T, M-&T. P^M, P x M y T z , .................. L540 3 . J 29 . S 133 . H60 33 5. 5. M,T,M/ n T,MiT.PM,P x M y T z , ........ : ..................... H60* 4 S. 5. M,T,M-/oT.pfm,p x M y T z ,p x m y t 2 , ........................ L540 5 . H6(P 5. 5. M,T,M/ T.PM, ................................................ L540 2 5. 5. M,T,m T 8 7 t,mift, MffT, mfft.p^t,2P x M y T z , .................. P80 1 ' 5. 5. M,T_, m + t.p x M, P X M X T Z , p + m y t z , ........................... ...... Ly28'' 5. 5. M,T_, 2m x t. p x m, P X T, p x m y t z , .................................... Ly28' J 5. 5. M,T_,m_t.p x T, ...................................................... Ly27 2 5. 5. m,T_, m_t. p x m, P.M^T,, ............................................. Ly27 4 5. 5. M,T , m_t. P X M, p x M y T z , p + m y t z .................................. Ly28 7 ' 8 242 49. LIEVRITE. Ilvaite. Per calcareo-siliceux. Cleavage = t, m|jt.pm. 3. 3. P + ,M|T J mft.pfm,pft, P Jm9t, ...... P 4.mlt a , ...... L529 6 . H110 3. 3. p,Mf T, MJT, M|T. P|M, p|t, P^Mf T, ........................... Ly69 7 . 4. 3. M|T.pfMT, ...... p-JLmStg, ...J in 539'. MH415 1 . L528 2 . H110 238 . 5. 3. MJT.PJM, ......... p|m a 6 t^, ........................... Elba ...... HHO 234 . 5. 3. M|T.pfM, ......... p+m^, ............... Mii415 2 . L528 1 . H110 237 . 5. 3. MfT.pfT, ......... p4-m|tS, ........................... L528 3 . H110 239 . 5. 3. M|T.p|T,p|mft, p^LmgtS, ........................... L528 4 . H110 240 . 5. 3. M|T,mt.pfm,p| T ,p*mt, ...... p4_mt a , ............ L529 5 . HHO 241 . 5. 3. M|T.p|t,p|MT, ...p4in5tS, ...... S 489 . D379- Mii 4 . Ly69 3 . Rose 88 . 5. 3. MfT,mt.p|M|T,...p^m a t, .................................... Ly69' 2 . 5. 3. MfT,m|t.p|m,p|m|t 5 .......................................... Ly69 4 . 5. 3. MST,M| T ,Mg T .pfM,P|M6T, .................................... Ly69 5 . 6. 3. M6T,M^ T ,M^T.pM,pft,P|M5T, .............................. Ly69 t; . 50. ALLANITE. Prismatic Cerium Ore. Cleavage = m, ^m 2 t. 5. 5. M,iM 2 T,im|t.ip_m,4(ip x m y t z ) P264. D365. A 133 . M ii 51. DICHROITE. lolite. Cordierite. Peliom. Cleavage = m,T, m 2 tif , M|f T 2 . 1. 5. P_,T, M||T 2 , Model 7. L467 1 . H 1. 5. P_,m,T, m 2 t}f , M|JT Model 10. L467 2 . H76 3. 5. P_,m,T,m 2 t|f,Mif T 2 .p|it,piJmift 2) L467 3 . P42. H76 194 . Ly38 2 . 3. 5. P_,m,T,m 2 tJf, M}|T 2 .3(p x t,p x mJ-ft 2 ) Ly38 3 . / u . 193 MINERALS OF THE PRISMATIC SYSTEM. 75 52. SPODUMEN. Triphane. Cleavage = T,Mf|T. 1. 5, P,T, M}T? 53. SCORODITE, Skorodit. Cleavage = m,t, m^-J-t. 4. 2. M,T, P X MJT, .............................. Cornwall ............ Ly7l 2 . 4. 5. M,T,m| 7 -t.P x MJJT, ........................ Peru. Saxony ...... Ly7l 3 . 4.5. M,T,miJt.p + m,P x M^T, ............... Saxony ............ Ly7l 4 . 4. 5. M,T, mjjt, mft. P X M J JT, P_M \ JT, p+mt_ .......... Brazil. . . Ly7 1 5 . 54. PREHNITE. Cleavage = P,mft. 1. 3. P,MT, ... Ratschinges. Tyrol... L471 1 . MH218 1 . Ly v 1. H75 187 . 1. 3. Pz,M|T, .................. (Koupholite) ...... J 70 . A 83 . P23 1 . Hii606 a . 1. 5. Pz,T,MfT, ............... Dumbarton ...... L471 2 .J 72 . Lyv2. H75 189 . 1. 5. Pz,m,T,mft, .............................. L471 3 . J 71 .. Mil 218 s . H75 191 . 3. 5. P,MT.P X M, ....................................... Fahlun ...... L471 5 . 3. 5. P_,MfT. PJT, ........................... Fahlun ...... L471*. H75 188 . 3. 5. P+,MT. PyT, ........................... Fahlun ...... L471 4 . H75 190 . 3. 5. P,MfT.P x M,P x T, ................................. Fahlun ...... L471 6 . 3. 5. P,T,MfT,pUT, ....................................... S 339 . Mil 13 . D286. 3.5. P,m, T, Mf T. p + m, p x m, p x m y t z , ............ Ratschinges ...... Ly37 2 . 5. 5. m,t,Mf T. P^M, 2p + m, p + t, ................................. S 340 . P23 2 . 55. PYROPHYLLITE. Cleavage = P,M_,T? 56. HARMOTOME. Cross Stone. Kreuzstein. Two varieties: 1. POTASH HARMOTOME. Kalikreuzstein. 2. BARYTES HARMOTOME. Barytkreuzstein. Cleavage = M,T. py^m. 3. 2. P +> M_,T.p/ -M 5 p x m y t z , ............................................. Mii 22 . 4. 2. M_,T.P x M y T z , ............ Oberstein ...... S 245 . P44 6 . H83 271 . Ly43 2 . 4. 2. (M_,T.P x M y T 2 ) x 2, ... Andreasberg...J 84 . P44 5 . H83 272 . Ly43 3 . 4. 5. m_,t.P x M y T z , ......................................................... P44 7 . 5. 2. M_,T.P T 7 oM, ......................................................... P44 2 . 5. 2. M^T.p/^m^M/r,, ................................................ P44 4 . 5. 2. M_,T. P T 7 -M, p x ra y t,, ..................... Oberstein ...... P44 3 . Ly43 4 . 5. 2. (M_,T. P T 7 oM, p x m y t z ) X 2, ............... Andreasberg ...... Ly43 5 . 5. 2. (M.jT.p/oMjp^m, p x m y t z ) X 2, ...... Strontian...M ii 40 . Ly43 6 . 5. 5. M_,T. P T 7 oM, P X M V T Z , ............... Strontian ...... Hiii 146". Ly43 7 . 5?. THOMSONITE. Cleavage = M } T. 3. 5. P+,m,M^ T - rM > ................................................ Ly45 2 . 3. 5. P + ,m,T,M{-gfT.PM, ............................................. Ly45 2 . 3. 5. P + ,M,T, MigaT.pm, ......... Brooke, Ann. Phil.xvi. 104. T1314. 3. 5. P + ,M,T, MJgfT, m_t, ra + t.pm, pt,pmt, ............... L208. P125. 58. DESMINE. Stilbite. Radiated Zeolite. C Si s +3 ASi 3 +6 Aq. Cleavage = m,T. 1. 2. Pf, Mf,T, ............................................................ H84 278 . 76 MINERALS OF THE PRISMATIC SYSTEM. 3. 2. p + ,M,T. P^Mf- T, ...... Sim. Model 43. D 268 . Lyv3. H84 289 . M ii lo . 3. 5. p+,M,T, mjjjt. Pf Mf T, ............................................. P24 3 . 4. 2. M,T. P^MfT, ............... Ly vl. HS4 279 . Mii239 l . A 96 . Rose 95 . 4. 5. M, T, mlJt. P^Mf T, ................................................ Ly43 2 . 59. EPISTILBITE. Cleavage = T. 5. 3. M&f.P^M.p^p^m^ .......................................... Ly44 2 . 5. 5. t 60. POLYHALLITE, Polyhalit. 1. 5. P,T_,MftT, ............................................................ P204. 61. CALEDONITE. Cupreous Sulphato-carbonate of Lead. Cleavage = T, m\\i. 3. 5. P_ 5 T,M^T.p 3 M,p_m,pjt, 3p x m{^t, ......... Lead Hills... P360 2 . S 122 . OS. WHITE TELLURIUM. Weisstellurerz. Cleavage = m^t. 3. 5. P_,m,t, mlt. P 3 M, P}T, p x m y t z , ........................... L688. P342. 63. SCHILFGLASERZ. Sulphuret of Silver and Antimony. 5. 3. Mjit, 3m_t. p x m, SP X T, ......... Cleavage = mft ................ P299- 64:. FLUELLITE. 5. 3. p + .P + M_T, ................................................ MiiilOl. P76. 65. POLYMIGNITE. 4. 5. M,T + , 3m x t. P_MT + . p_|_m.Lt a , ........................... D369. P262. 66. BROOKITE. Cleavage = t. 4t. 5. M_, M^T. p+t, p x M y T 2> ............................................. P256. 5. 5. M_, t, Mf T. 2p x m, p + t, 3p x m y t z , ........................... L725. Mii 190 . 67. LENTICULAR COPPER ORE. Linsenerz. Cuivre arseniate octaedral. Cleavage = m T 7 ^t. pjjm. 5. 3. M&T.PliM, ....................................... . ........ Ly65 2 . P329 1 . 5. 3. M, 7 2 T. piX 2p x m y t z , ................................................ P329 2 . 68. LAZULITE. Azurite. Cleavage = p. 5. 5. m,M||T.p^m,pjT,pft,pjM||T,pfmf|t, ......... P159 3 . Ly38 2 . 6O. CHILDRENITE. 4. 5. M, Mi^T. P_MT +) p+m_t, .......................................... Ly80 2 . 5. 3. p.p^fm,p_t,P!M^T, .................................... Pi58. D188. 7O. FORSTERITE. 3. 5. P,T,mJt.P 2 MiT, ................................................ A 132 . P87. MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 77 VI. SILLIMANITE. 1. 3. P,M*T, D320. 1. 3. P,MJT, P73. !?2. MENGITE. Mengit. Monazite ? 1. 3. P,M_T? R175. ?3. KOENIGITE. Konigine. Cleavage = P. 1. 3. PjMlaT, Levy, Ann. Phil, xxvii. 194 1 . 1. 5, P,M,M1T, idem. fig. 2 . 3.5. P,M,Mi!Tr.p + T, idem. fig. 3 . ff4L MONTICELLITE. 4. 5. T, M_T. P X T, P x M y T z , P403. 75. HERDERITE. Cleavage = m^t. 3. 5. P_,M T Vr.Pfr,pifm T 9 B t, P172. 76. HOPEITE. Cleavage = M. 3. 5. p+,M, T_,M^fT. P] 9 r t,p T 9 T m_t, P377 ' CLASS V. MINERALS BELONGING TO THE OBLIQUE PRISMATIC SYSTEM OF CRYSTALLISATION. The AXES of all Combinations belonging to this Class are = p x m y t*. The constituent FORMS of the Combinations of this Class are as follow : Zones. Homohedral Forms. Hemihedral Forms. Prismatic, M, M_T, M + T, T, pl_T, M + T. North, P, P_M, P + M, M, JP-M, JP+M. East, P, P_T, P+T, T, iP_T, 1P + T. Octahedral, P x M y T z , ^P x M y T z . The homohedral Forms of the prismatic zone, M, M_T, M+T,T, or some of them, occur upon almost every Combination in this Class. The homohedral Form P occurs very seldom. The hemihedral prismatic Forms ^M_T,^M + T, occur more frequently than would appear from the following list, for many Forms indicated as homohedral, consist of two pair of planes of unequal size. The homohedral Form P x M y T z occurs seldom, perhaps not at all. It is a Form that properly belongs to the Prismatic System, so that the examples of P x M y T z given in the following Table, should probably be rendered ^P x M y T z , |p x M y T z . The homohedral Forms P_M,P + M,P_T,P + T, are also of rare, per- haps of doubtful, occurrence. 78 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. The greater part of the Combinations of this Class fall into two divisions : a.) Those which have M X T, l P X M Zn and $ P x M y T z Zne Znw. b.) Those which have M X T, P X T Zw and P x M y T z Znw Zsw. These Combinations are commonly called oblique prisms, and it is said of those of the first division, that the terminal plane, namely, the Form JP X M Zn, is set on the obtuse lateral edge of the prism; and of the com- binations of the second division it is said, that the terminal plane, namely, the Form |P X T Zw, is set on the acute lateral edge of the prism. The hemihedral Forms |M_T, M + T, P_M, iP+M, |P_T, P+T, always consist of a pair of parallel planes. The hemihedral Form iP x M y T 2 , consists of two pair of parallel planes. Hence, half the planes of any given hemihedral Form are placed on different sides of any given meridian. 1. REALGAR. Arsenic sulfure rouge. Ruby Sulphur. Cleavage = m,t, Mf T. |PM Zn. 5. 5. m,T, MfT.-JPMZn,lp + M_ T Zn 2 wZn 2 e, ..................... Ly74 2 . 5. 5. M,T, MfT,M + T,MjT4PiMZn,^ P+ mZs, 3Qp x m + t_)Znw 2 Zne 2 , b+M_T Zn 2 w Zn 2 e, ............... Ly74 6 . 5. 5. M,T,Mf T, M + T, MjT. PM Zn, 2(ip+m) Zs, 3 (p x m + t_) Znw 2 Zne 2 , P+M_T Zn 2 w Zn 2 e, ......... Ly74 7 , 5. 5. t, Mf T, m+t, nrft. PM Zn, p x ro + t_ Znw 2 Zne 2 , ............ Ly74 3 5. 5. T, Mf T, M + T. P|M Zn, p + m Zs, |P X M + T_ Znw 2 Zne 2 , ...Ly74*. 5. 5. T,Mf T, m + t, mjt. iP^M Zn, ip + m Zs, |P X M + T_ Znw 2 Zne 2 , Ly74 5 . 9. PLAGIONITE. Cleavage = m^t. 5. 3. MfT.iP_MZn,ip + mZs,p x m y t z , ..................... P346. Ti580. 3. MYARGYRITE. Hemi-prismatic Ruby- Blende. Ag Sb 2 S 4 . Cleavage = t. |p x m y t z Zirw. 6. 3. Mff T.^T Zw, Jp|Jt Ze, ................................. D 97 . M ii I83 . 4. RED ANTIMONY. Rothantimonerz. Antimoine oxide sulfure. Antimonblende. Rothspiesglaserz. Cleavage = m,T. 5. 5. T, M_T. p x m y t z Zn 2 e Zs 2 e, |PT Zw, pf t Ze. Axes : p4_mlt a . L608. 5. TUNGSTATE OF IRON. Wolfram. Scheelin ferrugine. Tungstene, Cleavage = m,T. 5. 2. M,T.P$MZn, H118 324 ? Ly7i 6. 2. M_,T.^MZn, p x m y t z ? H118 3 5. 3. MT.P 7 6 T,p x m y t z ? H118 325 . 5. 5. M_, M}T.JPiMZn,JpimZ8,^t, D373. Mii 49 . P326. ft. 5. M_,T,m 3 t.^MZn,p + m, p x m y t z ? H118 327 . 5. 5. M_, M?T,M^T.^MZn, )pim Zs, P 7 6 T, ^m.t, Zne Znw, ... Ly79". MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 79 ft. 5. M_, MJT, mft. ^P 2 M Zn, pirn Zs, P 6 7 T. |p x m y t z Zne Znw, ^p x m y t z ~ Zne 2 Znw 2 , ............ Ly79 7 . ft. 5. M_,t, M'T, mf t. P 2 M Zn, PJ'T. |,p x m y t z Zne Znw, ............... Ly79 4 . ft. 5. M_,t, M^T, m^t. ^P 2 M Zn, ip 2 m Zs, PJT, p x m y t z Zne Znw, ...Ly79 5 . 5. 5. M_,t, M?T, m?t. ^P 2 M Zn, IT>~M Z 2 s, 2 p 2 m Zs 2 , PJT, lp x m y t z Zne Znw, ^ Px m y t z Zne 2 Znw 2 , .................. Ly79 8 . [Measurements gathered from Phillips and Mohs.] 6. CHROMATE OF LEAD. Rothbleierz. Red Lead Ore. Plomb chromate. The prism M T has an obtuse angle of 93, Hauy = Mjf T; of 93 30', Levy, Phillips, Brooke = Mff-T; of 93 40', Mohs = M}|T. Cleavage = m,t, M-ff t. 4. 3. MgT. ^p x m Zn, iPftMgT Znw Zne, !p_m*t Z 2 sw Z 2 se, ^P + M_T Zs 2 w Zs 2 e, ......... D233 1 . ft. 3. MJT, mJSt.^P + M Zs^Pf 7 M! 7 6 T Zne Znw, ..................... Ly53 3 . ft. 3. Mj?T, mlJt.Jp + m Zn, JP, 4 7 M{JT Zne Znw, ................. . ...... Ly53*. 5. 3. M1T, mgt. \p? 7 m Zn, >P + M Zs, ^P JMgT Zne Znw, ............ Ly53 r> . ft. 3. M}?T, m\lt. ^P + M Zn, ? 2 P + M Zs ; ^pJailjT Zne Znw, ............ Ly53 r> . 5. 3. M}|T, mgt. ^P + M Zs, JPJMJJT Zne Znw, p + itZw,P|>M^TZnwZsw, ...(Augite) ...L500 7 . H66 92 . 5. 5. t. M 2 ?T. |pf,T Zw, Jpgt Ze, ^p + m x t ZnV Zs 2 w, ^pgm^t Z 2 ne Z 2 se, ^p+m 2 ?t ZnV Zs 2 e 2 , ...... (Fassaite) ...... L501 18 . H68 113 . 5. 5. t,M|T. 2 p 2 6 ,t Zw, 2 P 2 6 ,T Ze, ^m 2 ?t Z 2 nwZ 2 sw, iPXMJJT ZnV ZsV, biX?T Zne Zse, ......... Mii 73 . D306 3 . 5. 5. M^^^T.bfitZw^pe.TZe^pSm^tZ^wZ^w, iPif MgT Zn 2 w 2 Zs 2 w 2 , MM^T ZnV ZsV, ............ H68 176 . MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 81 5. 5. M,T, i 177 . 5. 5. M,T,M 2 ? T .P9T, (Hypersthene) L516 2 . H69 125 . 5. 5. m,T,i4?T.;P&TZw, (Sahlite) L500 3 . H66 93 . 5. 5. M,T,MST.iPgM8TZnwZ8w, Axes: p4_m a tl, Model 98. H67 95 . Lyv 1 . J 114 . Mil 72 . L500 8 . D306 1 . S. 5. (M,T,ngT.iP I 6 l Ml?T Znvv Zsw) X 2, Model 99. Lyv 2 . H67 98 . tf. 5. m^i^r.ipjMifr^lPSMJJTNneNse, H67". J 116 . 5. 5. m,T^Cr.JpgtZw,iP8Mj}TZBwZiw ) R 103 . Lyv 3 . H67 105 . 5. 5. m,T l mSt,m/ 1 t.JPgTZw, (Sahlite) L500 5 . H68 106 . 5. 5. m,T,mlit,m 2 7 l t.|P 2 6 I TZw, Jpf,m^t ZnwZsw...(Diopside) H68 108 . 5. 5. m^X^P^M^TZnwZsw^p'XttZnwZsw, J 116 . Lyv 8 . L501 20 . H68 109 . 5.5. M,T,m|?t. X 6 iT Ze, IP^M^ Znw 2 Zsw 2 , jgp x M y T z ZnV ZsV, L500 18 . H68 111 . 5. 5. M^XjT.^TZwj^tZe^P^M^TZnwZsWjKmfitZneZse, L500 13 . Lyv 16 . H68 1 ". 5. 5. M,T,M|?T.JpgtZw, ^.TZe^pHm^tZ^wZ^w^P^M^TZnVZsV 2 , MM|?T Zne Zse L500 10 . H68 114 - 115 . 5. 5. M,T,M|?T. ipfitZw, ^TZe^P.^M^TZnVZsV^^m^tZ^wZ^w, Jpifmjt Z 2 neZ 2 se, 1 p+m + tZne 2 Zse 2 ...(Alalite)...L500 u . H69 118 . 5. 5. m,T, M|?T. P 2 6 ,MT Znw Zsw,... Axes: p.m a t4_, Lyv 1 . H67 9S . 5. 5. M,T, M 2 ?T. iP|[Mi?T Znw Zsw, L500' 5 . H67 109 . 5. 5. M,T, M 2 ?T. JPgT Zw,^p 2 6 lT Ze, Lyv 6 . H67 103 . 5. 5. M,T,M 2 2 ?T^Pf 1 TZw,^M^TZneZse...(Baikalite) L501 19 . H67 104 . 5. 5. m,t,MT.iPT, (Diallage) H70 128 . Levy gives figures of many other varieties of Augite, and some of them of very compli- cated combinations. It is, however, impossible to represent them in symbols, owing to the want of measurements. They do not differ in character from the combinations described above. Ha'uy's measurements are very numerous, but many of them are useless for my purpose, while several angles necessary for completing the above symbols are unfortunately omitted. 10. WAGNERITE. Fluophosphate of Magnesia. Cleavage = t, M-^T. 5. 3. Mj^T. IP^MZn, (assumed primitive) P186. 5. 5. m^n^t, M T 7 n T,m}at.lPiMZn,AP_M x TZ 2 neZ 2 nw,lp + ui x t Zne 2 Znw 2 , |p x m y t z Z' ? se Z 2 sw, |p x M y T z Zs 2 e ZsV, 2(lp + m x t) Zse 2 Zsw 2 , Mil 19 . 11. LITHIA MICA. Lithionglimmer. Lepidolite. 5. 5. t, m|t.iP/ 7 MZn,...Axes: pm.|_t a , ? IS. MALACHITE. Green Carbonate of Copper. Fibrous Malachite. Cuivre carbonate verte. Cleavage = mjt. |PM Zn. 5. 5. m, MfT.iPpIZn, L155 1 . . 5. (m, Mf T. iP^M Zn) x 2, Ly62 2 . Mii 78 . I 82 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 5. 5. m, MT. PiM Zn, Jp x mZs, .................................... L155 2 . ft. 5. m,MT.PiMZn, p_m x tZne 2 Znw 2 , ........................ L155 3 . 13. CARBONATE OF SODA. Soda. Cleavage = T, mf t. ft. 5. M,MfT4pfMfTZnwZsw,...p4.mt% ........................... H54 182 . 5. 5. M, t, M|T. pf t Zw, sPf M| T Znw Zsw, ............... Artificial crystal. 14. TRONA. Sesqui- carbonate of Soda. Urao. Cleavage = JMJT. ft. 3. ^MJTnw, ^MfTne.^P_TZw,iP + TZe,...p4Lmt a , ...... PJ97. A 36 . 15. PHOSPHATE OF COPPER. Phosphorsaures Kupferoxyd von Rheinbreitenbach. Hydrous Phosphate of Copper. Cleavage = t, |p_t. ft. 3. M T 6 7 T. 2 PJTZ 2 w, 2 p + tZw 2 , ....................................... Ly62 8 . 5. 5. t,M T 6 7 T. 2 P_TZ 2 w,p + tZw 2 , .................................... Ly62 3 . ft. 5. ^M^T.aP.TZV^P+MJTZnwSZsw 2 , ..................... Ly62 5 . 5. 5. t,M^ r T.^P_TZ 2 w^P + TZw 2 ,|p + m_tZnw 2 Zsw 2 ? ......... Ly62 6 . 5. 5. m,t,M T 6 T T.^P_TZV,^P + M_TZnw 2 Zsw 2 , .................. Ly62 4 . [P + M_TZnw a on Zsw 8 = 117 50', and on t = 123 30'. Levy.] 16. OBLIQUE PRISMATIC ARSENIATE OF COPPER. Cleavage = P T 2 T T. 5. 3. M^-T-iP^-TZw, ...... (assumed primitive) ......... P331 1 . Ly65 ! . . 3. M r ^T4PATZ 2 w,|PVTZe, ....................................... A 83 . 5. 5. M&T.^PTZV,^ T tZw 2 ,PyTZe, ..................... Ly65 2 . 1?. An Isomorphous Group, 1, 2: 1. VIVIANITE. Phosphate of Iron. Per phosphate. Blue Iron Ore. Cleavage = T. ft. 5. m,M T 8 T T. 2 P + MZn,'P + MZs, Ly70 2 . ft. 5. M, T,M^ T T,m + t4P_MZn, Ly70*. ft. 5. M, T, M^ T, m+t. 2 p_M Zn, 2 p_m x t Zne 2 Znw 2 , 2 P_M X T Zse Zsw, Ly70 6 . ft. 5. m, t, M T 8 T T. 2p_m Zn, P_M X T Z 2 ne Z 2 n w, 2 p + m x t Zn 2 e Zn 2 w, 2 P_M X T Z 2 se Z 2 sw, 2 p+m x t Zs 2 e Zs 2 w, Ly70 7 . 2. COBALT BLOOM. Kobaltbliithe. Cobalt arseniate. Arseniate of Cobalt. Red Cobalt. Cleavage = M,t. ft. 2. M,T4P|TZw, L161 1 . ft. 5. M,T,m^t. 2 P|TZw, L162 2 . ft. 5. M,t,M 2 T,m + t. 2 PTZw,...Axes: p4.m^t a ,,..L162 3 . P289. Ly73. MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 83 18. HURAULITE. Phosphate of Iron and Manganese. 5. 3. MfT4P + M x TZneZnw, Cleavage = 0, T1518. P247. 1 9. HETEROSIDERITE ? 20. PHARMACOLITE. Chaux arseniatee. Arseniate of Lime. Cleavage = T. 5. 5. M,T,m/ot.ip_M x TZneZnw, S 334 . 5. 5. M,T,m/ot.Pjyf x TZ 2 neZ 2 nw,|p + m x tZn 2 eZn 2 w, P181. 5. 5. M,T,mVflt.3(}PM y T I )ZneZnw, S 335 . 21. STRAHLERZ? 22. TINCAL. Tinkal. Borax. Soude boratee. Cleavage = m, t, Mj-f T. 5. 5. T, M|f T. ^P T 5 7 T Zw, ^p_m x t Z 2 nw Z 2 sw, lp+m x t Zn 2 w ZsV, P_M X T Zne Zse, Ly25 2 . 5. 5, ra,T,M}fT4P 1 ^TZw,^p_m x tZ 2 nwZ 2 sw^p + m x tZn 2 wZs 2 w, p_m x t Zne Zse, S 72 . P199. 23. GLAUBER'S SALT. Glaubersalz. Soude sulfatee. Cleavage == t (Leonhard), = |P r 4 jM (Levy). 5. 3. MiiT.|P T 4 7 M, (primary) P198. Lyi328. 5. 5. m,T, mif t/2 (Jp^m) Zn, 2 (Jp,m) Zs, 2 (Jp t) Zw, 2 gp,t) Ze, 2(^p x m y t z ) Znw Zsw, ^p x m y t z Zne Zse, Mii 58 . D173. 24. GYPSUM. Gyps. Selenite. Hydrous Sulphate of Lime. Chaux sulfatee. Fraueneis. Cleavage = m, T, |pf m Zn. 5. 2. M_,T. |Pf M Zn, (Haiiy's Primitive, but with p a and t a changed.) Wolfach, Model 79. L119 1 . H29 1 . 5. 5. M,t, Mj^T. |P_M+T Znw Zsw, iPM + T_ Zne Zse, H30 9 . 5. 5. m,T_, m T 9 5 t. ipy^m^t Zne Znw, |p T 6 3m|t Zse Zsw, H30 11 . 5. 5. T_, M^ 7 T. IP^M^T Zne Znw, |P T 6 3M| T Zse Zsw, (Hatiy's equivalente) Model 75. H30 6 . R". Ly v 3 . 5. 5. 'T_, M T 9 ^T. iP T ^M^ 5 T Zne Znw, p4.m a t,...(trapezienne elargie) Montmartre, Model 115 R 100 . Lyl4 2 . H29 3 . 5. 5. T_, M r 9 ? T. 1P M & T Z 2 ne Z 2 nw, |p+m x t Zne 2 Znw 2 , H30 5 . 5. 5. T_,M T %T.iP T %M T %TZneZnw,pfm4.t a , (trapezienne)Lyv 1 .H29 2 . 5. 5. (T_, M^T. iP/^M^T Zne Znw) x 2,...(Heinitrope)...Hi535- 5. 5. T_, MifT.|P T ^M^T,ZneZnw,...pm4.t a , H29 4 . 5. 5. T_,Mf|T. |p T 5 ^m T 9 3 t Zne Znw, Jp^mfjt Zse Zsw, H30 7 . 5. 5. T^m^M-JfT.iP^M^TZneZnw, H30 8 . 5. 5. T_, M T %T, mf Jt. ^P X M Zs, ^P T 5 ^M T %T Zne Zuw, ... Bex...H31 42 . . 5. T^M-^Tjmiftjmfft.iP.MZs, IP/^M-^TZneZnw, Lyv 5 .H31 13 . 84 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 25. SULPHATE OF IRON. Eisen vitriol. Fer sulfate. Green Vitriol. Cleavage = mf t. \ p/ot, |p + m x t Znw Zsw. 5. 3. MfT.iP2 5 oTZ 2 w,iPiTZw 2 ,iPiTZe,...p4_m a 6 t?. 5. 3. MfT.iP/oTZXIpxtZw, |p-lg-tZw 2 ,ip|tZe,...p4.ra^. 3. 3. MfT.iP/oTZV,iP-fg-TZw 2 ,iP;-gTZe,3(ipXt z )Zsw, ^p x m y t z Znw, ...... p+m^t 5. 3. Mf T. iP/o T ZV, iPfJT Zw 2 , iP||T Ze, 3(> x m y t z ) Znw Zsw, . 3. Mf T. *P^T Z 2 w, |P-}T Zw 2 , *PjjT Ze, 3(ip x ra y t z ) Znw, 3(ip x m y t z ) Zsw, ...... p^mSty. A great variety of factitious Crystals answer to the last two formulae, the planes of the 3 Forms |p x niy t z varying considerably in size on the north and south sides of the crystal. 26. BARYTO-CALCITE. Barytocalcit. Cleavage = JPM Zs, IPfMfT ZneZnw. 5. 3. MlfT.iPjMZn, ................................................... Mii 44 . . 3. M|f T. |PfM Zn, ipm Zs, |pfM| T Zne Znw, ... D202. A 54 . Pi 89- 5. 5. t, M|$T, mj Jt. |P|M Zn, |P_M Zn, |pm Zs, iPf M|T Z 2 ne Z 2 nw, lp + m x t Zne 2 Znw 2 ,.... ..... Mii 188 . S 53 . 2!f. AZURE COPPER ORE. Kupferlasur. Cuivre carbonate bleu. Cleavage = m, T, Mf T. ip x m y t z Zne 2 Znw 2 . 5. 3. Mf T. JP&M Zn, ip + m_t Zne Znw, ............... L152 1 . H101 140 . . 3. Mf T. IP/^M Zn, ^P + M_T Zn 2 e Zn 2 w, ip_m + tZne 3 Znw 2 , L152 2 . H101 141 . 5. 3. MfT.|P T ^MZn,|p + m_tZneZnw, ^p + M_TZseZsw,Hl01 143 .Ll52 4 . . 5. m, MfT.iP^MZn, |p + m_t Zne Znw, ............ L152 5 . H101 U4 . 5. 5. m, Mf T. |P T \M Zn, |P + M_T Zn 2 e Zn 2 w |p x m y t z Zne 2 Znw 2 , Model 103. L152 6 . HlOl 145 . 6. 5. M, Mf T. IP/^M Z 2 n, ip^m Zn 2 , ip + m_t Zn 2 e Zn 2 w, ip_ra + t Zne 2 Znw 2 , lp + m_t Zse Zsw, ......... H102 146 . 5. 5. T, Mf T. iP T ^M Zn, ip_m + t Zne 2 Znw 2 , ............ L152 3 . HlOl 143 . 28. TRIPHYLINE. 1. 3. P,M|T? ......... Cleavage = P,m|t, .............................. D219- 29. VAUQUELINITE. Chromate of Lead and Copper. 5. 5. m,M_T.|P|MZn, ................................................ Ly52 2 . 5. 5. (m, M_T. |P^M Zn) x 2, ..................... P369.D234.Ly52 3 . 3O. TITANITE. Sphene. Titane calcareo-siliceux. Menakerz. Cleavage = MfT.iP^MZn. 5. 3. MfT.PfM, ......................................................... H118 321 . 5. 3. Mf T. iP^ 5 M Zn, |Pf T Zs, ...Sim. Model 81. Mii '. H118 320 . J 168 . 5. 3. MfT.PV 5 M,p + m x t, ............................................. H118 322 . 5. 3. MfT. iP^MZn, IrfuZs, ^P+M X T Zne 2 Znw 2 , ...... P259 3 . Mii 47 . A 134 . D360 2 . MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 85 5. 3, MfT.lPk 5 MZ 2 n,ip+mZn 2 , ipfmZs, R">i. 5. 5. MfT,iMfTnesw.PfM,iP + M x TZneNsw, H118 323 . 31. EPIDOTE, comprehending four varieties : 1. ZOISITE. 2. PISTACITE. Pistazit. Thallite. 3. MANGAN-PIDOT. Epidote manganesifere. 4. BUCKLANDITE. Cleavage = M, P|M Zn. 5. 2. M_,T. P|MZn, ... Model 79 b . (Haiiy's primitive form, but in a different position,) H74 172 . P29 1 . 5. 5. M_.mMZn, iplimZs, iP+MT_Zne 2 Znw 2 , ModeMOl. Ly36 2 . H74 174 . 5. 5. M. ipim Zn, JPJJM Zs, |P + MT_ Zne Znw, J" 9 . H74 173 . 5. 5. M, m r 8 T t. iP|M Zn 2 , iP^M Z 2 n, 1PJJM Zs, 4(ip x m y t z ) ZneZnw, 3(ip x m y t.) Zse Zsw, M ii 77 . Mr 93 . 5. 5. M_, t, MifT. |PiM Zn, ipjm Zs, |P + MT_ Zne 2 Znw 2 ,... Mod. 101'. J 120 . Sim. H74 176 . 5. 5. M,T, m}ft JPiM Zn, IP^M Zs, |p x M yTz Zne 2 Znw 2 , ip x m y t z Zse 2 Zsw 2 , J 121 . H74 175 . 5. 5. M,t, M T 9 T T. |P|M Z 2 n, |pjm Zn 2 , ip-^m Zs, ip x m y t z Z 2 ne Z 2 nw, ip x m y t z Zne 2 Znw 2 , ip x m y t 2 Zse Zsw, J 122 . H74 177 - 5. 5. M,T, m|f t, m/ T t. iPJJM Z 2 n, Ip^m Zn*, lp x m y t z Z 2 ne Z 2 nw, |p x m y t z Zne 2 Znw 2 , p x M/r z Zse Zsw, J 123 . H74 178 - 32. COUZERANITE. Cleavage = t. 5. 3. M i a y T^P_M x TZneZnw, P122. L731 2 . S153. 33. EUCLASE. Euklas. Prismatic Emerald. Cleavage = M,T. ipf m Zn. 5. 5. m, MJf T, m/ J t.3(ip x m y t z ) ZneZnw, |P x M y T z Zse Zsw, ...Ly33 3 . 5. 5. m, MJf T, m/-5-t.4(ip x m y tj Zne Znw, iP x M y T z ,Zse Zsw, ...Ly33 4 . 5." 5. m, t, MJf-T, m/jt. 2(ip x m y t z ) Zne Znw, |P x M y f z Zse Zsw, Ly33 2 . 5. 5. m,t, 13m_t.ipfmZn, iP x M 7 TZneZnw,5(ip x m y t z )ZseZsw, P98. [An imaginary combination, representing, the planes of many Crystals.] 5. 5. T,m/ y t,MJfr,mjft.ipfMZn, iP x M 7 y TZne Znw, |P + M_T Zn 2 e Zn 2 w, |P x M y T z Zse Zsw, H72 152 . . 5. T, m/yt, m JJt, mjft. 6(|p x m y t z ) Zne Znw, 3(Jp x m y t z ) Zse Zsw, Mii 54 . H72 153 . 34. TWO-AXED MICA. Zweiaxiger Glimmer, Bi-axial Mica. Cleavage = iP T 3 7 M Zn. 5. 5. T, Mf T. |P T 3 7 M Zn, Vesuvius. Greenland Ly42 2 . 5. 5. T, Mf T. IP^VM Zn, |p_m^t Zn 2 e Zn 2 w, Siberia Ly42 3 . 5. 5. T,M^T. P/ 7 MZn, P+ n4tZs 2 eZs 2 w, Siberia Ly43 4 . 5. 5. t, M|T. |P r 3 7 MZn, p + m x t Zne 2 Znw 2 , Siberia Ly43 5 . 86 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 35. ACMITE. Akmit. Achmite. Cleavage = m, T, m}|t. 5. 5. m,T, mift. rV 6 Mf T, T i 479. 5. 5. m,T, mift. T^M^T, Ip/om^t Z 2 nw Z 2 sw, ... M ii 1S6 . P152. D315. 5. 5. mjTjMJfT.lpjVrZw, Ip^m^-t Znw Zsw, Ly32 2 . 5. 5. m,T, m}|t. Jr^Mf T Zn 8 w Zs 2 w, ip&m&t Z 2 nw Z 2 sw, ip ] / mft Zne Zse, Ly 32 3 . 36. HORNBLENDE. Amphibole, comprehending five varieties : 1. TREMOLITE. Grammatite. Amphibole blanc. 2. ACTYNOLITE. Strahlsteiii. Actinote. 3. ARFVEDSONITE. 4. BASALTIC HORNBLENDE. Amphibole noir. 5. ANTHOPHYLLITE. Cleavage = m,t, M}gT. 5. 3. M|gT.lP T 4 J MZn,...(Hauy's primitive form)... Model 84. H64 68 . P55 1 . Ly29 l . 5. 3. MjT. IP^-M Zn, 2(ip x m y t z ) Zse Zsw, Ly29 2 . 5. 3. M^T.JPiMyTZneZnw, D309 1 . S 243 . H64 71 . 5. 5. m,MfOT.ip T 4 3 mZn,iP*M^ 5 TZneZnw, S 215 . H64 73 . 5. 5. M,T,M|gT4P*MJ 5 TZneZnw, S 247 . Ly v 2 . H65 80 . M ii 72 . 5. 5. m^M-JgT.ip/jmZn^P^M^TZneZnw, D309 2 . H65 81 . 5. 5. m,T, M|gT. iP T 4 jM Zn, Jp^m Zs, |p x m y t 2 Zn 2 e Zn 2 vv, ip 4 m y 5 t Zse Zsw, H65 83 . 5. 5. m.T, MfgT, mfjt. ip^m Z 2 n, lp + m Zs 2 , 3(|p x m y t z ) Zne Znw, 3(|p x m y t z )ZseZsw, Mii 76 . 5. 5. T,M|jT.JP*MTZneZnw, S 244 . Ly v \ H64 72 . 5. 5. ^MfgT.iP^MZn^P^M^TZseZsw, Md. 112. H64 74 ' 79 . Mii 74 . P55 2 . S 246 . D309 3 . Ly v 3 . J 109 . 5. 5. T,M|T.}P^MN,|PiM?TZ, P55 3 . J 110 . H65 76 . S. 5. (T, MfT.iP T 4 5 -MZn,|PxM y 5 TZseZsw) x 2, Md. 113. H65 78 . 5. o. T,M^T.|P T 4 3-MZnNnNs,iP 4 M y 5 T Zse Zsw, J 110 ' 3 . H65 77 . 5. 5. t, MfJT. IP T ^M Zn, ip + m x t Zne 2 Znw 2 , Jp 4 M y 5 T Z 2 se Z 2 sw, ip+m^t Zse 2 Zsw 2 , S 248 . Ly v 7 . H65 82 . 5. 5. T, MJT. IP^M Zn, |p x m y t z Zn 2 e Zn 2 w, ip x m y t z Zne 2 Znw 2 , ip x MV 5 T Z 2 se Z 2 sw, ip x m y t 2 Zse 2 Zsw 2 , Ly v 8 . H65 8i . 5. 5. (T, M|T. |P T 4 jM Zn, ip x m y t z Zn 2 e Zn 2 w, |p x m y t z Zne 2 Znw 2 , MM y 5 T Z 2 se Z 2 sw, |p x m y t z Zse 2 Zsw 2 )x 2, Ly v 9 . H65 85 . 5. 5. T, MfjT. 1PA-M Zn, |p x m y t z Zne Znw, ip x m y t z Zse Zsw, Ly29 3 . 5. 5. ^MlgT.iP^MZn^^m^ZseZsw, Ly29 4 . 5. 5. T, M[JT. Jp&m Zn, 3(ip x m y t z ) Zne Znw, 2(ip x m y t,) Zse Zsw, D309 1 . MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 87 35*. An Isomorphous Group, 1, 2: 1. FELSPAR. Feldspath. Potash- Felspar. Common Felspar. Adularia. Orthoklas. 2. RHYACOLITE. Rhyakolith. Glassy Felspar. Eisspath. Cleavage = T, Jmgt ne sw, PM Zn Ns = (Model 105.) 5. 2. T.^MZn,'p 4 mZs, L426 3 . J 98 . H79 230 . R 102 . 6. 3. MgT.JP&MZs, H79 232 . Ly39 2 . 5. 3. MgT.iPiMZii, Mil 44 . Ly39 ! . 5. 3. MgT.ipJmZn.JP&MZs, Ly39 3 . 5. 3. MET. ipjm Zn, JPfcM Zs, JpJraSt Zne Znw, Ly39 4 - 5. 3. MgT.JPiM Zn, JPfcM Zs, ... Model 81. L426 5 . J 90 . M ii '. H79 234 . 5, 3. JM8Tne,XeTnw.JPJMZn, Model 81 a . L426 2 . J 91 . H79* 1 - 5. 5. T, iMT ne, Xfr nw. P|M Zn, L426 1 . J 92 . H79 233 . 5. 5, T, MJgT. JPiM Zn, Jp&m Z 2 s, Jpjm Zs 2 , Jp_m x t Zse 2 Zsw 2 , L426 8 . J 96 . H80 243 . R 100 ' 106a . 5. 5. T, M^T.^M Zn^P&M.Zs, ... Model 109, with n and s reversed, J 93 . Ly39 5 . L426 4 - Mil 252 s . H80 237 . 5. 5. T,M^r.^ M Zn,-P 4 MZs, M ii 6l . H80 236 . Ly39 6 . 5 . 5. T, MgT. ^P.^M Zn, Jp^ Z 2 s, ^m Zs 2 , L426 7 . J 94 . H80 238 . 5. 5. T, MIJT. JPJM Zn, M M Zs, ^p_m x t Zne 2 Znw 2 , H80 MI . 5. 5. (T,MgT.iPJMZn, K Z 2 s, ^m Zs 2 ,^p_m x t Zse 2 Zsw 2 ) x 2, Ly40 17 . 5. 5. (T,MJT.P^MZn,^MZs) x 2, Mii 80 . Mil 81 . 5. 5. (T,m8t.PM Zn, JP|M Zs, ^p_m x t Zne 2 Znw 2 ) x 2, Ly40 19 . 5. 5. T, IMIST ne sw. P^M Zn Ns, Position and planes of Hauys primitive form, the parallelepipede obliquangle, Model 1 05, having its marked planes placed as follows: T = PgM Zn ; M = M^Tne; P = Te, ...Puy-de-Ddmcj...?!!^. H79 229 . 5. 5. t,MJST.JP7 5 MZ8, H79 235 . 5. 5. T, M&T, m^t.^M Zn, P^M Zs, . . . L426 6 . J 49 . A 102 .H80 242 .Ly40 8 .R 105 . 5. 5. T ? M^r,mSt.^^iZn,5PiMZs, H80 240 . 5. 5. T, M^T, mfit. JP^M Zn,JPf 6 M Z 2 s, ^m Zs 2 , Ly40 9 . 5. 5. T, Mfe m 2 ^t. WM Zn, Jpjm Z's.iP&M Zs 2 , Ly40 10 . 5. 5. T,M^T,m 2 ft.^MZn J ^P 1 7 5 M Zs, p_m x t Zse 2 Zsw 8 ,...Ly40 n . H81 243 . 5. 5. T, M&T, m 2 ft. ^M Zn, JPf 5 M Z 2 s, Jp|m Zs 2 , p_m x t Zne 2 Znw 2 , L426 9 . H81 24tJ . Ly40 12 . 5. 5. T,M^T.m^t.HMZn,^MZs,^p_m x tZne 2 Znw 2 , H81 244 . 5. 5. T, M^T, M^T. JP|M Zn, ^P^M Zs, ^P_M X T Zne 2 Znw 2 , ^p_m x t Zse 2 Zsw 2 , J 67 . H81 247 . Ly40". . 5. T, M^T, m 2 |t. JP^M Zn, JP7 B M Z 2 s, Jp$m Zs 2 , ^p_m x t Zne 2 Znw 2 , |p_m x t Zse 2 Zsw 2 , Ly40 15 . 5. 5. T, MT, rnRt ^P^M Zn, iP&M Zs, p_m x t Zne 2 Znw 2 , p_m x t Zse 2 Zsw 2 , H81 249 . 5. 5. T, MT, mf|t. JP|M Zn, ^P, 7 5 M Zs 2 , ^P^M Z 2 s, ^p+m x t Zs 2 e ZsV, ^p_m x t Zse 2 Zsw 2 , Ly40 16 . 5. 5. T, M^T, m^t. ^P|M Zn, ^p, 7 5 M Z 2 s, JpSm Zs 2 , ^p_m x t Zne 2 Znw 2 , Jp_m x t Zse 2 Zsw 2 , H81 250 . 88 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. FELSPAR, Continued: 5. 5. (T, mgt, mSt. ^M Zn, MM Zs,) x 2, ........................ Ly40 18 . 5. 5. T, MgT, m?t. ^PiM Zn, Ipjm Z 2 s, 'Ppf Zs, Jpjm Zs 3 , p_m x t Zne 2 Znw 2 , ^p_m x t Zse 2 Zsw 2 , ......... P116 2 . H81 251 . 5. 5. T, MJJT, m 2 *t. iPJM Zn, p}m Z 2 s, MM Zs, ^p*m Zs 2 , |p_m x t Zse 2 Zsw 2 , ............ Mil 62 . 5. 5. (T,i48T,mStiPiM Zn, K M Zs 2 , ^m Z 2 s,) x 2, ......... Ly4(P. 5. 5. (T, MHJT, mgt, aPgM Zn, ^P|M Zs, 2F-m x t Zne 9 Znw 2 , p_m x t Zse 2 Zsw 2 ,) x 2, ............ Ly40 21 . 5. 5. t, MiiT, milt. MM Zn, ^p^ra Z a s, ^P|M Zs 2 , ^p_m x t Zse 2 Zsw 2 , H81 248 . 5. 5. m, T, mit. MM Zn, ^M Zs, .................................... Ly40 7 . 6, 5. w, T, Mgr. JPiM Zn, KM Z 2 s, lp\m Zs 2 , ........................ H80 239 . 5. 5. m, T, m^t, m 2 ^t.|p 2 T M Zn, Jpjm Z 8 s, JP&M Zs 2 , ............... Ly40 13 . 38. GLAUBERITE. Sulphate of Lime and Soda. Brongniartine. Cleavage = m? t. \P&T Zw. 6. 3. MftT. JPf T Zw, ................................................... H55 183 . 5. 3. Mp MP4 TZw, Jp?oM? T Znw Zsw, ............ M ii 55 2 . H55 188 . L270 1 . 5. 5. t, M? O T. JP^T Zw, ^Pp Mp TZnw Zsw, Mii55 3 . Ti 138. P205. L270 2 . 5. 5. t, m? t. ^P, 4 T Zw, ^p + t Ze, ^PfoMfoT Znw Zsw, |p? m? t Zn 2 e Zs 2 e, ^p + m x t Zne 2 Zse 2 , ............ Mil 60 . . 5. JPJT Zw, JPSMftT Znw Zsw, .................. L270 3 . Mil 59 . H55 185 . 39. AZURE LEAD ORE. Bleilasur. Cupreous Sulphate of Lead. Cleavage = T, MT Zw. 5. 5. M_, t, M^T. ^P 2 T Zw, |p_t Z 2 e, ^p + t Ze 2 , ........................ Ly56 2 . 40. LEADHILLITE. Sulphato-tri-carbonate of Lead. Cleavage = IP^T Zw. 5. 3. MIT. ^P=T Zw, ^p + t Ze, ........................................ ..Ly57 2 . 5. 3. MIT. \P-T Z 2 w, ^p_t Zw 2 , ^p+t Ze, ....... ....................... Ly58 5 . 6. 5. ^PzT Zw,^P + T Ze, ^P_M X T Zn'w Zs 2 w, ..................... Ly57 4 . 5. 5. MIT. \PzT Z 2 w, ^p_t Zw 2 , ^>_m x t Zn 2 w Zs 2 w, .................. Ly57 3 . . 5. T/MJT.iPjjgTZw, ................................................... A 59 . 5. 5. m,t, M^T, mf 6 t. \PzT Zw, 3(|p x t) Zw, ^p x t Ze, 9Gp x m y t z ) Znw Zsw, 9 Gp x m y t z ) Zne Zsw, ............ M ii 171 . S. 5. t, mjt. IP^T Z 2 w, lp + t Zw 2 , J 2 p + t Ze, ^p + m_t Znw Zsw, p + m_tZne Zse, ............ P359. 41. LANARKITE. Sulphate-carbonate of Lead. Dyoxilite. Cleavage = M. . 5. M, M^T. ^P_M Zn, ^p_m Zs, ^p_m x t Zne Znw ? ............... P358. 42. GAY-LUSSITE. Carbonate of Lime and Soda. 5. 5. m, M 2 T. ^P_M Zn, .......................... ..T1139. MINERALS OF THE OBLIQUE PRISMATIC SYSTEM. 89 43. LAUMONITE. Efflorescent Zeolite. Lomonite. Cleavage = m,t, M}{jT. Iplt Zw. 5. 3. MigT. JPfT Zw, ...................................................... Ly43 1 . 5. 3. Mj*T.P 7 5 T Zw, ....................................... Mii 44 . A 23 . Ly43 2 . ft. 3. M{*T. Jpgt Zw, ^P 7 5 T Ze, ^p + m x t Znw 2 Zsw 2 , .................. S 272 . P 27 . 5. 5. M, Mirr.^P 2 T Zw, ^T Ze, ........................ M ii 235 3 . Ly43 3 . 5. 5. M, MJ*T. IrlT Zw, >P*T Ze, ^p + m x t Znw 2 Zsw 2 , ............... Ly43 4 . 5. 5. m, M{iT. IPIT Zw, ^T Ze, .................................... H84 278 . 5. 5. ra, t, Ml'T.^PfT Zw, iP 7 5 T Ze, .................................... H84 277 . 5. 5. ra, t, M!jrr.P 7 5 T Zw, .................................... Mii234 2 . S3 2 . 44. MESOTYPE. Needlestone, comprehending three varieties : 1. NATROLITE. Mesotype. N Si 3 -f- 3 A Si -f 2 Aq. 2. MESOLITE. Mesole. N Si 3 -f 2 C Si 3 + 9 A Si + 8 Aq. 3. SCOLEZITE. Needle Zeolite. C Si 3 + 3 A Si + 3 Aq. Cleavage = MfgT. 5. 3. MfgT.3p_m x t, ..................... . ................. Mii 174 . D270. A 91 . 5. 3. MJ- T. |p_m x t Zne Znw, ............................................. S 292 . 5. 3. MfgT.|P_M x TZneZnw,iP_M x TZseZsw, ............ Model 67. L205 1 . D271. S 293 . J 73 . A 92 . R 98 . H85 291 . Ly45 2 . 5. 3. MfT. ip_m x t Z 2 ne Z 2 nw, 1P_M X T Zne 2 Znw 2 , |p_m x t Z 2 se Z 2 sw, |P_M X T Zse 2 Zsw 2 , ............ Ly45 2 . 5. 5. M, MT. ip_m x t Z 2 ne Z 2 nw, ^P_M X T Zne 2 Znw 2 , |p_m x t Z 2 se Z 2 s w, iP_M x T Zse 2 Zsw 2 , ............ Ly45 3 . 5. 5. M, M|gT. |P_M X T Zne Znw, 1P_M X T Zse Zsw, ... Pi 24. H85 292 . Ly v 2 . S 295 . 5. 5. m, Mf9-T.5p x m y t z , ............................................. S 294 . P123. 5. 5. T, Mf9T. ir_M x T Zne Znw, |P_M X T Zse Zsw, ......... L205 2 . Ly45 3 . 5. 5. t,m + t,Mf9-T.p_m, P_M X T, p + m x t, ........................ S 296 . Pi 26. 5. 5. m, t, Mf gT. iP_M x T Zne Znw, 1P_M X T Zse Zsw, ...... J 7 *. Fuchs. 45. STILBITE. Heulandite. Foliated Zeolite. Blatterzeolith. C Si 3 + 4 A Si 3 + 6 Aq. Cleavage = T. 5. 2. M_, T. JPf T Zw, $pt Ze, ip x m y t z Znw Zsw, p_m x t Z 2 ne Z 2 se, ^p|m x t Zne 2 Zse 2 , ............ S 241 . Ly44 5 . P25 2 . 5. 2. M_, T. P T Zw, Jpjt Ze, ip x m y t z Znw Zsw, ...H84 M1 . A 97 . Ly44 2 . 5. 2. M_, T. iP|T Zw, ipit Ze, ^p x m y t z Znw Zsw, ^pim x t Zne 2 Zse 2 , Ly44 3 . 5. 2. M_, T. JPf T Zw, Jpit Ze, lp x m y t z Znw Zsw, ip_m x t Z 2 ne Z 2 se, Ly44 4 . 46. BREWSTERITE. (NC) Si 3 + 4 A Si 3 + 8 Aq. Cleavage = T. 5. 5. m,M|T.|P_M x T Zne Znw, .................................... T1348 1 . 5. 5. m, T, MfT. ipJjM Zn, lp_m x t ZneZnw, ...... . ........... A". P45. 5. 5. m, T, Mf T, m + t. ^P_M X T Zne Znw, .............................. Ly44 2 . 90 MINERALS OF THE OBLIQUE PRISMATIC SYSTEM . 5. m, T, Mf T, m+t. ^P^M Zn, |P_M X T Zne Znw, ............... Ly44 3 . 5. 5. m, T, nfct, mft, mjt, mft. p T Vm Zn, P__M X T Zne Znw, ...... S 77 . 47. DATHOLITE. Datolith. Humboldtite. Borate of Lime. Chaux boratee siliceuse. Borosilicate of Lime. Cleavage = raft, (but that of Humboldtite = t, Levy.) 3. 5. P, m, MfT. p+m, p+m x t Zn 2 w, .................................... Lyl4 2 . 3. 5. Pl,t,mft,m + t.p_|_m, p + m x t Zn 2 w, .............................. Lyl4 3 . 3. 5. P, m, Mf T. p+m, p_m, p + m x t Zn 2 w, p_m|5t Znw 2 , ............ Lyl4 4 . According to Levy, the above right rhombic or prismatic combinations are Datholite, and the following oblique or hemi-prismatic combinations, Humboldtite. Other mineral- ogists state, that all the combinations of Datholite contain oblique forms. 5. 5. M, Mf T. iplm Z 2 n, |P_M Zn 2 , 4(ip x m y t z ) Zne 2 Znw 2 , 3(|p x m y t z ) Zse 2 Zsw 2 , ......... Lyl5 2 . 6. 5. M, Mf T. |P_M Zn, 3(|p x m y t z ) Zne 2 Znw 2 , 3(p x m y t z ) Zse 2 Zsw 2 , Lyl5 s . 5. 5. M|T, Mf T. pf M Z 2 n, p||m, p|m Zn 2 , 2(|p x m y t z ) Znw Zsw, 7(lp x m y t z ) Zse Zsw, ......... M ii 70 . 5. 5. MT, mft.pfm, ^P^T Z 2 w, |P|T Zw 2 , |p x M y T z Znw Zsw, lp x m y t z Zne Zse, ......... Mii* 8 . 5. 5. m, Mf T, Mf T. pfm Zn 2 , Pf M Z 2 n, IP^T Z 2 w, iPJ T Zw 3 , 2(-Jp x m y t z ) Znw Zsw, 3(ip x m y t z ) Zne Zse, ......... M ii 69 . . 5. t, Mf T, MfT.pfm, IP^T Z 2 w, ipjt Zw 2 , |p x m y t z Znw Zsw, Mii 67 . S 156 . D284 1 . A 84 . . RED IRON VITRIOL. Rother Vitriol. Botryogen. Cleavage = MJT. . 3. M|fT,M||T.iP_MZn, |p_m x t Zne Znw, ............... P233. A 3t . 49. JOHANNITE. Uran Vitriol. Urane sulfate. . 3. M|T. ip_m x t Zne Znw, ....................................... P271. A 172 . 50. GRAPHIC TELLURIUM. Schrifterz. Schrift-Tellur. Cleavage = p,m, py t. 3. 5. p, M, T, mft, m|t. P V M, 3p x m y t z , ......... S 215 . D4 1 6. L690. P34 1 . 3. 5. p,M,T,M x T.p_ T .4p x m y t z , .......................................... Mii 35 . 51. FLEXIBLE SULPHURET OF SILVER. Argent sulfure flexible. Beigsamer Silberglanz. Cleavage = T. . 5. M, T, mjt, mt. JP^M Z 2 n, lp m Zn 2 , 3(ip x m y t z ) Zne Znw, L779 2 . P297. . 5. M, T, m|t, mt. iP^M Z 2 n, ip x m y t z Zne Znw, .................. L779 1 . &%. HUMITE. Cleavage = t. 3. 5. P, M, T, m&t, m / r t, m ^t, MJf T, M^T, mft, m|ft. P/^ pf t, 1 1 p_m x t, 3p + m x t, ......... P89 2 . MINERALS OF THE DOUBLY OBLIQUE PRISMATIC SYSTEM. 91 3. 5. P,M,T, 4m x t.piM,2p_m x t, ........................ . ................. Ly46 2 . 3. 5. P,M,T, 5m x t.piM,7p_m x t, .......................................... Ly46 3 . 53. MONAZITE. Mengite. Lanthanite. 5. 3. MJ2-T. }P|4 M Zn, ....................................... D448. Ti672. 54L TURNERITE. Pictite. Cleavage = m,T. 5. 5. t, m&t, M^-T, mjt. |PM Z 2 n, P_M Zn 2 , P_M Zs, Zn 2 e ZnV, 5(|p_ni x t) Zne 2 Znw 2 , 2p x M&T Zs 2 e ZsV, | P+ mtZse 2 Zsw 2 , .......................................... P84 2 . 5. 5. T, M^O-T, M^T, mft. PM Z 2 n, ip_m Zs, 3p_m x t Zne Znw, p_m x t Zse Zsw, ......... Ly82 2 . CLASS VI. MINERALS BELONGING TO THE DOUBLY OBLIQUE PRISMATIC SYSTEM OF CRYSTALLISATION. The AXES of all Combinations belonging to this Class are = p x m y t*. The constituent FORMS of the Combinations of this Class are as follow : Homohedral Forms, ............ M, T, M X T. Hemihedral Forms, ............ |M X T. Tetartohedral Forms, ......... P x M y T 2 . Every form consists of a pair of parallel planes, and none of the forms ever meet at a right angle. The combinations generally contain from 3 to 8 pair, but sometimes as many as 1 2 pair of planes, and always such as belong to the series M,T, M X T, |M X T. P x M y T z . Finally, there is, in every separate combination, at least two pair of planes of the prismatic zone, and one pair of scalene tetarto-octahedral planes. 1. BORACIC ACID. Sassolin. Acide boracique. 5. 5. t, im_t nw, m_t ne. JP x M y T z Znw. 9. DIASPORE. Diaspor. Cleavage = mft nw. 5. 5. t, |Mf T nw, iMf T ne. P_M X T Zsw, 3(Jp x m y t z ) Zne ? Phillips, Annals Phil. July, 1822, p. 17. 3. CYANITE. Kyanite. Sappare. Disthene. Cleavage = M, |M|T nw. ipl!m x t Znw. M on iMJT nw = 106 6'. P_M X T Znw on M = 106 55', on iM|Tnw = 9438'. Haiiy. 1. 5. P + ,M, im T 7 n t n 2 w, ^M^T nw 2 , |m|t ne, ... (A right prism)... H63 60 . 5. 5. M, iM|Tnw 2 .JP_M x TZ 2 nw, .................................... Mii 32 . 5. 5. M, |M|Tnw,lMjTne. iP_M x TZ 2 nw, ............... L407 1 . Ly29 2 . 5. 5. M, IM^THW, iM|Tne. |P_M x TZ 2 sw, .................. L407 1 . H63 57 . ft. 5. M, mftn 2 w, IM^Tnw 2 , imftne. JP_M X T Z 3 nw, ............ P74. 92 MINERALS OF THE DOUBLY OBLIQUE PRISMATIC SYSTEM. S. 5. M lm i 7 -tnV 5 lrMJTnw 2 ,imJtne.JP_M x TZ 2 nw,...Siin.Md.l07. Ly29 3 . H63 58 . S 296 . L407 2 . . 5. (M,Am/otn 2 w, iMjinw 2 , |mftne.lPJVl x TZ 2 nw) x 2, Ly29 6 . H63 59 . L407 4 . 5. 5. M, im/st n 2 w, Jm^t nw, JMT nw 2 , imjt ne. |P_M X T Z 2 nw, Ly29 4 . H63 81 . 5. 5. M,im T 7 n tnV, |M|Tnw 2 , |mf t ne. IP_M X T Zne, ip_m x tZnw, IP_M X T Zsw, ip_ra x t Zse, ......... H63 62 . 5. 5. M, im^t n 2 w, M|T nw 2 , |m|t ne. |P_M X T Z 2 nw, |P + M X T Z 2 ne, Ly29 5 . 4. BLUE VITRIOL. Kupfer vitriol. Cuivre sulfate. Sulphate of Copper. Cleavage = im^t ne, ^m|t nw. Jp_ra x t Znw. |P_M X T Znw on |M^T ne = 109 32 r , on |Mf T nw == 128 3T. iM^T ne on iMfTnw = 124 2'. Hauj. 5. 3. IMiTne, JMfTnw. iP_M x T Znw,... (Haiiy's Primitive) H102 153 . 5. 5. M, iMiTneaMfTnw.lP_M x TZnw, ..................... H102 155 . 5. 5. M, |MlTn 2 e, im 6 tne 2 , iMf-Tnw. JP_M X T Znw, ......... H103 156 . 5. 5. M, JMiTn 2 e4m 6 tne 3 , JMJTnV, im|tnw 2 . iP_M x TZnw,H103 167 . 5. 5. M, |M|T n 2 e, }m 6 t ne 2 , JMJ- T nw. |P_M X T Znw, ip_m x t Zsw, H103 158 . 5. 5. M, M*T n 2 e, im 6 tne 2 , iMf- T nw. |P_M x TZ 2 nw, lp_m x t Znw 2 , 2 , ......... H103 159 . 5. LATROBITE. Diploite. Cleavage = m, im 16 tnw 2 . $p_m x t Z 2 nw. 5. 5. T,lM T yTnw.iP_M x TZ 2 nw? ................................. PH8. O. An Isomorphous Group, comprehending four varieties : 1. ANORTHITE. MS + 2 CS + 8 AS. 2. LABRADORITE. Labrador. Labrador Felspar. 3. OLIGOCLASE. Oligoklas. Natronspodumen. 4. ALBITE. Soda-Felspar. Cleavelandite. Pericline. Cleavage = T, JMf T ne sw. |p x M y T z Z 3 s 2 e NVw. [P x M y T z Z3s 2 e upon Te = 93 50'.] 5. 5. T,MfT.ip x m y t z Znw 2 ,ip x m y t z Z 2 ne,iP x M y T z Z 3 s 2 e, ......... Mii 84 . 5. 5. T, mf t, m + t. |p x M y T z Znw 2 , ip x m y t z Z 2 nw, ip x M y T 2 Z 2 ne, lp x m y t z Zn 2 e,"iP x M y T z ZVe, ip x m y t z Zsw 2 , ...... M ii s5 . 5. 5. T,mft, m + t.|p x M y T z Znw 2 , ip x m y t z Z 2 ne,ip x m y t z Zne 2 , 1P X M X T Z ZVe, JP x M y T z ZVe, lp x m y t z Zsw 2 , ...... Mii 86 . 5. 5. (T, M? T, M + T. lp x M yTz Znw 2 , ip x m y t z Z 2 ne, 1 P x M y T z ZVe) X 2, Mii 87 . 5. 5. (T,MfT.iP x M y T z Z 2 ne,iP x M y T z Z 2 se) x 2, ............... Mii 88 . Examples of Albite, G. ROSE, Gilberts Annalen der Physik, 1823. 3. 5. T, IM*T nw, ^M^Tne. JP x M y T z Znw, ip x ra y t z Zse, |p x m y t z Zsw, Rose 3 18 ' 19 . MINERALS OF THE DOUBLY OBLIQUE PRISMATIC SYSTEM. 93 5. 5. (T, M^T.|P x M y T z Znw, ip x m y t z Zse, ir x M/r z Zsw) X 2, R 3 20 - 2I . 5. 5. [T 5 MfT.iPxM y T z Znw,ipxm y t z Zne,3(ip x m y t 2 Zse)] X 2, R3 22 . 5. 5. (T, M^T, m + t. ip x m y t z Znw, ip x m y t z Zne, ip x m y t z Z 2 se, |P x M y T z Zse 2 ) x 2, R3 24 . 5. 5. [T, Mf T,m + t. |p x M y T z Znw, 2(lp x m y t z Zne), 3(ip x m y t z Zse)] x2, R3 25 . Examples of Anorthite, G. ROSE, Gilbert's Annalen der Physik, 1823. 5. 3. T, iMf Tnw,lMf Tne. ip x m y t z Znw 2 ,iP x M y T z Z 2 ne,lp x m y t z Zn 2 e, ip x m y t z Zne 2 , ip x m y t z ZsV, ip x ra y t z Zs 3 e, ip x m y t z Zs 2 w 2 , R3 30 ' 31 . 5. 3. T, iMJT nw, jMf T ne. ip x m y t z Znw 2 , |P x M y T z Z 2 ne, 3(Jp x m y t z Zne), 4(ip x m y t z Zse), 2(ip x m y t z Zsw), R3 32 ' 33 . 5. 5. T, |M|Tnw, WfTne. ip x m y t z Znw 2 , iP x M y T z Z 2 ne, ip x m y t z Zn 2 e, |p x m y t z Zne 2 , R3 28 ' 29 . 5. 5. T^MI-TnV^m+tnwMMfTn^ira+tne^ip^^Znw 2 , !P x M y T z Z 2 ne, Jp x m y t z Zne 2 , 4(ip m y t z Zse), 3(lp x m y t z Zsw), R3 34 . Since the article on Felspar, page 87, was sent to press, I have been furnished, by the kindness of Dr. T. Thomson, with the volume of Gilbert's Annalen der Physik which con- tains G. Rose's account of the distinctions between the felspathic minerals. The measure- ments of Weiss, quoted in that article, give simpler formulae for the felspar forms than do the measurements of Ha'uy and Phillips, which I had employed in the calculations. For example, the form ^Py 7 jM should be |P|M (similar to the cleavage plane,) and, upon the same authority, the form gP^-M shoud be gP^M. Distinctions between Felspar, Albite, and Anorthite. 1.) In the Prisms: Felspar M|fT = 120 north angle, Albite MT = 121 north angle, Anorthite ^MJT nw, JMf T ne = 121 north angle. The planes ne nw of Albite have the same inclination upon the two planes of T, but those of Anorthite have different inclinations. 2.) In the Pyramids : The chief terminal planes of Felspar are |P X M ZnNs, of Albite iP x M y T z ZnwNse, of Anorthite !P x M y T z Zne Nsw. 3.) In the Cleavage: The two principal cleavages of Felspar produce an edge that has an angle of 90, but the edge produced by the principal cleavages of Albite has an angle of 93 50'. 9. PETALITE. Silicate of Alumina and Lithia. Cleavage = m, mjt. 5. 3. M|T.iP x M y T z Znw? 8. AXINITE. Thumerstone. Cleavage =r |mjt ne. p_m x t Z 3 nw, ip_m x t Z 3 se. 5. 3. MfT.|P x M y T z Z 2 nw, lp x m y t z Z 2 ne 2 , Ly34 2 . 5. 3. MfT.lP x M y T z Z 2 nw,ip x M y T z Zn 2 e 2 ,lp x m y t z Z 2 ne 2 , ... Model 81 b , with marked letters altered, H73 167 . Ly34 3 . *. 3. Mf T. JP X M 7 T Z Z 2 nw, lp x m y t z Zn 2 w 2 , Jp x m y t z ZnV, H73 169 . Ly35 4 . 5. 3. Mfr,m x tne 2 .lP x M y T z Z 2 nw,lp x m y t z Zn 2 w 2 , Jp x m y t z Zn 2 e 2 ,Ly35 5 . 94 MINERALS OF THE DOUBLY OBLIQUE PRISMATIC SYSTEM. AXINITE, Continued: &. 3. MfT. !P x M y T z Z 2 nw, ir x M y T z ZnV 2 , ip x M/r z ZnV, ip x m y t z Z 2 ne 2 , M ii 89 . Ly35 6 . 5. 3. MfT, Jm,t ne 2 . iPJM T z Z 2 nw, ip x M yTz ZnV 2 , |p x M y T z ZnV, ip x m y t z Z 2 ne 2 , Ly35 7 . 5. 3. MfT.lP x M y T z Z 2 nw,i Px m y t z ZnV, H73 166 . 5. 3. MfT,lm x tnw 2 .iP x M y T z Z 2 nw,lp x m y t z ZnV 2 , H73 170 . 5. 3. Mf T.lP x M y T z Z 2 nw, ib x M y T z ZnV 2 , !p x M/r z ZnV, 2(ip x m y t z Z 2 ne 2 ) Ly35 8 . 5. 3. MfT4P x M y T z Z 2 nw, ip x M/r z ZnV, 2(lp x m y t z Z 2 ne 2 ) ... H74 m . 5. 3. MfT. |P x M y T z Z 2 nw,ip x M y T z ZnV 2 , ip x M/r z ZnV, ip x m y t z Z 2 ne 2 , ip x in y t z Zse 2 , Ly35 9 . 5. 3. MfT.iP x M y T z Z 2 nw,|p x M y T z ZnV 2 ,ip x M y T z ZnV, Jp x m y t z Z 2 ne 2 , lp x ui y t z Zsw 2 , Ly35 l . 5. 3. MfT, lm x t nw 2 . iP x M y T z Z 2 nw, Jp x M y T z ZnV 2 , ir x M y T z ZnV, 2(ip x m y t z Z 2 ne 2 ,) Ly35. S. 3. MfT, m x tne 2 . JP x M y T 2 Z 2 nw, lp x M y T z ZnV 2 , ip x M y T z ZnV, ip x m y t z Z 2 ne 2 , |p x m y t z Zsw 2 , Ly35 12 . 5. 3. MfT.|P x M y T z Z 2 nw, lp x M y T z ZnV 2 ,ip x M y T z ZnV, 2(ip x m y t z Z 2 ne 2 ), ip x m y t z Zse, Ly35 !3 . 5. 3. MfT.|P x M y T z Z 2 nw, lr x M y T z ZnV 2 , |p x M y T z ZnV, lp x m y t z Z 2 ne 2 , ip x m y t z Zse 2 , ip x m y t z Zsw 2 , Ly35 14 . 6. 3. MfT. lp x m y t z Z 3 nw, |P x M y T z Z 2 nw, lp x M y T z ZnV 2 , lp x m y t z ZnV, ip x m y t z Z 2 ne 2 , ip x m y t z Z 3 se, ip x m y t z Zse 2 , Ly35 15 . 5. 3. MfT. lp x m y t z Z 3 nw, !P x M y T z Z 2 nw, |p x M y T z ZnV 2 , P X M/T Z ZnV, 2(lp :x m y t z Z 2 ne 2 ),ip x m y t z Zse 2 , ip x m y t 2 Zsw 2 ,...Ly35 18 . 5. 3. MfT, im x tne 2 . ip x m y t z Z 3 nw, iP x M y T z Z 2 nw, |p x M y T z ZnV 2 , ip x M yTz ZnV, 2(lp x m y t z Z 2 ne 2 ), Ly35 16 . . 3. MfT, |m x tnw 2 . ^P x M y T z Z 2 nw, lp x m y t z ZnV 2 , ir x M y T z ZnV, Jp x M y T z Z 2 ne 2 , |p x m y t z Zse 2 , ip x m y t z Zsw 2 , Ly35 17 . 5. 3. MfT, |ra x tne 2 , lm x tnw 2 . ip x m y t z Z 3 nw, |P x M y T z Z 2 nw, |p x M yTz ZnV 2 , |p x M yTz ZnV, 2(Jp x m y t z Z 2 ne 2 ), ip x m y t z Zse 2 , Jp x m y t z Zsw 2 , Ly35 19 . 5. 3. MfT. |m x t ne 2 , 2(|m x t nw 2 ). |P x M y T z Z 2 nw, |p x M/r z ZnV 2 , 3(ip x m y t z Zne), ip x m y t z Zse 2 , ip x m y t z Zsw 2 , L36 20 . 5."3. MfT, 3(|m x t). ip x m y t z Z 3 nw, |P x M y T z Z 2 nw,ip x M y T z ZnV 2 , 4(lp x m y t z Zne), lp x m y t z Zse 2 ,|p x m v t z Zsw 2 ,ip x m y t z Z 3 sw, Ly36 22 . 9. BABI^GTONITE. Cleavage = m, ^rn^tnw 2 . |P_M X T Z 2 ne. 5. 5. m,^miftnV,^m 1 /tnw 2 ,|MiiTne.iP_M x TZ 2 ne,lp+m x tZn 2 e, Ly30 2 . P53. 95 SECTION III. A SYSTEMATIC ARRANGEMENT OF THE CRYSTALS FOUND IN THE MINERAL KINGDOM, WITH A LIST OF THE MINERALS COMMON TO EACH CRYSTAL. THE Crystals comprised in this Catalogue are divided into Six Classes, and each Class into Five Orders, agreeably to the principles laid down in Part I., Section IV., page 71. CLASSES : 1. Complete Prisms. 2. Complete Pyramids. 3. Complete Prisms combined with Incomplete Pyramids. 4. Incomplete Prisms combined with Complete Pyramids. 5. Incomplete Prisms combined with Incomplete Pyramids. 6. Incomplete Pyramids. ORDERS common to every Class : 1. Square Equator. 2. Rectangular Equator. 3. Rhombic Equator. 4. Rhombo-Quadratic Equator. 5. Rhombo- Rectangular Equator. The Orders are divided into Genera, and these into smaller groups, which differ in number according to the extent of each Genus. EXPLANATION OF THE MINERALOGICAL CHARACTERS EMPLOYED TO DISCRIMINATE THE MINERALS THAT CRYSTALLISE IN THE SAME FORMS. The characters employed are of four kinds, which are arranged in four separate columns after the names of the Minerals, and serve to denote, 1.) The Lustre and degree of Transparency. 2.) The Hardness. 3.) The Colour of the Streak. 4.) The Specific Gravity. 1 ) Signs denoting the Lustre and Transparency : m signifies metallic lustre ; opaque, io imperfect metallic lustre; opaque, itl imperfect metallic lustre ; translucent. When the lustre is non-metallic, it is not described, and the degree of transparency alone is then attended to, in which case, o signifies opaque, tp transparent, tl translucent. 96 EXPLANATION OF THE EXTERNAL CHARACTERS USED. 2.) Signs denoting the Hardness: The scale of MOHS is adopted, and the hardness is expressed in num- bers. When the hardness of a mineral is variable, the middle term is taken. Thus, 6J means hardness varying from 6 to 61. The sign which expresses hardness follows immediately that which expresses lustre or transparency. MOHS'S SCALE OF HARDNESS.* 1. Talc. 6. Adularia. 2. Gypsum. 3. Calc Spar. 4. Fluor Spar. 5. Apatite. 7. Rock Crystal. 8. Topas. 9. Corundum. 10. Diamond. " Numerous experiments of determining the degree of hardness, by the mere scratching of one substance with the other, have completely established, that this process alone is not sufficient, if we intend to make a more sure and extensive application of the characters that may be taken from hardness, than that which has hitherto been common in Miner- alogy. " But if we take several specimens of one and the same mineral, and pass them over a fine file, we shall find that an equal force will everywhere produce an equal effect, pro- vided that the parts of the mineral in contact with the file be of a similar size, so that the one does not present to the file a very sharp corner, while the other is applied to it by a broad face. It is necessary, also, that the force applied in this experiment, be always the least " Every person, however little accustomed, will experience a very marked difference, if comparatively trying in this way any two subsequent members of the above scale, and thus the difference in their hardness will be easily perceived. A short practice is sufficient for rendering these perceptions more delicate and perfect, so that in a short time it is possible to determine differences in the hardness very much less than those between two subse- quent members of the scale. " Upon these observations is founded the application of the scale, the general principle of which consists in this, that the degree of hardness of the given mineral is compared with the degrees of hardness of the members of the scale, not immediately, by their mutual scratching, but mediately, through the file, and determined accordingly. " The process of this determination is as follows: " First, we try, with a corner of the given mineral, to scratch the members of the scale, beginning from above, in order that we may not waste unnecessarily the specimens repre- senting lower members. After having thus arrived at the first, which is distinctly scratched by the given mineral, we have recourse to the file, and compare upon it the hardness of this degree, that of the next higher degree, and of the given mineral. Care must be taken to employ specimens of each of them nearly agreeing in form and size, and also as much as possible in the quality of their angles. From the resistance these bodies oppose to the file, and from the noise occasioned by their passing over it, we argue with perfect security upon their mutual relations in respect to hardness. The experiment is repeated with all the alterations thought necessary, till we may consider ourselves arrived at a fair estimate, which is at last expressed by the number of that degree with which it has been found to agree nearest, the decimals being likewise added, if required. " The files answering best for the purpose are fine and very hard ones. Their absolute hardness is of no consequence ; hence every file will be applicable whose hardness is in the necessary relation with that of the mineral. For it is not the hardness of the file with which we have to compare that of the minerals, but the hardness of another mineral by the medium of the file. From this observation it appears, that the application of the file widely differs from the methods of determining the hardness of minerals which have hitherto been in use, as scratching glass, striking fire with steel, cutting with a knife, scratching with the nail, &c." Mohs, i. 305. * Small cabinets containing suitable specimens of all these minerals, excepting the diamond, with a file for making the trials, are prepared for sale in Germany, and may be procured of R. GRIFFIN & Co. in Glasgow, Price IGs. CLASS I. ORDER I. GENUS I. 97 3.) The colour of the streak. When the hardness of a mineral is determined by means of a file, a fresh surface is produced, which shows the colour of the mineral in powder while untarnished by contact with the air. This is the streak. The character is highly trustworthy, and is easily ascertained without injury to the specimens. w signifies white, gr grey. bl blue, bk black. br signifies brown. y yellow, gn . green, r red. Combinations of these letters indicate a mixed colour, or one colour passing into another. Thus, grw signifies greyish white ; ry, reddish yellow ; brbk, brownish black ; blbk, blueish black, &c. 4.) The specific gravity. This character is given in numbers, the standard or unity of which is water = 1. Arrangement of the Signs. The minerals which have a metallic lustre are placed first, those with an imperfect metallic lustre next, and those with a non-metallic lustre at the end of each list. Then the soft minerals are placed above the hard varieties, and the light minerals above the heavy varieties. The fractions which, in many cases, are appended to the names of minerals, are the characteristics of the crystallographic symbols that are quoted in the descriptions prefixed to the groups. See Class 1, Order 3, Genus 1. CLASS I COMPLETE PRISMS. Order \. Square Equator, / ^ enUS l " f Xes: ( Genus 2. Axes: Order 2. Rectangular Equator, Genus 1. Axes: Order 3. Rhombic Equator, Genus 1. Axes: Order 4. Rhombo- Quadratic Equator, Genus 1. Axes; Genus 1. Axes; Order 5. Rhombo-Rectangular Equator, ... -J Genus 2. Axes; p a m a t a . p a x m a t a . P a *m a t a . pt m a t a . C , ... < (. t*. CLASS 1, ORDER 1. GENUS 1. P,M,T. Model 1. The Cube. Native Bismuth m2jw 9f Sulphuret of Silver m 2 gr 7 Galena m2lgr 7f Seleniuret of Lead m2igr 7f Do. Lead & Cobalt m2| gr 7f Do. Lead & Mercury. . .m 2 \ gr 7f Do. Lead & Silver m2|gr 7f Genus 3. Axes: p Genus 1 Continued: Native Gold m2Jy 15 Auriferous Silver m 2 J y w 1 2 Argentiferous Gold m 2J wy 12 Native Silver m2|w 10| Native Copper m2|r 8J Purple Copper m3 gr 5 Sulphuret of Tin m4 bk 4J Platinum m4Jgr 17^ 98 CLASS I. ORDER III. GENUS I. Genus 1 Continued: Sulpho-antimonite of Nickel? m5|gr 6 Nickel Glance m5iw 6 Tin White Cobalt......m5| gr 6i Silver White Cobalt . . . m 5 \ gr 6i Tesseral Pyrites m5|w 6j Magnetic Iron Ore m6 bk 5 Iron Pyrites m6Jbrbk5 Titanium m8 r 5J Red Oxide of Copper . . .itl3 j r 6 Sulphur*, of Manganese io 3} gn 4 Chloride of Silver tl 1 J grbr 5f M uriate of Ammonia... tl IJw lj Chloride of Sodium ....tp 2 w 2^ Alum tp2|w 1| Arseniate of Iron tl 2^ gn 3 Zinc Blende tlSJwbr 4 Fluorspar tp'4 w 3f Aualcime tp5| w 2 Boracite tl 7 w 3 Diamond tplOw 3 CLASS L ORDER 1. GENUS 2. Divided into two groups : a.) The right square-based prism. P X ,M,T. Models 2, 3. b.) The right eight-sided prism. P X ,M,T, mt. Model 4. Group a. Black Tellurium m li grbk7 Iron Pyrites, m 6| bkbr5 Chloride of Silver o 1| grbr5J Muriate of Ammonia. . .tl 1 J w 1 i Chloride of Sodium . . . .tp 2 w 2 J Uranite tl 2J gn 3 Cryolite tl 2|w 3 Murio-Carbon. of Leadtp2Jw 6 Molybdate of Lead tl 3 w 6| Tungstate of Lead tl 3 gr 8 Apophyllite tp4f w 2^ Gehlenite o 5jw 3 Rutile t!6br 4J Idocrase tl 6} w 3 Group b. Black Tellurium ........ m li grbk? Uranite .................. tl 2i gn 3J Murio- Carbon, of Lead tp 2 J w 6 Molybdate of Lead ..... tl 3 w 6| Humboldtilite ........... tl 5 br 3 Wernerite ............... tp5i grw 2f Mellilite .................. o 6 ry 3i Rutile ..................... t!6ibr 4i Idocrase .................. tl Q\ w 3f CLASS 1. ORDER 2. GENUS 1. Right Rectangular Prism. P+,M_,T. Model 5. Antimonial Silver ...... m3|w 9i Iron Pyrites ............. m 6 bkbr5 Anhydrite ............... tp 3j grw 3 Desmine .................. tp3}w 21 Phospha.of Manganese o 5J ygr 3f Chrysolite ............... tp6jw 3f CLASS 1. ORDER 3. GENUS 1. Right Rhombic Prism. P X ,MJT. Model 6. [The fraction appended to the name of the mineral is the characteristic of the rhombic prism, that is to say, the equivalent of the sign -. Thus, the prism of Jamesonite is Jamesonite f ............ m2igr 5j Sulphuret of Bismuth/^m 2i gr 6f Arsenical Pyrites .. . .m 51 bk 6J White Iron Pyrites j m 6 J bk 4 J Muriate of Lead ...... o 2| y w 7 Sulphate of Barytes f tl 3J w 4f Do. of Strontian j- ...... tl 3 w 3| Arragonite f ............ tl 3 J grw 3 Red Oxide of Zinc J tl 4 ry 5f Mengite .................. o 5 w 5 Triphylinef ............ tl 5 grw 3| Chiastolite y 9 ^ ......... tl 5 w 3 Hypersthene | T ......... o 5i wgr 3^- Fergusonite x ............ o 5|br 5f CLASS I. ORDER V. GENUS II. 99 Genus 1 Continued: Amblygonite f ......... tl 6 w 3 Sillimanite J, J ......... o 6J br 3f Prehnitef ............... tl 6| w 3 Staurolite -$ ............ o 7 J w 3| Andalusite |g ........... o 7i w 3 P,MJJT,mIt - Pyrolusite, ............... m2f brbk4f CLASS 1. ORDER 4. GENUS 1. Right square prism with the vertical edges bevelled. P + ,M,T, m_t,m + t. Finite, ..................... o 2J w 2f Humboldtilite, ........... tl 5 br 3 3 6f P + ,M,T, MT, m_t, m + t. Idocrase, .................. tl 6 w P_, mt, m_t, Molybdate of Lead ..... tl 3 w Rutile, .................... tl6br CLASS!. ORDER 5. GENUS 1. Regular six-sided prism. P x ,T,Mtf T 2 . = P X ,V. Model 7. Sulphr'ofMolybden? 1 mlgr 4 Graphite .................. mlibk 2 Telluric Silver ......... ni2 gr 8 Polybasite ............... m2jbk 6 Vitreous Copper ........ m2jgr Sulphuret of Nickel... m 4 gr Magnetic Iron Pyrites m4 Antimonial Nickel ..... m5 Osmium-Iridium ........ m5 Palladium ................ m5 Copper Nickel ......... m5br Ice ........................ tpl w Talc ....................... tl Ijgnw2j Chlorite .................. tl Ignw2{ Pinite ..................... o 2 w 2} One-axed Mica ......... tp 2 wgr 3 Red Silver ............... o 2J r 5| Cinnabar .................. tl 2^r 8 Sideroschisolite ......... o 2^ gn 3 bk r gr gr ? 19 12 7f 1 Genus 1 Continued: Cronstedtite ............. o Vanadiate of Lead ..... o 2jw Calcareous Spar ......... tp3 grw Phosphate of Lead ..... tl 3J y w Arseniate of Lead ...... tl 3 f yw Fluoride of Cerium ..... o 4 y Carbonate of Iron ...... tl 4 brw Pyrosmalite .............. o 4Jbr Apatite ................... tl 5 w Galmei .................... tl 5 w Nepheline ............... tl 6 w Dichroite ................. tl 7Jw Beryl ..................... tp7Jw Corundum ............... tp9 w 6| 2J 7 7 4| 2| 2f- 2f 4 CLASS 1. ORDER 5. GENUS 2. Regular twelve-sided prism. Model 10. The ratio of the axes m a t a may be exactly 14 to 13, and necessarily must be very near to that sum ; since if t a is taken =13, then m a can only fall between 13 and 15. See page 43. For practical pur- poses it is sufficient to know, that pmutr3 is use( i to indicate the ratio of the axes of such combinations as are represented by Models 10 and 52. Vitreous Copper m2|gr 5J Magnetic Iron Pyrites m4 bk 4f Pinite o 2w 2| Cronstedtite o 2|gn 3^ Calcareous Spar tp 3 grw 2} Phosphate of Lead tl 3J yw 7 Fluoride of Cerium o 4 y 4J Apatite tl 5 w 3J Nepheline tl 6 w 2f- Dichroite tl 7^ w 25- Beryl tp7 J w 2f 100 CLASS II. ORDER I. GENUS I. CLASS 1. ORDER 5. GENUS 3. Rhombic prisms, with two or more of the vertical edges replaced. Divided into three groups : a.) containing P X ,M,M_T. b.) containing P X ,T,M_T. Md.8,9. c.) containing P X ,M,T,M_T. Group a. PyrolusitefS- m2f brbk4f Uranitel tl 2jgn 3J- KoenigiteJJ tp2gn 3i Sulphate of Barytes f tl 3J w 4 j P + ,M,3(lm x t). Cyanite tl 6 w 2f Group b. Sulph 1 . of Antimonyf Jm2 gr 4f Do. of Bismuth T V m 2J gr 6f Brittle Sulph 4 . Silver f m 2 bk 6 J Antimonial Silver ^ . . .m 3| w 9i Sulphate of Potash f . .tp 2| w If Do. of Barytes f t!3jw 4f Polyhallite / T o Sjrgr 2| Strontianite f tl 3| w 3f Prehnite tl 6| w 3 Spodumenif tl 6|w 3f Staurolite T 8 7 Md.9 o 7 J w 3| Group c. Sulph 1 . of Antimony ff m2 gr 4f Bournonite ^| m2jbk 5f Augite |5 o Prehnitef tl 6i w CLASS II. COMPLETE PYRAMIDS. an ( Genus 1 . Axes : p a m a t a . Order I. Square Equator, 1 \ Genus 2. Axes : p* m a t a . Order 2. Rectangular Equator, Genus 1 . Axes : p* m a t a . f Genus 1. Axes: p a m a t a . Order 3 . Rhombic Equator, < Genus 2. Axes : p* m a t a . v Genus 3. Axes : p| m a t*. TIT T r\ j + > n f Genus 1. Axes: p a m a t a . Order 4. Rhombo- Quadratic Equator < _ \ Genus 2. Axes : p a m a t a . T,, , T, 7 n f Genus 1. Axes: p a m, a 5 t,V Order 5, Rhombo- Rectangular Equator,... < Genus 2. Axes: t:- CLASS 2. ORDER 1. GENUS 1. Divided into four groups : a.) The Regular Octahedron. PMT. Model 15. b.) The Hemitrope or Twin Octa- hedron. PMT x 2. Model 16. c.) Various Triakisoctahedrons, as 3P 2 MT. Md. 17. 3P 2 MT X 2. d.) Combinations of the Regular Octahedron with the Triakis- octahedron, as PMT, 3p|mt. Group a. Native Lead mHgr 11^ Native Bismuth m2jw 9f Sulphuret of Silver m 2 gr 7 J Galena m2gr 7f Native Copper m2|r Sf Native Gold m2jy 15 Native Silver m2|w 10| Purple Copper m3 gr 5 Amalgam m3w 13 j Copper Pyrites m3f gnbk4 Native Iron m4gr 7 Sulpho-antimonite of Nickel m5gr 6 CLASS II. ORDER I. GENUS II. 101 Group a Continued: Tin White Cobalt m5| gr 6* Silver White Cobalt . . . m 5 J gr 6 Sulphuret of Cobalt . . . . m 5 i gr 6^ Magnetic Iron Ore m6 bk 5 Iron Pyrites m6 bkbr5 Sulphu 1 . of Manganese io 3 J gn 4 Red Oxide of Copper it!3f r 6 Chromate of Iron io5br 4 Chloride of Silver o I^grbr5f Oxide of Arsenic o 1^ w 3 Muriate of Ammonia... tl IJ vv 1^ Alum tp2jw IJ Zinc Blende tl 3 J wbr 4 Fluorspar tp4 w 3J- Pyrochlore tl 5 br 4J Leucite tl 5| w 2J Nosian tl 5| gr 2J Lapis Lazuli tl 5| bl 3 Yttrocerite o 6 w 3J Hauyne tl 7 w 2f Pleonaste (bk) tl 7J w 3f Automalite (bl, gn) ....o 8 w 4J Spinel (r, bl) tp 8 w 3J Diamond tplOw 3^ Group b. Native Silver m2Jw 10^ Purple Copper m3 gr 5 Magnetic Iron Ore m6 bk 5 Alum tp 2 w 1 J Zinc Blende tl 3J wbr 4 Spinel (r,bl) tp8 w 3J Automalite (bl, gn) o 8 w 4-J Group c. [The fraction is the characteristic of the triakisoctahedron.] Galenaf m2lgr 7f Magnetic Iron Ore |...m 6 bk 5 Iron Pyrites f m6^ bkbr5 Red Oxide of Copper f it!3 J r 6 Fluorspar^ tp4 w 3 Diamond \ tplOw 3 Group d. 7f Magnetic Iron Ore |...m 6 bk 5 Galenaf Group d Continued : Red Oxide of Copper J it!3| r 6 Fluorspar \ tp4 w 3f Spinel f- (r, bl) tp8 w 3} CLASS 2. ORDER 1. GENUS 2. Divided into three groups : a.) Square-based Pyramids contain- ing Forms of the North and East Zones. Example: P X M, P X T. Models 12, 13. b.) Square-based Pyramids contain- ing Forms of the Octahedral Zones. Example : P X MT. Models 12, 13. c.) Combinations of two or more square-based octahedrons of any Zones. Examples: P X M, P X T, Px mt. P X M, p x m, P X T, p x t. Md. 14. P x M,P x T,p x mt,p x mt. The pyramids of groups a and b are not distinguishable by their ex- ternal appearance. They may be discriminated by comparing their interfacial angles with the charac- teristics. Group a. Copper Pyrites f m3f gnbk4 Chloride of Mercury f tl Ij w 6J Molybdate of Lead f . . .tl 3 w 6f Tungstate of Lime f . . .tl 4J w 6 Hausmannite f io 5 J rbr 4 J Anatase | tl 5| w 3 Oxide of Tin f tl 6| br 7 Group b. Copper Pyrites 1 m3|gnbk4^ Mellitef tp 2J w if Uranite tl 2^ gn 3 Tungstate of Lead f ...tl 3 gr 8 Tungstate of Lime | . . .tl 4 J w 6 Apophyllite f tp 4| w 2 Rutilef tlG^br 4J Zircon f tp7iw 4 102 CLASS II. ORDER IV. GENUS I. Group c. Copper Pyrites m3fgribk4 Hausraannite io 5 J rbr 4| Braunite io6bk 4f- Molybdate of Lead ....tl 3 w 6| Tungstate of Lead tl 3 gr 8 Tungstate of Lime tl 4J w 6 Anatase tl 51 w 3f CLASS 2. ORDER 2. GENUS 1. Models 82% 82 b ; but these forms properly belong to Class 5, Ord. 3. CLASS 2. ORDERS. GENUS 1. Divided into two groups: a.) Icositessarahedrons. Example: 3PiMT. Model 22. b.) Hexakisoctahedrons and Hemi- hexakisoctahedrons. Group a. 3PMT. Model 22. Sulphuret of Silver ..... m 2J gr 7^ Amalgam ................. m3Jw 13! Iron Pyrites ............. m 6 bkbr5 Analcime ................. tp5w 2 Leucite ................... t!5f w 2* Garnet .................... tl 7 w 4 3PMT. Similar to Model 22. Native Gold ............ m2|y 15 Native Silver ............ m2f w 10J Muriate of Ammonia . . .tl J J w 1 1 3PMT X 2. Native Gold ............. m2|y 15 3PZMT, 3P|MT. Analcime ................. tp5Jw 2 pmt, 3PJMT. Native Gold ............ m2|y 15 Native Silver ............ m 2| w 10 J Group b. . Model 23. Hexakisoctahedron. 3| Group b Continued : 6P^MT. Similar to Model 23. Hexakisoctahedron. J (6PM$T.) Model 24. Hemihexakisoctahedron with inclined faces. PMT, 6pimit. Diamond tplO w Model 25, Hemihexakisoctahedron with parallel faces. Iron Pyrites m6Jbkbr 5 CLASS 2. ORDER 3. GENUS 2. PMT, P 2 M 3 T, P 2 MT 3 . Braunite io 6 bk 4 J CLASS 2. ORDER 3. GENUS 3. Scalene Octahedrons, or octahe- drons with a rhombic equator. Example : P+M_T, or P}gM T ^T. Model 21. Antimonial Silver ...... m 3 J w 9 Sulphur Md. 21 ......... tl 2 yw 2 Thenardite ............... tp2 w 2| Sulphur .................. tl 2 y w 2 CLASS 2. ORDER 4. GENUS 1. The regular octahedron with the solid angles replaced by complex equiaxed scalene octahedrons. PMX, 3p|mt. Sulphuret of Silver m2 gr 7^ Galena m 2 gr 7f Red Oxide of Copper it!3| rs 6 PMT, 3pimt. Native Gold m2| y 15 Magnetic Iron Ore m6 bk 5 Pleonaste, bk t!7Jw 3J PMT, 3pimit. Iron Pyrites m6ibkbr5 PMT, 6 Px m y t z . Red Oxide of Copper itl3| rs 6 CLASS II. ORDER V. GENUS I. 103 CLASS 2. ORDER 4. GENUS 2. Pf M, P|T, pf mt, J(pm_t + ,pm + t_). Also several similar square-based Combinations. Tungstate of Lime tl 4 w 6 Rj. Model 26 a . RJ. Model 26 b . CLASS 2. ORDER 5. GENUS 1. Divided into seven groups : .) Single Rhombohedrons. Exam.: !P x T,P x MifT 2 ,orR x . If. Model 26 C . 1 Q . Model 26*. b.) Combination of two equal rhom- bohedrons, forming a regular six-sided Pyramid. Example: P X T, P x M|f T 2 . = 2R X Zw Ze. 2R|f Zw Ze. Model 26. c.) Combination of two dissimilar Rhombohedrons. Example : JP_T, Jp+t, JP_M}f T 2 , Jp+mift 2 . d.) Combination of three dissimilar Rhombohedrons. Example : Rf Zw, R| Ze, r| Ze. Model 26 e . e.) Four Rhombohedrons combined. f.) Rhombohedrons combined with Scalenohedrons. Example : >-ptt, jpJmHl* J(3P+M+T). g.} Scalenohedrons only. Example: J(3P|MT) = B!. Model 26 f . Group a. Tetradymite \ mljgr 7J Crichtonite f- m4Jbk 4 Specular Iron | m6 brr 5 \ Specular Iron f m6 brr 5J Nitrate of Soda | tplfw 2 Cinnabar f , Md. 26 C . . .tl 2 J r 8 Sulphato-tricarbonate of Lead, | tp2Jw 6^ Calcar 8 .SpariMd.26 a Do. Md.26 b -f Md.26 d Do. Do. Do. Do. tp3 grw2f Group a Continued : Plumbo-Calcite | ...... Carbon 6 , of Magnesia j- o 3| grw 2|- Dreelite J ............... o 3| w 3J Carbon e .of Manganese^tl 3J w 3f Dolomite | ............... tl 3| grw 3 Mesitinspar 1 ............ Carbonate of Iron i~] Do. H tl 4 brw 3 Do. fj Brown Spar i ........... tl 4Jgrw3 Chabasite | .............. tp4Jw 2 Alunite | ................. tl 5 w 2| Galmei| ............... "1 Do. ............... Itl5 w 4 o. ............... QuartzJ .................. tp7 w 2| Corundum | ............. tp9 w 4 Group b. Vitreous Copper %...\ Do. -f.../ m2f g r 5| Calcareous Spar f ...... tp 3 grw 2| Phosphate of Lead jf tl 3| yw 7 Quartz | .................. tp7 w 2| Corundum & . . Do. 2R|, 2Rf. Vitreous Copper ........ m2| gr 51 Corundum tp 9 w 4 Group c. Crichtonite m4Jbk 4 Specular Iron m6 brr 5J Copper Mica tp 2 gn 2] Calcareous Spar tp 3 grw 2f Dolomite tl 3| grw 3 Carbonate of Iron tl 4 brw 3 Chabasite tp4J w 2 Galmei tl 5 w 4 Quartz tp7 w 2| Group d. Calcareous Spar tp 3 grw 2| Chabasite Md. 26 e tp 4 J w 2 104 CLASS III. ORDER I. GENUS I. Group e. Sulphato-tricarbonate of Lead ; ..tp2Jw 6* Calcareous Spar tp 3 grw 2| Group f. Specularlron m6 brr 5 Red Silver o 2J r 5-j- Calcareous Spar tp 3 grw 2| Chabasite tp4^w 2 Group g. Red Silver o 2| r 5f Calcar 8 . Spar Md. 26 f . tp3 grw 2| Carbon 6 , of Manganese tl 3 J w 3f- CLASS 2. ORDER 5. GENUS 2. Irregular six-sided Pyramids. P X T,P X M X T Z . Antimonial Silver m3Jw Sulphur tl 2 yw Nitre tp2 w Sulphate of Potash tp2f w White Lead Ore tl 3J w Witherite tl 3 J w 2 2 If 6* 4* Sulphur tl 2 yw 2 PJT,pft,PMfT, P Jmft. Witherite tl 3Jw 4J CLASS III. COMPLETE PRISMS COMBINED WITH INCOMPLETE PYRAMIDS. /v^ i ' o E f Genus 1. Axes: p a m a t a . Order 1. Square Equator, { . \ Genus 2. Axes: p x m a t a . Order 2. Rectangular Equator, Genus 1 . Axes : p a m t a . Order 3. Rhombic Equator, Genus 1 . Axes : p x m t a . Order*. Rhombo- Quadratic Equator, / ^ enus L f xes: ? m ta ' \ Genus 2. Axes: p x m a t a . Order 5. Rhombo- Rectangular Equator,... CLASS 3. ORDER 1 . GENUS 1 . This genus is divided into four groups : a.) P,M,T predominant The cube with its edges and solid angles variously truncated. Models 27, 29, 3 1 , 32, 35, 36, 38, 39, 40.. b.) MT.PM,PT predominant The rhombic dodecahedron with its edges and angles modified. Models 28, 34. c.) PM.T predominant. -The regu- lar octahedron with its edges and angles modified. Models 30, 33. d.) JPMT predominant. The tetra- hedron variously modified. Model 37. Genus 1. Axes: p|m a 5 t a 3 . Genus 2. Axes: p x m a 4 t a 3 . Genus 3. Axes : p x m a t a . Group a. P,M,T, mt. pm, pt. Model 27. Cube with edges replaced. Sulphuret of Silver m 2 J gr 7 j Native Copper m2f r 8f Native Silver m2f w 10| Native Gold m2|y 15 Arsenical Nickel m w Chloride of Silver o 1J grbr5f Muriate of Ammonia. . .tp 1 f w 1 \ Chloride of Sodium tp 2 w 2i Arseniate of Iron tl 2 gn 3 Fluorspar tp4 w 3f Diamond tp 10 w 3J P,M,T.pmt. Similar to Model 29. Cube with angles replaced. Native Lead mUgr 11J Galena m2^gr 7f CLASS III. ORDER I. GENUS I. 105 Group a Continued: Native Silver m2| w 10| Native Copper m2|r 8f Purple Copper m3 gr 5 Sulphr*. of Manganese m3f gn 4 Sulpho-antimouite of Nickel m5i gr 6 Tin White Cobalt m5|gr 6i Bright White Cobalt.. .m5| gr 6J Sulphuret of Cobalt ....m5| gr 6J Nickel Glance m5| w 6 Magnetic Iron Ore m6 bk 5 Iron Pyrites m6Jbkbr5 Arsenical Nickel m w Chloride of Silver o H grbr5f Chloride of Sodium ....tp2 w 2J Alum tp2jw 1| Arseniate of Iron tl 2| gn 3 Leucite tl 5| w 2J P,M,T. PMT. Model 29. Middle crystal between the cube and the octahedron. Sulphuret of Silver in 21 gr 7 Galena ni2Jgr 7f Native Copper m2| r 8f- Native Gold m2fy 15 Bright White Cobalt... m5|gr 6J Zinc Blende tl 3| wbr 4 Fluorspar tp4 w 3f P,M,T, nit. pm, pt, pmt. Md. 31. Cube with edges and angles re- placed. Sulphuret of Silver m 21 gr 7J- Galena m2gr 7f Native Copper m2|r SJ Native Silver m2|w 10i Native Gold m2I y 15 Arsenical Grey Copp r . m4 rgr 4f- Arsenical Nickel m \v Red Oxide of Copper itl3| r 6 Arseniate of Iron tl 2|- gn 3 Fluorspar tp4 w 3f P,M,T, mt. pm, pt, PMT. Md. 31. Galena m2| gr 7f Tin White Cobalt m5^gr 6\ Sarcolite tl 5} w 2 Group a Continued: P,M,T. |pmt. Model 38. Native Gold m2|y 15 Arseniate of Iron t!2jgn 3 P,M,T, mt. pm, pt, |pmt. Md. 36. Arseniate of Iron t!2^gn 3 Boracite. tl 7 w 3 P,M,T, MT. PM, PT, iPMT, ipmt. Model 3.5. Boracite tl 7 w 3 P,M,T. 3pjmt. Model 39- Sulphuret of Silver m 2J gr 7} Iron Pyrites m 6 \ bkbr5 Fluorspar tp4 w 3f Analcime tp5J\v 2 P,M,T. 3pjmt. Sim. Md. 39. Native Gold m2|y 15 Fluorspar tp4 w 3f P,M,T.pmt, Spjmt. P,M,T,PMT, 3p 2 mt, 3p 3 rnt. Galena m2lgr 7f P,M,T.pmt, 3p 2 mt. P,M,T.ipmt,i(3p 2 mt). Arseniate of Iron tl 2 \ gn 3 P,M,T. pmt, 3pjm|t. Iron Pyrites m6Jbrbk5 P,M,T. PMT, 6pimit. Diamond tplO w 3J P,M,T,mt.pm,pt, P,M,T.6pimit. Sim. Mod. 40. Fluorspar ................. tp 4w P,M,T,MT.PM,PT,pmt, \ (3p^mt) P,M,T, MT. PM,PT, J?rMT Znw, ipmt Zne, \ (3pjmt) Z 2 ne, Boracite .................. tl 7 w 3 Group b. p,m,t,MT.PM,PT. Model 28. Native Copper .......... m2|r 8f Amalgam ................. m.3^ w 13J Magnetic Iron Ore ..... m6 bk 5 106 CLASS III. ORDER I. GENUS I. Group b Continued : Red Oxide of Copper. . .itl3| r 6 Chloride of Silver o 1 J grbr5f Zinc Blende t!3}wbr4 Nosian tl 5|gr 2 Lapis Lazuli tl 5|bl 3 Haiiyne tl 7 w 2 Garnet tl 7 w 3J p,m,t, MT. PM, PT, pmt. Md. 34. Sulphuret of Silver ra 2J gr 7 Arsenical Grey Copp r . m 4 rgr 4 J- p,m } t, MT. PM, PT, pmt, 3p|mt. Amalgam m3jw 13J p,m,t, MT.PM, PT, Jpmt. p,m,t, MT. PM, PT, ipmt Znw, i(3pJmt)Z 2 ne. P,M,T, MT. PM,PT, 1PMT Znw, JPMT Zne, l(3pjmt) Z 2 nw. P,M,T, MT. PM, PT |PMT Znw, Boracite tl 7 w 3 p,m,t, MT. PM, PT, 3p Jmt Amalgam m3jw 13f Sodalite tl 5f w 2J Nosian tl 5f gr 2J Lapis Lazuli tl 5| bl 3 Haiiyne tl 7 w 2f p,m,t, MT. PM, PT, pmt. 3pjmt. Magnetic Iron Ore m6 bk 5 P,M,T, MT. PM, PT, PMT, Spimt, 3p|mt. Arsenical Grey Copper m 4 rgr 4f Group c. p,m,t. PMT. Model 30. Galena m2gr 7| Native Copper m2|r 8f Native Silver m2|w 10^ Native Gold m2|y 15 Purple Copper m3 gr 5 Copper Pyrites m 3| gnbk4l Arsenical Grey Copp r . m 4 rgr 4| Platin-Iridium m4Jgr 17 Tin White Cobalt m5gr 6i Sulphuret of Cobalt . . . .m 5 1 gr 6 Bright White Cobalt... m5| gr 6J Group c Continued : Magnetic Iron Ore ..... m6 bk 5 Iron Pyrites ............. m6i brbk5 Red Oxide of Copper itlSf r 6 Alum ..................... tp2i w 1| Chloride of Silver ...... o 1 i grbr 5f Zinc Blende ............. tl 3| wbr 4 Fluorspar ................. tp4 w 3f Diamond ................. tplOw 3^ p,m,t. rat. pm, pt, PMT. Sim.Md.33. Arsenical Grey Copp r . m4 rgr 4j- Franklinite ............... m6J br 5 Red Oxide of Copper it!3| r 6 Zinc Blende ............. tl 3| wbr 4 P,M,T, mt. pm, pt, PMT. Md. 33. Alum ..................... tp2i w 1| Fluorspar ................. tp4 w 3f P,M,T, PMT, 3p 2 mt. p,M,T,mt.pm,pt,PMT, 3p 2 mt. Galena .................... m2j gr 7| p.nijt, MT.PM,PT, PMT, 3pjmt. Tin White Cobalt ...... m5|gr 6| Tesseral Pyrites ........ m5|w 6| Red Oxide of Copper it!3f r 6 p,m,t, MT. PM, PT, PMT, 3pjmt. Magnetic Iron Ore ..... m6 bk 5 Zinc Blende ......... ....tl 3| wbr 4 Fluorspar ................. tp4 w 3f Pleonaste (bk) ......... tl 71 w 3f p,m,t, MT. PM, PT, PMT, i(3pimt). Zinc Blende .............. tl 3| wbr 4 Group d. p,m,t. |PMT. Grey Copper ............ m3^bk 5 Copper Pyrites ......... m 3| gnbk4 Boracite .................. tl 7 w 3 p,m,t.-|PMT, |pmt. Zinc Blende ............. tl 3| wbr 4 p,m,t, mt. pm, pt, |PMT. Md. 37. p,m,t, rnt. pm, pt, JPMT, ipmt. Grey Copper ......... m3to4bk 5 Do. Antimonial ...... m3to4bk 5 Boracite ............. tl 7 w 3 CLASS III. ORDER, IV. GENUS I. 107 CLASS 3. ORDER 1. GENUS 2. This genus is divided into two groups : .) The prism has four vertical planes. A right square prism with the terminal edges or the solid angles replaced. Exam.: Pf,M,T.pfmt. Md. 41. b.) The prism has eight vertical planes. A right square prism with the lateral edges and the terminal edges or the solid angles replaced. Example: P+) M,T,MT.P|M,P| T. Md.42. Group a. Black Tellurium mlJgrbkY Galena m2|gr 7 Copper Pyrites m3| gnbk4 Chloride of Mercury. . .tl 11 w 61 Mellite tp2l w If- Uranite tl 21 gn 3J Muriocarbon 6 . of Lead tp2| w 6 Molybdate of Lead tl 3 w 6| Tungstate of Lead tl 3 gr 8 Apophyllite Md. 41 ....tp4| w 21 Anatase tl 51 w 3f Rutile t!6Jbr 41 Idocrase tl 61 w 3f Group b. Black Tellurium. mil grbkT Uranite tl 21 gn 3J Muriocarbon 6 . of Lead tp 2| w 6 Apophyllite tp4| w 2J Oerstedtite o 51 br 3f- Anatase tl 5| w 3f Rutile t!61br 41 Oxide of Tin tl 61 br 7 Idocrase Md. 42 tl 61 w 3f CLASS 3. ORDER 2. GENUS 1. Rectangular prism with the terminal edges or the solid angles re- placed. Example : p+,M,T. PMf T. Md. 43. Bournonite m 2| bk 5| Anhydrite tp 3 J grw 3 Genus 1 Continued: Desmine Md. 43 tp3| w 2 Harmotome t!4|w 2f Olivine tp6| w 3f CLASS 3. ORDER 3. GENUS 1. Rhombic prism, with the terminal edges or the solid angles, or both, replaced. P X ,M_T. p x m, p x t, p x m y t z . Model 44 is P x) M_T.p x m. [The characteristic added to the name relates to the prism, and is the substitute for the sign - in the above general formula.] Arsenical Pyrites f ....m5| bk 6J White Iron Pyrites | m61bk 4J- Lievrite |, f io5|gnbk4 Thenardite i tp2 w 2| Celestine tp3l w 3J Heavy Spar f , f tp 31 w 4f White-Lead Ore f tl 3jw 6 Arragonite f tl 3| grw 3 Euchroitef tl 3| gn 3f Libethenite T % tl 4 gn 3| Fergusonite o 5| br 5J Andalusite Jg o 7i w 3 , J|, |, }-|, tp8 w 31 CLASS 3. ORDER 4. GENUS 1. This genus is divided into nine groups : a.) P,M,T predominant. The cube with its edges and angles re- placed by equiaxed combina- tions of unequiaxed forms, of the kinds enumerated at page 15. Examples: P,M,T, mt 2 . pm 2 , p 2 t, Spjm^t. P,M,T, mt 3 , m 3 t. pm 3 , p 3 m, pt 3 , p 3 t, Model 45. P,M,T.PMT,6pJmit. b.) MT.PM,PT, the rhombic dode- cahedron predominant. 108 CLASS III. ORDER V. GENUS I. Genus 1 Continued: c.) MT + .PM + ,P + T, the pentagonal dodecahedron, predominant. Examples : p,m,t,M|T.P|M,Pf T. Md. 47. p,m,t,MJT,PJM,PfT,PMT. Model 48. d.) MT+, M+T. PM+, P+M, PT+, P+T, the tetrakishexahedron, predominant. Similar to Md. 45, but having p,m>t smaller. e.) PMT, the regular octahedron, predominant. Example : P,M,T. PMT, Sp^mt. /) | PMT, the tetrahedron, predom*. p,m,t.JPMT,i(3pimt). g) 3P_MT, the icositessarahedron? predominant. Example : p,m,t. 3P|MT. h.) 3PJVIT+, the hemihexakisocta- hedron, predominant. Exam.: p,m,t. 3PJMJT. Model 46. f.) 6P_MT + , the hexakisoctahe- dron, predominant. Exam.: p,m,t.6PiM|T. Group a. Galena m2jgr 7f Native Copper m2|r 8f Bright White Cobalt... m5i gr 6J Iron Pyrites m6| brbk5 Red Oxide of Copper itlSf r 6 Chloride of Sodium tp2 w 2i Fluorspar Md. 45 tp4 w 3f Diamond tplOw 3J Group b. Amalgam m3j w 13| Group c. Bright White Cobalt... m5i gr 6i Iron Pyrites Md.47,48 m6| brbk5 Group d. Fluorspar tp4 w 3f Group e. Galena m2|gr 7| Native Gold m2| y 15 Bright White Cobalt... m5i gr 6i Group e Continued: Iron Pyrites m6| brbk5 Red Oxide of Copper itl3| r 6 Group f. Grey Copper ra 3i bk 5 Group g. Sulphuret of Silver m2| gr 7-j- Native Gold m2|y 15 Grey Copper m3jbk 5 Iron Pyrites m6i brbk5 Analcime tp5| w 2 Group h. Iron Pyrites Md. 46 ...m6i brbk5 Group i. Fluorspar tp4 w 3f CLASS 3. ORDER 4. GENUS 2. < Right square prism, with the verti- cal edges bevelled, and the ter- minal edges, or the solid angles, replaced by oblique planes. Ex. : P+,M,T, mt, m^t, mf-t. p|m, p|t. Uranite tl 2| gn ty Murio Carbo e . of Lead tp 2| w 6 Apophyllite tp4|w 2J Wernerite tp 5i grw 2J Somervillite - 5jy - Idocrase tl 6| w 3f CLASS 3. ORDER 5. GENUS 1. The regular six-sided prism, Model 7, with the terminal edges, or the solid angles, or both, replaced. Examples : P,T,M|f T 2 . pm, pm 2 t}|. Md. 56. P,T,M J-f T 2 . * P |m Zn, |p|m 2 tif . = P,V.r| Zn. Model 57. P,T,MjfT 2 .pft,pfmift 2 . Md.58. P,T,M-;fT 2 .pm, p ift,pm 2 tJ-i, p|fmift 2 . Md. 52 without the six rectangular vertical planes. , CLASS III. ORDER V. GENUS III. 109 Genus 1 Continued: Graphite ml|bk 2| Black Tellurium m2j w 6A Polybasite m2| bk 6i Vitreous Copper m2|gr 5| Magnetic Iron Pyrites m 4 bk 4f- Osmium-Iridium m5 gr 19 Specular Iron m6 rbr 5J One-axed Mica tp2| wgr 3 Red Silver o 2i r 5f Cinnabar tl 2| r 8 Calcareous Spar tp3 grw 2| Dolomite tl 3| grw 3 Phosphate of Lead tl 3| yw 7 Arseniate of Lead tl 3f yw 7 Coquimbite tl ? w- ? Pyrosmalite o 4J br 3 Apatite Md. 58 tl 5 w 3i Eudialyte o 5i w 3 Nepheline tl 6 w 2| Quartz tp7 w 2f Beryl Md. 56 tp7l w 2f Corundum Md. 57 tl 9 w 4 CLASS 3. ORDER 5. GENUS 2. Regular twelve-sided prism, Md. 10, P,m,T, m^tif , M|f T 2 , or P,V, v, with the terminal edges or the solid angles, or both, replaced. Example : P,m,T, m 2 t-{f, Mjf T 2 . pm, pj|t, Md.52. Vitreous Copper m 2| gr 5 1 Magnetic Iron Pyrites m4 bk 4f- Pinite o 2| w 2| Calcareous Spar tp 3 grw 2 J Phosphate of Lead tl 3| yw 7 Apatite tl 5 w 3| Eudialyte o 5w 3 Nepheline tl 6 w 2f- Dichroite tl 71 w 2| Beryl tp7| w 2 3 5 CLASS 3. ORDER 5. GENUS 3. Divided into five groups : a.) The prism has 4 vertical planes. Example: P_,M|-T. p|t. The planes pjt meet and form two edges on the equator. Hence the equator is six-sided, al- though the prism has only four vertical planes. b.) The prism has 6 vertical planes. Example: P + ,T, M T 8 7 T.pi|m. Model 55. c.) The prism has 8 vertical planes. Examples : P_,m,t + , Mf T. pf m, p|t. Md. 50. P+ ,M_,T,M|T.PfM,P\T, pfm|t, Model 51. Znw Zsw. Model 53. d) The prism has 9 vertical planes. Exam, of the 9 vertical planes : ,t, JM+T, m_t. M_T. _ e.) The prism has above 9 vertical planes. Example: p + ,M,T,mit,mit.P||T. Group a. Arsenical Pyrites ...... m 5 f bk 6J White Iron Pyrites ....m6J bk 4f Sulphur .................. tl 2 yw 2 Heavy Spar .............. tl 3| w 4f Herderite ................ tl 5 w 3 Prehnite .................. tl 6J w 3 Group b. Sulphuret of Bismuth m 2i gr 6f Brittle Sulphr'. of Silv r . m 2i bk 6 1 Antimoniai Silver ...... m 3^- w 9i Koenigite ................. tp 2i gn 31 Caledonite ............... tl 2f gnw6| Sulphate of Lead ....... tl 2| w 6J Celestine .................. tp3i w 3J Heavy Spar .............. tl 3J w 4| Witherite ................ tl 3J w 4J White Lead Ore ....... tl 3J w 6 110 CLASS IV. ORDER I. GENUS I. Group b Continued: Strontianite tl 3| w 3f Thomsonite tp5 w 2f Siliceous Oxide of Zinc o 5 w 3| Prehnite t!6|w 3 Forsterite tl 7 w - Andalusite o 7i w 3 Staurolite Md. 55 o 71 w 3| Topaz tp8 w 3| Group c. Graphic Tellurium m If gr 5 1 White Tellurium m2|ywlO Bournonite m2| bk 5| Pyrolusite m2|brbk4f Nitre tp2 w 2 Hopeite tl 2| w 2| Heavy Spar Md. 50 ...tl 31 w 4| Desmine tp3| w 2 Thomsonite tp5 w 2f Augite o 5i wgr 3^ Group c Continued : Prehnite tl 6i w 3 Olivine Md. 51 tl 6| w 3f Group d. Pinite o 21 w 2| Tourmaline tp7i w 3 Tourmaline o 71 bk 3 Group e, Graphic Tellurium m 1 1 gr 51 Bournonite m2| bk 51 Antimoriial Silver m 3 \ w 9J Tantalite io6 brbk6 Columbite 106 brbk6 Celestine tl 31 w 3J Heavy Spar tl 31 w 4f White Lead Ore tl 3i w 6i Euchroite tlSfgn 3f Thomsonite tp5 w 2f Olivine tp6| w 3f Humite tp6| y Topaz tp 8w 31 Order 4. Rhombo- Quadratic Equator, < CLASS IV. INCOMPLETE PRISMS COMBINED WITH COMPLETE PYRAMIDS. f .Genus 1. Axes: p a m a t a . Order 1. Square Equator, j Genus 2. Axes:p a m a t a . Order 2. Rectangular Equator, Genus 1. Axes: p a m a t*. f Genus 1. Axes: p a m a t a . Order 3. Rhombic Equator, J Genus 2. Axes : p a m a t a . ( Genus 3. Axes: pi m*. t a . Genus 1. Axes: p a m a t a . Genus 2. Axes: p a m a t a . Genus 1. Axes: p^m^t^. Genus 2. Axes: Pxm a 4 t a 3 . Genus 3. Axes : p a m a t*. c.) The Regular Octahedron pre- dominant. Examples : mt.pm,pt,PMT. Model 64. MT.PM, FT, PMT, 3pmt, mt.pm,pt,PMT,3p + mt. Group a. Sulphuret of Silver m 21 gr 1\ Bismuth m 2J w 91 Copper m2|r 8f Silver m2| w 10| Gold m2|y 15 Order 5. Rhombo-Rectangular Equator { a.) b.) CLASS 4. ORDER 1. GENUS 1. Divided into three groups : The Rhombic Dodecahedron alone. MT.PM, PT. Md. 63. The Rhombic Dodecahedron predominant, but combined with other forms. Examples : MT.PM, PT,pmt. Md. 65. MT.PM, PT,ipmt. MT.PM,PT,3pmt (MT.PM, PT) x 2. CLASS IV. ORDER I. GENUS II. Ill Group a Continued : Amalgam' m3| w 13| Grey Copper, Mixed... m3jbk 5 Do. Antimonial mS^bk 5 Do. Arsenical m4 rgr 4J Magnetic Iron Ore m6 bk 5 Red Oxide of Copper itl3| r 6 Muriate of Ammonia. . .tl 1 f w 1 i Chloride of Silver o 1 J grbr5f Chloride of Sodium ....tp2 w 2i Zinc Blende tl 3| wbr 4 Fluorspar tl 4 w 3^ Sodalite tl 51 w 2| Cancrinite tl 51 w 2J Nosian tl 5| gr 2i Lapis-Lazuli tl5|bl 3 Hauyne tl 7 w 2f Garnet tl 7 w 4 Rhodizite tl 7 w 3 Uwarowite tp7 w of Pyrope tl 7 w 3| Pleonaste, bk tl 71 w -3f Spinel, r,bl tp 8 w 3 Diamond tplOw 3^ Group b. Bismuth m2| w 91 Copper m2| r 8f Silver m2| w lo'l Amalgam m3|w 131 Grey Copper, Antim 1 . m3jbk 5 Magnetic Iron Ore m6 bk 5 Red Oxide of Copper it!3| r 6 Zinc Blende tl 3| wbr 4 Nosian tl 5| gr 21 Lapis Lazuli tl 5f bl 3 Hauyne tl 7 w 2f Rhodizite tl 7 w 3 Boracite tl 7 w 3 Pleonaste, bk tl 7i w 3|- Diamona 1 tplO w 3J Group c. Bismuth m2l w 91 Galena m2| gr 7f Copper m2f r 8f Gold m2|y 15 Amalgam m3l w 13! Group c Continued : Zinc Blende m3| wbr 4 Arsen 1 . Grey Copper m4 rgr 4f Magnetic Iron Ore m6 bk 5 Iron Pyrites m6Jbrbk5 Franklinite m6Jbr 5 Red Oxide of Copper itl3| r 6 Chromate of Iron io 5 J br 4| Fluorspar tl 4 w 3f Rhodizite tl 7 w 3 Pleonaste, bk tl 7^ w 3f Spinel, rbl tp8 w 3 Diamond tplO w 3} CLASS 4. ORDER 1. GENUS 2. Divided into two groups : a.) The prism has four vertical planes. Examples : MT. pm, pt. MT.PfMT. Model 61. (M,T. Pf M, P|T) x 2. Md.62. MT. pf m, pf t, P|MT. b.) The prism has eight vertical planes. Examples : M,T, MT.PfMT. Model 60. m,t, MT. p 2 mt, Pf MT, p 2 mt 3 , p 2 m 3 t. M,T,MT.PfM,PJT. Md.59. M,T, MT. Pf M, Pf T, pjmt. Group a. Sulphuret of Silver m 2J gn 7j Magnetic Iron Ore m6 bk 5 Chloride of Mercury... tl H w 6| Muriate of Ammonia. . .tl 1 j w 1 J Mellite tp2w If Murio-Carb e . of Lead tp2|w 6 Molybdate of Lead tl 3 w 6j Tungstate of Lead tl 3 gr 8 Apophyllite tp4|w 2J Phosphate of Yttria ....o 4J b 4 J Rutile t!6br 4i Oxide of Tin Md. 62 tl 6i br 7 Garnet tl 7 w 3 Zircon Md. 61 tp7w 4^ Spinel tp8 w 3$ 112 CLASS IV. ORDER IV. GENUS I. Group b. Chloride of Mercury. . .tl 1 J w 6| Murio- Carbn 6 . of Lead tp 2| w 6 Wernerite Md. 59 tpSJgrw 2f Oerstedtite o S^br 3f Anatase t!5fw 3J Rutile tl6Jbr 4J Oxide of Tin tl 6| br 7 Zircon Md. 60 tp7i w 4 CLASS 4. ORDER 2. GENUS 1. Scalene Octahedron, with the lateral angles replaced by the planes of a right rectangular prism. M_,T.P x M y T z . Desmine .................. tpSfw 2 Seorodite ................. tl 3 j w 3f- Harmotome .............. tl 4| w 2f CLASS 4. ORDERS. GENUS 1. Combinations containing either the Tetrakishexahedron or the Pen- tagonal Dodecahedron. Exam.: Model 68. Copper Md. 68 ......... m2|r Sf Gold Md. 68 ............ m2f y 15 Iron Pyrites ............. m6brbk5 Fluorspar ........ .......... tl 4 w 3f CLASS 4. ORDER 3. GENUS 2. , Mf T. PJMT. Rutile tl 6|br 4J CLASS 4. ORDER 3. GENUS 3. Rhombic prisms terminated by rec- tangular or scalene pyramids; or scalene octahedrons with the horizontal edges replaced by the vertical planes of rhom- bic prisms. Examples : MftT.Pipl&T. Model 66. M_T. P*M y T z , p x m y t. 7 . MfT.p|m,p|t. [The characteristic relates to the prism.] Sulph 4 . of Antimony |f m 2 gr 4f Antimonial Silver T 7 ^. . .m 3| \v 9J Manganite f ............. io 4 rbr 4J Lievrite f ................ io5|gnbk4 Sulphur T % Md. 66 ...... tl 2 yw 2 Sulph e .of Magnesia f^ tp 2 w 1 1 Sulphate of Zinc ^ tp2Jw 2 Chromate of Lead }-f o 2| ry 6 Celestinef ............... tpSjw 3J Heavy Spar J, f ......... tl 3J w 4 J Mesotype f Md. 67...tp5 w 2J Aeschynite \ ............ o 5^bk 5| ,Jf ............ tp8 w 3i CLASS 4. ORDER 4. GENUS 1. Divided into three groups : a.) MT.PM,PT predominant. Ex.: MT.PM, PT, 3pjmt. MT. PM, PT, PMT, Spjmt. MT. PM, PT,3PiMT,6pJmit. b.) PMT predominant. Examples: mt. pm, pt, PMT, 3p^mt. mjt, mf t. pirn, pf m, pjt, pf t, PMT. c.) 3P|MT predominant. Example: mt.pm, pt, 3PJMT. Md. 69. Group a. Amalgam ................. m3j w 13| Grey Copper, Arsen 1 . rn4 rgr 4|- Magnetic Iron Ore ..... m6 bk 5 Red Oxide of Copper it!3 J r Garnet .................... tl 7 w Pleonastebk ............ tl 7t w Spinel r, bl ............... tp8 w Group b. Magnetic Iron Ore ..... m6, bk Iron Pyrites ............. m6Jbrbk5 6 4 3| Red Oxide of Copper it!3| r Fluorspar tl 4 w Pleonaste bk tl 7i w Group c. Arsen 1 . Grey Copper... m 4 rgr Garnet Md, 69 tl 7 w 6 3f 3* 4! 4 CLASS IV. ORDER V. GENUS II. 113 CLASS 4. ORDER 4. GENUS 2. Combination of the right square prism with bevelled edges (the 12-sided or 16-sided prism of the pyramidal system), with square- based pyramids. Ex. : M,T, mlt, mftPfMT. m,t,MT,mJt,mft.p|m,pft, PfMT. Mellite tp2w If- Apophyllite tp4Jw 2J Wernerite tp 5 J grw 2 J Rutile tl6Jbr 4 Oxide of Tin tl 6|br 7 CLASS 4. ORDER 5. GENUS 1. Divided into four groups: a.) Regular six-sided prism with a rhombohedral or three-faced pyramid at each end. Exam. : T,M{fT 2 .iPMZn,iPM 2 Tj-f. or V.R,Zn. Model 71. T,M jf T 2 . iPT Zw, iPMif T 2 . or V. R! Zw. Model 72. b.) Regular six-sided prism with regular six-sided pyramid at each end. Examples: T,MifT 2 .PT,PfMifT 2 . or V. 2R . Model 73. T,MffT 2 .PifT,PifMjfT 2 . orV.2RJ-f. Model 74. c.) Regular six-sided prism termin- ated by two or more dissimilar pyramids, but without scaleno- hedrons. Examples : V. Rf Zw, r| Ze. T,M if T 2 . p|m, Pf T, pim 2 t}f , d.) Regular six-sided prism termin- ated by complex pyramids with scalenohedrons. Examples: T, M|f T 2 . K3 V. 2RZwZe,3s+. Group a. Red Silver o 2r 5f- Sulphato-tricarbonate of Lead, tp2Jw 6 Calcareous Spar tp 3 grw 2J Dioptase tp5 gn 3|- Phenakite tp6 w 3 Willelmine o 6 rw 4^ Corundum tp 9 w 4 Group b. Sulph*. of Molybdenum m 1 gr 4 J Zinkenite mS^gr 5J Magnetic Iron Pyrites m4 bk 4J- Calcareous Spar tp 3 grw 2 J Phos e . of Lead Md.74 tl 3|yw 7 Arseniate of Lead tl 3 J y w 7 Apatite tl 5 w 3| Phenakite tp6 w 3 Quartz Md. 73 tp7 w 2| Corundum tp 9 w 4 Group c. Red Silver o 2r 5J Calcareous Spar tp 3 grw 2| Dolomite tl 3 1 grw 3 Arseniate of Lead tl 3 J yw 7 Carbonate of Iron tl 4 brw 3f Chabasite tp4w 2 Phenakite tp6 w 3 Quartz tp7 w 2| Beryl tp7|w 2f Group d' Red Silver o 2|r 5f Calcareous Spar tp3 grw 2j Carbonate of Iron tl 4 brw 3f Chabasite tp 4J w 2 Quartz tp7 w 2| CLASS 4. ORDER 5. GENUS 2. The vertical planes of the twelve- sided prism, m,T,m 2 t||,Mj|T 2 , Model 10, terminated by rhom- bohedral pyramids, or by rhom- bohedrons and scalenohedrons. Vitreous Copper m 2| gr 5| Calcareous Spar *tp 3 grw 2| Apatite tl 5 w 3 114 CLASS V. ORDER I. GENUS I. CLASS 4. ORDER 5. GENUS 3. Divided into three groups : a) The prism has either three or nine vertical planes, and the pyramidal terminations are rhombohedral, and generally different at each end. Exam. : sesw, M|JT Zn,lr 2 Zn,!R,Zs, b.) The prism has six vertical planes (= T,M_T), and the termina- tions are scalene octahedrons. Examples : T_,M|T.p|m,p3t,pfm|t. Md. 75. c.) The prism has any other num- ber of vertical planes than 3, 6, or 9 ; and the terminations are scalene pyramids. Exam.: m_,P T gM&T. Model 70. Group a. Tourmaline ............... tpTJw 3 Tourmaline ............... o 71 w 3 Group b. [The characteristic relates to the form M_T.] Sulph 1 . of Antimony |f m 2 gr 4f Antimonial Silver ^ . . .m 3 J w 9 J Group b Continued : Nitre |4 tp2 w 2 Gypsum -^ Md. 75 ... .tp 2 w 2 Sulphur tl 2 yw 2 Sulphate of Zinc -f^ . . . tp 2 1 w 2 Heavy Spar f tl 3J w 4f Witherite f , tl 3 w 4J White Lead Ore f tl 3i w 6* Strontianite f tl 3 1 w 3f Arragonite f tl 3} grw 3 Childrenite {j tl 4| w ? Prismatic Iron Ore f f tl 5J ybr 4 Monticellite tp 5| y ? Augitef^ o 5iwgr3j Brookite | tl 5| yw ? Group c, Sulphuret of Antimony m 2 gr 4f White Iron Pyrites . . ..in 6 bk 4f Tantalite 106 brbk6 Columbite 106 brbk6 Polymignite io 6Jbr 4| Sulphur tl 2 yw 2 Sulphate of Magnesia tp2w If Desmine tp3| w 2J Scorodite tl 3| w 3f Harmotome tl 4} w 2| Prismatic Iron Ore tl 5Jybr 4 Chrysoberyl tp8|w 3| CLASS V. INCOMPLETE PRISMS COMBINED WITH INCOMPLETE PYRAMIDS. n C Genus 1. Axes: Order I. Square Equator < . V Genus 2. Axes: Order 2. Rectangular Equator, Genus 1. Axes: Order 3. Rhombic Equator -, Genus 1 . Axes : Order 4. Rkombo- Quadratic Equator, / ^ enus l ' f xes : X Genus 2. Axes: OrderS. Rhombo- Rectangular Equator,... f Genus L Axes: I Genus 2. Axes: p a m a ^ m a p* m a p a m a p*m a CLASS 5. ORDER 1. GENUS 1. Combinations in which the tetrahe- dron is predominant. Exam. : mt.pm,pt, ^PMT. Md. 78. Bismuth m2Jw Grey Copper m3^bk Helvine tl 6J w t :i . t a . tt t\ t\ t a . 91 5 3 CLASS V. ORDER III. GENUS I. 115 CLASS 5. ORDER 1. GENUS 2. Divided into two groups: a.) Quadratic pyramids combined with the horizontal planes P. Examples : P_. Pf M, P^T. Model 76. p. pm, pt, PMT. Model 77. b.) Pyramidal forms combined with vertical prismatic forms. Group a. Black Tellurium mljgrbk? Copper Pyrites Md. 77 m3f gnbk4j Braunite io 6J bk 4 Mellite tp2w If Uranite ...*. tl 2J gn 3 Tungstate of Lead tl 3 gr 8 Molyb 6 . of Lead Md.76 tl 3 w 6f Tungstate of Lime tl 4 w 6 Apophyllite tp4J w 2| Anatase tl 5|w 3| Group b. Edingtonite tl 4w 2| Rutile tl 6br 4J CLASS 5, ORDER 2. GENUS 1. All with vertical prismatic planes, except Bournonite. Examples: M x ,T.PfM. Model 79 a . M_,T.iPfMZn. Model 79. M_,T. |P|M Zn. Model 79 b - Bournonite m 2 j bk 5f Pyrophyllite ? tp 1 1 gnw2f Gypsum Md. 79 tp2 w 2$ Cobalt Bloom tl 21 r 3 Heulandite tl 3f w 2 Harmotome tl 4J w 2| Tungstate of Iron ...... o 5 J rbr 7 j Augite o 5Jwgr Felspar tl 6 grw 2f Epidote Md.79" tl GJgrw 3i Olivine tp6w 3| Chrysoberyl Md. 79 a ...tp8| w 3j CLASS 5. ORDER 3. GENUS 1. Divided into nine groups : cz.) Scalene octahedrons with the apex replaced by the horizontal planes P. Example: p + . P|gM T %T. Model 80. The planes P X M, P X T may also be present. b.) Type : M_T. P X M. A right rhombic prism with homohe- dral oblique terminations be- longing to the north zone. PxMyTj, may also be present, but no form belonging to the east zone. Example : c.) Type : M_T. iPJMZn. A right rhombic prism, terminated by hemihedral oblique forms be- longing to the north zone. It may also have Jp x m y t z ZneZnw, or |p x m y t z Zse Zsw, but not p x m y t z Znw Zsw, nor |p x m y t.j Zne Zsw, nor any form belong- ing to the east zone. Exam.: . IP^MZn, ^P-/ 5 MZs, Model 81. n. Md. 84. d.) Type: M_T. P X T. A right rhombic prism, terminated by homohedral oblique forms be- longing to the east zone. It may also have the form P x M y T z but no form belonging to the north zone. Examples : MfT.PfT. Axes: pt 5 a . , Model 82. Mf T. Pf T. Axes: pim? fl ta, Model 82 b . M T G -T. P T 7 n T. Model 82 a . 116 CLASS V. ORDER III. GENUS I. e.) Type : M_T. P X T Zw. A right rhombic prism terminated by hemihedral oblique forms be- longing to the east zone. It may also have ip x m y t z ZnwZsw, or |p x m y t z Zne Zse, but not p x m y t z Zne Znw, nor ip x m y t z Zse Zsw, nor any form belong- ing to the north zone. Exam. : MfJT. Ps 6 T T Zw. Model 87. /) Type : M_T.P X M,P X T. A right rhombic prism terminated by a combination of homohedral ob- lique forms belonging both to the north and east zones. It may also have the form P x M y T z . Example : MfT. pf m, P T 3 n T. g.) Type: M_T. |P x M y T z Zne Znw Nse Nsw. A right rhombic prism terminated by a hemi- hedral scalene octahedron hav- ing, the positions Zne Znw Nse Nsw. It must have no forms belonging to the north and east zones, but it may have other hemioctahedrons in the posi- tions Zse Zsw Nne Nnw. Ex. : MfT. P+M X T Zne Znw. h.) Type: M_T. iP x M y T z ZnwZsw Nne Nse. It may also have p x m y t z Zne Zse Nnw Nsw. It must have no forms belonging to the north or east zones. i.) Type: M_T.iP x M y T z ZnwNse. A doubly oblique combination, containing three pair of parallel planes, of which two pair must belong to the prismatic and one pair to the octahedral zone, and none of which must meet at a right angle. Several other , varieties of lp x m y t z may be present. The prism M_T is frequently M_T, IM_T. Example : MfT. iP x M y T z Z 2 w, ip x M yTz ZnV, y t z Z 2 ne. Group a. Sulphur Md.80 tl 2 yw 2 Childrenite tl 4 w ? Fluellite tp ? w ? Group b. [The characteristic relates to M_T.] Arsenical Iron| m5ibk 7j Lievrite | io5|gnbk4 Lenticular Cop r .Ore -?$ tl 2i blgnS Celestine f tpSjw 3J Heavy Spar | tl 3i w 4f Wavellitef tl 3}w 2J Felspar *, Md. 81 tl 6 grw 2f Group c. [The characteristic relates to M_T.] Plagionite f m 2 J bkgr5f Glauber's Salt f tplfw H Red Iron Vitriol |-f . . .tl 2 j y 2 Chromate of Lead |f o 2} ry 6 Azure Copper Ore f ...tp3f bl 3| Baryto-Calcite |f tp 4 w 3f Wagnerite |J tl 5i w 3J Titanitef tl 5^w 3i Hornblende |g, Md.84 o 5i gr 3 Felspar J, Md.81, 81 a tl 6 grw 2f Monazite-J-^ tl 6 rw 4| Rutile \ t!6ibr 4i Gadolinite T 7 T o 6J gngr4^ Group d. [The characteristic relates to M_T.] Galena T 7 n m2|gr 7f Silver^ m2| w 10| Amalgam -^ m 3i w 1 3 J Sulphr*. of Cobalt -&. . .m 5| gr 6 Tin White Cobalt ^...m 5 igr 6* Arsenical Pyrites ,...m5Jbk 6|- Magnetic Iron Ore ^ m6 bk 5 White Iron Pyrites | . . ,m 6| bk 4f Red Ox. of Copper / ff itl3| r 6 Lievrite f io5Jgnbk4 Oxide of Arsenic ^ . . . o H w 3| Nitre fj tp2 w 2 Sulphfof Leadf ,Md.82 b tl 2| w 6J Celestine f tp3J w 3f Heavy Spar f ,|, Md.82 tl 3i w 4| CLASS V. ORDER IV. GENUS I. 117 Group d Continued: White Lead Ore T 6 T) , Model 82 a ......... tl 31 w 6} Muriate of Copper f . . .tl 31 gn 4f Olivenite 3 9 ............. tl 31 gnbr4| Junkeritef ............... o3|y 3 Arragonite f ............ tl 3| grw 3 Libethenite T 9 .......... tl 4 gn 3j Topas f| Md. 90 ...... tp8 w 3i Spinel T 7 P X T Zw. ar.) M,T, M_T. ^P x M y T z Zne Znw. y.) M,T, M_T. ^P x M y T z Znw Zsw. T M, aM x T. JP x M y T z \ doubly oblique Zt \ T, ^M X T. !P x M y T J combinations. Compare these groups with those of Class 5. Order 3. Genus 1. The same additional Forms are admissi- ble here that are stated there to be admissible, and the same Forms are excluded here that are said there to be excluded, from the groups of particular types. Thus, group c must have no Forms belonging to the east zone, and group g must have none belonging to the north zone. Groups containing planes of either of these zones may have any number of Forms belonging to the same zone. When 2P x M y T z is part of the type, the combination must have neither P X M nor ^P X T. When ^P x M y T z is not part of the type, it may be present on the combinations that contain \P X M or IP^T in any number of varieties. The combina- tion must contain all the Forms that are named in the type. EXAMPLES: Group< M_,M*T.PM Model 100 c M,M*T.P|-T 104 d T,MIT.PT Ill g T_,Mp T.p 7 4 m,P7T....r 110 h . 7 oT 97 j , ...... ^P + MT_Zne 2 Znw 2 ...... 1 Zn 2 e Zn 2 w,^p x m y t z V 103 m Zne 2 Znw 2 J T^MgTnesw.iPJMZnNs, 105 q 109 q ZseZsw :.../ 112 q T_,M9 3 T4P&M? 3 TZ 2 neZ 2 nw 115 s k^.ZnVzaW 1 .;....} ^ Oli M M,T,Ml?T4P 2 G iM|?T ZnwZsw 98 y 2 ............... / y M, 3 (Xt). 1P_M + T Z 2 ne ... 1 07 z Group a. Sternbergite ml^bk 41 Bournonite m2|bk 5j Antimonial Silver m3| w 9i Sulphur tl 2 yw 2 Group b. Arsenical Pyrites f . . ..m 5 1 bk 6 J White Iron Pyrites J...m6i bk 4f White Lead Ore f tl 3J w 6J Andalusite fg o 7J w 3 Group c. Allanite io6 gr 4 Heavy Spar J Md. 100 tl 31 w 4f Group d. Sulph e .of Leadf Md.l04tl 2} w 6 Celestine | tp3l w 3| Heavy Spar J, | tl 31 w 4f Libethenite fa tl 4 gn 3| Topas|9 tp8 w 3J- 120 CLASS V. ORDER V. GENUS II. Group e* Sulphate of Lead J ..... tl 2i w 6J Heavy Spar} ........... tl 31 w 4f Brochantitef ............ tp3jgn 3j LazuliteJU ............... tl 5 bi 3 AndalusiteJJ ............ o7Jw 3 ! Group f. Heavy Spar j ............ tlSjw 4f Wavellite | .............. il 3| w 2J Sil c . Ox. of Zinc J ...... o5 w 3| Grottpg. Arsenical Pyrites f ..... m 5} bk 6 J Nitre |4 .................. tp2 w 2 White Antimony f ..... tl 2| w 5f Sulphate of Lead | ..... tl 2| w 6 Witheritef .............. tl3w 4J HeavySpari ............ tl 3w 4| White Lead Ore f ...... tl 3 J w 6| Muriate of Copper $. . , tl 3 J gn 4f Arragonite Md 111 tl 3}grw 3 Libethenite^ ........... tl 4 gn 3| Sil c . Ox. of Zinc ...... o 5 w 3J Prismatic Iron. Ore jjtt 51 ybr 4 Group A. Needle OreJ ............ m2Jgr 6 Mascagnine.. ............. tl 1 y ? Celestinef ............... tp3jw 3J Heavy Spar | ............ t!3^w 4J White Lead Ore J Model 110 ............ tlSJw 6J Epistilbite ^ ............ tp4w 21 SiK Ox. of Zinc J ...... o 5 w 3 J -tp8 w 3* Group i. Picrosmine tl 2| w 2 Chrysoberyl $y tp 81 w 2| 3f Orpimentf, | Anhydrite ^ White Lead Ore f Muriate of Copper | . Arragonite | Md. 97. Hypersthene |J Chrysoberyl ^ tl If y 3* tp3i grw3 tl 3J w 6J . .ti 3i gn 4J .. tl 3| grw 3 o 5 J wgr 3i tp8j w 3| Group ft. Glauber's Salt ^ tplfw H Haidingerite tp2Jw 2} Heavy Spar J tl 3 J w 4f Olivenite -& tl 31 gnbril Brochantite J tp 3| gn 3| Brookite f tl 5| yw Prehnite $ tl 6w 3 Olivine$ tp6| w 3? Chrysoberyl ^ tp 8 J w 3i Group f. Epidote Md. 101 tl 6* grw 31 Group m. Vivianite & tl If bl 2f Gay-Lussite f tp2*gr 2 Lanarkite tl 2^w 7 Vauquelinite tl 2| gn 51 Chromate of Lead | o 21 ry 6 Azure Copper Ore f Model 103 tpSlbl :>! Malachite J tl Sign 4 Wagnerite^y tl 51 w 3J Tungstate of Iron f , f o 51 rbr 7 Hornblende j o 5 gr 3 Datolitef U5Jw 3 Epidote ft tl 61 grw 3i Gadolinite /j- o 6| gngr41 Group w. Laumonite^J tl 4 w 2J Group o. Chromate of Lead o 21 ry 6 Mesotype Jg tpSJw 21 Brewsterite f tp5 J yw 2 Euclase Jf, fa tp71 w 3 Group p. Carbonate of Soda J . , .tp 1 J w If Group q. Realgar f tl Ijyr 3J Gypsum-^ tp2 w 2 Lepidolite tl 21 rw 2f Two-axed Mica ^ tp2l wgr 3 Chromate of Lead o 2f ry 6 Azure Copper Ore f ,,.tp3l bl 3J Baryto-Calcite ff ...... tp 4 w 3| Turnerite & tl 5 grw ? CLASS V. ORDER V. GENUS II. 121 Group q Continued: Mesotype j jj ............ tp5w 2J Hornblende |g Md. 1 1 2 o 5 \ gr 3 Felspar Jf Md.109,105 tl 6 grw 2f Gadolinite ............... o 6Jgngr4J Euclase ................... tp7l w 3 Group r. Red Antimony .......... ollrbr 4J Tincal |f ................. tl 2-1 w If Leadhiliite T ............ tl 2Jw 6J Glauberite ^ ............ tl 2| w 2f Phosph 6 . of Copper T 6 7 tl 4f gn 4} Augite ff ................ o 5| wgr 3J Group s. Gypsum r 9 5 Md.75, 115 tp 2 w 2j Mesotype f-g ............ tp51 w 21 Hornblende { .......... o 5Jgr 3 Euclase if, / 5 .......... t P 7|w 3 Group t. Augite wgr 3J Group u. Flexible Sulphuret of Silver i Vivianite ^ T Realgar f Chromate of Lead {f Brewsterite f Tungstate of Iron f , f Hornblende -J-g- Felspar Jf Euclase m5 yr tl If bl tllfyp o 2f ry tp51yw o 51 rbr o 5| gr tl 6 grw tl 6 1 grw tp7|w Group v. Carbonate of Soda j . . . tp 1 1 w If Cobalt Bloom | tl 2r 3 Tincal }f t!2iw If Group v Continued: Leadhillite tl 2 J w Azure Lead Ore f o 2| bl Laumonite }J tl 4 w Phosphate of Cop r . T 6 T tl 4| gn Augite |f o Sahlite ff o 5|wgr 3^ Group w. Glauber's Salt |f tpl| w 1J Group x. Gypsum -f^ tp2 w 2& Pharmacolite /o o 2w 2| Mesotype | g tp5^-w 2J Hornblende {0 o 5|gr 3 Euclase J tp7Jw 3 Group y. Gypsum -$ tp2 w 2^ Augite |f Md. 98, 99 o 5 1 wgr 3i Acmite jf o 61 ygr 31 Group z. [Including several complex combinations likely to be mistaken, when held in cer- tain positions, for doubly oblique combina- tions.] Allanite io6 gr 4 Boracic Acid o l|w 1 $ Blue Vitriol tl 2jw 2J Tabular Spar t!4Jw 2f Titariite tl 5J w 3 Babingtonite o 5f bk 3fc Felspar Md. 105 tl 6 grw 2f Latrobite 06 rw 2| Diaspore 06 gr 2f Cyanite Md. 107 tl 6 w 3f Albite tl 6 w 2f Petalite tl 6Jw 2 Axinite Md.81 b tl 6| w 3 122 CLASS VI. INCOMPLETE PYRAMIDS. ^ Genus 1. Axes: p a m a t a . Order I. Square Equator, j ^^ ^ ^^.^^ Order 2. Rectangular Equator, Genus 1 . Axes : ? Order 3. Rhombic Equator, Genus 1 . Axes : p a m a t a . Ocfer 4. Rhombo- Quadratic Equator, Genus 1. Axes: p a m a t a . Order 5. Rhombo- Rectangular Equator, ... Genus 1. Axes: p^m* t*. CLASS 6. ORDER 1 . GENUS 1 . Divided into two groups : a.) The regular tetrahedron. Ex. : 1PMT. Model 117. b.) The right and left tetrahedron. Ex.: 1PMT, Ipmt. Md. 118. Group a. Silver m2|w 10J Gold m2|y 15 Grey Copper m 31 bk 5 Copper Pyrites m 3| gnbk4^ Bismuth Blende. tl 3| ygr 6 Zinc Blende tl 3| \vbr4 Helvine t!6w 3 Spinel, r, bl tp 8 w 3 Automalite o 8 w 4 Group b. Bismuth m2w 9| Grey Copper m 31 bk 5 Copper Pyrites m 3| gnbk4 J Magnetic Iron Ore m6 bk 5 Zinc Blende tl 3| wbr 4 Helvine tl 6J w 3 Spinel, r, bl tp8 w 3^ Automalite o 8 w 41 CLASS 6. ORDER 1. GENUS 2. JPMT or ipigfiMT predominant. Copper Pyrites m3| gnbk4 CLASS 6. ORDER 3. GENUS 1. J(3P_MT). Example: l(3PiMT). Model 119. Grey Copper m3jbk 5 Copper Pyrites m 3|gnbk4^ Bismuth Blende tl 3f ygr 6 CLASS 6. ORDER 4. GENUS 1 . Divided into two groups : a.) |PMT predominant. Example; 1PMT, l(3plmt). b.) 1(3P^MT) predominant. Ex. : IPMT Znw, i(3PiMT) Z 2 nw. Group a. Grey Copper m 31 bk 5 Group b. Grey Copper m 3J bk 5 Arseniate of Iron tl 21 gn 3 Bismuth Blende tl 3| ygr 6 CLASS 6. ORDER 5. GENUS 1. pJ-gt, P|gM T %T. Model 120. Sulphur tl 2 yw 2 123 SECTION IV. A DESCRIPTIVE CATALOGUE OF THE MODELS OF CRYSTALS EMPLOYED TO ILLUSTRATE THIS SYS- TEM OF CRYSTALLOGRAPHY. THE Models are one hundred and twenty in number, and, with two or three exceptions, they all represent crystals that have been found among Minerals. They are arranged in this catalogue according to the classi- fication of crystals adopted in PART II. SECTION III. I have added to the description of every model, the name of a mineral which it particularly characterises; but as the same crystal often indi- cates a variety of minerals, I have prefixed to each model the number of the Class, Order, and Genus, in which an account is given of all the minerals that have been found crystallised in the shape of that model. The positions of the axes p a m a t a in every model, are indicated by'the letters P, M, T, stamped upon each, as already explained at page 2, PART I. Since the models were stamped, I have changed my views regarding the positions in which some of them should be held for examination and description. In such cases, new numbers are given to the models, and the alterations are fully described in the catalogue. To render the positions of the equator and of the north and east meridians perfectly evident, I have marked portions of them on many of the models with coloured inks. The reader will be so good as to remark that a brown line indicates the equator, a blue line indicates the north meridian, a purple line indicates the east meridian. The axis p a of every combination is situated in the intersection of the north meridian with the east meridian ; the axis m a , in the intersection of the north meridian with the equator; and the axis t% in the intersection of the east meridian with the equator. Consequently, the meeting of a blue line with a purple line upon a model shows the position of a pole of the axis p% the meeting of a blue line with a brown line, shows the posi- tion of a pole of the axis m a , and the meeting of a purple line with a brown line, shows the position of a pole of the axis t a . In general, the three poles thus indicated by the coloured lines on the models, are Pz, m, t a w . It is also necessary to remark, that since the brown line is on the prismatic zone, all planes that cross it evenly belong to the prismatic series, and are expressed by the symbols M, M X T, or T ; that since the blue line is on the north zone, all planes that cross it evenly are expressed by the symbols P, P X M, or M ; and that since the purple line is on the east zone, all planes that cross it evenly are expressed by the symbols 124 DESCRIPTION 'OF THE MODELS OF CRYSTALS. P, P X T, or T. There being no lines drawn to indicate the octahedral zones, the planes belonging to those zones are to be sought for in the octants or open spaces enclosed by the three coloured lines. The angles of inclination of the planes of the different forms upon one another, are given in the " Table of Angles,'' in PART I. It may be considered desirable in some instances to extend the coloured lines completely round the models, or to make other marks with the same colours. I think it, therefore, not improper to state in what manner the coloured inks are prepared. Blue Ink. This is prepared by dissolving pounded indigo in concentrated sulphuric acid, and diluting the solution with thick gum-arabic water. Brown Ink. Boil catechu in water, in such proportion as will make a pretty strong decoction. To this, when cold, add a solution of Bichromate of Potash. The weight of the catechu should be about four times that of the Bichromate of Potash. It is unneces- sary to add gum arabic to this ink, because the decoction of catechu is of itself sufficiently thick. Purple Ink. Boil logwood in water in such quantity as will make a strong decoction. To this, when perfectly cold, add a solution_of Protomuriate of Tin. Thicken with strong gum-arabic water. Write with a clean quill pen, having a fine point. Black lines may be written on the models with japan writing ink, or with common writing ink thickened with gum, and these can be effaced by diluted muriatic acid. The black lead pencil may also be employed to write with, and its traces yield to the action of caoutchouc. If durable marks are desired, it is best to mix amber varnish with turpentine, and colour it with lamp black or vermillion. Those who wish to remove the coloured lines, need only wash the brown lines with diluted muriatic acid, or the blue lines and purple lines with a solution of bleaching powder, to effect what they desire. The models can be cleaned from dust by caoutchouc, and from grease by soap and water. The coloured marks can also be removed by a solu- tion of caustic potash. B 5 N0 . OF COMPLETE PRISMS. g g i MODEL - 1. 1. 1. 1. P,M,T. The cube Galena. 1. 1. 2. B. P^,M,T. A short right square prism Entile. [According to the letters stamped upon this model, it is P T 5 T ,M,T.] 1. 1. 2. 3. PJ,M,T. A long right square prism Apophyllite. 1. 1. 2. 4. P + ,M,T, mt. A right square prism, M,T, with the lateral edges replaced by another right square prism, mt. Egeran. 1. 2. 1. 5. P + ,M_,T: or P},M T %,T. A right rectangular prism Anhydrite. 1 . 3. 1 . 6. P_,M| T. A right rhombic prism Heavy Spar. 1. 5. 1. 9. P x ,T,MjfT 2 :orP x ,V. The regular six-sided prism. Apatite. 1. 5, 3. 8. P X ,T,M_T. A right rhombic prism, M_T, with the acute lateral edges replaced by the form T. Axes: p x n4t a . Altered to No. 79 a Chrysoleryl. I- 5. 3. 9. (P + ,T,M T 8 T T) X 2. A twin crystal of a right rhombic prism, M T 8 yT, with the acute lateral edges replaced by the form T. Axes : p a x mt a Staurolite. DESCRIPTION OF THE MODELS OF CRYSTALS. 125 C. O. G. NO. 1. 5. 2. 1O. PLm, T,m 2 tf|,MJfT 2 : or P x ,V,T,pJm|t. A right rectangular prism, No. 5, with its vertical edges replaced by the rhombic form M|T, its long terminal edges by the form PJMF, its short terminal edges by the form P^T, and its solid angles by the scalene octahedron p|m^t Olivine. 3. 5. 2, 5S. P,m,T, m 2 t-Jf, Mi|T 2 . pm, pjt, pm 2 tif, pjgmjjt,: or P, V, v. 2r, Zn Zs, 2r|f Zw Ze. The regular twelve-sided prism, No. 10, with its terminal edges replaced by a regular six-sided pyramid belonging to the east zone, 2rJ J Zw Ze, similar to No. 26, and its solid angles re- placed by the six-sided pyramid 2^ Zn Zs, also similar to No. 26, but belonging to the north zone Beryl. 3. 5. 3. 53. p+,M,T, Mfo T . JP^MJOT Znw Zsw. A right rhombic prism, p + ,M|5T, with its obtuse vertical edges replaced by the form M, its acute vertical edges by the form T, and its four terminal edges Znw Zsw Nne Nse by the obtuse scalene hemioctahedron iP^MJJT Augite. 54. Altered to No. 101 a . 3. 5. 3. 55. P + ,T,M T 8 7 T.p}|m. A right rhombic prism, P + ,M T 8 7 T, with its acute vertical edges replaced by the form T (constituting the individuals of No. 9)? and its obtuse solid angles replaced by the form pj^m, which occupies the positions Zn Zs Nn Ns Staurolite. 3. 5. 1. 56. P,T, M}f T 2 . pm, pm 2 tif : or P,V. 2^ Zn Zs. The regular six-sided prism, No. 7, with its solid angles replaced by the planes of a regular six-sided pyramid, similar to No. 26, but in a different position Beryl. 3. 5. 1. 5*. P,T,M}fT 2 .-ip|mZn,ip|m 2 t}-f : or P,V.r|Zn. The regular six-sided prism, No. 7, with its alternate solid angles replaced by the planes of an acute rhombohedron, whose zenith planes have the positions Zn ZseZsw Corundum. DESCRIPTION OF THE MODELS OF CRYSTALS. 129 C. O. G. NO. 3. 5. 1. 58. P,T,M{fT 2 .pf-t, pmlft 2 : or P,V. 2ri Zw Ze. The regu- lar six-sided prism, No. 7> with its terminal edges re- placed by the planes of an obtuse regular six-sided pyramid, similar to No. 26 Apatite. INCOMPLETE PRISMS COMBINED WITH COMPLETE PYRAMIDS. 4. 1. 2. 59. M,T, mt. Pf M, Pf T. A right square prism, No. 3, with its lateral edges replaced by the planes of another right square prism, No. 4, and its terminal edges and terminal planes replaced by the planes of an obtuse quadratic octahedron similar to No. 12, but composed of forms belonging to the north and east zones Wernerite. 4. 1. 2. 6O. M,T, mt. PfMT. A right square prism, No. 3, with its lateral edges replaced by another right square prism, No. 4, and its solid angles and terminal planes replaced by the planes of the obtuse quadratic octahedron, PfMT, No. 12 Zircon. 4. 1. 2. 61. MT. PfMT. The obtuse quadratic octahedron, PfMT, No. 12, with its horizontal edges replaced by the vertical planes of a right square prism, MT Zircon. 4. 1. 2. 6. (M,T. PfM, Pf T) X 2. A hemitrope or twin crystal, each individual of which is an obtuse quadratic octahedron similar to No. 12, having its horizontal edges replaced by the vertical planes of a right square prism, M,T, forming a combination similar to No. 6 1 . The plane of junction is P_MT Nse Oxide of Tin. 4. 1. 1. 63. MT. PM, PT. The rhombic dodecahedron Garnet. 4. 1. 1. 64. mt. pm, pt, PMT. The regular octahedron, No. 15, with its edges replaced by the planes of the rhombic dodeca- hedron, No. 63 Red Oxide of Copper. 4. 1. 1. 65. MT. PM, PT, pmt. The rhombic dodecahedron, No. 63, with its obtuse three-faced angles replaced by the planes of the regular octahedron, No. 15 Amalgam. 4. 3. 3. 66. M T 8 (T T. PjgM T 8 T. The acute scalene octahedron, No. 21, with its equatorial edges replaced by the vertical planes of a rhombic prism, M r %T Sulphur. 4. 3. 3. 6?. M|9-T.|P_MA9TZneZnw,iP_M|gTZseZsw. A rhombic prism nearly square-based;, with its terminal edges and terminal planes replaced by the planes of an obtuse scalene octahedron, or by the planes of two obtuse scalene hemioctahedrons differing a little in size Mesotype. 4. 3. 1. 68. MT, MfT.PlM,PfM, IHT,PfT. A tetrakishexahedron. Native Copper, J50 DESCRIPTION OF TUE MODELS OF CRYSTALS. c. 4. 4. 1. 69. mt. pm, pt, 3P|MT. The icositessarahedron, No. 22, combined with the rhombic dodecahedron, No. 63. The planes of the latter are distinguished by their rhombic shape, whereas the modified planes of the icositessara- hedron are hexagonal Garnet. 4. 5. 3. ffO. m_. P|$M T 8 (T T. The scalene octahedron, No. 21, with its obtuse solid angles replaced by the planes of the prismatic form m Sulphur. 4. 5. 1. 1. T,M|fT 2 .lPMZn,JPM 2 T{4: or V. R, Zn. The regu- lar six-sided prism, No. 7, terminated by the obtuse rhombohedron, No. 26 a , but having the latter situated on the north instead of the east zone Calcareous Spar. 4. 5. 1. 7%. T,MjfT 2 . |PTZw,iPM}fT 2 : or V. R, Zw. The regu- lar six-sided prism, No. 7, terminated by the obtuse rhombohedron, No. 26% the planes of both forms retaining their usual positions. Or, we may describe No. 72, as the obtuse rhombohedron, No. 26% with its acute lateral angles replaced by the planes of the regular six-sided prism, No. 7 Calcareous Spar. 4. 5. 1. 73. T,M|fT 2 . PfT, PJMf|T 2 : orV.2RZwZe. The regu- lar six sided prism, No. 7, terminated by a regular six- sided pyramid similar to No. 26 Quartz. 4. 5. 1. ?4. T,M}fT 2 .PjfT.P|fM|fT 2 : or V. 2 Rjf Zw Ze. The obtuse regular six-sided pyramid, No. 26, with its equa- torial edges replaced by the planes of the regular six- sided prism, No. 7 Phosphate of Lead. 4. 5. 3. 75. T_, M&T. IPf^M^T Zne Znw, &&XL&T Zse Zsw. A right rhombic prism, M-&T, with its acute lateral edge deeply replaced by the planes of the form T, four of its terminal edges, Zne Znw Nse Nsw, replaced by the planes of the obtuse scalene hemioctahedron ^P^V^^T, and its other four terminal edges, Zse Zsw Nne Nnw, by the planes of the obtuse scalene hemioctahedron, FJ^M}T. Gypsum. INCOMPLETE PRISMS COMBINED WITH INCOMPLETE PYRAMIDS. 5. 1. 2. 76. P_.PJM, PT. An obtuse quadratic octahedron, similar to No. 12, but belonging to the north and east zones, with its two obtuse solid angles replaced by the planes of the form P Molybdate of Lead. DESCRIPTION OF THE MODELS OF CRYSTALS. 131 C. O. G. NO. 5. 1. 2. ff. p. pm, pt, PMT. A square-based octahedron, very nearly the same as the regular octahedron, with its summits replaced by the planes of the form p, and its oblique edges by the planes of the form pm belonging to the north zone, and pt belonging to the east zone; all of which forms, except p, are nearly if not absolutely equi- axed. See part II. page 36 Copper Pyrites. 5. 1. 1. *8. mt.pm,pt,iPMT. The regular tetrahedon, No. 117, with its solid angles replaced by the planes of the rhombic dodecahedron, No. 63 Mixed Grey Copper. 5. 2. 1. ?. M_,T. ^PfM Zn. A right rectangular prism, M_,T, terminated by the planes of a hemihedral form of the north zone, |PfM Zn Ns Gypsum. 5. 2. 1. !79 a . M x ,T.PfM. Aright rectangular prism, M X ,T, with dihe- dral terminations belonging to the north zone, and con- sisting of the form P^M Chrysoberyl. This model, held in the position indicated by the stamped letters, shows a very common variety of the regular six-sided prism, in which the form T predom- inates. Example: Phosphate of Lead, Vanadiate of Lead, Arseniate of Lead, &c. Symbol P X ,T_, M||-T 2 . 5. 2. 1. 79''. M_,T. |P|M Zn. A right rectangular prism, M_,T, ter- minated by the planes of the hemihedral form |P|M Zn Ns Epidote. This model, held in the position indicated by the stamped letters, represents a right prism with a rhombo- idal base, which form was adopted by Haiiy as the primitive form of Epidote. Compare No. 79 b with Nos.101 and 101 a . o. 3. 1. SO. p + . P-J-gMy^T. The scalene octahedron, No. 21, with the summits replaced by the horizontal form P Sulphur. 5. 3. 1. 81. M^T.iPlMZn^PJL-MZs. A right rhombic prism,MJT, terminated by twohemihedral forms of the north zone; the obtuse angles Zn Ns being replaced by the form JP^M, and the obtuse angles Zs Nn, by the form JP/jM. But according to some crystallographers, (Weiss and Hose), these oblique forms are JP|M Zn, Iri-M Zs, or forms of the same kind but of different magnitude Felspar. 5. 3. 1. 81 a . JMifTne, JM-]$TIIW. JPiM Zn. A right rhombic prism similar to MJfT in No. 81, but consisting of two pair of planes of unequal size. It is terminated by a hemihedral form of the north zone, the planes of which replace the Zn Ns obtuse angles of the rhombic prism. The planes 1P/-M, which in No. 81 replace the Zs Nn obtuse angles of the rhombic prism, are absent from this com- bination Felspar. ]32 DESCRIPTION OF THE MODELS OF CRYSTALS. C. O. G. NO. 5. 3. 1. Sl a . Continued: Compare No. 81 a with No. 105, which contains the same forms, excepting that iMjf T nw is replaced by T. Model 81 a also nearly represents some of the forms of Felspar that are described by the symbols MJT. JPiM Zn. Axes: pJjLmlt a . -However, the resemblance of this model to either of the forms described is only approxi- mate, the interfacial angles being different from those of Felspar. 5. 3. 1. 81 b .MfT.JP x M y T z Z 2 nw,ip x M y T z ZnV,ip x m y t z Z 2 ne 2 . Aright rhombic prism, terminated by three tetarto-octahedral forms having the positions indicated in the symbols. This is one of the doubly oblique combinations, which all consist of one homohedral or two or more hemihedral vertical prismatic forms, with one or more tetarto-octa- hedral forms, forming three or more pair of planes situ- ated obliquely to one another Axinitc. 5. 3. 1. 82. MfT. PfT. Axes: p? 2 m a t a . Aright rhombic prism, MJT, terminated by an oblique form of the east zone, P|T, the planes of which are placed on the Zw Ze Ne Nw acute angles of the prism. The eidogen of the east zone is much larger than that of the prismatic zone, so that the combination has an acicular or columnar appearance Columnar Heavy Spar. 5. 3. 1. 82 a . M T 6 n T. P T 7 oT. Axes: p a m a t a . A right rhombic prism, Mj-^T, terminated by an oblique form of the east zone, p_7_x, the planes of which have the positions Zw Ze Ne Nw Carbonate of Lead. This form is commonly considered to be a rectangular octahedron = P^M, P^T. See note at page 66. 5. 3. 1. 82 b .MfT.PfT. Axes: p| 5 m^. A right rhombic prism, Mf-T, terminated by an oblique form of the east zone, Pf T, the planes of which have the positions Zw Ze Ne Nw .... Sulphate of Lead. This model, held in the position indicated by the let- ters stamped upon it, is a rectangular octahedron. Nos. 82 a and 82 b are combinations of the same kind, only differing in dimensions. Compare No. 82 b with No. 104. 83. M_T. JP_M Zn. Altered to No. 26 a . 5. 3. 1. 84. MjgT. IP^-M Zn. Axes: p a m a t a 9 . A right rhombic prism, M| T, terminated by an oblique hemihedral form of the north zone, iP-^M, whose planes are situated ZnNs Hornblende. 85. M_T. 1P_M Zn. Altered to No. 26 b . 86. M_T. |P_M Z 2 n, Ip+mZn 2 . Altered to No. 1 14 l . DESCRIPTION OF THE MODELS OF CRYSTALS. 133 C. O. G. 5. 3. 1. 8*. M$T. iP^-T Zw. Axes: pgm? t|,. A right rhombic prism, Mf Q-T, terminated by an oblique hemihedral form of the east zone, IP^T, the planes of which occupy the posi- tions Zw Ne Augite. (Mussite). 88. M_T. |P_T Zw. Altered to No. 26 C . SO. M_T. |P_T Zw. Altered to No. 26 d . 5. 3. 1. 90. Mi|T, m|t. P|f T, p|f mj|t. Axes: p.|.mlt a . A right rhombic prism, M^g-T, which has its acute vertical edges bevelled by another right rhombic prism, m}|t, its acute solid angles replaced by an oblique form of the east zone, PffT, the planes of which are situated Zw Ze Ne Nw, and its obtuse solid angles bevelled by the planes of the obtuse scalene octahedron, p|fm^t Topas. 5. 4. 1. 91. M|T. P|M, PfT. A pentagonal dodecahedron, the planes of which consist of three equal and similar rhombic forms, the first belonging to the prismatic zone, the second to the north zone, and the third to the east zone. See Part I., page 35 Iron Pyrites from Elba. 5. 4. 1. 93. M^T.PIM, PfT, PMT. The middle crystal between the pentagonal dodecahedron, No. 91, and the regular octa- hedron, No. 15 Iron Pyrites from Elba. 5. 4. 1. 93. mjt. p|m, pf t, PMT. The regular octahedron, No. 15, with its solid angles bevelled by the planes of the pen- tagonal dodecahedron, No. 91 ; a combination resembling No. 92 in the number and positions of its planes, but differing in respect to their relative magnitude Bright White Cobalt from Tunaberg. 5. 4. 1. 94. mt.pm,pt, |PMT, |(3p|mt)Z 2 nw. Axes: p a m a t a . The regular tetrahedron, No. 117, with its solid angles re- placed by the planes of the rhombic dodecahedron, No. 63, and its edges bevelled by the planes of the triakis- tetrahedron (or hemiicositessarahedron), No. 119 Grey Copper. 5. 4. 1. 95. MT.PM,PT, ipmtZnw, -J(3pJmt)Z 2 ne. The rhombic dodecahedron, No. 63, with four of its obtuse three-faced angles replaced by the planes of the right tetrahedron, No. 117, whose positions are Znw Zse Nne Nsw, and with its six acute four-faced angles replaced by the planes of a left triakistetrahedron, a form similar in figure to No. 119, but in different positions, having its two upper planes situated Z 2 ne Z 2 sw, instead of Z 2 nw Z 2 se, as marked upon the model Zinc Blende from Kapnik. 5. 5. 1. 96. p_.PifT,Pji-MjfT 2 : or p_.2RffZwZe. The obtuse regular six-sided pyramid, No. 26, with the summits re- placed by the planes of the form p Phosphate of Lead from Johann-Gcorgcnstadt. 134 DESCRIPTION OF THE MODELS OF CRYSTALS. C. O. G. NO. 5. 5. 2. 9!?. M,T, Mf T. P T 7 oT. A right rhombic prism, Mf T, with its obtuse vertical edges replaced by the form M, its acute vertical edges by the form T, arid its acute solid angles by an oblique form of the east zone, P T 7 T, the planes of which are situated Zw Ze Ne Nw. Compare this Model with No. Ill Arragonite from Piedmont. 5. 5. 2. 98. M, T, M|T. \ P 2 6 T M^T Znw Zsw. Axes: p.|_m a tL. A combination similar to No. 53, excepting that it is without the horizontal planes P Augite. 5. 5. 2. 99. (M,T, M|^T. iP^M^T Znw Zsw) X 2. A hemitrope or twin crystal of the combination represented by No. 98. If the latter were divided into two pieces by a sec- tion passing through the north meridian, and the halves were joined together by the same plane, after in verting one of them, the result would be the same as No. 99. Augite. 5. 5. 2. fiOO. M_, M|T. P|M. Axes: p a m:Lt_|.. A right rhombic prism, MfT, No. 6, with its obtuse vertical edges replaced by the planes of the prismatic form M, and its obtuse solid angles by an oblique form of the north zone, P|M, the planes of which are situated Zn Zs Ns Nn. . .Heavy Spar. 5. 5. 2. 1O1. M_. iP^VlZn, |p||m Zs, P + MT_ Zne 2 Znw 2 . Axes: p a m!_tj|_. A right rectangular prism, M_,T, No. 5, with its Zn Ns edges replaced by the planes of an oblique hemihedral form of the north zone, ^PiM Zn Ns, No. 79 b > its Zs Nn edges replaced by another oblique hemi- hedral form of the north zone, pj-m Zs Nn, and its east and west vertical planes replaced by the scalene hemioctahedron, iP_j_MT_, whose planes are situated Zne 2 Znw 2 Nse 2 Nsw 2 , and meet at the east and west poles Epidote. 5. 5. 2. 101 a . M_, t, Mj*T. JPJMZn, ipJmZs, T + MT_ Zne 9 Znw 2 . Axes: p a mLt+. A right rectangular prism, M_,T, No. 5, with its vertical edges replaced by the planes of a right rhombic prism, M|f T, its Zn Ns edges by the planes of an oblique hemihedral form of the north zone, IP^M; its Zs Nn edges by the planes of an oblique hemihedral form of the north zone, ip|m; and its Ze Zw Nw Ne edges by the planes of a scalene hemi- octahedron, |P + MT_ the positions of which are Zne 2 Znw 2 Nse 2 Nsw 2 , Epidote. 1O2. m, M_T. iP_M Z 2 n, |p +m Zs 2 , |r x M y T, Zn 2 e Zn 2 w, |p x m y t z Zne 2 Znw 2 . Altered to No. 26 C . 5. 5. 2. 103. m,MT, |Pf 4 -MZn, } P + M_T Zn 2 e ZnV, |p x m y t z Zne* Znw 2 . A right rhombic prism, Mf T, with its obtuse vertical edges replaced by the form m, its obtuse solid angles Zn Ns by an oblique hemihedral form of the DESCRIPTION OF THE MODELS OF CRYSTALS. 135 C. O. U. NO. 5. 5. 2. 1O3. Continued: north zone, \ Pt\M, its Zne Zn\y Nse Nsw terminal edges by the planes of a scalene hemioctahedron, |P + ;M_T Zn ? eZn 2 w Ns 2 e NsV, and its acute solid angles by the planes of a scalene hemioctahedron, ip x m y t j! Zne 2 Znw 2 Nse 2 Nsw 2 Azure Copper Ore. A comparison of this model with No. 102 (26 e ) will show how nearly the oblique prismatic combinations are related to the rhombohedral combinations. 5. 5. 2. 1O4. M, Mf T. Pf T. A right rhombic prism, Mf T, with its obtuse vertical edges replaced by the form M, and its acute angles replaced by an oblique form of the east zones, Pf T, the planes of which, situated Zw Ze Ne Nw, constitute dihedral terminations to the prism Sulphate of Lead. This combination is the same as No. 82 b , excepting that it has the form M additional. 5. 5. 2. 1O5. T, iMJ&Tnesw. JPJMZnNs. This model represents the cleavage form of Felspar, and is Hawjs assumed primitive form of that mineral. It consists of the form T, half the planes of the form M|T, the usual rhombic prism of felspar, and this half holding the positions nesw, being a parallel pair of planes, and finally, the oblique hemihedral form of the north zone, ?P|M Zn Ns, the form so characteristic of the felspar crystals, where T upon PjM measures 90. The relation of this combination to the ordinary rhombic crystals of felspar, is distinctly shown by the figure drawn on the north east quadrant of No. 109. On holding the west plane of No. 105 against the east plane of No. 109, and bringing the two Zn planes to the same level, the agreement of the planes M|f T and PiM, on the two models, may be both seen and felt. Felspar from Puy -de-Dome. 1O6. M_T. iPJM Zn. Altered to 8i a . 5. 5. 2. 10 . M,|m-/otnV, IMJTnw 2 , imjtne. PJVI+T Z ? ne. A doubly oblique combination, consisting of the form M, three hemi-rhombic vertical forms, |m x t, each of them being a parallel pair of planes, and one scalene tetarto- octahedron, occupying the positions Z 2 ne N 2 sw. Cyanite. 1O8. Altered to No. 81 b . 5. 5. 2. 1O9. T, MifT. iP^MZn, }P&M Zs. This combination is the same as No. 81, with the addition of the form T Felspar. The figure drawn upon the north east quadrant of the model, is explained in the description of No. 105. 136 DESCRIPTION OF THE MODELS OF CRYSTALS. C. O. G. NO. 5. 5. 2. 11O. T_, M T n T. pjm, P^T. Axes: pjf.m a tl. A right rhombic prism, M T 0T, similar to the vertical form of the com- bination, No. 82 a , with its acute vertical edges deeply replaced by the form T, its obtuse solid angles replaced by an oblique form of the north zone, pjm, whose planes occupy the positions Zn Zs Nn Ns ; and its acute solid angles, by an oblique form of the east zone, Py>T, the planes of which are situated Ze Zw Ne Nw White Lead Ore. 5. 5. 2. 111. T + , Mf T. P T 7 o-T. This combination is the same as No. 97, excepting that it. is without the form M. Arragonite from Piedmont. 5. 5. 2. 112. T,M|gT. IPy^MZn^PlM^TZseZsw. Aright rhombic prism, M|$T, with its acute vertical edges replaced by the form T, its Zn Ns obtuse solid angles replaced by an oblique hemihedral form of the north zone, ^P^M, and its Zse Zsw Nne Nnw terminal edges replaced by the planes of the scalene hemioctahedron, JrP*M y 5 T. Hornblende. 5. 5. 2. 113. (T,MjgT.iP&MZn,JPiM} 5 TZseZsw)x2. A hemi- trope. or twin crystal of the combination represented by No. 112. If we suppose No. 112 to be divided by the east meridian, one of the halves to be turned upside down, and the two to be joined together by the same planes as before, the result would be similar to No. 113. Hornblende. 5. 5. 1. 114. P_.P|T, |P|M}fT 2 : or P_. R|. An acute rhombo- hedron nearly similar toNo.26 d ,with its acute solid angles deeply truncated by the horizontal form P... Corundum. According to the letters stamped upon this model, the description of it would be, M_T. P_T Ze, P + T Zw. 5. 5. 1. 114 a .P_.iPT,JPMffT 2 : or P_. R,. An obtuse rhombohe- dron, R,, No. 26% with its obtuse solid angles truncated by the horizontal planes P Calcareous Spar. According to the letters stamped upon this model, it should be described as follows: M_T4P_MZ 2 n,|p + mZn 2 . 5. 5. 2. 115. T_,M T %T.lP T ^M r %TZ 2 neZ 2 nw. This combination is the same as that represented by No. 75, excepting that one of the obtuse scalene hemioctahedrons, namely, |P T 6 jMijT Zse Zsw Nne Nnw, which occurs upon No. 75, does not occur upon this model Gypsum from Montmartre. 116. M_T. P + T. |P x M y T z Z 2 ne Z 2 nw. Altered to No 26 f . ' DESCRIPTION OF THE MODELS OF CRYSTALS. 137 INCOMPLETE PYRAMIDS. C. G. O. NO. (>. 1. 1. 11 7. |PMT. The regular tetrahedron, or hemioctahedron, or right tetrahedron ; the four planes of which occupy the positions ZnwZse Nne Nsw Grey Copper. 6. 1. 1. US. PMT, ipmt. The right tetrahedron, with its solid angles replaced by the planes of the left tetrahedron, the latter occupying the positions Zne Zsw Nse Nnw. Grey Copper. 6. 3. 1. HO. 1(3P^MT). A triakistetrahedron, or hemiicositessarahe- dron; the hemihedral form of the combination repre- sented by No. 22 Grey Copper. 6, 5. 1. ISO. pig-t, Pf|}M T 8 (>T. An acute scalene octahedron, No. 21, with its acute terminal edges replaced by an oblique form of the east zone, pyt, which possesses the same relations to the axes p a and t a as do the planes of the octahedron itself Sulphur. INDEX. PAGE. PAGE. PAGK . Achmite ... 86 Antimonial Nickel 46 Babingtonite . 94 Acicular Bismuth Glance 72 Antimonial Silver 62 Baikalite . 81 Acide boracique 91 Antimonial Copper 64 Barytes Harmotome 75 Acmite 86 Antimony . 45 Baryte carbonated 65 Actinote . 86 Apatite 56 sulfatee 68 Actynolite Adularia 86 87 Aplome Apophyllite 29 42 Baryto-Calcite . Barytkreuzstein 84 75 Aeschynite Akmit 65 86 Aquamarine Arfvedsonite 58 86 Basaltic Augite Basaltic Hornblende 80 86 Alalite Allanite 80 74 Argent antimonial antimonie sulfur6 62 49 Berthierite Beryl 64 58 Alaun 31 muriat6 23 Berzelite . 65 Alaunstein 59 natif . 16 Bi-axial Mica 85 Albite Almandine 92 29 rouge . sulfur6 49 20 Biegsamer Silberglan Bismuth Blende 90 30 Alum 31 flexible 90 Bismuth natif 17 Alumine sulfatee 31 Argentiferous Gold 17 Bismuth sulfure 63 fluatee alkaline 38 Arragonite 66 Bismuth 17 hydro-phosphatee 71 Arseniate of Cobalt 82 Bitterspar . 55 magnesiee . 26 Copper 71 Bittersalz . 72 Alum-stone 59 Iron . 30 Black Copper 29 Alun 31 Lead . 57 Black Manganese 38 Alunite . . 59 Lime . 83 Black Garnet . 29 Amalgam . 18 Arsenic . . 46 Black Spinel 26 Amblygonite 73 Arsenic, oxide . 25 Black Lead 46 Ammonia Alum 31 Arsenic natif 46 Black Tellurium 42 Ammoniaque sulfatee 72 oxide . 25 Blatter-Zeolith . 89 muriatee 23 sulfure jaune 63 Tellur . 42 Amphibole 86 rouge 78 Blattererz 42 blanc . 86 Arsenical Bismuth 30 Bleiglanz 19 noir 86 Iron . 62 vitriol 70 Amphigene 30 Nickel 19 Blei . 18 Analcime . 30 Pyrites 64 Bleilasur 88 Analzim 30 Cobaltic . 64 Blende 19 Anatase Andalusite 35 67 Cobalt Grey Copper 18 28 Blue Carbonate of Co Iron Ore per 84 82 Anglesite . 70 Arsenik 46 Vitriol 92 Anhydrite 71 Arsenik-Nickel . 19 Boracite . , 27 Anhydrous Gypsum 71 Arsenikkies 64 Boracic Acid 91 Anorthite . 92 Arsenikwismuth 30 Borate of Lime . 90 Anthophyllite . Antimoine gris . 86 63 Arsenikbliithe Arsenikeisen 25 62 Magnesia . Soda . 27 83 natif . 45 Arseniksaures Blei 57 Borax 83 rouge . 78 Arsenikfahlerz . 28 Boraxsaures Natron 83 sulfure 63 Atakamit . 71 Borazit 27 oxide . 63 Augite 80 Borosilicate of Lime 90 oxide sulfure 78 Auriferous Silver 17 Botryogen . 90 Antimon 45 Auripigment 63 Bournonite 72 Antimonblende 78 Automalite 26 Braunite . 34 Antimonnickel . 46 Axinite 93 Brachytypous Manga ese Antimonsilber 62 Azure Copper Ore 84 Ore 34 Antimonfahlerz 29 Azure Lead Ore 88 Braunbleierz 57 Antimonglanz Antimonial Grey Cop 63 per 29 Azurite Azurestone 76 31 Braunspath Breunnerite 55 55 140 INDEX TO MINERALS. Brewsterite 89 Copper Glance 62 Emerald Copper . 57 Bright White Cobalt 25 Mica . 57 Emeraude . 58 Brittle Sulphuret of Silver 64 Brochantite . . 72 Nickel Pyrites 46 36 Endellione . . 72 Epidote ... 85 Brongniartine Bronzite 88 80 Red Oxide Uranite 24 42 Epistilbite . . 76 Epidote manganesifere 85 Brookite . 76 Coquimbite 57 Etain oxide . . 34 Brown Iron Ore 66 Cordi6rite . 74 sulfure . . 29 Spar . . . 55 Corindon . 47 Euchroite . . 71 Garnet 29 Corundum 47 Euclase . . 85 Bucklandite 85 Couzeranite 85 Eudialyte . . 58 Buntkupfererz . 26 Crichtonite 60 Euklas . . 85 Calamine . 55 Cronstedtite 61 Fahlerz . . 28 Calc Spar . 50 Cross stone 75 Fassaite . . 80 Calcareous Spar Caledonite 50 76 Cryolite Cube Ore . 38 30 Felspar . . 87 Feldspath . . 87 Cancrinite 32 Cube Spar 71 Feldspath apyre . 67 Carbonate of Barytes Copper, green G5 81 Cubicite Cubic Nitre 30 56 Fer arseniate . . 30 arsenical . . 64 Jrl 5 o blue 84 Cuivre ars6niate octat dral 76 calcardo-siliceux 74 Iron . 55 octaedre a gu 71 chromat6 . . 27 Lead . G6 carbonate bleu 84 muriate . . 58 Lime, . . 50, G6 vert 81 natif ... 18 Lime and Lead . Lime and Magnesia 56 55 dioptase gris 57 28 oligiste . . 47 hydro-oxide . 65 Lime and Soda . 88 arsenifere . 29 oxidule . . 27 Magnesia 55 muriat6 71 titane . . 60 Magnesia and Iron 55 natif . 16 phosphat6 . . 82 Manganese . 55 oxide rouge 24 spathique . . 55 Soda . 82 oxidu!6 24 speculaire . . 47 Strontian . G5 phosphat6 . 71, 82 sulfat6 . . 84 Zinc 55 pyriteux 36 sulfur6 . . 21 Celestine 70 sulfate 92 blanc . . 63 Cerium fluat6 . 47 sulfur6 62 magnelique 49 Ceylanite Chabasie Chabasite Chalkolite 58 58 42 vitreux Cupreous Sulphate of Sulphato-carbona Lead 62 Lead 88 teof 76 Fergusonite . . 38 Fibrous Malachite . 81 Fish-eye-stone . . 42 Fischaugenstein . 42 Chaux arseniat6e borat6e sijiceuse . 83 90 Cyanite Cymophane 91 74 Flexible Sulphuret of Silv. 90 Fluorcerium . . 47 carbonatee . 50 Dark Red Silver 49 Flucerine ... 47 magnesifere . 55 Datolith 90 Fluellite ... 76 datolit 90 Datholite 90 Fluoride of Cerium . 47 fluatee 23 Demant 17 Fluorspar ... 23 phosphatee . 66 Desmine 75 Fluophosphate of Mag- sulfat6e 83 Devonite 71 nesia ... 81 anhydre . 71 Diallage 80 Flusspath ... 23 Chiastolite 74 Diamant 17 Foliated Zeolite . 89 Childrenite 76 Diamond 17 Forsterite . . 76 Chloride of Silver 23 Diaspore 91 Franklinite . . 27 Sodium' 23 Dichroite 74 Fraueneis ... 83 Mercury 33 Diopside 80 Gadolinite . . 79 Chlorite 61 Dioptase 57 Gahnite '. . . 26 Chlorsilber 23 Diploite 92 Galena . . . 19 Chromate of Iron 27 Disthene 91 Galmei . . 55,72 Lead . 79 Dolomite 55 Garnet ... 29 Lead and Copper 84 Dreelite 61 Gay-Lussite . . 88 Chromeisenerz . 27 Dreelith 61 Gehlenite . . . 41 Chromsaures Blei 79 Dvoxilite , . 88 Gelbbleierz . . 39 Chrysoberyl 74 Edler Granat . 29 Gemeiner Augite . 80 Chrysolite . 67 Edingtonite 43 Gemeiner Granat . 29 Cinnabar . 46 Efflorescent Zeolite 89 Glassy Felspar . . 87 Cinnamon Stone 29 Egeran 41 Glaubersalz . . 83 Cleavelandite . 92 Einaxiger Glimmer 58 Glauber's Salt . . 83 Cobalt arseniate 82 Eisenvitriol 84 Glauberite . . 88 arsenical 18 Eisen 18 Glimmer . . 58, 85 gris 25 Eisenkies . 21 Gold ... 17 sulfur6 21 Eisenglanz 47 Grammatite . . 86 Cobalt Bloom . 82 Eisenspath 55 Granat ... 29 Cobaltine . Columbite . 25 65 Eisspath Eis . 87 47 Graphic Tellurium . 90 Graphite ... 46 Colophonite 29 Electric Calamino 72 Grau-Spiesglanzerz . 63 Copper . . Copper, Black, . 16 29 Electrum . Emerald 17 58 Green Carbonate of Copper 81 Vitriol . . 84 INDEX TO MINERALS. 141 Green Garnet . . 29 Kreuzstein . 75 Mercure sulfure . 46 Grenat ... 29 Kryolith . . 38 Mesitinspath . . 55 Grey Antimony . 63 Krysolith . . 67 Mesole ... 89 Copper . . 28 Cobalt . . 18 Kubizit . . 30 Kupfer . 16 Mesolite ... 89 Mesotype ... 89 Ore of Manganese 64 Oxide of Manganese 65 Kupferglanz . 62 glimmer . 57 Mica, one-axed . . 68 Mica, two-axed . . 85 Grossular ... 29 kies . . 36 Mispickel ... 64 Grunbleierz . . 57 lazur . . 84 Molybdate of Lead . 39 Gyps ... 83 nickel . 46 Molybdanglanz . . 46 Gypsum ... 83 uranite . 42 Molybdene sulfure . 46 anhydrous . . 71 smaragd . 57 Monazite . . 77,91 Haarkies ... 46 antimonglanz 64 Monticellite . . 77 Haidingerite . . 71 vitriol . 92 Muriate of Ammonia 23 Hallite ... 55 ) armotome . . 75 Kyanite . . 91 Labrador . . 92 Copper . . 71 Lead ... 65 Hausmannite . . 38 Labradorite . 92 Mercury . 33 Haiiyne ... 31 Labrador Felspar 92 Silver . . 23 Heavy Spar . .68 LabradorischeHornb nde 80 Soda ... 23 Hedenbergite . . 80 Lanarkite . . 88 Murio-carbonate of Lead 40 Helvine ... 31 Lanthanite . 91 Mussite ... 80 Hemi-prismatic Ruby- Blende ... 78 Lapis-Lazuli . 31 Latrobite . . 92 Myargyrite . . 78 Nadelerz ... 72 Herderite ... 77 Laumonite . 89 Nadeleisenerz . . 65 Heterosiderite . . 83 Lazurstein . 31 Nagyagererz . . 42 Heulandite . . 89 Lazulite . . 76 Natronspodumen . 92 Hexagonal Talc . 56 Leadhillite . 88 Natrolite ... 89 Hohlspath . . 74 Lead . . 18 Needle Ore . . 72 Honeystone . . 40 Lenticular arseniate of Needle Stone . . 89 Honigstein . . 40 Copper . . 76 Needle Zeolite . . 89 Hopeite ... 77 Lepidolite . 81 Nepheline . . 58 Hornsilber . . 23 Leucite . . 30 Nickel, arsenical . 19 Hornbleierz . . 40 Levyne . . 59 binarseniate . 1 9 Hornblende . . 86 Ldbethenite . 71 sulfure . . 46 Hornerz ... 23 Lievrite . . 74 Nickelglanz . . 25 Humboldtilite . . 42 Light Red Silver 49 Nickel Glance . . 25 Humboltite . . 90 Lime Uranite . 42 Nickeliferous Grey Anti- Hurnite ... 90 Lomonite . . 89 mony ... 25 Huraulite ... 83 Linsenerz . . 76 Nickel, Sulpho-Arsenide 25 Hyacinth ... 39 Lithia Mica . 81 Nickel,Sulpho-Antimonite 25 Hydrated Deutoxide of Manganese . . 65 Lithionglimmer 81 Made . . 74 Nickelantimonglanz . 25 Nitrate of Soda . 56 Hydrous Silicate of Iron 61 Magnesie boratee 27 Nitre ... 66 Hypersthene . . 80 sulfatee . 72 Nosian ... 31 Hydrous Phosphate of Magneteisenerz 27 Nosin ... 31 Copper ... 82 Magnetic Iron Ore 27 Oblique Prismatic Arse- Hydrous Sulphate of Lime 83 Ice .... 47 Pyrites . 49 Magnetkies . . 49 niate of Copper . 82 Octahedrite . . 35 Idocrase ... 41 Malachite . 81 Octahedral Titanium Ore 31 Ilvaite ... 74 Malakolith . 80 Oerstedtite . . 43 lolite ... 74 Manganblende . 19 Oktaedrisches phosphor- Iridium osmie . . 46 Iridosmine . . 46 Manganese carbonate 55 oxid6 hydrate 38 saures kupfer . 71 Olivenkupfer . . 71 Iron Pyrites . . 21 phosphate . 66 Olivenerz . . . 71 Chromate . . 27 sulfur6 . 19 Olivenite . . . 71 Arseniate . . 30 sulphuret . 19 Olivine ... 67 Iron .... 18 Mangan-Epidot 85 Oligoclase . . 92 Jamesonite . . 64 Manganspath . 55 Oligoklas ... 92 Junkerite ... 66 Manganglanz . 19 One-axed Mica . 58 Johannite ... 90 Manganite . 65 Ornatif ... 17 Kalkspath ... 50 Kalikreuzstein . . 75 Mangangranat . 29 Manganesian Garnet Orpiment ... 63 Oriental Ruby . . 47 Kalkuranite . . 42 Kaneelstein . . 29 Manganhaltiger Aug e 80 Mascagnine . 72 Orthoklas ... 87 Osmium-Iridium . 46 Kieselzinkerz . . 72 Meionite . . 41 Oxide of Arsenic . 25 Kobaltbliithe . . 82 Melanite . . 29 Tin ... 34 glanz . . 25 Mellilite . . 43 Oxydulated Iron . 27 kies ... 21 Mellite . . 40 Copper . . 24 Koenigine . . 77 Koenigite ... 77 Mellate of Alumina 40 Menakerz . 84 Palladium . . 60 Panabase ... 28 Kohlensaurer Kalk . 50 Mendipite . . 65 Paranthine . . 41 Kohlensaures Blei . 66 Mengite . . 77,91 Paulite ... 80 Korund Mercure argon tal 18 Pearl Spar . . 65 Koupholite . . 75 muriate . 33 Peliom ... 74 142 INDEX TO MINERALS. Pericline . 92 Radiated Zeolite 75 Seleniuret of Lead and Peridot 67 Rautenspath 55 Silver . . 20 Persulphate of Iron from Chili 57 Realgar Red Antimony 78 78 Palladium . . 60 Sesqui-carbonate of Soda 82 Petalite ,93 Cobalt 82 Sideroschisolite . 61 Pharmacolite 83 Copper Ore 24 Silber . . . 16 Phenakite . 56 Garnet 32 Silberglanz . . 20 Phosphate of Copper 71, 82 Lead Ore 79 Silberkupferglanz . 62 Iron 82 Manganese 55 Silicate of Magnesia . 72 Alumine . . 71 Oxide of Zinc 63 Alumina and Lithia 93 Lead . 57 Copper 24 Siliceous Oxide of Zinc 72 Lime . 56 Silver 49 Sillimanite . . 77 Manganese GO Iron Vitriol 90 Silver ... 16 Uranium 42 Zinc Ore . 63 sulphuret . . 20 Yttria 38 Rhodizite . 28 flexible . 90 Iron and Manganese 83 Rhodonite 80 brittle . 64 Iron, Manganese, and Rhomboidal Arseniate of chloride . . 23 Lithia Phosphorsaure Yttererde 84 38 Copper . Rhomb Spar 57 55 Silver White Cobalt . 25 Skorodit ... 75 Phosphorsaures Kupfer Picrosmine 82 72 Rhyacolite Rhyakolith 87 87 Smaragdite . . 80 Smaragd ... 58 Pictite .01 Right prismatic Arseniate Soda Alum . . 31 Pikrosmin Finite 72 61 of Copper Rock Salt . 71 23 Sodalite ... 30 Soda .... 82 Pistacite 85 Rock Crystal 48 Soda-Felspar . . 92 Pistazit 85 Rothantimonerz 78 Somervillite . . 43 Platin-Iridium . 17 Rothbleierz 79 Sommite ... 58 Plagionite 78 Rother Vitriol . 90 Soudeboratee . . 83 Platinum 17 Rothgultigerz 49 carbonated . . 82 Platin 17 Rothkupfererz . 24 nitratee . . 56 Pleonaste 2G Rothspiesglaserz 78 sulfatee . . 83 Plomb arseniate 57 Rothoffite . 29 muriat6e . . 23 carbonat6 . G(i Ruby Sulphur . 78 Soufre . . . 61 chromat6 7!) Ruby Silver 49 Spathose Iron . . 55 molybdat6 . 39 Rutile 34 Specular Iron . . 47 muriate 65 Sahlite 80 Speerkies ... 63 natif . 18 Sal-ammoniac . 23 Speiskobalt . . 18 phosphat6 . 57 Salpetersaures Natron 56 Sphene ... 84 phosphato-arseniate seleniure | . 57 20 Salmiak Salpeter 23 66 Spinel ... 26 Spinel zincifere . 26 sulfate 70 Salzkupfererz 71 Spinelle noir . . 26 sulfur6 1!) Salzsaures Kupfer 71 Spinellane . . 31 tungstate 89 Sappare 91 Spodumen . . 75 Plumbago . 40 Sapphire . 47 Sprodglaserz . . 64 Plumbo-calcite . 56 Sarcolite . 30 Staurotide . . 67 Polybasite . 4.9 Sassolin 91 Staurolite . . 67 Polyhallite 7G Scapolite 41 Steinsalz ... 23 Polymignite 7G Schaalstein 80 Sternbergite . . 64 Potash Harmotome . 75 Sch6elin calcaire 38 Stilbite . . 75,89 Potasse nitrat6e 68 ferrugin6 78 Strahlerz ... 83 sulfatee 68 Scheelbleierz 39 Strahlstein . . 86 Potash Alum 31 Scheelsaures Blei 39 Strontiane carbonatee 65 Potash-Felspar . 87 Schilfglaserz 76 sulfatee . . 70 Precious Garnet 2,9 Schorl 59 Strontianite . . 65 Prehnite . 75 Schrifterz . 90 Strontites ... 65 Prismatic Iron Ore . 65 Schrift-Tellur . 90 Strontspath . . 70 Corundum . 74 Schwartzerz 29 Sulphate of Ammonia 72 Cerium Ore 74 Schwefel . 61 Barytes . . 68 Emerald 85 Schwefel and Kohlensaure i Copper . . 92 Purple Copper . 26 Blei . . . 88 Iron ... 84 Pyramidal Garnet 41 Schwefelsaures Kali . 68 Lead ... 70 Pyramidal Felspar 41 Schwefel Kobalt 21 Lime, anhydrous . 71 Pyrgom 80 Schwerbleierz 48 hydrous . 83 Pyreneite . Pyrochlore 2.9 31 Schwerspath Scorodite . 68 75 Magnesia . . 72 Potash . . 68 Pyrolusite . 64 Scolezite . 89 Soda . . 68, 83 Pyrope 32 Selenite 83 Strontian . . 70 Pyrophyllite 75 Selenblei . 20 Zinc ... 72 Pyrosmalite 58 kobaltblei . 20 Lime and Soda . 88 Pyroxene . Pyroxene noir . 80 80 quicksilberblei . silberblei . 20 20 Sulphate-carbonate of Lead ... 88 Quarz Quartz 48 18 Seleniuret of Lead Lead and Cobalt 20 20 Sulphato-tricarbonate of Lead? . . 88, 48 Quecksilberhornerz . 33 Mercury 20 Sulphur . . . 61 INDEX TO MINERALS. 143 Sulphuret of Antimony Arsenic, yellow . 63 63 Titan Titanium 18 18 Weisstellurerz . tellur . 76 46 red 78 Titaneisenerz 48 antimonerz 63 Bismuth 63 Titanite 84 Wernerite 41 Cobalt 21 Titanic Iron Ore 48 White Antimony 63 23". : 62 19 Titane calcar6o-siliceu Topas x 84 73 Iron Pyrites Lead Ore . 63 66 Nickel 46 Tourmaline 59 Vitriol 72 Manganese 1.9 Tremolite . 86 Tellurium . 76 Mercury 46 Triphyline . 84 Willelmine 56 Molybdenum 46 Triphane . 75 Willemit . 56 Silver 20 Triplit 66 Wismuth . 17 Silver and Antimony 4.9 Triple Sulphuret 72 Wismuthglanz . 63 76, 78 Trona 82 Wismuthkieselerz 30 Copper 62 Tungstate of Iron 78 Witherite . 65 Tin ... 29 Lead . 39 Wolfram . 78 Zinc . 19 Lime . 38 Wollastonite 80 Silver and Arsenic 49 Tungstein . 38 Wurfelerz . 30 Silver, brittle flexible . 64 90 Tungstene Turmalin . 78 59 Yttria phosphate^ Yttrocerite 38 24 Tabular Spar 80 Turnerite . 91 Zeilanite . 26 Tafelspath 80 Two-axed Mica 85 Zeilanit 26 Talc, Talk 56 Uranglimmer 42 Zinc-Blende 19 Talkspath . 55 [Iran Mica 42 Zinc carbonat6 . 55 Tantalite . 65 Uran Vitriol 90 oxide . 63 Tellur 46 Urane sulfat6 90 ferrifere 27 Tellursilber 46 Uranite 42 silicifere 72 Telluric Silver . 46 Urao 82 sulfur6 19 Tellurium 46 Uwarowite 32 sulfate" 72 Tennantite 28 Vanadinbleierz 57 vitriol 72 Tessaralkies 19 Vanadiate of Lead 57 Zinciferous Spinel 26 Tessaral Pyrites 19 Vanadinsaures Blei 57 Zinkspath 55 Tetradymite 49 Variegated Copper 26 Zinkblende 19 Thallite 85 Vauquelinite 84 Zinkenite 64 Thenardite 68 Vermischtes Fahlerz 28 Zinnkies 29 Thomsonite 75 Vesuvian . 41 Zinnober 46 Thumerstone Tin Pyrites 93 29 Vitreous Copper Vivianite 62 82 Zinnstein Zinnerz 34 34 Tin Stone . 34 Wagnerite 81 Zinkoxyd 63 Tin White Cobalt 18 Wavellite . 71 Zircon 39 Tincal 83 Weissbleierz 66 Zoisite 85 Titane Anatase . 35 spiesglanzerz 63 Zweiaxiger Glimmer 85 oxid . 34 BELL AND BAIN, PRINTERS. GRIFFIN'S SCIENTIFIC MISCELLANY: AN OCCASIONAL PUBLICATION OF TREATISES RELATING TO CHEMISTRY AND THE OTHER EXPERIMENTAL SCIENCES. PART I. Illustrated with 43 Engravings, Price 2s. 6d. INSTRUCTIONS FOR THE CHEMICAL ANALYSIS OF ORGANIC BODIES. BY JUSTUS LIEBIG. TRANSLATED FROM THE GERMAN BY DR. WM. GREGORY. PARTS II. AND III. A SYSTEM OF CRYSTALLOGRAPHY, WITH ITS APPLICATION TO MINERALOGY. BY JOHN JOSEPH GRIFFIN. Translator of " Rose's Manual of Analytical Chemistry," Author of" Chemical Recreations." CONTENTS. Principles of Crystallography : The three Axes of Crystals. Synopsis of the Planes of Crystals. Of Prisms and Pyramids, and their combinations with one another. Classification of Crystals. Limit to the variety of Planes that can occur upon Crystals. Reduction of the Planes of Crystals to seven kinds. A new system of Crystallographic Notation, referring to a single system of three Rectangular Axes. Of Cleavage, Primitive Forms, Simple Forms, and Combinations. Of Zones. Investigation of the Law of Sym- metry. A Theory of Crystallisation. A Popular Explanation of the use of Trigonometry in Crystallography. An Inquiry into the Crystallographic Forms and Combinations which occur upon the Crystals of Minerals. Mathematical proofs of the separate identity of the different Forms. Logarithmic Tables. Formulae for Trigonometrical Calculations. Application of Crystallography^ to Mineralogy: A Tabular Arrangement of Minerals, ac- cording to ROSE'S Method of Six Systems of Axes of Crystallisation. A Catalogue of Crys- tallised Minerals, showing the Crystallographic Combinations that occur in Nature. A Systematic Arrangement of the Crystals found in the Mineral Kingdom, with a list of the Minerals common to each Crystal. Descriptive Catalogue of a set of 120 Models of Crys- tals, employed to illustrate this work. PART IV. With 27 Engravings of Apparatus, Price 3s. INSTRUCTIONS FOR THE MULTIPLICATION OF WORKS OF ART IN METAL, BY VOLTAIC ELECTRICITY: With an Introductory Chapter on Electro-Chemical Decompositions by Feeble Currents. BY THOMAS SPENCER. IN THE- PRESS. INSTRUCTIONS FOR THE DISCRIMINATION OF MINERALS, By Means of Simple Chemical Experiments. TRANSLATED FROM THE GERMAN OF FRANCIS VON KOBELL. THE GEOLOGY OF THE ISLAND OF ARRAN. BY ANDREW RAMSAY. Other Works are in course of Preparation for this Miscellany. SCIENTIFIC WORKS RECENTLY PUBLISHED. In One Volume, small &vo, with 300 Wood Cuts, Price Is. A TKEATISE ON CHEMICAL MANIPULATION, And on the use of the BLOWPIPE in Chemical Analysis. BY JOHN JOSEPH GRIFFIN. CONTENTS -Introduction-PulverisationSolutionApplication of Heat Different kinds 5, Supports for Apparatus-Testing-Precipitatlon-Filtration-Washing of Pre- cipitates -Evaporation -Crystallisation-Ignition-Sublimah^ use of the Blow- and Management of the Subordinate Blowpipe Apparatus-A Course of Experi- ments exemplifying Chemical and Mineral Analysis by the Blowpipe-pistillation-Ma- nlgement offeaseslweighing and Measuring-Glass Blowing-Mechanical Operations- Fitting up of a Laborato?y-The Author explains the nature of the different Chemical processes; describes the best Apparatus for use in each operation the causes of frequent fiilure the precautions necessarf to be taken to insure success, and the methods by which the operations can be simultaneously performed by large classes of Students in Schools. K Most of the Apparatus described in this work may be procured of R. GRIFFIN & Co. by whom Teachers of Chemistry may be supplied with complete Sets of Chemical Appar- atus, adapted to the performance of a moderate Course of Analytical or Demonstrative Ex- periments, at eight, twelve, or twenty Guineas the Set, containing from 200 to 300 articles. A MANUAL OF ANALYTICAL CHEMISTRY. BY HENRY ROSE. TRANSLATED FROM THE GERMAN BY JOHN JOSEPH GRIFFIN. One Volume Bvo, Price 16s. GEOLOGICAL MAP OF SCOTLAND. Just Published, with Corrections and Additions to 1840, on four large sUets, size 7 feet by 5 feet, accurately coloured, and accompanied bg a descriptive pamphlet, A GEOLOGICAL MAP OF SCOTLAND, BY DR. M'CULLOCH. Constructed by order of the Lords of the Treasury, Engraved by Mr. Arrowsmith, and Published by G. F. Cruchley, London. Price in Sheets, 3 13s. 6d., originally published at Five Guineas. In Case, 4 4s. On Rollers, varnished, 4 14s. 6d. BKONGNIAET'S SYSTEM OF GEOLOGY, Handsomely Engraved on Copper, size 28 by 21 inches, Printed on a Sheet of Atlas Drawing Paper, and Coloured. THEORETICAL TABLE of the most general European Succession and Disposition of the Strata and Rocks which compose the Crust of the Earth; or, a Graphical Exposition of the Section of the Earth, so far as it is now known, with Symbols indicating the Mineral Con- stitution of each Rock, and its Mechanical Structure, the Metals which are found in the various Veins, and the Organic Remains which characterise the Succession of Strata; ex- hibiting also the relative Dispositions of the Neptunic, Plutonic, and Volcanic Rocks, BY ALEX. BRONGNIART, Professor of Mineralogy in the Museum of Natural History of Paris. The Catalogue of Rocks and Formations is carefully Translated, and enlarged by many British Examples and Synonymes. Price of the Table. On a Sheet of Atlas, 34 by 37 inches, Coloured, 5s.; in a Case for Travellers, 7s.; Mounted on Rollers and Varnished, 10s. 6d.; in a Rosewood Picture Frame, French Polished, Gilt Border, 34 by 27 inches, the Chart Varnished, 21s. PUBLISHED BY RICHARD GRIFFIN & CO., GLASGOW. MODELS OF CEYSTALS. RICHARD GRIFFIN & CO. respectfully announce that they have on Sale the MODELS OF CRYSTALS which are described in Mr. J. J. GRIFFIN'S SYSTEM OF CRYSTALLOGRAPHY. These Models represent the most important Natural Crystals, both of simple and complicated forms. Their SIZE is from 2 to 4 inches in diameter. The MATERIAL of which they are formed is CREAM COLOURED BISCUIT PORCELAIN, which presents the following advantages : The Models are much stronger than those made of paper. Their edges are sufficiently sharp and their planes sufficiently even to permit very good approximate measurements to be taken by means of the goniometer. They can be written upon, either with a black-lead pencil, the marks from which can be effaced by india rubber, or with common ink, which is easily remov- able by muriatic acid. Consequently the names of the Crystals, or their symbols, or the angles across their edges, or the names of the Minerals they represent, can be written upon them and removed at pleasure. These properties are not possessed in the same degree by Models made of glass or wood. When soiled, they can be cleaned by soap and water. Finally, they are cheaper than Models made of any other material. Price of the Set of 120 Models in a Packing Case, 3 13 6 Price of a Hardwood Cabinet with Four divided Trays, or of Three Wooden Trays, with Covers, and 120 Paper Boxes, 1 1 The whole in a Packing Case, 4 14 6 Every Model in this collection is not merely described in Mr. GRIF- FIN'S work, but is made the subject of numerous mathematical problems, and the relation of the Models to Crystallised Minerals is illustrated in the most ample and practical manner. The Models will be found alike useful to private Students and to Teachers. Although great care has been taken to render these Models particu- larly useful to persons purposing to study Crystallography according to MR. GRIFFIN'S system, by marking upon the Models the Axes, Poles, and Zones peculiar to MR. GRIFFIN'S system, yet, as the marks are capable of removal, the Models can also be used to illustrate any other system of Crystallography. MR. GRIFFIN'S book contains very copious Tables of Reference to works on Crystallography and Mineralogy, containing figures or de- scriptions of the Crystals represented by each Model. It also contains lists of all the Minerals which have been found in the shape of each Model, or in a shape nearly related to it. The greatest pains have been taken to render these Models practically useful to Students of Miner- alogy. SUPPORTS FOE MODELS OF CRYSTALS. THESE supports are intended to hold Crystals in a proper position for Crystallographic examination. They are useful for the purpose of exhibiting the Models in a Museum, or when a series of Crystals is to be placed before a Crystallographer for study. It is ex- tremely convenient when a suite of Crystals, either belonging to one system, or to one mineral, are to be examined, to have the means of placing the whole in a proper position for examination, and yet to have the hands at liberty to attend to pen or book. Nos. 1 to 4 are made of Pale Blue Biscuit Porcelain ; No. 5 of Cedar. Price Sixpence each, or 5s. per dozen. No. 1 supports all kinds of square based pyramids; No. 2 supports pyramids that have a rhombic base; No. 3 supports rhombohedrons and six-sided pyramids; No. 4 supports all Crystals with dihedral summits; No. 5 (cedar) with the aid of brass pins, supports such Crystals as have unsymmetrical summits, and are not adapted to the porcelain supports. No. 6 represents the manner in which the Crystals are supported. MODELS OF CRYSTALS FOR LECTURERS. Now preparing for Sale, a Series of MODELS OF CRYSTALS, ON A LARGE SCALE, adapted for PUBLIC LECTURES. It will consist of eight or ten Models, from six to ten inches in length, and provided with tangible axes. Some of the Models will be solid, and others dis- sected, and provided with skeletons. They will be accompanied by several large sections of Crystals. These Models are intended to give a popular explanation of Crystal- lographic terms. They will show the Planes and Axes of Crystals ; their Equators, Meridians, Poles, Normals, and Zones. They will ex- hibit the difference between Prisms and Pyramids, and enable the teacher to explain the principles according to which Crystals are divided into systems of Crystallisation. The sections, skeletons, and solid tri- angles, will iUustrate the Law of Symmetry, the combination of Forms into Complex Crystals, the Rules of Trigonometrical Calculations, &c. The Price of the Set will be from Two to Three Guineas. RICHARD GRIFFIN & CO., GLASGOW, MANUFACTURERS OF CHEMICAL AND PHILOSOPHICAL APPARATUS. UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine IFeb'54D|f 7 1954 LU Om-12,'46(A2012s 6)4120 / - \