UC-NRLF //// .* .-/:' y '.'ff < FAMILIAR INTRODUCTION TO CRYSTALLOGRAPHY; INCLUDING AN EXPLANATION OF THE PRINCIPLE AND USE OF THE GONIOMETER. an CONTAINING THE MATHEMATICAL RELATIONS OF CRYSTALS J RULES FOR DRAWING THEIR FIGURES; AND AN ALPHABETICAL ARRANGEMENT OF MINERALS, THEIR SYNONYMES, AND PRIMARY FORMS. ILLUSTRATED BY NEARLY 400 ENGRAVINGS ON WOOD. By HENRY JAMES BROOKE, F. R. S. F. L. S. &C. LONDON : PRINTED & PUBLISHED BY W.PHILLIPS, GEORGE YARD, LOMBARD STREET; SOLD ALSO BY W. & C. TAIT, EDINGBURGH J AND R. MILLIKIN, DUBLIN. 1823. IOAN TO THE INVENTOR OF THE REFLECTIVE GONIOMETER, AN INSTRUMENT TO WHICH CRYSTALLOGRAPHY IS LARGELY INDEBTED FOR THE PRECISION OF ITS RESULTS, THIS INTRODUCTORY TREATISE IS RESPECTFULLY INSCRIBED BY THE AUTHOR, 310 INTRODUCTION. THE immediate purpose of the science of Crystal- lography, regarded as a branch of Mineralogy, is to teach the methods of determining the species to which a mineral belongs, from the characters of its crystalline forms. But the science itself is also capa- ble of being rendered more extensively useful. The crystalline forms of pharmaceutical prepara- tions will furnish a certain test of the nature of the crystallised body, although it will not determine its absolute state of purity. In chemical analysis, the forms of crystals will frequently supersede a more rigorous examination of the crystallised matter; and commercial transactions in the more precious mine- ral productions may frequently be guided by the crystalline form, or by the character of the cleavage planes, of those bodies. It does not appear in the works hitherto published that the connection between the crystal and the mine- ral has been any where so systematically explained as to enable the mineralogical student readily to connect the one with the other. The Abbe Haiiy's works on crystallography are the only ones in which a truly scientific exposition of the theory of crystals is to be found ;* but by designating * An interesting volume on Crystallization, founded on the Abbe Hauy's theory, was published in 1819, by Mr. Brochant de Villiers, and will afford the reader a clear view of that theory, connected with other interesting objects relating to the formation of crystals. vi Introduction* most of their forms by separate names, he has pre- sented those forms to the mind rather as indepen- dent individuals, than as parts of such groups as should render their relations to each other, and hence their mineralogical relations, apparent. I have been induced, therefore, to attempt such an arrangement of the various forms of crystals, as will indicate their constant relations to, or differences from, each other, for the purpose of more readily re- ferring from the crystal to the mineral ; and this ar- rangement is contained in the Tables of Modifications which will be found in the following pages.* The best illustration of the manner in which some of the forms of crystals may be conceived to be allied to others, is afforded by the Abbe Haiiy's theory of decrement. That theory appears, however, much encumbered by his adoption of two kinds of molecules, and by the forms which he has assigned to particular molecules of one of those kinds ; in consequence of this, I have ventured to propose a new theory, in re- ference to several of the classes of primary forms, which may in some respects be regarded as more simple, and which forms the subject of the section on molecules. t * Since these tables were constructed I have learned from Mr. Konig of the British Museum, that he had for some time entertained an inten- tion of framing a set of tables nearly on the same principle: and he has shewn me a considerable number of drawings of the figures of crystals which were made partly with a view to this object, yet serving at the same time as records of many of the crystalline forms of minerals con- tained in that rich collection, upon which his attention is so constantly and so assiduously bestowed. + The theory of spherical molecules which has been entertained by some distinguished philosophers, has not been alluded to in this treatise, as the laws of decrement appeared more readily explicable on the sup- position of the molecules of crystals being solidb contained within plane surfaces. * Introduction. vii I have also attempted to supply some rules for studying the forms of crystals, and for what may be termed reading them ; which, although they may not enable the learner to trace at once the relation of the different crystalline forms to each other, they will certainly assist him in his examination of the minerals themselves ; and it is from an attentive study of these that he must at last derive his best information. From the very elementary nature of some of the definitions, it is evident that the reader of the earlier part of the volume is supposed to be unacquainted with the first rudiments of geometry. By being thus elementary I have been inclined to hope that crystal- lography may be rendered more familiar, and its prin- ciples be more easily acquired: and that the young- collectors of minerals may be led by these first and easy steps in the path of science, to make their collec- tions subservient to the cultivation of higher sources of amusement. The description of the principle and of the method of using the reflective goniometer, has been mi- nutely detailed on account of the importance of the instrument to the practical mineralogist, and with a view to remove the impression of its application to the measurement of crystals being difficult. The Abbe Haiiy has used plane trigonometry in his calculations of the laws of decrement. The sub- stitution of spherical for plane trigonometry in this volume, was made at the recommendation of Mr. Levy ; from whom I have also received many other valuable suggestions relative to the methods of cal- culation employed in the section on calculation ; which it will be apparent to the reader, is little more than an outline of a method which must frequently be filled up by the exercise of his own judgment. In this as well as in other respects the mathematical v i i i Introduction part of the volume differs much from the analytical processes contained in the Abbe Haiiy's more com- prehensive works.* When I first began to examine the crystalline forms of minerals I was much assisted by a large collection of the drawings of crystals, which was very kindly lent me by my friend Mr. William Phillips. The number and variety of the figures contained in this collection was the immediate cause of the attempt to reduce the forms of crystals into classes, and of the construction of the tables of modifications already al- luded to. Since that period, and particularly during the printing of this volume, Mr. Phillips has fre- quently assisted me by his communications relative to the forms, the cleavages, and the measurements of crystals, which I had not other means cf immediately acquiring. During the course of my investigations I have fre- quently found it necessary to consult larger collec- tions of minerals than my own. On these occasions I have generally referred to my friend Mr. Heuland, and lam happy in the opportunity of acknowledging the readiness and the liberality with which he has invariably assisted my views, by permitting an access at all times to his large and valuable cabinets, and very frequently by contributing rare and interest- ing specimens to mine which I could not otherwise have acquired. I have sometimes sought information from the ex- tensive cabinet in the British Museum, and have * Mr. Levy is at present engaged in an examination and description of one of the finest collections of minerals in the kingdom, which be- longs to C. H. Turner, Esq. The opportunity which this examination will afford him of connecting a knowledge of the forms of crystals with his well-known mathematical attainments, will enable him to convey valuable instruction in this department of science to others, to which object he intends to devote some portion of his time in future. Introduction. it always found the utmost facility afforded to research by the habitual urbanity and friendly attention of Mr. Konig. And I have frequently been indebted also te Mr. G. B. Sowerby for illustrative specimens of mine- rals of unfrequent occurrence. Since the committal of a considerable portion of these pages to the hands of the compositor, and in- deed subsequently to the printing a large part t>f the volume, a new edition has appeared of the Abbe Hairy 's treatise on crystallography; this event was very soon followed by the decease of the learned author; and subsequently to his decease three vo- lumes of a new edition of his treatise on mineralogy have been published : events so intimately connected with the subject of this volume that I cannot well .,!_ T pass them over in silence. I am perfectly disposed to concur in the public eulogium which has been so deservedly passed upon the deceased philosopher, for having been the first to elevate crystallography to the rank of a science, and to trace out a secure path to its attainment; but I regret that I cannot agree in that unqualified ap- probation of his recent works which some of his sur- viving friends have so liberally bestowed upon them. For those works will be found to contain errors of so remarkable a character, as to excite our surprise when we recollect the generally accurate and enlight- ened judgment of the author. Upon these, as criticism can no longer reach the ear of the author, I shall offer but few remarks. One of his sources of error may be discovered in an apparently groundless notion which his theory em- braces, that nature has imposed limits to the angles x Introduction at which the primary planes of crystals incline to each other. And some of the mistakes which originate from this supposition are so important, as to cast a shade of disconfidence over his determinations relative to the primary forms of crystals. His inaccuracy with respect to the angle of carbo- nate of lime is a well known example of one of these theoretic errors. His inaccurate measurements of many of the angles of crystals, have probably been occasioned by the comparatively imperfect instrument with which those measurements were taken. That he should have con- tinued to prefer this, to the more perfect goniometer invented by Dr. Wollaston, may possibly have been owing to the decay of sight incident to his period of life, and to that dislike to change which so frequently accompanies advanced age. But some of his inaccuracies are independent both of his theory and his goniometer, and it would almost appear that he had occasionally written from the dic- tates of his fancy, without examining the minerals he has described. The resemblance he imagines to exist between the crystals of bournonite and those of sulphuret of anti- mony is an instance of this nature; and on some of his figures, as those of wolfram, and some of those which he has still retained as stilbite, although they belong to a distinct species of mineral to which I have given the name of heulandite, he has placed imagi- nary planes which have no existence on the crystals themselves. tlis persisting in the identity of the angles of the primary forms of carbonate of lime, bitter spar, and carbonate of iron, if he has really been deceived by his goniometer, evinces a carelessness in the use of > Introduction. xi that instrument which must still further diminish our confidence in his results. Dr. Wollaston in the year J812 discovered that these species differed from each other, and noticed their differences in a paper published in the Philoso- phical Transactions for that year. He found carbo- nate of lime to measure 1055' Bitter spar 10615' Carbonate of iron 107 Notwithstanding the discovery of these facts, which' have been so frequently verified by subsequent mea- surements, the Abbe Haiiy has not only continued to insist on the superior accuracy of his own measure- ments, but discusses through several pages how it could have happened that the iron should have dis- placed the lime in the crystals of carbonate of iron, which his original error has led him to regard aspseu- domorphous I With all the faults, however, which the late Abbe's works contain, many of which we must in justice to his better judgment ascribe to feelings of a personal nature, those works present to the reader truly phi- losophical views of the sciences of which they treat, and they cannot be perused without frequently afford- ing him both gratification and improvement. riip V.JL , .*. * *..... ;': ' TABLE OF CONTENTS. DEFINITIONS. . . The object of the science . .... ....... .,. . >** '-twi* <: I A crystal, what 1 Its planes, edges.., and angles 1 3 The value of an angle how determined . . 2 Crystals formed of homogeneous molecules ......... 5 The same mineral crystallises in a variety of forms .w^.itt These forms are primary * * v**:*i..v4'4 * ' or secondary -.. <., .**,&* y. * . . . 24 Tangent plane 24 OF THE GONIOMETEH AND THE METHODS OF USING IT 25 32 GLNEUAL VIEW or THE SUBJECT 33 MOLECULES OF CKYSTALS 36 are particular solids .; * 3$ their fonns , 3852 STRLXTLRE, illustrated by the crystallisation of common , ; CcS salt . .. 53 5 xiv Table of Contents. PAGE CLEAVAGE 56 takes place in the direction of the natural joints of crystals 56 different sets of cleavages 56 the mutual relation of the solids obtained by cleavage 59 66 DECREMENTS. simple, mixed, and intermediary .... 67 73 causes producing these not understood 73 influenced by the pyroelectricity of bodies . . 76 SYMMETRY, its laws 77 PRIMARY FORMS 79 may sometimes be developed by cleavage 79 frequently require secondary forms to determine them 81 SECONDARY FORMS 86 HEMITROPE & INTERSECTED CRYSTALS 88 EPICENE & PSEUDOMORPHOUS CRYSTALS 93 ON THE TABLED OF MODIFICATIONS < 95 TABLES OF MODIFICATIONS OF the cube , 106 regular tetrahedron 112 regular octohedron 116 rhombic dodecahedron 120 octahedron with a square base 1 26 rectangular base 138 rhombic base 146 right square prism 162 rectangular prism 166 rhombic prism 170 oblique-angled prism 176 oblique rhombic prism 180 doubly oblique prism 1 90 hexagonal prism 1 96 rhomboid 200 TABLES OF SECONDARY FORMS 214 222 ON THE APPLICATION OF THE TABLES OF MODIFICATIONS 223 method of discovering the forms of crystals, or of reading (hem 223228 method of describing crystals by means of the tables 229 ON THE USE OF SYMBOLS FOU DESCRIBING THESECONBARY FORMS OF CRYSTALS 233 252 ON THE RELATION OF THE LAWS OF DECREMENT TO THE DIFFERENT CLASSES OI MODIFICATIONS . , . 253 of Contents. ** ';^ APPENDIX. PAGE CALCULATION OF THE LAWS OF DFCREMENT 253 How to determine the decrement from the mea- sured angles of the crystal 293374 How to determine the angles when the law of decrement is known 374 378 Application of the proposed method of calcula- tion to a particular case 378 392 ON THE DIRECT DETERMINATION OF THE LAWS OF DE- CREMENT FROM THE PARALLELISM OF THE SECONDARY EDGES OF CRYSTALS . 393 METHODS OF DRAWING THE FIGURES OF CRYSTALS 402 ON MINER ALOGICAL ARRANGEMENT , 439 ALPHABETICAL ARRANGEMENT OF MINERALS WITH THEIR SYNONYMES AND PRIMARY FORMS 451 497 TABLE OF PRIMARY FORMS ARRANGED ACCORDING TO THEIR CLASSES ,,....., 498 501 SOI ^<^..,;.\^-.**+*.i>*.~*~*.&^$'**tyte " ,.- 6U ' ; >--* ' ' ' s --^ DI- CORRIGENDA. The reader is requested to make the following corrections with the most of which are important to the accuracy of the text. The lines reckoned do not include the running titles, or the word Fig. standing over the diagrams. Page 39, line 8, for direction of c, ... read . ... e. 40, 1, line ..................... lines. 43, 21, tetrahedrons ............ tetrahedron. 80, 7 from the bottom, for class n ... class o. 92, 2, .for deduced ................. derived. 97, 2, together with ...... ..: and. 5, after symbols ..... add .. . . will follow the tables of modifications. 107, 9, for 120 1 5/52".... read ....... 125 8 15'52" i.<.:113, 9, . 7031'43" ............... 70'31'44'' 115, 15, 12015'52" ............... 12515>52" 177, 6, and a rhombic base ...... with a rhombic base. 181, 6 from the bottom, for F ....... E. 2(Mj 6,forF ....................... E. 207, ~. 6, after intersect .... add ...... two of. 209, 10, for* .... ......... read .... /. 213, 9 from the bottom, after diagonals, add, or edges. 218, 4, for m .......... ... read.... o. 257,- 5, <.,., ................... > P i> 260, 3 from the bottom, for P ...... A P P 270, S, forGp ..................... Gq 6, Bq ...................... Hq 277, 2, BpB/p ................. BpB/q 278, 19, C .............. \ ......... O 298, -- 13, if ...................... H. SOI, 15, Ep ...................... Fp 810, 10 from the bottom, for B/qB^r... B'/qB'r 311, 9, after = ........ add ....... R. 12, - = ..................... R. 3 from the bottom, for B/qB'^ read B'/qB'r Sl5, 12, for cos.Ag ................. cos. A 3 16,afersin.(l20 A 2 )..add.. : 316, at the bottom, after = in both formulae, add R, S17, 9from the bottom, for alec read ab o d. 319, 3 - after (7 8 /, ) add : 322, in both formulae, after =r... add R. 323, 10 from the bottom, fora ... read c. 324, 2, for , and o, , ............... d, and J. S2G, 5 from the bottom, for cos. / 2 .. cot. 7 2 349, 5, fora ...................... c. 351, 6, -- 351 .................... 353. 11, in the formula, for sin. J t ... sin. / f 353, at bottom, for a e ... ........... ac. 354, 1, for a e ...... . ............ ac. 5, R ..................... R: 6, after 1 ......... add . . ----- : 377^ _ at bottom, for tang. ( 7=45) read tang. (^1 45") 390, --- 39 2 / 30' / = ........ 392'3O"~ 391, 5 from the bottom, for 4I5'32" . . 4l5 / 33" 392, 4, for4'15'32"... ............... 415 / S3 / ' 6, 28 1 0>26" ................ . 210'27" DEFINITIONS. "' THE object of the science of Crystallography, regarded as a branch of mineralogy, is to trace and to demonstrate those relations and differences between the various crystalline forms of minerals, by means of which we are enabled generally to discriminate the different species of crystallized minerals from each other. A crystal^ in mineralogy, is any symmetrical mineral solid, whether transparent or opaque, contained with- in plane, or sometimes within curved surfaces. These surfaces, as , 6, c, fig. 1., are called planes or faces. Fig- I- The exterior planes of a crystal as they occur in nature, are called its natural planes. Crystals may sometimes be split in directions parallel to their natural planes, and frequently in other directions. The splitting a crystal in any direction, so as to obtain a new plane, is termed cleaving it, and the crystal is said to have a cleavage in the direction in which it may be so split. The planes produced by cleaving a crystal are called its cleavage planes. An edge, as d fig. 1, is the line produced by the meeting of two planes. ' A 2 DEFINITIONS. A plane angle, or as it is more commonly termed, an angle, is formed by the meeting of any two lines or edges. The angles do e, dog, fig. 1. are formed by the meeting of the lines do, oe, and do, og. A solid angle is produced by the meeting of three or more plane angles, as at o, fig, 1. Fig. 2. The measure, or, as it is sometimes termed, the value of an angle, is the number of degrees, minutes, &c. of which it consists ; these being determined by the portion of a circle which would be intercepted by the two lines forming the angle, supposing the point of their meeting to be in the centre of the circle. For the purpose of measuring angles the circle is divided into 360 equal parts, which are called degrees; each degree into 60 equal parts, which are called minutes ; and each minute into 60 seconds ; and these divisions are thus designated; 360% 60', 60", the ' signifying degrees, the ' minutes, the " seconds. If | of the circle, or 90, be intercepted by the two lines a o, ob, fig. 2, which meet at an angle a o b in the centre, those lines are perpendicular to each other, and the angle at which they meet is said to measure 90, and is termed a right angle. If less than | of the circle be so intercepted, as by the lines ob,oc, the angle b o c will measure less than 90, and is said to be acute. If it measure more than 90, as it would if the angle were formed by the lines a o, o c, it is called obtuse. 4 DEFINITIONS. 3 The lines a o, o b, or a o, o c, or b o, o c, are sometimes said to contain a right, an obtuse, or an acute angle. In fig. I, the plane a, and that on which the figure is supposed to rest, are called summits, or bases, or terminal planes, and the planes b and c, with those parallel to them, are termed lateral planes'. The edges of the terminal planes, as d, e, m, n, fig. I, are called terminal edges. The edges/, g, h, produced by the meeting of the lateral planes, are termed lateral edges. The planes of a crystal are said to be similar when their corresponding edges are proportional, and their corresponding angles equal. Edges are similar when they are produced by the meeting of planes respectively similar, at equal angles. Angles are similar when they are equal and con- tained within similar edges respectively. Solid angles are similar when they are composed of equal numbers of plane angles, of which the corre- sponding ones are similar. Fig. 3. An equilateral triangle, fig. 3, is a figure contained within three equal sides, and containing three equal angles. Fig. 4. An isosceles triangle, fig. 4, has two equal sides, b, which may contain either a right angle, or an A2 DEFINITIONS, acute, or obtuse angle. If the contained angle be less than a right angle, the triangle is called acute, but if greater, it is called obtuse. The line on which c is placed is called the &ase of the triangle. Fig. 5. A scalene triangle, fig. 5, has three unequal sides, and contains three unequal angles. Fig. 6. A square, fig. 6, has four equal sides, containing four right angles. Fig. 7. A rectangle, fig. 7, has its adjacent sides, a and b, unequal, the four contained angles being right angles. Fig. 8. A rhomb, fig. 8, has four equal sides, but its ad- jacent angles, a and b, unequal. DEFINITIONS. 5 Fig. 9. An oblique angled parallelogram,* fig. 9, has its opposite sides parallel, but its adjacent sides , b, and its adjacent angles, c, d, unequal. Where certain forms of crystals are described with reference to the rhomb as the figure of some of their planes, they are termed rhombic.^" A parallelopiped is any solid contained within three pairs of parallel planes. Crystals are conceived to be formed by the aggre- gation of homogeneous molecules., which may be again separated from each other mechanically, that is, by splitting or otherwise breaking the crystal. These molecules, which relate properly to the crys- tal, must be carefully distinguished from the elemen- tary particles of which the mineral itself is composed. Sulphur and lead are the elementary particles, which, by their chemical union, constitute galena; but the molecules of galena are portions of the com- pound crystalline mass, and are therefore to be re- garded as homogeneous, in reference to the mass itself. * A parallelogram is any right lined quadrilateral plane figure, whose ofifioiite sides are equal and parallel. f What is here called rhombic, most writers on this subject have, in imitation of the French idiom, denominated rhomboidal; but as the term rhomboid has been used in works on geometry to signify an oblique angled parallelogram, and as the same term has also been already appropriated in crystallography to a solid contained within six equal rhombic planes, the application of the term rhomboidal to any other solid seems to involve a degree of ambiguity. The term rhombic is, besides, more conformable to the practice of our own language. 6 DEFINITIONS. AM minerals which are composed of similar ele- mentary particles combined in equal proportions, and whose molecules are similar in form, are said to belong to one species. The same species of mineral is frequently observed to crystallize in a great variety of forms. From among the variety of crystalline forms under which any species of mineral may present itself, some one is selected as the primary , and the remainder are termed secondary forms. A primary form is that parent or derivative form, from which all the secondary forms of the mineral species to which it belongs, may be conceived to be derived according to certain laws. The primary forms are at present supposed to con- sist of only the following classes. Fig. 10. The cube^ fig. 10, contained within six square planes. Fig. 11. The regular tetrahedron, fig. 11, contained within four equilateral triangular planes. The solid angle at #, is sometimes called its summit. DEFINITIONS. Fig. 12. The regular octahedron, fig. 12, resembling 1 two four-sided pyramids united base to base. The planes are equilateral triangles, and the common base of the two pyramids (which will hereafter be denominated the base of the octahedron) is a square. Fig. 13. The rhombic dodecahedron, fig. 13, contained with- in twelve equal rhombic planes, having six solid angles, consisting each of four acute plane angles, two opposite ones as a, b, being sometimes called the summits, and eight solid angles consisting each of three obtuse plane angles. 8 DEFINITIONS. An octahedron with a square base, fig. 14, contained within eight equal isosceles triangular planes ; the bases of the triangles constitute the edges of the base of the octahedron. When the plane angle at a measures less than 60% the octahedron is called acute. When the angle at a is greater than 60 , the octa- hedron is called obtuse. The square base serves to distinguish this class from the two which follow, it. The isosceles triangular planes distinguish it from the regular octahedron. Fig. 15. d An octahedron with a rectangular base, fig. 15. The planes of which are generally isosceles triangles, but not equal. The plane angles at c and d of the planes a and a' being more obtuse than those of the planes b and b' '; and the planes a, and a', inclining to each other at a different angle from that at which those marked b] and b', meet. DEFIMTIONS. Fig. 16. An octahedron with a rhombic base, fig. 16, contained within eight equal scalene triangular planes. The solid angles at a, b, fig. 12 and 14, and c, d 9 fig. 15 and 16, are sometimes called the summits of the octahedron. Fig. 17. A right prism with a square base*, fig. 17, or right square prism, the edge, a, being always greater or less than b : if a, and b, were equal, the figure would be a cube. Fig. 18. A right prism with a rectangular base, fig. 18, or right rectangular prism, whose three edges a, b, c, are unequal. For if any two of those were equal, the prism would be square. * A prism is a solid whose lateral edget are parallel, and whose terminal planet are also parallel. Those prisms which stand perpendicularly when resting on their base, are called right prisms. Those which incline from the perpendicular > are called oblique prisms. B 10 DEFINITIONS, Fig. 19. A right rhombic prism, or right prism whose base is a rhomb, fig. 19, and whose lateral planes a, b, are equal These planes may be either square or rectan- gular. Fig. 20. A right oblique-angled prism, or right prism whose base is an oblique-angled parallelogram, fig. 20, and whose adjacent lateral planes a, b, are unequal. One of these planes must be rectangular, the other may be either a square or a rectangle. Fig. 21. An oblique rhombic prism, or oblique prism whose base is a rhomb, fig. 21, and whose lateral planes d, e, are equal oblique-angled parallelograms if they were equal rhombs the solid would be a rhomboid. DEFINITIONS. 11 Fig. 22. A doubly oblique prism, fig. 22, whose bases and whose lateral planes are general!?/ oblique-angled paral- lelograms. The only equality subsisting among these planes, is between each pair of opposite or parallel ones. Fig. 23. The rhomboid, fig. 23, a solid contained within six equal rhombic planes, and having two of its solid angles, and only two, as a, b, composed each of three equal plane angles ; these are sometimes called the summits. Fig. 24. The regular hexagonal prism, fig. 24, or right prism whose bases are regular hexagons. The secondary forms of crystals consist of all those varieties belonging to each species of mineral, which differ from the primary form. These, although extremely numerous, may be re- duced to a few principal classes, as will appear in the sequel. B 2 12 DEFINITIONS, tig. 25. A line, as 06, or c tf, fig. 25, drawn through two opposite angles of any parallelogram, and dividing the plane into two equal parts, is called a diagonal of that plane. In the oblique rhombic prism, the doubly oblique prism, and the rhomboid, fig. 21, 22, and 23, the line a c, which appears to lean from the spectator, will be termed the oblique diagonal, and the line fg of the oblique rhombic prism, and df of the rhomboid, the horizontal diagonal. The line dfof the doubly oblique prism, may also for the sake of distinction be termed its horizontal diagonal; although from the nature of the figure, that line must be oblique when the lateral edges are perpendicular. The diagonal plane of a solid, as a b c d, fig. 25, is an imaginary plane passing through the diagonal lines of two exterior parallel planes, dividing the solid into two equal parts. The axis of a crystal, generally, is an imaginary line passing through the solid, and through two oppo- site solid angles. In prisms, this may be termed an oblique axis, to distinguish it from another line which passes through the centres of their terminal planes, and may be termed a prismatic axis. The axis of a pyramid, passes through its terminal point and through the centre of the base. DEFINITIONS. Fig. 26. In the cube, an axis passes through the centre and through two opposite solid angles, a , fig. 26 ; from the perfect symmetry of its form, the cube has a simi- lar axis in four directions, or passing through its centre and through each pair of opposite solid angles. Fig. 27. The axis of the regular tetrahedron passes through the centres of the summit and base as a b, fig. 27, and it has a similar axis in four directions in consequence of the symmetrical nature of its form. Fig. 28. In all octahedrons the axis passes through the two summits and through the centre of the base, as a b, fig. 28; the regular octahedron, having all its solid 11 DEFINITIONS. angles similar, may be said to hare a similar axis in three directions. But the lines c d, e f, joining the opposite lateral solid angles of irregular octahedrons, may be called the diagonals of their base. Fig. 29. The rhombic dodecahedron has two dissimilar sets of axes passing through its centre ; one set, as a b, fig, 29, passes through the pairs of opposite solid angles, which consist each of four acute plane angles, and may be called the greater axes ; another set, as c d, passes through the solid angles which consist of three obtuse plane angles each, and may be called the lesser axes of the crystal. Fig. SO. Fig. 31. The right square, and right rectangular prisms, have each an axis in four directions similar to a b, fig. 30 and 31, but as prisms they have an additional prismatic axis, c d. DEFINITIONS. 15 Fig. 32. Fig. 33. The right rhombic prism, fig. 32, and right oblique angled prism, fig. 33, have each two greater and two lesser axes. The greater axis, a b, passes through the solid angles which terminate the acute edges of the prism, and the lesser, c d, through those which ter- minate the obtuse edges of the prism. They have also the prismatic axis, ef. Fig. 34. The oblique rhombic prism has, besides the pris- matic axis, i k, fig. 34, a greater, a lesser, and two transvere axes. The greater axis is that which passes through the two acute solid angles of the prism a, b ; the lesser that which passes through the two obtuse solid angles of the prism c, d, and the tranverse, those which pass through the lateral solid angles, 16 DEFINITIONS. Fig. 35. The doubly oblique prism has four unequal axes passing through the pairs of opposite solid angles, a by &c. fig. 35 ; it has also the prismatic axis i A*. Fig. 36. The line a b, fig. 36, which passes through the summits of the rhomboid, may be called the perpen- dicular axis, and those lines, c d, e /, g h, which pasa through the opposite pairs of lateral solid angles may be termed the transverse axes. But the lines a by and c d, are sometimes called the greater and lesser axes of the rhomboid. Fig. 37. The line a b, fig. 37, passing through the opposite solid angles of the hexagonal prism, may be termed an axis ; but the prismatic axis, c d of this form, is that which is most generally regarded as its axis. DEFINITIONS. 1^ t The diagonals and axes of crystals are imaginary lines, by means of which the secondary planes of crys- tals may frequently be described with greater pre- cision than could be attained without their assistance; they also facilitate the mathematical investigations into the relations which subsist between the primary and secondary forms. The diagonal planes are imaginary planes of a similar character. A crystal is said to be in position, when it is so placed, or held, as to permit its being the most easily and precisely observed and described. For this purpose tetrahedrons are made to rest on one of their planes, as in the figure already given. Octahedrons are supposed to be held with the axis vertical, and in this position the plane angles at a and 6, fig. 28, are called the terminal angles, and the edges a c, a d, a e, af y the terminal edges, or edges of the pyramid. The edges e d, df, &c. may be termed edges of the base ; and the angles a e d, a df, lateral angles. The angles of the base are the angles c e d, or e df> The cube stands on one of its planes, and all prisms on their respective bases. Rhombic dodecahedrons are supposed to be held with a greater axis vertical, as in the former figure. The rhomboid is also supposed to be held with its perpendicular axis vertical, Crystals are supposed to be first formed by the aggregation of a few homogeneous molecules, which arrange themselves around a single central molecule in some determinate manner ; and they are conceived to increase in magnitude, by the continual additions of similar molecules to their surfaces. 18 DEFINITIONS. In these additions, the molecules appear to arrange themselves so as to form laminae, or plates, which successively, either partially, or wholly, cover each other. These plates are theoretically supposed to be either single, that is, of the thickness of single molecules, or to be double, treble, &c. that is of the thickness of two, three, or more molecules. Fig. 38. Fig. 38 represents a single plate of molecules, Fig. 39. X^x / / y >r ,.' Fig. 39 represents a double plate. Fig. 40. When such additions envelope the whole of a smaller crystal, its original form is preserved through every increase of size. Fig. 40 represents a right rectangular prism which has increased in magnitude without change of figure. When the additions do not cover the whole surface of a primary form, but there are rows of molecules omitted on the edges, or angles of the superimposed plates, such omission is called a decrement. 4 DEFINITIONS. J9 The terra decrement has been adopted to express these omitted rows of molecules, because, in conse- quence of such omissions, the primary form on which the diminished plates are successively laid, appears to decrease as it were, on the edge or angle on which such omissions take place. Decrements are said to begin at, or to set out from, the particular edge or angle at which the omission of molecules first takes place ; and to proceed along that plane on which the defective plate of molecules is conceived to be superimposed. And they are said to take place either in breadth or in height. Decrements in breadth are those which result from the reduction of the superficial area of the super* imposed plate, by the abstraction of rows of molecules from its edges or angles. Decrements in height relate to the thickness of the plate from which the abstraction of rows of molecules takes place. Fig. 41. Let c d fig. 41, represent an edge of a primary form, and let a b represent an edge of a double plate of molecules, from which one row has been abstracted ; the decrement, or omitted portion of this superimposed plate, would be stated to consist of one row in breadth, or one row omitted upon the terminal surface of the primary crystal, and two rows in height, signifying that the omitted row belonged to a double plate of molecules \abcd would be the position of the new plane produced by this decrement. 20 DEFINITIONS. Fig. 42. Decrements have been divided by the Abbe Haiiy into three principal classes simple, mixed, and inter- mediary. The simple and mixed may however, in strictness, be regarded as varieties of the same class. Simple decrements are those which consist in the abstraction of any number of rows, in breadth of single molecules, or of single rows, belonging to plates of two or more molecules in thickness. Fig. 42 exhibits a simple decrement by one row in breadth on the edge c d of the primary form. Fig. 43. Fig. 43 exhibits a simple decrement by one row in breadth, on the angle c of the primary form. For the sake of rendering the expression rows of molecules generally applicable to decrements both on the angles and edges of a primary form, the term row is applied to express the single molecule first abstracted from the angle of any plate. Fig. 44. In fig. 44, the single molecule a, b, is regarded as the first row to be abstracted from the angle of the DEFINITIONS. imaginary plate; the two molecules c, rf, as the second row; the three molecules e,/J as the third row, and so on. Fig. 45. Fig. 45 shews a simple decrement by two rows in height on the edg-e of the primary form. Fig. 46. Fig. 46 shews a simple decrement by two rows in height on the angle of the primary form. It is observable in these figures, that each successive plate is less by one row of molecules than the plate on which it rests. It is by this continual recession of the edges of the added plates, that the crystal appears to decrease on its edges or angles, and that new planes are produced. The edges of the new planes which would be produced by the four preceding decrements, are shewn by the lines a b c d, fig. 42 and 45, and by the lines a b c, fig. 43 and 46.* A mixed decrement is one in which unequal numbers of molecules are omitted in height and in breadth, neither of the numbers being a multiple of the other ^ such as three in height and two in breadth, or four in * The molecules of crystals are so minute, as to render those in- equalities of surface imperceptible which are occasioned by decrements. 22 DEFINITIONS. height and three in breadth ; for if either number were a multiple of the other, as would be the case if the supposed decrement took place by two rows in height and four in breadth, or three in height and six in breadth, the new plane thus produced would be perfectly similar to that which would result from a decrement by one row in height and two in breadth, and would therefore belong to the planes produced by simple decrements. Fig. 47. Fig. 47 shews a mixed decrement on an edge of the primary form by two rows in breadth and three in height, and the lines abed mark the position of the new plane produced by this decrement. It has been found convenient to express mixed decrements by fractions, of which the numerator, or upper Jigurt, denotes the number of molecules in breadth, and the denominator, or lower figure, the number in height, abstracted from the edge or angle of the superimposed plates; thus, a decrement by | would imply a decrement by three molecules in breadth and four in height. Intermediary decrements affect only the solid angles of crystals, and may be conceived to consist in the abstraction of rows of compound molecules from the successively superimposed plates, each compound mo- lecule containing unequal numbers of single molecule* DEFINITIONS. 23 in length, breadth, and height. Thus if we suppose the compound molecule abstracted in an intermediary decrement to belong to a single plate, it must consist of some other numbers of molecules in the directions d, and e, fig. 48.* Fig. 48. In fig. 48 the compound molecule consists of a single molecule in height, two on the edge d, and three on the edge e, producing the new plane a b c. Fig. 49. Fig. 49 exhibits an intermediary decrement in which the compound molecule consists of three single molecules in height, four on the edge d^ and two on the edge e, producing the new plane a b c. In the simple and mixed decrements upon an angle, as shewn in fig. 43 and 46, the number of molecules * It may be remarked that the planes produced by simple and mixed decrements, intersect one or more of the primary planes in lines parallel to one of their edges or diagonals. The term intermediary has been used to express this third class of decrement, because the line at which the secondary plane produced by it, intersects any primary plane, is never parallel to either an edge or diagonal of that plane, but is an intermcdiatt line between the edge and the diagonal, as may be observed by com- paring the figures 42, 43, and 48. 24 DEFINITIONS. abstracted in the direction d, will always be equal to the number abstracted in the direction e. Thus if it be a simple decrement by one row in breadth, one molecule will apparently be omitted on each of the edges cf, and e, as in fig 43. But in an intermediary decrement, the numbers are obviously unequal in the direction of those edges, and the number in height will also differ from both the numbers in the direction of the edges, as in fig. 48 and 49. The new planes produced by decrements are deno- minated secondary planes, and the primary form, when altered in shape by the interference of secondary planes, is said to be modified on the edges or angles on which the secondary planes have been produced. And such edges or angles are sometimes also said to be replaced by the secondary planes. The law of a decrement is a term used to express the number of molecules in height, and breadth, abstracted from each of the successively superimposed plates, in the production of a secondary plane. When an edge, or solid angle, is replaced by one plane, it is said to be truncated. When an edge is replaced by two planes, which respectively incline on the adjacent primary planes at equal angles, it is bemlled. If any secondary plane replacing an edge, and being parallel to it, incline equally on the two adjacent primary planes, or if replacing a solid angle, it incline equally on all the adjacent primary planes, it is called a tangent plane. OF THE GONIOMETER. The instruments used for measuring the angles at which the planes of crystals meet, or, as it is frequently expressed, incline to each other, are called goniometers. Let us suppose the angle required at which the planes a, and Z>, of fig. 50, incline to each other. The inclination of those planes is determined by the portion of a circle which would be intercepted by two lines ed, e f, drawn upon them from any point e of the edge formed by their meeting, and perpendicular to that edge the point e being supposed to stand in the centre of the circle. Fig. 51. Now it is known that if two right lines as gf, d h, fig. 51, cross each other in any direction, the opposite angles def, geh, are equal. 26 OF THE GONIOMETER. If therefore we suppose the lines gf, dh, to be very thin and narrow plates, and to be attached together by a pin at e, serving as an axis to permit the point / to be brought nearer either to d, or to h; and that we were to apply the edges e d, ef, of those plates, to the planes of the crystal fig. 50, so as to rest upon the lines ed, ef, it is obvious that the angle g e h, of the moveable plates fig. 51, would be exactly equal to the angle def of the crystal fig. 50. Fig. 52. Fig. 53. The common goniometer is a small instrument cal- culated for measuring this angle geh, of the move- able plates. It consists of a semi-circle, fig. 53, whose edge is divided into 360 equal parts, those parts being half degrees, and a pair of moveable arms dh, gf> fig. 52. The semicircle having a pin at i, which fits into a hole in the moveable arms at e. The method of using this instrument is, to apply the edges de, ef, of the moveable arms, fig. 52, to the two adjacent planes of any crystal, so that they shall accurately touch or rest upon those planes in directions perpendicular to the edge at which they meet. The arm dh, is then to be laid on the plate m n of the semicircle fig. 53, the hole at e, being suffered to drop on the pin at /, and the edge nearest to h of the arm ge, will then indicate on the semicircle, as in fig. 54, the number of degrees which the measured angle contains. OF THE GONIOMETER, 27 Fig. 54. When this instrument is applied to the planes of a crystal, the points d and/, fig. 52, should be previously brought sufficiently near together for the edges e?e, efy to form a more acute angle than that about to be measured. The edges being then gently pressed upon the crystal, the points d, and /, will be gradually separated, until the edges coincide so accurately with the planes, that no light can be perceived between them. The common goniometer is however incapable of affording very precise results, owing to the occasional imperfection of the planes of crystals, their frequent minuteness, and the difficulty of applying the instru- ment with the requisite degree of precision. The more perfect instrument, and one of the high- est value to Crystallography, is the reflective gonio- meter invented by Dr. Wollaston, which will give the inclination of planes whose area is less than To^Vso of an inch, to a minute of a degree. This instrument has been less resorted to, than might, from its importance to the science, have been expected, owing* perhaps to an opinion of its use being attended with some difficulty. But the ohserv- ance of a few simple rules will render its application easy. D 2 OF THE GONIOMETER. The principle of the instrument may be thus ex- plained. Let us suppose a b c, fig. 55, to be a crystal, of which one plane only is visible in the figure, attached to a circle, graduated on its edge, and moveable on its axis at o ; and a and b the two planes whose inclina- tion we require to know. And let us further suppose the lines o e, og, to be imaginary lines resting on those planes in directions perpendicular to their common edge, and the dots at i and 7z, to be some permanent marks in a line with the centre o. Let us suppose the circle in such a position, that the line o e would pass through the dot at h, if ex- tended in that direction as in fig. 55. Fig. 56. If we now turn round the circle with its attached crystal, as in fig. 56, until the imaginary line o g, is OF THE GONIOMETER. 29 i brought into the same position as the line o e is in fig. 55, we may observe that the No. 120 will stand opposite the dot at t. This is the number of degrees at which the planes a and b incline to each other. For if we suppose the line og, extended in the direction oz, as in fig. 56, it is obvious that the lines oe, oz, which are perpen- dicular to the common edge of the planes a and 6, would intercept exactly 120 of the circle. Hence an instrument constructed upon the principle of these diagrams, is capable of giving with accuracy the mutual inclination of any two planes, if the means can be found for placing them successively in the relative positions shewn in the two preceding figures. Fig. 57. When the planes are sufficiently brilliant, this pur- pose is effected by causing an object, as the line at /w, fig. 57, to be reflected from the two planes , and &, successively, at the same angle. It is well known that the images of objects are reflected from bright planes at the same angle as that at which their rays fall on those planes; and that when the image of an object reflected from a horizon- tal plane is observed, that image appears as much 30 OF THE GONIOMETER. below the reflecting surface, as the object itself is above it. If therefore the planes #, and b, fig. 57, be suc- cessively brought into such positions, as will cause the reflection of the line at ??z, from each plane, to appear to coincide with another line at n, both planes will be successively placed in the relative positions of the corresponding planes in figs. 55 and 56. Fig. 58. To bring the planes of any crystal successively into these relative positions by the assistance of the reflec- tive goniometer, the following directions will be found useful. The instrument, as shewn in the sketch fig. 58, should be first placed on a pyramidal stand, and the stand on a small steady table, placed about 6 to 10 or 12 feet from ajlat window. The graduated circular plate should stand perpen- dicularly from the window, the pin x being horizontal^ with the slit end nearest to the eye. * Place the crystal which is to be measured, on the table, resting on one of the planes whose inclination is " This goniometer is sometimes drawn with the pin x in the direction of its axis, in which position of the pin, the instrument may be regarded as nearly useless. OF THE GONIOMETER. 31 required, and with the edge at which those planes meet, the farthest from you, and parallel to the win- dow in your front. Fi- 69. Attach a portion of wax about the size of d, to one side of a small brass plate e, fig. 59 lay the plate on the table with the edge/ parallel to the window, the side to which the wax is attached being uppermost, and press the end of the wax against the crystal until it adheres ; then lift the plate with its attached crys- tal, and place it in the slit of the pin jr, with that side uppermost which rested on the table. Bring the eye now so near the crystal, as, without perceiving the crystal itself, to permit your observing distinctly the images of objects reflected from its planes; and raise or lower that end of the pin ,r which has the small circular plate affixed to it, until one of the horizontal upper bars of the window is seen reflected from the upper orjirst plane of the crys- tal, which corresponds with plane , fig. 55 and 56, and until the image of the bar is brought nearly to coincide with some line below the window, as the edge of the skirting board where it joins the floor. Turn the pin x on its own axis, if necessary, until the reflected image of the bar of the window coincides accurately with the observed line below the window. Tutn now the small circular plate a on its axis, and from you, until you observe the same bar of the window reflected from the second plane of the crystal corresponding with plane &, fig. 55 & 56, and nearly coincident wiili the line below ; and having, in adjusting 32 OF THE GONIOMETER. the first plane, turned the pin x on its axis to bring the reflected image of the bar of the window to coincide accurately with the line below, now move the lower end of that pin laterally, either towards or from the instrument, in order to make the image of the same bar, reflected from the second plane, coincide with the same line below. Having assured yourself by looking repeatedly at both planes, that the image of the horizontal bar reflected successively from each, coincides with the same line below, the crystal may be considered as adjusted for measurement. Let the 180 on the graduated circle be now brought opposite the o of the vernier at c, by turning the middle plate b ; and while the circle is retained accurately in this position, bring the reflected image of the bar from the Jirst plane to coincide with the line below, by turning the small circular plate a. Now turn the graduated circle from you, by means of the middle plate b, until the image of the bar reflected from the second plane is also observed to coincide with the same line below. In this state of the instrument the ver- nier at c will indicate the degrees and minutes at which the two planes incline to each other. SECTION I. GENERAL VIEW. THE regularity and symmetry observable in the forms of crystallized bodies, must have early attracted the notice of naturalists ; but they do not appear to have become objects of scientific research, as a branch of natural history, until the time of Linnaeus. He first gave drawings and descriptions of crystals, and attempted to construct a theory concerning them, somewhat analogous to his system of Botany. We are indebted however to Rome de L'Isle for the first rudiments of crystallography. He classed together those crystals which bore some common resemblance, and selected from each class some sim- ple form as the primary, or fundamental one ; and conceiving this to be truncated in different directions, he deduced from it all its secondary forms ; and it was he who first distinguished the different species of minerals from each other by the measurements of their primary forms. The enquiries of Bergman were nearly contem- poraneous with those of the Abbe Haiiy, and both these philosophers appear to have entertained at the same time nearly the same views with regard to the structure of crystals ; both having supposed that the production of secondary forms might be explained by the theory of decrements on the edges or angles of the primary. E 34; GENERAL VIEW. Here, however, Bergman's investigation appears to have terminated, while the Abbe Haiiy proceeded to complete this theory, by determining the forms and dimensions of the molecules of which he conceived the primary forms were composed, and by demonstrating mathematically the laws of decrement by which the secondary forms might be produced. He also established a peculiar nomenclature, to designate individually each of the observed secondary forms of crystals; the nomenclature consisting of terms derived from some remarkable character or relation peculiar to each individual form. But the disadvantage accruing to the science from encumber- ing each individual crystal with a separate name, must be immediately apparent, when it is considered that the rhomboid of carbonate of lime alone is capa- ble of producing some millions of secondary crystals by the operation of a few simple laws of decrement. The number of names requisite to designate all these, if they existed, would form an insuperable obstacle to the cultivation of the science of crystal- lography, even if it were practicable to devise some sufficiently short and simple terms for the purpose. To obviate the inconvenience arising from the use of so many individual names, the Gomte de Bournon adopted a much simpler method of denoting the secondary forms. He numbered all the individual modifications he had observed, from one onwards, and as the secondary forms are produced either by a single modification, or by the concurrence of two or more single modifications, any secondary form what- ever might, according to his method, be expressed by the numbers which designate all the particular modi- fications which it is found to contain. Mr. Phillips has adopted this method in his papers on oxide of tin, red oxide of copper, &c. published in GENERAL VIEW. 35 the Transactions of the Geological Society, and has thus proved its utility for the purpose of crystallo- graphical description. The descriptive system of the Comte de Bournon, with some alterations, will be adopted in this volume, as well as the theory of decrements which constitutes the basis of the Abbe Haiiy's System of Crystallo- graphy. SECTION II. MOLECULES. The homogeneous molecules which are aggregated together in the production of crystals, are supposed to be minute, symmetrical, solid particles, contained within plane surfaces. They are also conceived to be again separable from each other by mechanical divi- sion, which however stops very short of the separa- tion of single molecules from the mass which has been formed by their union. For, however minutely we may divide a piece of carbonate of lime, we cannot imagine that we have ever obtained any single portion or molecule contain- ing only one atom or proportion of carbonic acid, and one atom or proportion of lime. This effect of mechanical division merely implies that the molecules are separated at their surfaces by cleavage, and are not divided or broken. And it thus serves to distinguish them from the elementary par- ticles or atoms which enter into their composition, and which cannot be separated from each other but by chemical agency.* * Although it is not immediately connected with Crystallography, I am induced to state an observation here which has occurred to me relative to the forms of the homogeneous molecules of minerals, when com- pared with the forms of the atoms , or elementary particles, of which those molecules are composed. We certainly know nothing of the forms of the atoms of those elemen- tary substances which do not occur crystallized, such as oxygen, hydro- gen, and many others. But we infer from analogy that the atoms of sulphur, carbon, the metals, and such other elementary substances as MOLECULES. 37 The figures of the solid molecules require to be explained in reference to each of the five following classes of primary forms. 1. The cube and all the other classes of parallelo- pipeds, or solids contained within six planes. 2. The regular octahedron and all the other classes of octahedrons. 3. The regular tetrahedron. 4. The rhombic dodecahedron. 5. The hexagonal prism. If we attempt to fracture a piece of galena, it will split into rectangular fragments. But we find by observing the secondary forms of galena, that its primary crystal may be a cube, and we know also that by supposing this cube to be composed of cubic molecules, the angles at which the secondary planes incline upon the primary, may be computed and de- termined with mathematical precision. We are there- fore led to infer, that if the rectangular fragments ob- tained bij cleavage could be reduced to single molecules , those molecules would be cubes.^ are found crystallized, are similar in form to the molecules of other crys- tallized substances which present similar primary forms. Now according to two suppositions, the first being that entertained by the Abbe Haiiy, the other arising out of a theory which will be pre- sently stated, the molecule of sulphur may be an irregular tetrahedron, or a right rhombic prism , and the molecule of silver a regular tetrahedron^ or a cute. But the compound of sulphur and silver crystallizes in the form of a cuke. Hence the molecule of sulphuret of silver, arising out of the chemical union of irregular tetrahedrons with the regular tetrahedrons or cubes t accord- ing to one supposition, or of right rhombic prisms and cubes y according to the ether supposition, performs the function of a cube. If this subject were pursued it might be shewn that the cubic function is performed by molecules very variously composed. f Whether these little cubes would consist of one or more atoms of lead and of sulphur, or how these elementary particles would be com- bined in the production of a cubic molecule, are circumstances not im- mediately relating to Crystallography, however interesting they may be as separate branches of enquiry. 38 MOLECULES. If we reduce a crystal of carbonate of lime to frag- ments, the planes of those fragments will be found to incline to each other at angles which are respectively equal to those of the primary rhomboid. We there- fore infer that the molecule of carbonate of lime is a minute rhomboid similar to the primary form. Sulphate ofbarytes may be split into right rhombic prisms, whose angles are respectively equal to those of the primary crystal. It is therefore supposed that the primary crystal and the molecules of this substance are similar prisms. Having thus found that crystals belonging to seve- ral of the classes of parallelepipeds may be split into fragments resembling their respective primary forms; and having assumed that these fragments represent the molecules of each of those forms respectively, it has been concluded that the primary forms and the molecules of all the classes of parallelepipeds are respectively similar to each other. This similarity does not however exist between the other classes of primary forms and their respective molecules. Fig. 60. Fig. 61. If a regular hexagonal prism of phosphate of lime be split in directions parallel to all its sides, it may be divided into trihedral prisms whose bases are equi- lateral triangles ; these may be regarded as the mole- cules of this class of primary forms. Fig. 60 shews the hexagonal prism composed of t MOLECULES. 39 trihedral prisms; and fig. 61 shews the trihedral prism separately. Fig. 62. If we reduce a cube of jluate of lime to fragments, we shall find that it does not split in directions paral- lel to its planes as galena does, but that it splits obliquely. If we suppose fig. 62 a cube of fluor, and we apply the edge of a knife to the diagonal line a b y and strike it in the direction of c, we may remove the solid angle a b e c. Fig. 63. If we again apply the edge of the knife to the same line a b, and strike it in the direction of f, we may remove another solid angle a bfd; applying the knife again in the direction of the line c c?, and strik- ing successively in the directions g, and h, we may remove two other solid angles. The new solid produced by these cleavages is re- presented by fig. 63. MOLECULES. Fig. 64. If we apply our knife again to the line i A*, k I, I m, m 2, fig. 64, and strike in the direction of n, we may remove the remaining solid angles of the cube, and we shall then obtain the regular octahedron iklmno. Fig. 65. The position of this octahedron in the cube is shewn by fig. 65. This octahedron is the primary form of Jluate of limey and it may obviously be cleaved in a direction parallel to its own planes. Fig. 66. To illustrate more perspicuously the relation we are about to trace between the octahedron and tetra- MOLECULES. 41 hedron, it will be convenient to place the octahedron of fluor, which we have just obtained, in the position represented in fig. 66, resting on one of its planes. Fig. 67. In this position of the crystal, if we suppose the three lines a b, c d, e f^ to be drawn through the centre, and parallel to the edges, of the now upper- most plane, and if we apply our knife to the line a b, we may cleave the crystal parallel to the plane g, and may detach the portion a b g /, fig. 67. Fig. 68. By cleaving again from the lines c d, and ef, paral- lel to the plane ^, and to the back plane of fig. 66, and by also cleaving parallel to the plane on which the figure rests, beginning at the line i k, we shall obtain a regular tetrahedron as seen in fig. 69. In fig. 68 this tetrahedron is exhibited in the position which it occupied in the octahedron. MOLECULES. The tetrahedron thus obtained, may be reduced again to an octahedron, as shewn in fig. 69, by removing a smaller tetrahedron, as a b c d, from each of its solid angles. And all the fragments separated from the octa- hedron by the cleavages just described, may also be reduced, by cleaving in the proper directions, to regular octahedrons and tetrahedrons. In this case two distinct solids are obtained from the cleavage of an octahedral crystal ; and the Abbe Hauy has chosen to assume the tetrahedron as the molecule of the octahedral crystal, upon the supposition that if the cleavage were continued until only single molecules remained to be separated, these molecules would be tetrahedrons ; and the octahedron is, according to his theory, conceited to be composed of tetrahedral solids united by their points, and octahedral spaces. * From considerations analogous to these, the Abbe Haiiy has concluded that the tetrahedron, when it occurs as a primary form, is constituted also of tetra- hedral molecules and octahedral spaces. * The same imaginary structure has also been supposed by the Abbe Haiiy to exist in every class of octahedrons, the molecules peculiar to each being distinct irregular tetrahedrons, varying in their angles and relative dimensions in each particular case. But it will be attempted to be shewn presently that this imaginary structure does not belong to the octahedron, and that the tetrahedral solid dees not rtprestnt the molecule of that form* MOLECULES. 45 The regular dodecahedron may be cleaved into obtuse rhomboids, obtuse octahedrons, and irregular tetrahedrons, as will be shewn in the section on clea- vage. Of these the Abbe Hauy has chosen the irregu- lar tetrahedron for the molecule of the dodecahedron, and he has supposed that the decrements on this form are produced by the abstraction, not of single molecules, but of masses of single molecules packed into the figure of those obtuse rhomboids which are produced from its cleavage.* The very complicated system of molecules which the Abbe Haiiy has, by this view of the structure of the octahedron and dodecahedron, introduced into his otherwise beautiful theory of crystals, arid the apparent improbability that the molecules of the cube, the regular octahedron, tetrahedron and dode- cahedron, among whose primary and secondary forms so perfect an identity subsists, should really differ from each other, have induced me to propose a new theory of molecules in reference to all the classes of octa- hedrons, to the tetrahedrons, and the rhombic dode- cahedron, which 1 shall now state. Fluate of lime, as we have seen, has for its primary form a regular octahedron, under which it sometimes occurs in nature ; but it is generally found in the form of a cube, and sometimes as a rhombic dodeca- hedron, and it has a cleavage in the direction of Us primary planes. Galena, whose primary form is a cube, is also found under the forms of an octahedron, and rhombic dode- cahedron, with a cleavage parallel to its cubic planes. * Under the head of cleavage I shall endeavour to explain the nature of the relation which the different solids obtained by cleavage from the tetrahedron, octahedron, and rhombic dodecahedron, respectively bear to those primary forms, and to each other; and to shew that they da not in eithtr case represent th: molecules of those farms. 44 MOLECULES. Grey copper, whose primary form is a tetrahedron, occurs under the forms of the cube, octahedron, and rhombic dodecahedron. Blende is found sometimes, though rarely, crystal- lized in cubes, sometimes in octahedrons, tetrahedrons, and rhombic dodecahedrons. Fig. 70. If we attempt to fracture a cube of blende, we find it will split in directions parallel only to its diagonal planes. These cleavages will truncate the edges of the cube, and if continued until all the edges are removed, and the face of the cube disappear, a rhom- bic dodecahedron will be produced, which has been considered the primary crystal of blende. If a cube of blende, fig. 70, be cleaved in directions parallel to its diagonal planes, beginning at the lines a b, c d, fig. 71 will be produced. Fig. 71. If fig. 71 be further cleaved in directions corres- ponding to a b c d e, so as to remove all the perpen- dicular edges, and to obliterate the remainder of the perpendicular planes of the cube, fig. 72 will remain. MOLECULES. Fig. 72. i'jfo: Fig. 73. Fig\ 73 exhibits the dodecahedron contained in fig. 72 ; this may be obtained by cleavages in direc- tions corresponding with the lines a df, fig. 72, which will remove the solid angles of the base on which fig. 72 and 73 rest. Fig. 74. Fig. 74 shews the position of the rhombic dodeca- hedron in the cube. Having thus observed that the cube, the regular tetrahedron and octahedron, and the rhombic dodeca- hedron are common as primary or secondary forms to different crystallized substances, we may reasonably infer that they are produced in each instance by mole- 4.6 MOLECULES. cules of a form which is common to all; and let us suppose this common molecule to be a cube. Fig. 75, 76, 77, and 78, shew the arrangement of the cubic molecules in each of these forms. Fig. 75. Fig. 75 in the cube. Fig. 76. Fig. 76 in the tetrahedron. Fig. 77. Fig. 77 in the octahedron, , MOLECULES, Fig. 78. 47 Fig. 78 in the rhombic dodecahedron. These arrangements of cubic molecules cannot be objected to on account of any supposed imperfection of surface which would be occasioned by the faces of all the primary forms, except the cube, being con- stituted of the edges, or solid angles, of the molecules* For as we observe that the octahedral and dodeca- hedral planes of some of the secondary crystals of galena, which are obviously composed of the solid angles, or edges, of the cubic molecules, are capable of reflecting objects with great distinctness, it is evident that the size of the molecules of galena is less than the smallest perceptible inequality of the splendent surface of those planes, and hence we in- fer generally that there will be no observable difference in brilliancy between the surfaces of the planes obtained by cleavage parallel to the sides of molecules, and of those which would expose their edges or solid angles. This theory may be reconciled with the cleavages which are found to take place parallel to the primary planes of the tetrahedron, the octahedron, and the rhom- bic dodecahedron, as well as to those of the cube, if we suppose the cubic molecules capable of being held to- 48 MOLECULES. gether with different degrees of attractive force in dif- ferent directions. * I shall call this force molecular attraction. Fig. 79. When this attraction is least between the planes of the molecules, they will be more easily separated by cleavage in the direction of their planes, than in any other direction, as shewn in fig. 79, and a cubic solid will be obtained. Fig. 80. When the attraction is least in the direction of the axis of the molecules, they will be the most easily se- parated in that direction, as in fig. 80, and the octa- hedron or tetrahedron will be the result of cleavage. * It is possible to conceive that the nature, the number, and the par- ticular forms, of the elementary particles which enter, respectively, into the composition of these three species of cubic molecules, may vary so much as to produce the variety of character which I have supposed to exist. MOLECULES, 49 Fi s . 81. And if the attraction be least. in the direction of its diagonal planes, the edges will be most easily sepa- rated, as in fig. 81, and a rhombic dodecahedron will be the solid produced by cleavage. This supposition of greater or less degree of mole- cular attraction in one direction of the molecule than in another, is consistent with many well known facts in Crystallography. - The primary form both of corundum, and of car- bonate of lime, is a rhomboid ; and the crystals of these substances may be cleaved parallel to their pri- mary planes, the carbonate of lime cleaving much more readily than the corundum. But the corundum may also be cleaved in a direction a 6, fig. 82, perpen- dicular to its axis, which carbonate of lime cannot be. This cleavage would either divide the rhombic mole- cules in half, or,' the cleavage planes would expose the terminal solid angles of the contiguous molecules. G 50 MOLECULES. But it is contrary to the nature of molecules that they should be thus divided, and we may therefore infer from this transverse cleavage that the molecular attraction is comparatively less in the direction of the perpendicular axis of the molecules of corundum, than it is in the same direction of those of carbonate of lime. And from the greater adhesion of the planes of corundum, than of those of carbonate of lime, we infer that the attraction is comparatively greater be- tween the planes of the molecules of the corundum, than between those of carbonate of lime.* This supposition of the existence of a greater or less degree of molecular attraction in one direction of the molecule than in another, appears to explain the nature of the two sets of cleavages which occur in Tungstat of lime : one of these sets is parallel to the planes of an acute octahedron with a square base, which we will call the primary crystal ; the other set would produce tangent planes upon the terminal edges of that crystal. If we suppose the molecules to consist of square prisms whose molecular attraction is greatest in the direction of their prismat^p axis, and nearly equal in the direction of their diagonal planes , and of their oblique axes, the first set of cleavages may be conceived to expose the edges of the molecules, and the second set to expose their solid angles. * I am aware of an objection that may be made to this view of the subject, by supposing all the cleavages which are not parallel to the primary planes of a crystal, to be parallel to some secondary plane, and to be occasioned by the slight degree of cohesion which frequently sub- sists between the secondary planes of crystals and the plates of mole- cules which successively cover them during the increase of the crystal in size; but althpugh the second set of cleavages may sometimes be connected with the previous existence of a secondary plane, it may also be explained! according to the theory I have assumed. Those cleavage planes which would not expose the planes, edges or solid angles of the molecules, must be considered to belong always to , MOLECULES. 51 This theory may, by analogy, be extended to the form of molecules of every class of octahedron. For we may conceive the molecules of all the irre- gular octahedrons to be parallelepipeds, whose least molecular attraction is in the direction of their diagonal planes. Thus the molecules of octahedrons with a square, a rectangular, and a rhombic base, would be square, rectangular, and rhombic prisms respectively; the dimensions of such molecules being proportional re- spectively to the edges of the base and to the axis of each particular octahedron. According to the view here taken, the following table will exhibit the form of the molecules belong- ing to each of the classes of primary forms. The cube rhombic dodecahedron , J all quadrangular prisms molecules, similar prisms. -\ Proportional r i i in dimensions octahedron with a square base 5 mo e ? u e> a to the edges rectangular C molecule, a rectan- I ^ !L ^ S ^' base I gular prism f , , - rhombic base $ lecule,a rhombic ^ocSe- t P nsm dron,respec- J lively. rhomboid molecule, a similar rhomboid , . C molecule, an equilateral triangular hexagonal prism } . Having thus advanced a new theory of molecules in opposition to one that had been long established, and possibly without a much better claim to general the class of /danes of composition, a term which Mr. W. Phillips has plied to those cleavage planes which result from cleavages parallel secondary planes only. 52 MOLECULES. reception than the former theory possessed, I cannot avoid observing that the whole theory of molecules and decrements, is to be regarded as little else than a series of symbolic characters, by whose assistance we are enabled to investigate and to demonstrate with greater facility the relations between the primary and second- ary forms of crystals. And under this view of the subject, we ought to divest our notions of molecules, and decrements, of that absolute reality, which the manner in which it is necessary to speak of them in order to render our illustrations intelligible, seems generally to imply. SECTION III. STRUCTURE. THE structure of crystals, or the order in which their molecules are arranged, may be inferred from an experiment with common salt. If we dissolve a portion of this salt in water, and then suffer the water to evaporate slowly, crystals of salt will be deposited on the sides and bottom of the vessel. These will at first be very minute, but they will increase in size as the evaporation proceeds; and if the quantity of salt dissolved be sufficient, they will at length attain a considerable bulk. If the forms of the small crystals be examined, they may be found to consist of en- tire, or modified cubes. If we continue to observe any of these cubic or modified crystals during their increase in bulk, we may find that the forms of some of them undergo a change, by the addition of new planes, or the extinction of some that had previously existed. But we shall also frequently find that both the cube, and the modified crystal, when enlarged, preserve their respective forms. The increase of a crystal in she appears therefore to be occasioned by the addition of molecules to some, or all. of the planes of the smaller crystal, whether these planes be primary or secondary. STRUCTURE. Fig. 83. ,7 7t If we apply the edge of a knife to the surface of any one of these cubic crystals of common salt, in a direction parallel to an edge of the cube, as at a b, fig. 83, the crystafmay, by a slight blow, be cleaved parallel to one of its sides. If we apply a knife in the same manner successively to the other lines c d, e f, g h, and to the other sur- faces of the crystal, so that its edge be parallel in each instance to the edge of the cube, we shall find that there are cleavages parallel to all the planes of the cube; and if the crystal be split with perfect accuracy, a cubic solid may be extracted ; and the rectangular plates which have been removed by these cleavages, may be also subdivided into smaller cubes. From these circumstances we infer that the mole- cules which have successively covered the planes of the small crystals, are cubes, and that they are so arranged as to constitute a series of plates, as shewn in p. 18. And we further conclude that the molecular attraction is least, in common salt, between the sur- faces of the molecules. This regular structure is supposed to belong to all regularly crystallized bodies. It frequently happens that the regular crystal- lization of bodies has been prevented by some dis- STRUCTURE. 55 turbing cause, in which case the crystalline mass will be curved or otherwise irregular, or it may even present a granular character. This granular cha- racter would be presented if the solution we have supposed of common salt were rapidly evaporated and suddenly cooled. SECTION IV. CLEAVAGE. The spilling a crystal in the manner already des- cribed, is, in the language of Mineralogy, termed cleaving it. The direction in which the crystal can be split is called the direction of the cleavage , or the natural joint of the crystal. The direction of the natural joints may depend, according to the preceding theory, upon the com- parative degrees of molecular attraction existing in the different directions of the molecules. This may be so proportioned in different directions, as to occa- sion other cleavages than those which are parallel to the planes which we may assume as the primary planes, as in the instances already cited of the corun- dum, and tungstat of lime. When this occurs the crystal is said to possess two or more sets of cleavages. Those which are parallel to the planes of the primary form, are called the primary set, and those which are not parallel to those planes are termed supernumerary sets. The oxide of tin, described by Mr. Phillips in the Geological Transactions, has three sets of cleavages ; one parallel to the planes of an obtuse octahedron with a square base, which is considered the primary set, and two others which are sitperminierftr?/, and CLEAVAGE. fift are parallel to the edges, and to the diagonals, of the square base, being at the same time perpendicular to the plane of that base. If all the planes of any primary form be similar, as those are of the cube, rhomboid, and some other forms, the primary cleavages will generally be ef- ected with equal facility in the direction of each of those planes, and the new planes developed by this cleavage will be similar in lustre and general cha- racter. This may be illustrated by cleaving galena and carbonate of lime. Where the planes of a primary form are not all similar, as in all prisms, and some octahedrons, the .. primary cleavage is not effected with equal facility in all directions, nor do the new planes all agree in their general characters. Hence the cleavage planes of a mineral will frequently enable us to determine what is not its primary form, by their similarity or dissimilarity; but, as will be seen in the section on primary forms, the cleavage is not sufficient to deter- mine what the primary form really is. Felspar, cyanite, and sulphate of lime, afford in- stances of the greater facility with which a cleavage takes place in one or two directions than in any other. The Abbe Haiiy has supposed that these unequal cleavages are occasioned by the unequal extension of the different primary planes. The broader planes, presenting more points of contact than the narrower ones, may, he imagines, be held together with greater force than the narrower ones are. This may possibly be the cause of the observed inequality of cleavage, or possibly where the planes are unequal, the degree of attraction between point and point is unequal also. 58 CLEAVAGE. There are among minerals some substances which yield readily to mechanical division in one or two directions, but do not admit of distinct cleavage in a third direction, so as to produce a regular solid. This circumstance has introduced into mineralogy the terms single cleavage, or double, triple, fourfold, &c. cleavage, which are sometimes perplexing to a learner, as they may be confounded with the different sets of cleavages before spoken of. But these terms single, double, triple cleavage, &c. are intended to refer strictly to the sets of primary cleavage only. When a mineral can be split in only one direction, the cleavage is said to be single ; when in two direc- tions, which may be conceived to give four sides of a prism, it has a double cleavage. When there is a cleavage in . three directions, such as to produce either the lateral planes of the hex- agonal prism, or a solid bounded by six planes which are parallel when taken two and two, it is termed a triple cleavage. A four-fold cleavage, or one in four directions, will produce a tetrahedron, an octahedron, or a perfect hexagonal prism ; the two latter solids consisting of four pairs of parallel planes, lying in as many dif- ferent directions. The rhombic dodecahedron possesses six pairs of parallel planes lying in different directions, and may be said therefore to have a sixfold cleavage. Sometimes the natural joints of a crystal may be perceived by turning it round in a strong light, al- though it cannot be cleaved in the direction of those joints. Different specimens of the same substance will also yield to the knife or hammer with unequal degrees of facility; and even carbonate of lime, which splits CLEAVAGE. 59 readily in general, will sometimes present a con- choidal fracture.* As a crystalline solid cannot be contained within planes lying in less than three directions, it is obvious that it cannot be produced by a single or double cleavage. The solids obtained by cleavage may therefore, according to what has preceded, consist either of primary forms produced by triple, fourfold, or sixfold primary cleavages, or of other forms resulting from the supernumerary cleavages, either alone or com- bined with the primary. But another class of solids may also result from cleavage when that takes place parallel to some only of the primary planes of those forms which possess fourfold or sixfold cleavages. From a primary triple cleavage it. is clear that only a single solid can be produced, that solid being a parallelepiped . But from either a fourfold or sixfold primary cleavage, more than one solid may result, according as the cleavage takes place parallel to all, or only to some, of the primary planes. * Some practice is necessary in order to cleave minerals neatly, and some experience in the choice of the instruments to be used for this purpose. In many instances, the mineral being placed on a small anvil of iron or lead, a blow with a hammer will be sufficient for dividing it in" the direction of its natural joints ; and sometimes a knife or small chissel may be applied in the direction of those joints, and pressed with the hand, or struck with a hammer; or the crystal may be held in the hand and split with a small knife; or it may be split, by means of a pair of small cutting pincers whose edges are parallel. A small short chissel, fixed with its edge outward in a block of wood, is a convenient instrument for resting a mineral upon which we are desirous of cleaving. CM, RAVAGE. Fig. 81. If we cleave an octahedron parallel to six only of its planes, omitting any two opposite ones, as b and e, fig. 84, and if we continue the cleavage until only the central points of the planes b and e remain, a figure of six sides will evidently be produced. This figure is a rhomboid whose plane angles are 60 and 120. Fig. 85. Fig. 85 shews the position of this rhomboid in the octahedron, from which it is evident that the cleavage would be continued as far as the lines f k, I m, n o, and those which are parallel to them on the opposite plane. CLEAVAGE. Fig. 86 exhibits the same rhomboid separately, the planes being marked with the same letters as are placed on such planes of the octahedron as are pa- rallel to those of the included rhomboid. Fig. 87. Our rhomboid may thus be regarded as an imperfect octahedron, two of its planes being concealed, or covered by small tetrahedrons p r s, and t u x, as in figure 86. These tetrahedrons consist of masses of cubic molecules, and by their removal, as in fig. 87, we shall obviously reproduce the perfect octahedron. Fig. 88. If we now cleave the octahedron parallel to any four alternate planes, as c d, ef, fig. 84, and continue the cleavage as far as the lines i k, I m, no, fig, 88, and until only the central points of the four planes a, b, g, h, remain, we shall produce a regular tetra- hedron, as shewn by the interior lines in the figure. CLEAVAGE. Fig. 89. Fig. 89 exhibits this tetrahedron separately, its planes being marked with the same letters as appear on the planes of the octahedron, fig. 84, which are parallel to those of the included tetrahedron. * * The tetrahedron thus obtained may be regarded as an imperfect octahedron, four of its planes being con- cealed, or covered by smaller tetrahedrons, p q r s, p q u t, u q x T, q x r #, as in fig. 90, and it is capable of being reduced again to the perfect octa- hedron by the removal of those masses of cubic molecules which constitute the tetrahedrons by which the concealed planes are covered. The tetrahedron and octahedron have thus ob- viously the same set of cleavages, and if the tetra- hedron be the primary form, the octahedron may be regarded as an imperfect tetrahedron, requiring cer- tain additions to complete that form.* * The student is advised to trace the relation of the octahedron to the acute rhomboid and tetrahedron, by means of an octahedron of fluor produced by cleavage or otherwise. Let him place this on a table, and by the assistance of a small hammer and a knife, he may procure from it, by well observing the figures as he proceeds, the acute rhomboid, a#d tetrahedron, and from them he may re-produce the octahedron. CLEAVAGE. Fig. 91. If the rhombic dodecahedron, fig. 91, be cleaved parallel to the planes, , 6, c, d, and to the four planes opposite to these, until the four remaining planes of the dodecahedron disappear, an obtuse octa- hedron will be produced. Fig. 92. Fig. 92 exhibits this octahedron separately, the planes being marked by the same letters as appear on the corresponding planes of the dodecahedron. Fig. 93. If the cleavage be effected parallel only to the planes 0, d, e, and 7z, b, k 9 until the other primary planes disappear, an obtuse rhomboid will result, as seen in fig. 93; this rhomboid measures 120 over the edges at which the planes #, and e, meet. CLEAVAGE, Fig. 94. If the cleavage take place parallel only to the planes c d and i k, and be continued until only the four cleavage planes remain, an irregular tetrahedron, fig. 94, will be produced, whose planes meet at an angle of 90 at the edges n o, p q, and at an angle of 60 at the other edges. Thus an obtuse octahedron, d on m, fig. 92, mea- suring 60 % an obtuse rhomboid of 120", and an" irregular tetrahedron , obtained by partial cleavages from the rhombic dodecahedron, may be regarded as imperfect dodecahedrons, to which figure the?/ may be reduced, by detaching from each solid the portions of cubic molecules which respectively cover the obscured dodecahedral planes. But it is very obvious that these imperfect forms may be obtained as well by cleaving at once through the interior of the crystal, in directions parallel to 8 to 6 or to 4 only of the primary planes, as by beginning to cleave from the outside, and arriving by degrees at the new figure in the manner already described. Hence it appears that a dissection of the octahedral and dodecahedral crystals by cleavages, parallel to some only, of the primary planes, will yield only the imper- fect solids above described, not any of which will represent the molecules of which the crystals are com- posed.* * Blende will afford the student an opportunity of producing, by cleavage, the solids represented by fig-. 91 to 94. CLEAVAGE, 65 The relation of the tetrahedron to the octahedron in reference to the theory of cubic molecules, may be explained in the following manner. The Abbe Haiiy's theory, it will be recollected, supposes that if the tetrahedron obtained by cleavage from the octahedron, were to be successively reduced to an octahedron and four still smaller tetrahedrons, we should at length arrive at a tetrahedron consisting of four single tetrahedral molecules enclosing only an octahedral space, instead of an octahedral solid. But according to the structure assigned to the octahedron by the theory of cubic molecules, that figure is an entire solid; and the smallest tetra- hedron that can be imagined to exist, will contain an octahedral solid, and would be reduced to an octahedron by the removal of four cubic molecules from its four solid angles, and not of four tetrahedrons. Fig. 95. Fig. 96. Let fig. 96 be supposed to represent the smallest octahedron that can be imagined to exist, formed of seven cubic molecules, and let fig. 95 represent a i 66 CLEAVAGK. tetrahedron containing this minute octahedron. The tetrahedron would obviously be reducible to the oc- tahedron, as other tetrahedrons are, by the removal of all its solid angles. But it is apparent that the solid angles to be re- moved in this instance, are the small cubes a e g t, and by their removal the octahedral solid shewn in fig. 96 will remain. This octahedron is supposed to rest on one of its planes, and the molecules b c, c h, /c, c d, may be conceived to constitute four of its edges. Thus the necessity of adopting the tetrahedron as the molecule of the octahedron is removed, and in consequence a more simple theory of the structure of the octahedron, may be substituted for that which has been established upon the adoption of tetrahedral molecules. By a similar mode of reasoning, the compatibility of the cubic molecule with the solids obtained by cleavage from the rhombic dodecahedron, might be shewn; and by adopting the cubic molecule, a more simple theory of decrement, in relation to the rhom- bic dodecahedron, may be substituted for that which has been established upon the assumption of the irregular tetrahedron as the integrant molecule, and the obtuse rhomboid as the subtractive molecule. SECTION V. DECREMENTS. Decrements have been already defined to be either simple, mixed, or intermediary ; and the simple decre- ments have been divided into two classes, according as they take place in breadth, or in height : see Defi- nitions, page 19. The manner in which all the classes of decrements operate in the production of new planes, has also been explained. Simple and mixed decrements take place either on the edges of crystals, or on the angles, and produce new planes which intersect one at least of the primary planes, in lines parallel to one of its edges or diagonals. Fig. 97. If either a simple or mixed decrement take place on the edge a b, of any primary form whatever, a new plane is produced, whose intersection c d, with the pri- mary plane along which the decrement may be conceived to proceed, is parallel to the edge a b, from which it may be said to begin or set out. Fig. 97 shews the character of the secondary plane produced by a simple or mixed decrement on the edge of a rectangular prism. i 2 DECREMENTS. Fig. 98 shews the character of a similar plane on the edge of a tetrahedron. Fig. 99. Fig. 99 shews the character of a similar plane on the edge of an octahedron. If a simple or mixed decrement take place on the angle of a tetrahedron or octahedron, one new edge c' d', of the secondary plane, will be always parallel to an edge a b, of the primary form, as shewn in figures 98 and 99. If either a simple or mixed decrement take place on the angle of any primary form, except the tetrahedron and octahedron, a new plane is produced, whose in- tersection with the primary plane, along which the DECREMENTS, decrement may be conceived to proceed, is parallel to the diagonal of that plane. Fig. 100. Fig. 100 exhibits a plane produced on the angle of a rectangular prism, by a simple or mixed decrement; the edge c d, of the secondary plane, being parallel to the diagonal a b, of the primary form. Fig. 101, Intermediary decrements may be said to take place only on the solid angles of crystals, by the omission of unequal members of molecules in the direction of the edges which meet at such solid angle. And a plane is thus produced, none of whose lines of intersection, a by c b, ac, with the primary planes, are parallel to any edge, or diagonal of those planes. Fig. 101 shews the position of a plane produced by an intermediary decrement on the angle of a rectan- gular prism. 7Q DECREMENTS, Fig. 102. Fig. 102 contains a similar decrement on the angle of a tetrahedron. Fig. JOS. Fig. 103 shews the effect of a similar decrement on the angle of an octahedron. In the illustrations hitherto given of the nature of decrements, and of the characters of the secondary planes produced by them, we have considered the effect produced upon only a single edge or angle of the primary form. But as decrements generally take place equally on all the similar edges and angles of any primary form, it will be necessary now to enquire into the manner in which these similar edges and angles are affected, when they are all operated upon at the same time, by any given decrement. From the definition already given in page 3, of the nature of similar edges and angles, it will appear that in the rectangular prism, those edges only are similar DECREMENTS. 71 which arc parallel to each other. And if we refer to the tables of modifications of that form, which will be found in a subsequent section, we shall observe it is only the parallel edges which are affected by the modifications 6, c and d. Fig. 104. Let us now suppose a modification belonging to the class c to have taken place on a right rectangular prism, and let us suppose the secondary crystal pro- duced by this modification represented by fig. 104. The upper part of this figure shews the manner in which the new planes are conceived to be produced, by the continual abstraction of single rows of mole- cules on both the edges a 6, and c, of each of the superimposed plates, until the last plate consists of only one row, forming the new edge of the secondary crystal. Fig. 105. The square prism has all its terminal edges similar^ and 'all its terminal angles also similar, and con- 72 DECREMENTS. sequently when one of those edges or angles is affected by any decrement, they will generally all be so. Fig. 105 exhibits modification c, of the square prism, the upper part of the figure shewing the man- ner in which the secondary planes are conceived to be generated, by the continual abstraction of single rows of molecules from each edge of each of the suc- cessively superimposed plates. Fig. 106. Fig. 106 exhibits the effect of a decrement by one row of molecules on the angle of a square prism, producing a secondary form belonging to the class a of the modifications of that figure. See tables. The cube has its three adjacent edges similar, and consequently they are all affected equally by decre- DECREMENTS. 73 ments upon the edge of that form, except in some particular cases which will be referred to in a future section. Fig\ 107 shews the effect of a decrement by one row of molecules on the edges of a cube ; producing the planes of the rhombic dodecahedron. Fig. 108. Fig. 108 shews the manner in which an inter- mediary decrement, taking place at the same time upon the three adjacent angles of the cube, is con- ccived to produce six planes on the solid angle. The causes which occasion decrements do not ap- pear at present to be understood : crystals so minute as to be seen only by the aid of a microscope, are found variously modified ; hence the circumstance, whatever it may be, which occasions the modifica- tion, begins to operate very soon after the crystal has been formed. Perhaps it may influence the arrangement of the first few molecules which combine to produce the crystal in its nascent state ; and as we find that crys- tals during their increase in magnitude, sometimes undergo a change of form, by the extinction of some modifying planes, or the production of others, it is evident that the cause which occasions a decrement, may be suspended, or may be fii'st brought into ope- ration, at any period during the increase of a crystal in size. 74 DECREMENTS. For the purpose, however, of affording a clearer illustration of the theory of decrements, it has been found convenient to imagine that the primary form of any modified crystal had attained such a magni- tude, before the law of decrement had begun to act upon it, as to require for the completion of the mo- dified crystal, the addition of only those defective plates of molecules by which the modifying planes were produced. A primary form of this magnitude, is evidently the greatest that could be inscribed in the given secondary form. And a primary form so related to the secondary, is in theory termed the nu- cleus of the secondary form. This nucleus may frequently be extracted from the secondary crystals by cleavage. If we take a crystal of carbonate of lime of the variety called dog-tooth spar (the metastatique of the Abbe Haiiy,) and begin to cleave it at its sum- mit, we shall first remove those molecules which were last added in the production of the crystal ; and by continuing to detach successive portions, thus pro- ceeding in an order the inverse of that by which the modified crystal has been formed, we may remove the whole of the laminae, which enclose or cover the theoretical primary nucleus. As far as we have proceeded with the theory of decrements, we have supposed the diminished plates of molecules to be laid constantly upon the primary form, in order to produce the modifications which are found to exist in nature. But from a comparison of the angles at which some secondary planes incline on the primary, and on each other, it is probable that the decrements sometimes take place on second- ary crystals. Thus, for example, we may conceive decrements to take place on any secondary rhomboid DECREMENTS. 75 of carbonate of lime by the abstraction of secondary molecules similar in form to that secondary rhomboid. These secondary molecules would consist of certain numbers of primary ones arranged in the same order as they would be in the production of the entire secondary crystals, and they would in fact be minute secondary crystals. There is an interesting paper on this subject by the Abbe Haiiy, in the 14th vol. of the Annales du Museum, p. 290, where, in order to express the laws of decrement in as low numbers as possible, he has in several instances conceived the decrements to take place on secondary forms. Another circumstance apparently influenced by a cause in some degree similar to that which produces decrements, is the colour which occasionally covers some of the modifying planes of a crystal while the other planes remain uricoloured. A specimen of carbonate of lime, from St. Vincent's rocks near Bristol, which now lies before me, affords an instance of this. Fig. 109. In this specimen the planes , , of some of the crystals and those planes only^ are covered with par- ticles of, I believe, oxide of iron, upon which no molecules of carbonate of lime appear to have been subsequently deposited; but a thin plate of that sub- stance is observed on some crystals as at c, to cover 76 DECREMENTS. part of the coloured plane, having apparently begun to form at the edge / p ,-"7 v^ > x - \/ V \AX 'ii ; - V x x\ /X y Modifications. Class a. \ ITS MODIFICATIONS. 121 This may be easily conceived, if we recollect that the octahedron itself is a figure resulting from the modification a of the tetrahedron; and that the edges of the modifying plane 0, of the tetrahedron, are not affected by modification e, or f y which replace the primary edges of the tetrahedron ; and that the pri- mary edges of the tetrahedron are not affected by the modifications , and c, which would replace the edges of the modifying plane a. Three classes of modifica- tion must therefore concur upon the tetrahedron, to produce most of the secondary forms of the octa- hedron. ITS MODIFICATIONS. Primary form. The rhombic dodecahedron. Plane P on P', 90. P on P", 120. For the sake of avoiding a frequent repetition of the description of the two kinds of solid angles of this figure, those contained within four acute plane angles will be called the acute solid angles, and those con- tained within three obtuse plane angles, will be called the obtuse solid angles. Modifications. Class . Acute solid angles replaced by tangent planes. The new figure would be the cube. Plane a on P, 135. THE RHOMBIC DODECAHEDRON, AND Class b. Class c. Class d. Class e. V r y ^X ' ^ty : -.. - ITS MODIFICATIONS. 123 Class b. Acute solid angles replaced by four planes resting on the primary planes. The new figures would be contained within 24 isosceles triangular planes. Class c. Acute solid angles replaced by four planes resting on the primary edges. The new figures would be contained within 24 trapezoidal planes. Class d. Acute solid angles replaced by eight planes. The new figures would be contained within 48 triangular planes. Class e. Obtuse solid angles replaced by tangent planes. The new figure would be the regular octahedron. Plane e on P 144 44' S". 124; THE RHOMBIC DODECAHEDRON, AND Class/ Class g. Class h. Class i. ITS MODIFICATIONS. 125 Class/. Obtuse solid angles replaced by three planes resting on the primary planec. The new figures would be contained within 24 isosceles triangular planes. Class g. Obtuse solid angles replaced by three planes resting on the primary edges. The new figures would be contained within 24 trapezoidal planes. Class //. Obtuse solid angles replaced by six planes. The new figures would be contained within 48 triangular planes. .- _,, Class t. Edges replaced by tangent planes. The new figures would be contained within 24 trapezoidal planes. THE RHOMBIC DODECAHEDRON, AND Class L THE OCTAHEDRON WITH A SQUARE Primary form. f '*$*!', Modifications. Class a. ITS MODIFICATIONS. 127 Class k. Edges replaced by two planes. The new figures would be contained within 48 triangular planes. It may be remarked that the secondary forms be- longing to the four preceding classes of primary forms, are nearly similar to each other. And, with the exception of the tetrahedron, each primary form is found to be also a secondary form to each of the other primary. The table of secondary forms, which will be found at the end of the tables of modifications, exhibits the relation to each other of each of the preceding classes of secondary forms. BASE, AND ITS MODIFICATIONS. Primary form. The octahedron with a square base. The individuals belonging to this class will differ from each other in the inclination of P on P", and consequently of P on P'. Modifications. Class a. Terminal solid angles replaced by tan- gent planes. As this modification contains only two parallel planes, those planes can never efface the primary planes and produce a solid. The same observation 128 THE OCTAHERON WITH A SQUARE Class b. Class c. Class d. BASE, AND ITS MODIFICATIONS. 129 t will apply to all those classes which produce only two, or four ', parallel planes upon the primary form. In these cases, as no entire secondary form can result from the secondary planes alone, no new figure is described. Class b. Terminal solid angles replaced by four planes resting on the primary planes. The new figures would be more obtuse octa- hedrons. Class c. Terminal solid angles replaced by four planes resting on the primary edges. Another series of octahedrons more obtuse than the primary would result from this class. Class /. Terminal solid angles replaced by eight planes. The new figures would be double eight-sided pyra- mids united at a common base, and measuring un- equally over the two pyramidal edges of each of the planes. The surfaces of the new planes would generally be scalene triangles. 130 THE OCTAHEDRON WITH A SQUABE Class e. Class/. Class g. ^ adT ..< Otitt BASE. AND ITS MODIFICATIONS. 131 ' , Class e. Solid angles of the base replaced by tan- gent planes. This modification would produce only four sides of a prism. Class/. Solid angles of the base replaced by two planes resting on the edges of the summits. This modification would produce a series of octa- hedrons more acute than the primary. Class g. Solid angles of the base replaced by two planes resting on the edges of the base. n 2 THE OCTAHEDRON WITH A SQUARE Class h. Class i. Class k. BASE, AND ITS MODIFICATIONS. 133 Class h. Solid angles of the base replaced by four planes resting on the primary planes, and having their edges parallel to the edges of the pyramids. Class i. Solid angles of the base replaced by four planes inclining more on- the terminal edges than modification h. Class k. Solid angles of the base replaced by four planes inclining more on the edges of the base than modification h. A series of double eight-sided pyramids might result from class h y , and , analogous to those re- sulting from class d, but more acute. 134r THE OCTAHEDRON WITH A SQUARE Class /. Class Class n. BASE, AND ITS MODIFICATIONS. 135 Class /. Edges of the pyramids replaced by tan- gent planes. The new figure resulting from this modification would be an octahedron with a square base, but more obtuse than the primary. Class m. Edges of the pyramids replaced by two planes. Class m would produce eight-sided pyramids simi- lar in character to those resulting from class d. Class n. Edges of the base replaced by tangent planes. 136 THE OCTAHEDRON WITH A SQUARE Class o. BASE, ^AND ITS MODIFICATIONS. 137 Class o. Edges of the base replaced by two planes. The new figures would be octahedrons, more acute than the primary. The character, and number, of the modifications of this, and some other of the following classes of pri- mary forms, arise from the dissimilarity between the edges and angles of the summits, and those of the base, in the octahedrons ; and between the angles of the terminal planes, or the terminal and the lateral edges of the prisms, and sometimes between the la- teral edges themselves. The primary crystals belonging to this class of primary forms, are distinguishable from regular octa- hedrons by the unequal inclinations of the plane P on P', and P on P'' ; and the secondary forms may be distinguished by the modifications taking place on some only of the edges or angles, and not on all, as they do on the regular octahedron. If two of its edges measure over the summit more than 90, the octahedron of this class is called obtuse ; if less than 90, it is called acute. In the regular octahedron the edges measure exactly 90 over the summit. The secondary octahedrons belonging to this class, have, like the primary, square bases. THE OCTAHEDRON WITH A RECTANGULAR Primary form, Modifications. Class a. Class b. BASE, AND ITS MODIFICATIONS. Primary form. An octahedron with a rectangular base. In this figure the broad planes P P', meet at the edge of the base at a more obtuse angle than the nar- row ones MM 7 . The edge D may therefore be denominated the obtuse edge of the base, and the edge F the acute edge; or they may be termed the greater and lesser edges of the base. The individuals belonging to this class of primary forms will differ from each other in the inclination of Pon P', or ofM on M'. Modifications. Class . Terminal solid angles replaced by single planes. Class b. Terminal solid angles replaced by two planes, resting on the broad planes. 140 THE OCTAHEDRON WITH A RECTANGULAR Class c. Class d. i Class e. Class/ BASE, AND ITS MODIFICATIONS. 141 Class c. Terminal solid angles replaced by two planes, resting on the narrow planes. Class d. Terminal solid angles replaced by four oblique planes. The new figures would be octahedrons with rhombic bases. Class e. Solid angles of the base replaced by single planes, whose edges are parallel to the edges of the pyramids. Class f. Solid angles of the base replaced by single planes, inclining on the greater edges of the base more than those of modification e. 142 THE OCTAHEDRON WITH A RECTANGULAR Class g. Class h. Class i . Class L BASE, AND ITS MODIFICATIONS. 143 Class g. Solid angles of the base replaced by single planes inclining more on the lesser edges of the base than those of modification e. Classes e/and g, would give the lateral planes of a series of right rhombic prisms. Class h. Solid angles of the base replaced by two planes, one edge of each new plane being parallel to a pyramidal edge of the broad primary planes. Class t. Solid angles of the base replaced by two planes, one edge of each new plane being parallel to a pyramidal edge of the narrow primary planes. Class k. Solid angles of the base replaced by two planes, neither of whose edges are parallel to any edge of the primary form. 144: THE OCTAHEDRON WITH A RECTANGULAR Class /. Class m. Class n. Class o. BASE, AND ITS MODIFICATIONS. 145 Class /. Edges of the pyramids replaced by single planes. A series of octahedrons with rhombic bases would result from Classes h, i, k, and /. Class m. Greater edges of the base replaced by single planes. Class n. Greater edges of the base replaced by two planes. Class minal edges of a primary form ; if we find them so, we should not make that the vertical series. . If there be a series of vertical planes, and a horizontal plane, we should observe whether any of the vertical planes are at right angles to each other, and whether there be any oblique planes lying between some of the vertical planes, and the horizontal plane. We should remark the equality, or inequality, of the angle at which any of the vertical or oblique planes incline on the several adjacent planes. We should notice whether there be any such sym- metrical arrangement of the vertical planes, or of the oblique planes, if there be any, as would induce * When the series of planes with parallel edges are held verii^Uyn the plane at right angles to them will of course be horizontal. These may therefore be called the vertical and horizontal planes ; all other planes ivill be termed oblique ; and the edges of the horizontal and vertical planes, will be termed horizontal and vertical edges. SF 226 ON THE APPLICATION OF THE us to refer our crystal to any particular class of primary forms ; and by comparing the characters we thus observe with those described in the tables, we shall probably discover the class of primary forms to which our crystal belongs. Let us now suppose our crystal to be contained within any series of vertical planes, and to be terminated, not by a horizontal plane, but by a single oblique plane, the crystal may then belong to the class of oblique rhombic prisms, doubly oblique prisms, or rhomboids. If there be four oblique planes, inclining to each other at equal angles, the crystal may belong to the class of square prisms, or of octahedrons with square bases. If there be four oblique planes, each of which in- clines on two adjacent planes at unequal angles, the crystal will probably belong to the class of right rectangular prisms, right rhombic prisms, or octa- hedrons with rectangular or rhombic bases. If the series of vertical planes consist of 6, 9, 12, or some other multiple of 3, and if there be a single hori- zontal plane, the crystal may belong to one of the classes of right prisms, rhomboids, or hexagonal prisms. If there should be three oblique planes, the primary form is a rhomboid. But if the termination consists of six oblique and equal planes, the crystal may belong to the class of rhomboids or hexagonal prisms. Crystals not falling within any of the preceding descriptions, may yet be found to resemble some of the secondary forms given in the tables. Those which belong to the class of doubly oblique prisms, are sometimes very difficult to be understood; and the relation between the primary and secondary forms of this class, can be learned only by a coin- TABLES OF MODIFICATIONS. 227 parisoii of the crystals themselves with each other, assisted by the tables of modifications already given. A circumstance, which has not yet been alluded to, will also frequently render it very difficult to read a crystal. This is the unequal extension of some of its parallel planes. A very remarkable instance of this character prevails in copper pyrites, and has been the occasion of the erroneous opinions entertained until very lately, respecting the primary form of that substance. In all the works on mineralogy, except that by Professor Mohs, its primary form is stated to be a regular tetrahedron. Mohs, however, discovered that its form was an octahedron with a square base. The two following figures, for the drawings of which, made from crystals in his own possession, I am obliged to Mr. W. Phillips, exhibit crystals containing equal numbers of similar planes ; fig. 300 having these planes regularly placed on the primary form, and fig. 301 representing the same crystal as it frequently occurs in nature, with some of its planes considerably enlarged. The same letters are placed on the corresponding planes of each. Fig. 300. Fig. 301. Mr. Phillips had discovered from the cleavage of this substance, before the publication of Professor Mohs's book, that its primary form was not a tetra- 2 F2 228 ON THE APPLICATION OF THE hedron, as appears by a paper in the Annals of Phi- losophy, new series, vol. 3. p. 297. When crystals of this irregular character occur, it is generally, only by cleavage, and by using the goniometer, that we can be led to an accurate deter- mination of their true forms. Having ascertained the class of primary forms to which our supposed crystal belongs, our next step would be to measure the angles of its primary or secondary planes, in order to determine the species to which the mineral itself belongs. If the crystal belongs to one of the four regular solids, whose angles, when the forms are similar, are always equal, its hardness, or specific gravity, or some other character, will the most readily lead to a de- termination of its mineral species. But it may happen that the secondary crystal we are examining, may be referred with equal propriety to either of the two, or more, classes of primary forms. If we turn to the modifications of the octahedron with a square base, and to those of the square prism, and imagine the modifying planes of the square prism much enlarged, we shall observe such a resemblance between them, that either form may be taken as the primary, in reference to the secondary forms of both. The same remark will apply to the octahedron with a rectangular or a rhombic base, and the right rect- angular, or rhombic prism. In these cases it has been usual to adopt that as the primary form which is developed by cleavage. But if there be no practicable cleavage, or if there be two sets ,of cleavages, parallel to the planes of two primary forms, we are then at liberty, as it has been already stated, to adopt either of these, and TA15LES OF MODIFICATIONS. 229 our choice would probably fix on that which was most predominant among the secondary forms. If a crystal is to be described by the assistance of the preceding tables, we must suppose the primary form to be known ; this must be first described according to its class, and if necessary by its angles also. Its modifications, if they are single, may then be denoted by the letters under which they are arranged in the tables. But as each of the classes of modifications, except those which consist of tangent planes, compre- hends an almost unlimited number of individual modi- fying planes, differing from each other in the angles at which they respectively incline on the primary planes, it becomes necessary to add to the tabular letter which expresses the modification, the value of the angles at which the plane we have observed inclines on the adjacent primary planes. We have already seen that modification a of the right rectangular prism, comprehends a considerable number of planes varying in their relative inclinations on P, M, and T. Let us suppose the crystal we are examining, to belong to the class of right rectangular prisms, and to be modified by a plane a, and let the inclination of the plane we have observed, on P, M, and T, be called m, n*^ and o", these letters signifying any number of degrees and minutes whatsoever. A crystal containing the primary planes of the right rectangular prism, and a set of planes belong- ing to modification a might then be thus described. Right rectangular prism, Modification a, m on P. n M. o T. The character of the plane being thus established, Vve may in future, in order to avoid the repetition of ON THE APPLICATION OF THE the measurements, describe the plane as Modification c, plane 1, and it may be marked in the figure of the crystal as a 1. Let us now suppose we find on another crystal, another plane modifying the same solid angle, and inclining on P, M, and T, at j?', 9, and r% and a third plane, also modifying the same solid angle, and inclining on P, M, and T, at s, t* 9 v, we should de- scribe these planes as we did the first, by Modification a, p on P plane 2. p on Jf ^ '-M \\ r' T J Modification c, s* P plane 3. r r > r-M I tf T J And having thus recorded the character of the planes, we may in future describe them as Modification a 2, Modification a 3. This method of description may be applied, whe- ther the three planes have occurred on the same crystal, or on different crystals. The inclination of the modifying plane on too of the primary planes, is generally sufficient, when a solid angle is modified, for determining ttye law of decrement; but the third inclination serves as a check upon the accuracy of the other two. If the edge of any prism be modified by one or more planes, it will be sufficient to give the incli- nation of each plane, on either of the primary planes, where the inclination of the primary planes to each other is known, as the inclination on the other pri- mary plane may be readily ascertained. But when an edge is modified, of any crystal whose adjacent primary planes do not meet at a right angle, and when their mutual inclination is unknown, the in- TABLES OF MODIFICATIONS. 231 clination of the modifying plane on both the primary should be given. This method of description maybe readily extended to all the classes of primary forms ; and although it may sometimes be rather tedious in its application, it will convey an accurate description of the planes to which it is applied. It may frequently happen that we are examining a crystal whose primary form is unknown to us, and whose secondary planes do not enable us to deter- mine that form; we can in such case describe the crystal only by giving a drawing of it, accompanied, by the inclinations of its several planes to each other. It will perhaps be found convenient, where it can be done, to number the observed planes, belonging to each class of modifications, in some certain order; when there is a series of secondary planes whose edges are parallel, that plane may be denoted by No. 1, which forms the most obtuse angle with thq primary plane to which the series may be referred. SECTION XIV. ON THE USE OF SYMBOLS FOR DESCRIBING THE SECONDARY FORMS OF CRYSTALS. IN Section 11, p. 102, it vyas stated that certain letters had J)een appropriated by the Abbe Haiiy, as symbols, to designate the similar and dissimilar edges, angles, and planes of each of the classes of primary forms. And in the tables of modifications, these let- ters are placed on the figures of the primary forms, to denote in each its similar, and dissimilar, edges, angles, and planes. The order in vyhich they are placed on the figures is obviously that of the alphabet ; and they are ar- ranged also according to the ordinary method of writing, beginning at the upper part of the figure, and then proceeding from left to right until the seve- ral parts of the crystal are marked by the appropriate letters. This will be very apparent, if we refer to the pri- mary form of the doubly oblique prism. The letters P M T are retained to designate the primary planes of crystals, although the term primir tive, from which those letters were derived, is not used in this treatise. The letter A is, by the Abbe Hauy, placed, gene- rally, on an obtuse angle of the primary form ; but according to the positions in which the primary forms ON THE USE OF SYMBOLS. 233 are drawn in the preceding tables, the letter A will not necessarily stand on an obtuse angle, excepting on the rhombic dodecahedron, the right rhombic, the right oblique-angled, and the hexagonal prisms. The edges and angles of that terminal plane of the prism, on which the figures appear to rest, and the edges and planes which constitute the back of the figures, are supposed to be denoted by a series of small letters, corresponding with the capitals, by which the diametri- cally opposite edges, angles, and planes, exhibited in the front ofthejigure, are designated. The representation of the secondary forms of crys- tals by means of these symbols, is effected by annex- ing numbers, expressive of the particular laws of decrement by which the secondary planes are con- ceived to have been respectively produced, to the letters which denote the edges or angles on which the decrements have taken place. These numbers will be termed the indices of the secondary planes, and will generally be represented by the letters p q r s. Before we proceed, however, to explain the man- ner in which these symbols may be applied to the representation of the secondary forms of crystals, we shall for a moment consider the theory of decrements more particularly in reference to the descriptive cha- racter it affords. As this character is to be regarded as little else than a symbol, indicating the change of figure which the primary form has undergone, if there be two laws of decrement which will equally well express this change of figure, we are obviously at liberty to adopt either law, as the generator of the new plane by which the figure of the primary form is altered ; but it will be found convenient to be guided by some rule in our choice. 234 ON THE USE OF SYMBOLS. Fig. 302. If, for example, we find a secondary plane, such as a b, fig. 302, on the terminal edge of any prism, pro- duced by the abstraction of three rows of molecules in the direction of the lateral edges, and of one row in the direction of the terminal edges, such a plane might be conceived to be produced by a decrement proceeding along the plane a c, consisting of three molecules in height and one in breadth, or of three molecules in breadth, if we suppose it to proceed along the planes b d; and the symbol denoting either of these decrements, might therefore with equal pro- priety be used to describe the new plane; and it would be indifferent, as far as the descriptive charac- ter of the symbol is regarded, which of the two we should adopt. The rule which it will be more convenient to fol- low, is, to suppose all planes on the terminal edges of prisms, to be produced by decrements proceeding along the terminal planes ; and the planes replacing the lateral edges of prisms, and the edges of all the other classes of primary forms, may be conceived to result from decrements proceeding along those planes in the direction of which the greatest number of molecules appear to have been abstracted. And any intermediary decrement may be conceived to proceed ON THE USE OF SYMBOLS. 235 along that plane in the directions of whose edges the greatest number of molecules have been abstracted. It may not be useless to remark, that when two or more planes replace the solid angle of a crystal, if an edge at which the secondary planes intersect each other be parallel to the edge at which one of them intersects the primary plane, they will generally both result from simple or mixed decrements. Intermediary decrements however sometimes pro- duce a series of planes whose intersecting edges are parallel to each other, and when this happens, the symbols of those planes will have two of their corre- sponding indices in the same ratio to each other. The edges of such secondary planes as replace the edges of crystals, and which result from simple or mixed decrements, are always parallel. From what has been already stated it will appear, that if we are about to describe a secondary crystal, belonging to any species of mineral whose primary form is known, and upon several of whose edges, or angles, similar decrements have produced similar planes, it will be sufficient, generally , to describe one only of the new planes, produced upon one of those edges or angles. And if two or more laws of decrement have con- curred in the production of any secondary crystal, we should be required, generally ', to describe only one of the planes produced by each particular law. For the change of figure which any primary form has undergone would be, generally, thus indicated. And in drawing the crystal we might construct planes, similar to those which are described, upon all its similar edges or angles. 236 ON THE USE OI 1 SYMBOLS. As it sometimes, however, occurs that all the simi- lar edges or angles of crystals are not similarly modified, it will riot be sufficient in all cases to indi- cate the decrement which has taken place on one edge or angle, but our representative symbol should also indicate the absence of the modifying plane from some other edge or angle, where according to the law of symmetry, it might be expected to appear. This necessity of distinguishing the symmetrical modifications of crystals, from those which are not so, will render the symbols rather more complicated than they would be otherwise. The new theory of molecules which has been in- troduced into this treatise, will render it necessary to vary the character of some of the symbols employed by the Abbe Haiiy in reference to the tetrahedron, to all the classes of octahedrons, and to the rhombic dodecahedron ; and as these changes will occasion some other slight deviations from his system of no- tation, it will conduce to perspicuity if we consider the application of the symbols to each of the classes of the primary forms separately. This will be done in a table subjoined to this section, where the order of the primary forms will correspond with that adopted in the tables of modifications. The general nature of this system of notation will be best illustrated by its application to one of the least regular of the primary forms. Let us suppose that we are about to represent a secondary crystal belonging to the class of doubly oblique prisms, according to the theory of decre- ments, and by means of the symbolic letters already alluded to, the primary form being known, and the law of decrement by which the secondary plane has been produced, having been also ascertained. ON THE USE OF SYMBOLS. 237 The crystal is supposed to be held with the plane marked P, horizontal, and with that edge or angle nearest to the eye on which the decrement we are about to describe has taken place. Let us suppose this crystal to be modified by an individual plane, belonging to the series of modifi- cations of that figure comprehended under class b. The planes belonging to this class of modifications, may incline more or less on either of the adjacent primary planes, and may result from a decrement on either of the adjacent plane angles which constitute the solid angle on which O is placed. If the modifying plane be produced by a simple or mixed decrement, beginning at the angle O, and proceeding along the terminal plane, consisting of one row of single molecules, it should be expressed thus, O, and be read, one over O, signifying that the abstraction of molecules from the superimposed plates took place above, or receding from O, in the direction O A. If the decrement be simple, and by two rows of molecules in breadth, it would be expressed by O, and if it be a mixed decrement by three rows in 4 breath and two in height, it would be denoted by O, and so of any other decrement acting in that di- rection. Jf the modifying plane be occasioned by a simple or mixed decrement, beginning at the angle of the plane M adjacent to O, and proceeding along the plane M, by p rows of molecules, p signifying any whole number or fraction, it would be denoted by r O, and be read p on the left of O. If the new plane were produced by a simple or mixed decrement by p rows on the angle of the plane T, adjacent to O 5 and proceeding along that plane, it 238 ON THE USE OF SYMBOLS. would be denoted by O p , and be read p on the right ofO. In either of the preceding cases, the intersection of the new plane with the primary plane along which the decrement is conceived to proceed, will, as we have already seen, be parallel to the diagonal of that plane. Let us now suppose an intermediary decrement to have taken place on the angle O, of such a nature, that the mass of molecules abstracted should belong to a double plate, or be two molecules in height, or as it might be otherwise expressed, 2 molecules in the direction of the edge H, 3 in the direction of the edge D, and 4 in the direction of the edge F. The appropriate symbol to denote such a decre- ment, ought obviously to represent this threefold character ; which it does by combining the indices expressive of the particular law of decrement, with the letters which represent the edges and angles af- fected by it, in this manner, (D3 H2 F4). This symbol is placed in a parenthesis to distinguish it from a combination of three simple or mixed decre- ments, and it would be read thus, 3 on the edge D, 2 on the edge H, 4 on the edge F.* Jf instead of the angle marked by O, we no\v imagine the solid angle on which A is placed to be modified simlarly to that denoted by O ; before we describe the modifications of A, the crystal is con- ceived, to be turned round, until the angle on which * This mode of representing intermediary decrements differs from that adopted by the Abbe Haiiy, in referring the decrement to the adjacent edges ; whereas he refers them to two edges and the angle they include. But the form of the symbol here given will best accord with the results obtained by the methods of calculating the laws of decrement, which will be given in the Appendix. ON THE USE OF SYMBOLS. 239 A is placed is nearest to the eye; or we may be sup- posed to pass round the crystal, until we place our- selves opposite the angle at A ; and while the eye and that angle are in these relative positions, we should proceed to describe the new planes, as we did those on the solid angle at O. If two or more planes, resulting from simple or mixed decrements, are found modifying the same solid angle of any crystal, the symbols representing them are to be placed immediately following each other. Thus if the three planes we have supposed on the angle O, should occur on the same crystal, its change of figure would be thus represented, p o 6 o p . These symbols not being placed in a parenthesis, are understood to represent three separate planes. If three intermediary decrements should occur on the same solid angle, their symbols would also be placed following each other, thus, (D3 H2 F4) (Dl H3 F2) (D4 HI F3). Here, each of the three sets of characters being included within a separate parenthesis , three varieties of intermediary decrement are implied ; and as they stand singly , it is implied that they are independent of each other-, and they are evidently produced by dif- ferent laws of decrement. Let us now suppose we are about to describe a decrement on a terminal edge of a doubly oblique prism. The prism is again supposed to be placed or held with that edge nearest to us, the plane P con- tinuing horizontal. And first let us suppose a terminal edge F to be replaced by a plane resulting from a decrement by p 240 ON THE USE OF SYMBOLS. rows of molecules proceeding along the plane P ; p, meaning^ as before, any whole number or fraction. The symbol to denote this decrement, would be F, and be read as before, p over F. If the decrement be supposed to have proceeded along- the plane T by three rows of molecules, as in figure 303, the general symbol used to represent the new plane would still be F, but p would in this case represent the fraction y, and the particular symbol T would be F. If we suppose p to be a fraction , it is evident from what has been already slated, that the numerator of that fraction may be either greater or less than the denomi- nator^ according as the decrement in breadth exceeds or falls short of that in height. ' If either of the other terminal edges be modified, the modified edge is supposed to be the nearest to the eye, when the modifying plane is described. This change of position must be understood to take place in every instance where the position of the modified edge or angle requires it. If two dissimilar planes occur on the same terminal p P edge of a crystal, the symbol is repeated thus F F, which expresses the coexistence of the two planes on one edge. If the lateral edge H of a doubly oblique prism be modified, and if it has been found that the decrement producing it has proceeded along the plane M, by p rows of molecules, its characteristic symbol would be 1} H, and it would be read^? on the left of H. If the decrement appears to have proceeded along the plane T by p rows of molecules, its symbol would be H p , or p on the right of H ; p being either a ON THE USE OP SYMBOLS. whole number or fraction, expressive of the particu- lar law of decrement, in reference to each plane respectively, as it is supposed to have been ascer- tained by calculation. If the two planes occur on the same crystal, they would be denoted by the two symbols being used to- gether, thus, P H H p . If it should be required to describe any decrement acting upon an edge or angle of the lower plane of the crystal, upon which the small letters are supposed to be placed, the crystal is imagined to be turned with that plane upwards, the edge or angle on which the decrement has taken place is to be brought the nearest to the eye, and we are then to describe the plane or planes in the manner already directed, only using the small letter, instead of the capital, to indi- cate the edge or angle which is modified. And if it should be necessary to describe a decrement upon the back planes of the crystal, we are supposed to pass round it, and to substitute small letters in the symbol for the capitals which designate the cor- responding front planes. 3 The preceding explanations will render sufficiently intelligible the general method of representing the secondary planes of crystals by means of symbolic characters. Before we proceed, however, to apply this method to the different classes of primary forms, it will be necessary to separate the secondary forms of the crystals to be represented into three principal classes. 1. Those which are strictly symmetrical, as modifi- cation , , c, d, e, or f, of the cube, where similar decrements take place on similar edges or angles, and proceed along similar planes. 242 ON THE USE OF SYMBOLS. 2. Those which are partially modified, or on which the same modification does not occur on all the similar edges or angles ; as in modification g, h, i, A:, of the cube. These may be termed defective modifications, and they may be again subdivided into two portions. a. Those in which an edge or angle is replaced by only half the number of planes which the law of symmetry would require. b. Those in which only one of two similar edges or angles is modified, while the other remains entire. 3. Those in which two or more similar edges or angles are affected by different laws of decrement. And the symbols, to be perfect, ought to represent each of these divisions clearly and perspicuously. In the table subjoined to this section, which will point out the relation of the theory of decrements to the different classes of modifications, the various modes of adopting the symbols to particular cases will be fully explained. Whence it will not be necessary here to give more than an outline of the general principle which will regulate their application. To represent the secondary forms belonging to the first of these divisions, it may not appear strictly necessary to do more than indicate the character of a single plane belonging to any set of similar planes occurring upon the same crystal ; but it may tend to prevent ambiguity if we construct our symbol so as to indicate that the secondary planes occur sym- metrically on certain edges or angles of the crystal. We may here remark, that the sets of planes which, in the tables of modifications, replace the solid angles '! : ON THE USE OP SYMBOLS, 243 of the cube, tetrahedron, and rhomboid, and rest, as it is said, upon the planes of those primary forms, are distinguished from those which are said to rest upon their edges. But in reference to the theory of decre- ments, both these sets of planes are similar in character, and result from simple or mixed decrements on an angle of the primary form. The planes which are said to rest upon the primary planes, are produced by decrements in which the number of molecules abstracted in breadth, is greater than the number in height, while those which are said to rest upon the edges, result from decrements where- in the number in height exceeds the number in breadth. The numbers or fractions expressing ihejirst of these sets will be always greater than unity, as 2, 3, 4, -f, f , &c. ; those expressive of the latter set, will be al-? ways less than unity, as , -f, f, &c. ; and the planes in this latter case are conceived to be carried, as it were, over the solid angle, and made to replace a portion of the adjacent edge. The Cube > Fig. 303 . A A' .sfs\ -o JS '/& R V A ^X ! ( j i ^:---' ^ Let us now suppose a cubic crystal, modified on the angles by three planes belonging to class b of the modifications of that form ; and let us suppose that the modifying planes result from a decrement by two rows in breadth on the angles of the cube. The sym bol denoting these planes would be 2 A% and if this ON THE USE OF SYMBOLS. be unaccompanied by any other symbol, it would be implied that all the solid angles were similarly mo- dified. The symbol representing class C might be The planes of modification a of the cube might be denoted thus. A, but for the sake of uniformity with the preceding symbol, they will be represented by i the symbol * A * . The planes belonging to class i of the cube, do not differ from those belonging to class d, except in being three single similar planes, instead of three pairs of similar planes, as there are in class d. To distinguish class d therefore, by its symbol, it will be requisite that the symbol should represent one of the pairs of planes, and not merely a single plane, as might have been sufficient if class i had not existed. Suppose an individual modification belonging to class d is to be denoted, and if the decrement pro- ducing it be by three molecules on the edge B, one on the edge B',* and two in height on the angle A, the symbol would be (B3 B'"2 B'l : Bl B'"2 B'3), which would imply a pair of planes resting on the plane P. And the two symbols being both included within a parenthesis, and separated from each other by two dots, implies that both the planes represented result from the same law of decrement, but acting in two different directions. If two similar planes belonging to class f of the cube, resulting from a decrement by three rows in breadth, occur on all the edges of a cubic crystal, the symbol B will be used to denote their existence on one of the edges ; and their existence on the other * B*, B", &c. is read B dash t B two dash, &c. ON THE USE OF SYMBOLS. 245 edges is implied, unless their absence be denoted by the characters which will be presently given and ex- plained. This symbol implies that the edge B is replaced by two planes, one of which results from a decrement by three rows in breadth proceeding along the terminal plane, and the other by three rows in breadth proceeding along the lateral plane. The symbol B might be sufficient to denote the planes of modification e, but fur the sake of conformity with the general system of notation, it should be written i B. i When the lateral edge of a prism is modified by two similar planes, the symbol representing them will be P G P . The G standing single, implies that the symbol refers to a single edge. The planes belonging to class d of the modifications of the right rectangular prism, may be readily con- ceived to result from decrements proceeding along either of the planes M or T. If along the plane M 5 the symbol would be G /p P G ; but if the decrement be supposed to have proceeded along the plane T, its symbol would be P G' G p . The Tetrahedron. Fig. 304. Simple and mixed decrements on the angles of the tetrahedron producing planes belonging to class b, 246 ON THE USE OF SYMBOLS. are supposed to proceed along the plane P ; and the symbol by which they are to be represented is P A P p The symbol representing a pair of planes of any particular modification belonging to class d would be (B P B' q B"r : B P B'r B" q ) The edges of this primary form are neither per- pendicular nor horizontal, and the decrements by which they become modified might therefore be ex- pressed by the symbols which represent the modifi- cations upon either the terminal or lateral edges of prisms. But as the edges of the tetrahedron are more analagous to the lateral, than to the terminal edges of prisms, the symbol P B P will be used to de- note the modifying planes belonging to classy. The Octahedrons. The laws of decrement which produce the modify- ing planes of the octahedrons, are, according to the Abbe Haiiy's theory, supposed to take place on paral- lelepipeds, which would be formed by adding two tetrahedrons to two opposite planes of the octahedron. In the appendix to this treatise, rules will be given for determining directly the laws of decrement on the octahedrons, independently of these added tetra- hedrons. And the symbols representing the secondary planes will therefore vary from those adopted by the Abbe Haiiy. ON THE USE OF SYMBOLS. 247 The regular Octahedron. Fig. 305. The simple and mixed decrements on the angles, which would produce the planes belonging to class b, may be represented by the symbol P A P ; which im- plies that similar planes occur on the three adjacent angles, and consequently on the fourth: The intermediary decrements are of two kinds, 1. Those which produce the planes compre- hended under class c. The general symbol to represent these would be (Bp B' q b' q br). 2. Those which produce the planes compre- hended under class d. The general symbol to represent these would be\B P B' q b'r b s : B P B'r b' q b s ). These symbols denote the abstraction of /?. 1 ; equal to J, . p 1 ; or less than 1, p <^ ]. The mark ^> signifying greater than, = . . . equal to, <^ . . . less than. The letter p will generally be used to denote the greater edge of the defect, the letter q the next, and r the least, when there are only three edges to be denoted. But when the symbol represents a modifi- cation on the solid angle of an octahedron, s is intro- duced to denote the fourth and least edge of the defect. Hence the relative values of p, q, r, and s, may be thus expressed, p ^> q ^> r ^> s. The Cube. Fig. 306. 1. Symmetrical modifications. Simple and mixed decrements on the angles, produce the planes belonging to modifications , b, and c. The general symbol to represent these is P A P . MODIFICATIONS. 255 If p ^> 1, class b is represented, and as the value of p increases, the planes b incline more and more on the primary planes. p 1, mod. a is represented. p <^ 1, class c is represented, and as the value of p diminishes, the planes c incline more and more on the primary edges. Intermediary decrements produce the planes be- longing to class d. The general symbol representing these, is (Bp B'q B'"r : Bq B'p B"'r). Decrements on the edges, produce the planes be- longing to the classes e and /. The general symbol representing these is fi p If p = I, class e is represented. p ^> 1, class fis represented. 2. Modifications not strictly conformable with the law of symmetry ; or, such as have been termed defective modifications. The following are the symbols representing these classes. o 04/0 A' 'A 1 'a" a represents class g. A' P A P V a . . . . class h. (B' q B P B // r)(B" q B'p Br)(B'r B" P Bq) represents the planes at A' belonging to class i. (B q B'" P B'r)(B'"q B'p Br)(B ; q B P B //7 r) represents the planes at A belonging to the same class. 256 RELATION OP DECREMENTS TO When the symbolic character is not accompanied by a figure of the crystal, both the preceding- symbols should be given ; but when there is a figure, it will be sufficient to use the second only. P * B B' B //p P B'" is the symbol representing class /:. The regular Tetrahedron. Fig. 307. Simple and mixed decrements on the angles. General symbol P A P . p Ifp ^> 1, the symbol represents class b. p zz 1, mod. a. p < 1, class c. Intermediary decrements. General symbol, (B P B' q B"r)(B P B'r B" q ) repre- sents class d. Decrements on the edges. General symbol, P B P . 1, the symbol represents mod. e. p ^> 1, class f. MODIFICATIONS. 257 The regular Octahedron. Fig. 308. Simple and mixed decrements on the angles. General symbol, P A P . p If p 1, the symbol represents mod. a. p <^ 1, class b. Intermediary decrements Are of two kinds, and require two general symbols. 1st. (Bp B'q b'q br) represents class c; Id. (B P B'q bV bs : Bp B'r b' q bs) represents class d. Decrements on the edges. General symbol, P B P . If p zz 1, the symbol represents mod. e. P > 1, class/. The planes belonging to classes a and k of the cube, sometimes occur on the same crystal, and when the planes a are much enlarged, the secondary form pre- sents the figure of the octahedron modified by two only of the planes c of that figure. The secondary crystal may however be referred properly to the cube, so long as it retains any portion of the planes A*. 258 RELATION OF DECREMENTS TO The rhombic Dodecahedron. Fig\ 309. Simple and mixed decrements on the obtuse solid angles. v General symbol P A P , represents the classes e,f,&g. Ifp^> 1, the symbol represents class f. p = 1, . mod. e. p < 1, . . . . . . . . class g. Intermediary decrements on the obtuse solid angles produce the planes of class h, which class may be generally represented thus : (B' P B q B" r : B'q Bp B"r). Simple and mixed decrements on the acute solid angles, produce the planes belonging to classes a and b. General symbol P E P . p If p zz: 1, the symbol represents mod. a. p > 1, class b. Intermediary decrements on the acute solid angle consist of two kinds, producing the planes of classes c and d. The general symbol representing class c, is (B P B'q b' q br). The general symbol representing class d, is (Bp B'q b'r bs : B P B'r b'q bs). MODIFICATIONS. 259 Decrements on the edges, may be represented by the general symbol P B P , If p 1, the symbol represents mod. i. p ^> 1 class k. Some of the secondary crystals of Blende are pro- duced by defective modifications of this primary form, and are such as might result from regular modifi- cations of the tetrahedron. The Octahedron with a square base. Fig. 310. Simple and mixed decrements on the terminal edges. General symbol, P A P . p If p =. 1 ? the symbol represents mod. a. p ^> 1, class b. Intermediary decrements on the terminal solid angles. These are of two kinds, and require two general symbols. 1st. (Bp B' q b'q br) represents class c ; 2d. (B P B x q b'r bs : B P B'r b' q bs) represents class d. Simple or mixed decrements on the lateral angles. General symbol P E P . 1, the symbol represents class e. p > 1, . class h. 260 RELATION OF DECREMENTS TO Intermediary decrements on the lateral solid angles. These are of three kinds, and require three general symbols. 1st. (Bp D q D'q B"r) represents class f. 2d. (Dp B q B" q D'r) .... class g. 3d. (B P D q D'r B" s : B P Dr D' q B" s ) represents class i when p ]> q; The same symbol represents class k when p <^ q. Decrements on the edges of the pyramids. General symbol P B P . Ifp zz 1, the symbol represents mod. I. p ^> 1 3 ....... class m. Decrements on the edges of the base. P General symbol D p Ifp = J, the symbol represents mod. n. p ^> 1, ../... class o. The Octahedron with a rectangular base. Fig. 311. Simple and mixed decrements on the terminal angles of the planes P. p General symbol P. Ifp 1, the symbol represents mod. a. p ^> 1, class b. MODIFICATIONS. 261 Simple and mixed decrements on the terminal angles of planes M. General symbol P A P . Ifp I, mod. a is again represented, because plane a results from a decre- ment by one row on all the ter- minal angles. p ^> 1, class c is represented. Intermediary decrements on the terminal solid angles. General symbol (B P B' q b'r bs) represents class d. Simple and mixed decrements on the lateral angles of the plane P. General symbol E /p P E. l?p zn 1, the symbol represents mod. e. p ^> 1, class h. Simple and mixed decrements on the lateral angles of the 'plane M. General symbol P E' E p . If p =: 1, mod. e is again represented. p ^> 1, class i is represented. Intermediary decrements on the lateral solid angles. These are of two kinds. 1. Such as produce the single planes on each angle, which are comprehended under class gi or class f. The general symbol is (D P B q Bf' q Fr). If p ^> r, the symbol represents class f. I>O - class g. 2. Such as produce two planes on each angle belonging to class k. General symbol (D P B q B"r Fs.) In this symbol, the particular values of either of 262 RELATION OF DECREMENTS TO the indices may be greater or less than either of the others, in reference to particular modifying planes. Decrements on the terminal edges. The planes produced by these decrements are all comprehended under class /, although they may be said to consist of three varieties. Jst. When the decrements proceed along the plane P. 2d. When the edge at which the planes / inter- sect each other at the base, is parallel to a diagonal of that base. 3d. When the decrements proceed along the plane M. The general symbol of the 1st, is B' p P B. 2d, 'B'. Decrements upon the edge D of the base. P General symbol D. p If p zz 1, the symbol represents mod. m. p > 1, ....... class n. Decrements upon the edge F of the base. General symbol F. p If p zz 1, the symbol represents mod. o. p > 1, ....... class p. MODIFICATIONS. 263 The Octahedron with a rhombic base. Fig. 312. Simple and mixed decrements on the terminal angles. Genera] symbol P A P . p If p =z I, the symbol represents mod. a. p ^> 1, class d. Intermediary decrements on the terminal solid angles. These are of four kinds, and require four general symbols. We suppose the edges of the upper pyramid, which are opposite to those marked with B and C, to be denoted by b and c. Class b is represented by (Op B q b p c r ). Class c (B P C q c q br). Class e . . . (C P B q br c s : C P Br b q c s ). ClaSS f . . . (BpCqCr bs : Bp CrCq bs). Simple and mixed decrements on the angle E at the base. General symbol ^E 1 '. If p m 1, the symbol represents mod. n. p ^> 1 , class q. 264 RELATION OF DECREMENTS TO Intermediary decrements on the acute solid angle at E. These are of four kinds, and require four general symbols. Class o is represented by (B P d' q D q B'r). Class p ...... (Dp B' q D q d'r). Class r . . (Bp D q d'r B's : B P d' q I) r BV). Class rWj . (Dp B' q Br d's : D P B'r B q d's). i Simple and mixed decrements on the angle 1 at the base. General symbol p l p . If p ~ 1, the symbol represents mod. g. p ^> 1, . . ..... class k. Intermediary decrements on the obtuse solid angle at I. These are of four kinds, and require four general symbols. Class h is represented by (C P D q D' q C'r). Class i . ..... (Dp C q C'q D'r). Class I . . (C P D q D'r jC ; s : C P D' q Dr C's). Class m . . (Dp C q C'r D's : D P C' q Cr D' s ). Decrements on the acute terminal edges. General symbol P B P . If p 1, the symbol represents mod. v. p ^> I, ....... class x. Decrements on the obtuse terminal edges. General symbol P C P . 1 ? the symbol represents mod. t. p ^> 1, ....... class u. Decrements on the edges of the base. P General symbol D. p If p 1, the symbol represents mod. y. p ^> 1, ....... class z. MODIFICATIONS, 265 The right Square Prism. Fig. 313. Simple and mixed decrements on the terminal angles. General symbol A, represents class a generally. In this symbol p may be ^> 1. or= 1. or<1. Ifp =z I, an individual plane belonging to the class is represented, whose three edges would be respectively parallel to the diagonals of the adjacent pri- mary planes. This plane may, from, its station in the series, be denomi- nated the middle plane. p ^> 1, the planes represented would incline more on P than the middle plane does. p <^ 1, the planes represented would incline more on the edge G. Simple and mixed decrements on the lateral angles. General symbol P A P , p being ^> 1. This symbol represents a series of planes belonging to class by whose intersections with the planes M and M', are parallel to the diagonals of those planes. 266 RELATION OF DECREMENTS TO Intermediary decrements. General symbol (B P G q B'r : Br G q B' P ), repre- sents the remainder of the series of planes belonging to class b. Decrements on the terminal edges. p General symbol B. If p =. 1, an individual plane belonging to class c is represented, which may be termed the middle plane of the series com- prehended under that class ; and it would intersect the lateral planes in lines parallel to one of their diagonals, p ^>1 3 the symbol would represent that part of the series of class c, which inclines more on the terminal plane than the middle plane does. p <^ 1, the same symbol would represent that part of the series which inclines more on the lateral plane than the middle plane does. Decrements on the lateral edges. General symbol P G P . If p =: 1, mod. d is represented. p^> 1) class e is represented. MODIFICATIONS. 267 * The right Rectangular Prism. Fig. 314. M Decrements 'on the angles. The planes belonging to class a, comprehend the following varieties. 1st. Those which result from simple and mixed decrements on the P terminal angles, of which the general symbols is A. angles of plane M, P A. angles of plane T, A 1 '. Ifp = 1, in either of these symbols, the same individual plane belonging to the class is represented by each ; the three edges of which are, respective- ly, parallel to the diagonals of the planes P M and T. This may be termed the middle plane of the series. p ^> 1, the planes represented will incline on the plane P, or M, or T, more than the middle plane does. p <^ 1, the plane represented will incline less on the respective primary planes than the middle plane does. 268 RELATION OF DECREMENTS TO 2d. Those which result from intermediary decre- ments, which may be represented by this general symbol, (Cp Gq Br). in which p, q, and r, will vary relatively to each other as the decrements proceed along the plane P, M ? or T. Decrements on the terminal edges. p General symbols, C represents class b. p B . . class c. Ifp == 1, in either of these symbols, a middle plane will be represented belonging to each class respectively. And the planes of each class would respec- tively incline more on the terminal, or on the lateral plane, than its cor- responding middle plane does, as p is ]> 1, or <^ 1. Decrements on the lateral edges, producing the planes of class d. These are of three kinds, and require three general symbols. 1. G /p P G, when the decrement proceeds along the plane M. 2. P G' G p , when the decrement proceeds along the plane T. 3. When p zz 1, in either of the two preceding symbols, the middle plane of the series will be represented. MODIFICATIONS. The right Rhombic Prism. Fig. 315. 269 \ M Simple and mixed decrements on the acute terminal angles. General symbol E, represents class c, generally. Ifp i=i 1, the symbol represents the middle plane of the series. p ^> 1. or <^ 1 ? the planes represented incline more on plane P, or on the edge G, than the middle plane does. Simple and mixed decrements on the obtuse terminal angles. P General symbol A represents class a generally, i Ay represents the middle plane. Simple and mixed decrements on the lateral angles. 1. On those adjacent to E. General symbol P E P , represents one series of planes belonging to class d, which intersect the lateral planes parallel to one of their diagonals. 2< On those adjacent to A, General symbol P A P , represents a similar series of planes belonging to class b. 270 RELATION OF DECREMENTS TO Intermediary decrements on the acute and obtuse solid angles. General symbol (B' P G q Br : B'r G P BP) repre- sents a further series of planes belonging to class d. (B P H q B'r : Br Bq B'p) repre- sents a further series of planes belonging to class b. Decrements on the terminal edges. p General symbol B, represents class e, generally. i B 5 represents its middle plane. Decrements on the lateral edges. 1. On the acute edges. *G* represents mod. h. *G V .... class i. 2. On the obtuse edges. 1 H 1 represents mod. f. P H P . . . class g. The exposition which has been given, in reference to the preceding classes of primary forms, of the re- lations of the laws of decrement to the several classes of modifications, will, it is presumed, have been suf- ficiently full, to render more than an outline of those relations unnecessary, in reference to the classes of primary forms which are to follow. MODIFICATIONS. 271 The right Oblique-angled Prism. Fig. 316. Decrements on the acute solid angles, are all comprised within class b. 1. Simple and mixed. E represents the middle plane. E . . the planes which intersect the plane P parallel to a diagonal. P E . . . the planes which in the same manner intersect the plane T. E p M. 2. Intermediary. General symbol (B P Gr C q ). Decrements on the obtuse solid angles are all comprised within class a. The symbols representing the planes corresponding in character with those above described, are, A. A. P A. A p . (Cp Hr Bq). . 272 RELATION OF DECREMENTS TO Decrements on the terminal edges. C, symbol of middle plane 1 C, general symbol . . J B, symbol of middle plane 1 B, general symbol . . J of class c. Decrements on the lateral edges. 1 G 1 symbol of middle plane "\ 'g p ; ; ; } other planes J of class 1 H 1 . . . middle plane ~\ *W '. '. '. } other planes The Oblique Rhombic Prism. Fig. 317. Decrements on the acute solid angles. 1. Simple and mixed decrements on the angle A, P The general symbol A, represents class c. i A represents the middle plane of that class. P A P represents part of the series of class d. MODIFICATIONS. 273 2. Intermediary. (B'p h q Br : B P h q B'r) represents another part of the series of class d. The corresponding decrements on the obtuse solid angles are, P O. i O. p o p . (D'p H q Dr : D'r H q Dp). Decrements on the lateral solid angles, are all com- prised within class e. 1 . Simple and mixed. i E represents the middle plane of the class. E is the symbol, when the intersection of the planess e and P, is parallel to the oblique diagonal of P. When the lateral planes are intersected by the planes e, parallel to a diagonal, the symbol will.be either P E or E p . 2. Intermediary. General symbol (Bp Gq Dr). In this symbol the comparative values of p, q^ and r, will vary according to the positions of the planes represented. Decrements on the acute terminal edges. \ B represents the middle plane of class g. P B is the general symbol of that class. 2 M 274 RELATION OF DECREMENTS TO Decrements on the obtuse terminal edges. D represents the middle plane of clas'kf. P D is the general symbol of that class. i Decrements on the edges of the prism. 1. On the lateral edges G. O 1 G 1 represents mod. k. r G p .... class I. 2. On the oblique edges H. 1 H 1 represents mod. h. H P . . class ?'. The doubly Oblique Prism. Fig. SIS. Decrements on the solid angles. The planes comprehended under class a, may b^ represented by the following* symbols. P O, when the decrement proceeds along the plane P. P O . M. O p T. (Dp Ilq Fr) is intermediary. MODIFICATIONS. 275 The corresponding planes belonging to the other classes, may be represented as follows. Class ^ A. Class c, E. Class d, I. P A. P E. p l. A p . E p , P. (Bp llq Cr). (Bp Gq Dr). (Fp G'q Cr). Decrements on the edges may be expressed as follows. P P Class i, P H. Class e, B. Class g, D. H p . p P Class f> C. Class //, F. Class k, P G. G p . Hexagonal Prism. Fig. 319. Decrements on the angles. i A represents the middle plane of the series be- longing to class a. P A ... the other planes belonging to that class. P A P . . . those planes belonging to class b, whose edges intersect the planes M parallel to a diagonal. 276 RELATION OF DECREMENTS TO (Bp Gq BV : Br Gq B'p) represents those planes of class b which are produced by intermediary decrements. 13 represents class c. 1 G 1 . . . mod. d. P G P . . . class e. In some crystals of phosphate of lime, the planes belonging to class b occur singly. If they result from simple or mixed decrements, their symbol would be A P or P A, according as they lie on the left or right of the modified angle. And if they are produced by intermediary decrements, their symbol might be (Bo Go B'o : B q G P B'r.) The Rhomboid. Fig. 320. Simple and mixed decrements on the superior angles. General symbol P A P . p Ifp n: 1, the symbol represents mod. a. p ^> 1, . . . . . . . . class b. p <^ J, class c. MODIFICATIONS. 277 Intermediary decrements on the superior angles. (B'p B q B"r : B P B'p B"r) represents class d. The inferior plane angle at O, and the lateral plane angle at E, both belong to the lateral solid angles, all of which are similar, according to the definitions already given. The planes modifying the angles at E are therefore similar to those modifying the angle at O, but are reversed in their position on the crystal. The laws of decrement producing both are consequently simi- lar. But if we refer the decrements producing the planes belonging to any of the classes e, J\ g, h, i, k, /, to the solid angle at E, the symbols representing them will differ from those which would represent the same planes, if we refer the decrement to the solid angle at O. A single example will sufficiently illustrate this observation. Let us imagine the lateral solid angles of a rhomboid to be modified by two planes, which inter- sect the primary planes parallel to their oblique diago- nals. If the decrement producing these planes were referred to the angle at E, it would appear as a sim- ple or mixed decrement, and its symbol would be E /p P E. But if it be referred to the angle at O, it might be regarded either as a simple or mixed, or as an intermediary decrement, of which latter the symbol would be (D' P D q b /; P : Dp D' q b /; P ). If we regard these symbols with a little attention, we shall per- ceive that the variation in their form, does not alter the identity of their character, which is derived from the parallelism of one edge of each of the secondary planes to an oblique diagonal of the primary. But this character is implied in the supposition of a sim- ple or mixed decrement, which the first symbol 278 RELATION OF DECREMENTS TO represents; and it is directly indicated in each branch of the second symbol, by those indices which denote the abstraction of equal numbers of molecules in the direction of the edges D' b", and D b". The Abbe Hauy has referred some of the planes which modify the lateral solid angles, to the angle at E, and others to the angle at Q. It may therefore be convenient to possess the symbols representing those planes, in reference to both angles, and they will accordingly be given below. The symbol on the left represents the modification when the decrement is referred to the angle at O, and that on the right represents the same modification when the decrement is referred to the angle at E. 1. Simple and mixed decrements on the lateral solid angles. 1. Producing one plane on each solid angle. General symbols. In reference to angle C. q In reference to angle E. (Dp Bq D"p). If p zz 2 q, in either of these symbols, mod. e is represented. P cfaw J. Producing two planes on each solid angle. General symbols. P O 1J . or, E p \Jly (D' P b" P D q : Dp b"p D' q ). These symbols represent the series of planes be- longing to class h. MODIFICATIONS. 279 * 2. Intermediary decrements on the lateral solid angles. The general symbols to represent the modifications produced by these, are, (D' P b"r D q : D' q b"r Dp). | (Dp D" q Br : D q D% Br). Ifri=j9, the symbol represents the planes be- longing to class h, as described above, r ^> p, the planes of class i are represented. r <^ p, the planes of class I, and class f, are represented. Although these two classes are represented by a common symbol, there is this distinction between them ; that the edge produced by the intersection of the planes belonging to class/, is always paral- lel to the vertical axis of the rhomboid ; and that their indices p, q^ and r, are in a constant ratio to each other, as will be shewn in the appendix; while the planes belonging to class k do not intersect each other parallel to the axis of the rhomboid, nor is there any constant ratio between their indices. Decrements on the superior edges. General symbol P B P . If p =z 1, the symbol represents mod. m. p ^> 1, class n\ Decrements on the inferior edges. General symbol P D P . If p 1, the symbol represents mod. a. p ^> 1, class p. It may be remarked, that several of the classes of modifications on the angles of some of the primary 280 RELATION OF DECREMENTS TO forms, comprise planes which are produced by very different laws of decrement. And it may possibly appear to some of my readers, that different classes ought to have been established for the planes pro* duced by the several varieties of laws. But this would have rendered the tables of modifications less gene- rally applicable to the description of secondary forms, independently of the theory of decrements, than they are at present. This will become very obvious if we refer to the classes a, &, c, or J, of the modifications of the doubly oblique prism. All that can be known of any individual plane belonging to either of these classes, independently of calculation, is that it be- longs to such a class, and inclines on two of the adjacent primary planes at particular angles ; and this enables us to record the particular plane. If we refer to p. 274, we may perceive that the planes belonging to either of those classes might be produced by four different kinds of decrement. Let us suppose that we have observed a plane upon a doubly oblique prism produced by one of those decrements. As it replaces the solid angle O, we refer it without hesitation to our present class, a. But if class a had been divided into four classes, we could not, without previous calculation, know to which of those the observed plane ought to be re- ferred ; and the measurement of the crystal would not in such case, enable us to describe the secondary form, by the assistance of the tables only. The symbols used in this volume to represent planes produced by intermediary decrements, contain indices which are always whole numbers ; whereas the symbols used by the Abbe Haiiy to represent similar planes, frequently contain fractional indices. CODIFICATIONS. 281 Fig. 32 L There is, however, no real difference in the charac- ter conferred on the plane by the two methods of representing it. The only difference between them consists in this ; the indices used in this volume simply give the cha- racter of the compound molecule by whose continual abstraction the new plane is produced, while the Abbe Hatty's symbol supposes this molecule com- pounded of several other compound molecules. This will be readily understood by a reference to the above figure, which we shall suppose a doubly oblique prism, with an intermediary decrement on the solid angle at O. Let dec represent a compound molecule consisting of three molecules in height, four in the direction a d, and six in the direction b c, and let us suppose this the molecule abstracted from the first plate super- imposed on the terminal plane, and let us also sup- pose that two of these would be abstracted from the second plate, and so on, as explained in p. 22 and 23. The edge d a, in the above figure, corresponds with the edge D of the primary form, b e with F, and c/with H. The symbol we should use to represent the plane produced by this decrement, would be (D4 H3 F6), 2N 282 RELATION OF DECREMENTS, &C. X but the Abbe Haiiy's symbol would be (D2 O F3), and would be understood to imply that the compound molecules abstracted in the production of the new plane, consisted of smaller compound ones, each of these being three molecules in height, and two in breadth, repeated twice on the edge D, and three times on the edge F. From this exposition of the difference between the two symbols, it will be readily perceived that if in the Abbe Haiiy's symbol, we substitute for the letter denoting the angle on which the decrement is con- ceived to take place, that which denotes the edge upon which the angle of the new plane may be said to rest, and place the number used by him to express the decrement in height, which is in this case the denominator of his fraction, after it as its proper index ; and if we multiply at the same time his other indices by the number he uses to express the decre- ment in breadth, which is in this case the numerator of his fraction, the new symbol will be similar in character to those which are contained in this volume. This method of converting the form of the one symbol into that of the other, may be considered general, and by reversing the process, the symbols given in the preceding pages, may be converted into the form of those which he has used. APPENDIX. APPENDIX. CALCULATION OF THE LAWS OF DECREMENT. IN the preceding sections, some general rules have been given for determining the class of primary forms to which any given secondary crystal belongs, and for describing the secondary crystal by means of the position of its secondary planes, and of the angles at which those planes respectively incline on the pri- mary. The following is an outline of the method of apply- ing the theory of decrements to determine the rela- tions between the secondary and primary forms of crystals.* The application of this theory will embrace the following problems. First, To determine the law of decrement by which any secondary plane is produced, the elements of the primary form being known, and the angles at which the secondary plane inclines on the adjacent primary planes, being also known. * The reader of this appendix is supposed to be acquainted with th* elements of plane and spherical trigonometry, and with the use of the tables of logarithms. 286 APPENDIX CALCULATION OF THE Second, To determine the angles at which the secondary plane inclines upon the adjacent pri- mary planes, the elements of the primary form, and the law of decrement by which the secondary plane is produced^ being known. Or, to determine the particular values of the gene- ral indices given in the tables at p. 254, the inclination of the secondary planes to the pri- mary being known ; and to determine those in- clinations when the indices are known. The elements of the several classes of primary forms consist of 1st. The angles at which the primary planes in- cline to each other. These may be ascertained by means of the goniometer, if not already known. 2d. The plane angles of the primary planes. When these angles cannot be ascertained by other means, they may be deduced by spherical trigonometry, from the known inclination of the primary planes to each other. 3d. The comparative lengths of the primary edges, and of such other lines upon or within any crystal as maybe required for facilitating our calculations of the laws of decrement, or for delineating its primary or any of its secondary forms. The methods of de- ducing: such of these elements as cannot be ascer- O tained by measurement of the crystal, will be des- cribed where the elements of the several classes of primary forms are described. In the tables at p. 254, &c. the letter p is used to re- present any whole number ', or fraction. But it will be more convenient for our present purpose to represent simple and mixed decrements by the general fractional index fL; p expressing the decrements in breadth, and q q the decrements in height. LAWS OF DECREMENT. 287 It has been already remarked, thai one half the number of planes by which any crystal is bounded, are generally shewn in front of the engraved figure of that crystal. And as we know that the opposite angles, edges, and planes, which are supposed to form the back of the engraved figure, are respectively similar to those which appear on its front, if the decrements on these be described, the decrements on the hidden or back planes may be conceived to be described also. And again, as the law of symmetry requires that all similar angles and edges shall be similarly modified, if among the modified angles and edges, which are supposed to be in front of the figure, there be two or more, similar to each other, it is obviously sufficient to investigate the decrement upon one of these, in order to determine the character of the modifying planes upon the others. Decrements, as we have already seen, take place on the ^edges or angles of crystals, and are of two principal kinds ; one of which produces planes inter- secting the primary planes, in lines, of which one at least, is parallel to an edge or diagonal of one of those planes ; and the planes produced by the other intersecting the primary planes in lines, not any of which are parallel to an edge or diagonal of any of those planes. The effect of both these classes of decrements upon the primary form, is similar to that which would take place, if we conceive the enlarged crystal to have been completed, and the whole of the omitted molecules to have been then removed from it in one, mass. This will be readily perceived, if we refer for an example to modification a of the rhomboid. Let us conceive a rhomboid of a given dimension to hare been formed; and during its further increase in 288 APPENDIX CALCULATION OF THE bulk, a row of molecules to have been abstracted at the angle A of the primary form, from the first plate of molecules added to the plane P, and an additional row to have been abstracted from each succeeding plate. As the three plane angles which concur to produce the solid angle at A, are similar, a similar abstraction of molecules would take place simultaneously from the plates superimposed on each of the three adjacent planes, and the result would be the production of a tangent plane, presenting at its surface the terminal solid angles of the molecules belonging to those plates which had been added to the smaller rhomb- oid.* Let us next suppose that instead of any abstraction of molecules from the superimposed plates, those plates had been added entire, and a perfect enlarged rhomboid had been produced. * It will be recollected that the molecules are so small, as to occasion no perceptible difference iu the character of the secondary plane, LAWS OF DECREMENT. 289 We may conceive it possible to reduce this entire rhomboid to the state of the modified one, by remov- ing, in one mass, the triangular pyramid of molecules d c f g, fig. 322, in which the supposed modified crystal is deficient. The mass of molecules, therefore, in which any secondary form is deficient) when compared with its primary form, is equal to the number of molecules abstracted in the production of that secondary form, arranged in the same order as they would have been, if they had completed the enlarged primary form. This mass, so arranged, being all the addition to the secondary form which would be required to com- plete the primary, will be called the defect of the primary form, and it will be shewn presently that the edges of this defect may be used in every instance, to determine the decrement by which the secondary plane is produced. Fig. 323. For the purpose of illustrating this proposition further, let us observe the change which would have taken place, if a parallelepiped of any kind, either right or oblique, as a b c d e, fig. 323, had been modified on one of its edges, by a decrement consisting of a single row of molecules. whether that plane exposes the edges, solid angles, or planes of the molecules. 290 APPENDIX CALCULATION OF THE Let us suppose the edge i A*, of the primary form, to be to the edge i d, in the ratio of m to n, m and n being any numbers whatsoever. It follows from what has been before stated, that the ratios of the corresponding edges of the molecules which compose this form, will also be as m to n; and consequently that the primary edges are composed of equal numbers of edges of molecules, and may there- fore be regarded as multiples of m and n, by some indefinite whole number. Let us further suppose that the decrement had begun to act at the edge a b, and had proceeded along the plane a b c. In the first plate of molecules superimposed on that plane, the row 1 would have been omitted. In the second plate, the additional row 2. In the third plate, the additional row 3, and so on. Now the evident result of these abstractions from the several superimposed plates, would have been the production of a new plane, a b g f, replacing the edge h , of the enlarged crystal ; and the triangular prism, whose base is the triangle b if, represents the defect of the primary form occasioned by this decre- ment. But it is obvious from the figure, that the ratio of the lines i f to i b of the defect, is as 3 m to 3 n, or as m to n. Hence when a decrement by 1 row of molecules takes place on the edge of any parallele- piped, the ratio of the edges of the defect, correspond- ing to if, i b, is similar to the ratio of those edges of the primary form, of which these are respectively supposed to be portions. And as the edge b f, of the new plane, coincides with a diagonal of the molecules, it is evidently parallel to a diagonal of the plane d i k e. LAWS OF DECREMENT. 291 Fig. 324. Let us now suppose a decrement by 2 rows in breadth to have taken place on the edge of a similar parallelepiped. If we imagine the first plate of molecules which is superimposed on the primary plane to be deficient in two rows of single molecules ; and if we imagine two additional rows of molecules abstracted from the second plate, and so on, the plane i k would be produced, and the lines i g, g &, would be the edges of the defect of the primary form occa- sioned by this decrement. But it is evident that the line g k is to the line g /, as 4 m is to 2 ft, or as 2 m to ??, this being the ratio which the number of molecules abstracted in the direction g f, bears to the number deficient in the direction of g d. From these examples we find that whenever a decrement takes place on the edges of any parallele- piped, replacing that edge by a plane, the edges of the defect will be to those edges of the primary form, of which they are respectively parts, in the ratio of the numbers of molecules abstracted from each su- perimposed plate in the direction of the same edges respectively; and that such ratio will express the law of decrement by which the new plane has been produced. 2o2 292 APPENDIX- CALCULATION OF THE We may perceive from the figures 323 and 324, that if we had conceived the new plane to be pro- duced by the superposition of a single plate of mole- cules, the edges of the defect would still be in the same ratio to each other as if the new plane were produced by a series of decreasing plates. We may therefore express the character of this plane by the ratio of the edges of the defect of the first plate of molecules, consisting of one or more molecules in thickness, according to the nature of the decrement. Fig. 325. Let us derive another illustration of this propo-r sition from a decrement on the angle of a parallel o- piped ; and, to render the example more general, let us suppose an intermediary decrement acting on that angle to have produced a plane a b c, fig. 325, by the abstraction of a compound molecule, consisting of three molecules in height, two in the direction of i h, and four in the direction of i k. If we suppose the lines i #, and i d, to be to each other in the ratio of m to n, and the line i h, to be as o, the corresponding edges of the compound molecules would consequently be in the same ratio, and the edges, i c, i b, i a, of the defect, would be as 4 m 9 3 n, and 2 o, and would, when divided by m, n 9 and o, express the law of de- crement by which the new plane is produced. L.AWS OF DECREMENT. 293 From these examples it appears that the edges of the defect of the primary form are multiples of the cor- responding edges of the molecules / and the ratios of the edges of the defect are consequently multiples of the ratios of the corresponding edges of the primary form. For let the ratio of i k : id, which is that of m : /z, be represented by the fraction - n and the ratio of i b : i c being that of 4 m : 3 n, be represented by the fraction t??. a n It is evident that the ratio is a multiple of o n by t n 7 3 Hence the problem of ascertaining the law of decre- ment producing any secondary plane, is reduced to that of ascertaining the ratios of the edges of the^ defect of the primary form occasioned by such decrement, and dividing these ratios by the ratios of the correspond- ing edges of the primary form. We may also discover the law of decrement in some particular case?, by dividing the ratios of the edges of the defect, by the ratio of an edge to some other line upon the crystal. Whatever ratio we may use for this purpose, will be termed the unit of comparison. This unit of comparison is, generally, the ratio of certain edges or other lines, either on the surface, or passing through the interior of crystals, of which, pro- portional parts would be intercepted by any new plane, resulting from a decrement by one row of molecules. According to the theory already explained, the molecules of all parallelepipeds are similar parallejo- pipeds, and their edges are consequently propor- tional to the corresponding edges of the primary form. APPENDIX CALCULATION OF THE Hence, when through the operation of a decrement on an edge of any parallelepiped, a single row of molecules is abstracted, the parts which are removed from the two edges adjacent to that on which the decrement has taken place, will be proportional to those edges respectively ; and the ratio of those edges may therefore constitute the unit of comparison for decrements on the edges of parallelepipeds. If a decrement take place by one row on an angle of a parallelepiped, a single molecule is first abstracted from its solid angle; and the parts thus abstracted from the three edges which meet at the solid angle, are respectively proportional to those edges. The ratios of the three adjacent edges of any paral- lelopiped, therefore, may be taken as the units of comparison for determining the various laws of de- crement on the angles of that class of primary forms. And, by analogy, we may take the ratios of the three or four edges adjacent to the solid angles of any class of primary forms, to express the ratios of the edges of the defect occasioned by a decrement by one row of molecules. But other lines may be traced on some of the classes of primary forms, proportional parts of which will also be intercepted by decrements by one row of molecules. The ratios of these may therefore be taken as the units of comparison, if we find them more convenient for our calculations than those of the primary edges. When the edges, or other lines, from which the unit of comparison is to be derived, are equal, their ratio will be = I, and in this case the lowest whole numbers which will express the ratios of the edges of the defect of the primary form, will also express the law of decrement. And whenever an edge of any primary form is re- placed by two similar secondary planes, as in LAWS OF DECREMENT. 295 of the cube or regular octahedron, or g, or i, of the right rhombic prism, &c. the lines whose ratio con- stitutes the unit of comparison in such cases, will always be equal. And the units of comparison for determining any law of intermediary decrement, will always be the ratios of the edges which meet at the solid angle on which the decrement has taken place. Fig. 326. To ascertain the ratio of the edges of the defect of the primary form, when a decrement takes place on an edge of any parallelepiped, fig. 326, we must suppose the inclination of the primary planes to each other to be known, and the inclination of the modi- fying plane a b cf, to the primary planes P and T. Fig. 327. Now to determine the ratio which the line i , bears to i c, we require the angles of the plane tri- angle i b c. These may be obtained by means of a spherical triangle, fig, 327, whose angle A is the 296 APPENDIX CALCULATION OF THE supplement of the inclination of P on the plane a b c fy the angle B the inclination of P on T, and the angle C the supplement of the inclination of T on the plane a b cf. From this spherical triangle we deduce the side , containing the required angle i b c, by the known formula sin, i - R -t /-cos. \ (A+B+C). cos. \ (B-f C=A) y sin. B. sin. C Fig. 328. and by applying the same formula to a second spherical triangle, fig. 328, whose angle C is similar to that of the preceding, B is the inclination of M on T, A the supplement of M on the plane a b c f\ derived from actual measurement, or deduced from the known inclination of P on the plane a b c /, and of P on M, we may again obtain the sidec, which contains the other required angle, i c b. Having thus determined the two plane angles, i be, i c by the ratio of i b to i c is known from the analogy between the sines of the angles of triangles, and the sides subtending those angles, thus, i b : ic : : sin. \y i c b : sin. \/ i ' b c.* Let us suppose i k : id:: m : n ; m and n being any whole numbers whatever, and being already known by means which will be pointed out in a later part of this appendix. * This mark V ' s used to denote the word angle. LAWS OF DECREMENT. 297 If the new plane has resulted from a decrement on the edge h ', by one row of molecules, the lines i 6, f c, must also be to each other as m to n, and we should then have sin. \/ icb : sin. \/ i'b c :: m : n. But if the new plane has resulted from a decrement by unequal numbers of molecules in height and breadth, the ratio of i b to i c, should be as p m to q n ; the letter p representing the number of mole- cules abstracted in the direction of the edge i k, and q representing the number abstracted in the direction of the edge i d. The ratio of p m : q w, may be expressed by the fraction of P m . which is evidently the product of qn ' jf _ 1 We may therefore obtain the values of p q ' n and q, whatever may be the particular values of ff m and z , if we divide P m , which expresses the q n n q n ratio of the edges of the defect by , which expresses the ratio of the corresponding edges of the molecules, or, which is the same thing, of the corresponding edges of the primary form. There are two methods by which this division may be effected, Thejirst is by finding the absolute values of m and w, and of p m and q n^ by means of the tables of na- tural sines, &c. and then reducing those ratios to their lowest denominations in whole numbers; and after dividing the one fraction by the other, reducing the quotient to its lowest denomination in whole numbers. The quotient so reduced would express the law of decrement. 298 APPENDIX CALCULATION OF THE As an example of this method, let us suppose we have found i k : id :: 11 : 8 therefore = i!. n 8 And let us suppose the ratio of ib : ic :: sin. y icb : sin. \/ ibc :: 33 : 16; then we should have P m zz . q n 16 If we divide the second fraction by the first, the quotient will be |? X ~ = ~ = -, which would give a law of decrement by three rows in breadth, or in the direction of i k, and two in height, or in the direction of the edge i d. If we now suppose the edges i k, and if, to be equal, it is evident that becomes equal to 1. n Under this supposition the ratio of i b to ic might be expressed by a fraction of the form . . q Let us now imagine the ratio of i b : ic to have been found as 1 : 3. This would indicate a decrement proceeding along the terminal plane by 3 rows of molecules in height. If we find i b : i c : : 4 : 3, the law of decrement producing the plane from which that ratio is deduced, is by 4 rows in breadth, and 3 in height, on the termi- nal plane. The second method of dividing the proposed frac- tion -J? by is by means of the logarithms of the q n n quantities from whence those ratios are deduced. LAWS OF DECREMENT. Let us suppose we have found i k : i d : : R : sin. a, we should then have m n and . . . Log. = Log. R Log. sin. a. n Let us also suppose ib : ic *: s'm.\/icb :\/sin.ibc, then P m = shl ' V ' c b q n sin. N/ ib-c; and Log. ?_???_ Log. sin. Y/ 2 c ^ Log. sin.yf 6 c. qn The division of J!? by is effected by sub- q n n tracting the logarithm of the latter fraction from that of the former. And the natural decimal number corresponding to the resulting logarithm, will bear the same ratio to 1-0, 1-00, 1-000, &c. according to the number of decimal planes in the number found, as the decrement in breadth bears to that in height. Examples of the application of this method of deducing the values of p and q, will occur in the course of this appendix. Fig. 329. Let us now enquire how we may determine the ratios of the three edges, i ft, i c, i , fig. 329, of the defect^ occasioned by a decrement on one of the angles of a parallelepiped. 2 p 2 300 APPENDIX CALCULATION OF THE We are supposed to know the inclination of the primary planes to each other, and let P on M be called I- P T . . . /, M T . . . J 3 We may from these readily deduce the plane angles at t by means of a spherical triangle ; and having measured the inclination of the plane a b c on P, M, and T, we may discover the plane angles at , #, and c, by means of the three spherical triangles marked on fig. 330. Fig. 330. In these triangles we known only the angles, which are those at which the primary planes incline to each other, and the supplements of those at which the secondary plane inclines on the adjacent primary planes. The plane angles at 0, b, and e, being found, we may readily discover the ratios of ib : i c, and ib : ia, which will give the law of decrement by which this modifying plane has been produced. Let us still suppose, i k : i d : : m : n y and i k : i h :: m : o, our units of comparison here would be - and n <> and let i b : i c : : p m : q n i b : i a :: p m : r o. LAWS OF DECREMENT. 301 After effecting our division of P-2H by , and of q n n , we should find r o o i b = p, I c = q, i a =. r, which would imply a decrement by p molecules in the direction of i k, q molecules in that of i rf, and r molecules in the direction of f h. If we suppose fig. 329 to be a doubly oblique prism, the letter to denote the edge i h would be D, ik . / . F, id . . . H, and the symbol of the plane a b c, would then be (Dr Ep Hq). It may be remarked here, that tangent planes are generally the result of a decrement by 1 row of molecules, whether they replace the angles or edges of those classes of the primary forms in which they occur. By this general method of proceeding we may, when we know the inclination of the primary planes to each other, and of the secondary plane on one or more of the primary, discover the law of decrement by which any secondary plane has been produced on any of the classes of parallelepipeds ; and it may be adapted also to all the other classes of primary forms. We shall now apply it to the several classes of those forms in succession, and the calculation will be found to become much more simple in its application to many of those classes. 302 APPENDIX CALCULATION OF THE As it will not be necessary to repeat even the for- mulaB in all the cases which are to follow, it may not be useless again to observe that when a law of decre- ment producing any plane is to be determined, the general symbol of that plane is to be first discovered, and then the particular values of its indices to be found. In simple or mixed decrements^ these values are deduced from the ratio of radius to tangent a, or of sin. a to sin. &, as we have already seen ; a and b representing the particular angles in each particular case. The following may be regarded as the general pro- cess for determining the law of an intermediary decre~ ment. 1st. To measure the inclination of one of the secondary planes on two of the adjacent primary planes. 2d. To determine the two plane angles at the ter- mination of the greater edge of the defect of the primary form occasioned by the plane we have measured. 3d. From a knowledge of these plane angles, and of the plane angles of the primary planes, to deduce the ratios of the edges of the defect. 4th. When the primary edges are unequal, to divide these ratios by the ratios of the correspond- ing edges of the primary form, and thus to deduce the law of decrement. 5th. If the intermediary decrement has taken place on an octahedron, to determine the fourth edge of the defect by a method which will be described when we apply our calculations to the regular octahedron. LAWS OF DECREMENT. 303 It will be recollected that in framing the general symbol of any secondary plane, we are generally to consider p^> #> r> s. By carefully observing the position of the plane we have measured, and whose law of decrement we require to know, we shall feel no difficulty in adapt- ing an appropriate symbol to it. And having found our general symbol, we may readily find the particu~ lar values of these indices by the methods already described, or by such as will be detailed in the suc- ceeding part of this appendix. From what has preceded, the method will be rea- dily perceived by which we may determine the ratios of the primary edges of crystals, if we assume some observed secondary plane replacing an edge of those forms whose terminal edges are equal, or replacing an angle of those whose terminal edges are unequal, to have been produced by some given law of decre- ment. If we assume that a plane replacing an angle or edge of any primary form, has resulted from a decre- ment by one row of molecules, we determine the ratio of the primary edges by discovering the r^tio of the edges of the defect occasioned by that plane. And if we assume any other law of decrement to have produced the given plane, the ratios of the primary edges may evidently be determined, by dividing the ratios of the edges of the defect by the assumed law of decrement. 304 APPENDIX- -CALCULATION OF THE THE CUBE. Its elements. The inclination of any two adjacent planes at their common edge zz 90. Plane angles rz 90. Edges all equal. Inclination of an edge to an axis 54 44' 8 /; . Ratio of an edge : \ a diagonal : : 2 : VcJ an edge : an axis : : 1 : Vg t Its units of comparison. In reference to decrements on the edges, the unit is nz 1. ..... simple and mixed decrements on Vo I the angles, it is = _ = y=. Simple and mixed decrements on the angles. The law of a simple or mixed decrement on any angle of a cube, may be computed by means of an edge of the primary form, and half a diagonal of one of its planes. The diagonal being used to measure the decrements in breadth, and the edge those in height* Ing. 331. Let us suppose a cube represented by figure 331, and let us jmaginc a simple decrema|it to have taken LAWS OF DECREMENT. 305 CUBE. place by 1 row of molecules on the angle afb. The edges of the defect of the primary form, would, in this case, be, as we have already seen, proportional to the corresponding edges of the primary form, and might consequently, if the secondary plane were suf- ficiently enlarged, be equal to those edges. The edges of the new plane might therefore coincide with the lines a b, a c, b c. If we now draw the diagonal g /", on the terminal plane, we shall observe that one half of it is inter- cepted at the point /*, by the edge a b of the second- ary plane. As a decrement by 1 row therefore intercepts the half diagonal fh, at the same time that it intercepts the whole of the edges f a,fb,fc, the ratio offh :fc may be assumed as the unit of comparison for deter- mining the law of a simple or mixed decrement on the angle afb. For let us suppose a decrement to have taken place on that angle, by 2 rows in breadth ; if this decrement be conceived to be continued until the edges af, and b f, are again intercepted by the new plane, it is obvious from what has been already stated, that only one half of the edge/c would be intercepted by the same plane. Here then the law of decrement would be ex- pressed by the ratio of fh : |/c, or 2fh : fc; and if f h : fc be represented by m : n, the ratio of fh : f d, should be as 2 m : n, and would thus give the required law of decrement by 2 rows in breadth on the angle afb. But the ratio of fh :fd, is that of radius : tang, of the angle fhd- } yuidjh d is the supplement of the 2Q 306 APPENDIX CALCULATION OF THE CUBE. angle g h d, the inclination of the primary to the secondary plane. The planes belonging to classes b and c of the modifications of the cube, result from simple and mixed decrements. Let the inclination of P on the plane b adjacent to it, or on the plane c which rests on the edge between P' and P",* be measured and called /, . If p, as before, be used to represent the decrement in breadth, and q the decrement in height, then p- R q tang. ( ISO -/, ) Intermediary decrements. The general symbol representing a single plane belonging to class d, would be (B P B //7 q B'r). And the law of decrement producing a particular plane of that class, would be discovered by finding the values of />, q, and r, in relation to that particular plane. Let us suppose q ^> r. If we refer to the tables of modifications, we shall observe that two of the planes which have the d placed upon them, rest on the edge between P and P 7 . Let that which inclines most on P', be measured on P and P'. Let Pond / 2 . P 1 ..d = I 3 . Let the plane angle of the defect corresponding to i a c, fig. 329, be called A , , and that corresponding to i a b of the same fig. be called A z . * See the tables of modifications whenever the classes are referred to. LAWS OF DECREMENT. 307 CUBE. Tir i IT i. & COS. (180 / 2 ) We shall have cos. A, = : i =-? sin. (180 / 3 ) A R. COP. (180 /,) and cos. A 2 = _ -A ^ sin. (180 / 2 ) The plane angles being thus found, we have a i : i c : : p : q : : R : tang. A l a i : i b : : p : r : : R : tang. -4 2 . We may determine these particular values of the ratios of p : q, and p : r, by means of the tables of natural tangents, or by logarithms. Decrements on the edges of the cube. Let the inclination of the plane P on the plane/, or &, adjacent to it, be called / 4 . ?= *- q tang. (180/ 4 ). Throughout the remainder of this appendix, when the law of a simple or mixed decrement is expressed by ^, the letter p will always be understood to denote the decrement in breadth, and q the decrement in height. SOS APPENDIX CALCULATION OF THE THE REGULAR TETRAHEDRON. Fig. 332. Its elements. The mutual inclination of any two adjacent planes at their common edge zz 70 31' 44", and may be called /,. Plane angles zz 60, and may be denoted by A t . Edges all equal. Inclination of an edge a do a perpendicular, a b = 54 44' 8", and may be called A 2 . Inclination of an edge to an axis zz 35 15' 52", and may be called A 3 . Inclination of a perpendicular a b to an axis zz 19 28' 16", and may be called A 4 . Ratio of a perpendicular a b : an edge a c : : ^3 ' 2. Its units of comparison. Vg The unit is zz , in relation to simple or mixed decrements on the angles. . . . . zz 1, in relation to decrements on the edges. LAWS OF DECREMENT. 309 REGULAR TETRAHEDRON, Simple and mixed decrements on the angles. To determine the law of a simple or mixed decre- ment on an angle of the regular tetrahedron, we may assume as the unit of comparison, the ratio to an edge a c, of a perpendicular a b upon the base, fig. 332, drawn from the angle . The line a b measuring the decrement in breadth, and the edge a c measuring the decrement in height. The ratio of ab : ac is known, from the relation of the tatrahedron to the cube, to be as *3 : 2. Let/g-, fig. 332, represent a secondary plane be- longing to class &, whose inclination to the primary plane, which is obviously equal to the angle b d e, has been determined by measurement; we may call this angle f 2 . In the triangle a d e, we have the following angles, \/ ade=(lSO /,), V d a e zz (90 |I t ) nr 54 44' 8", which we have called A,. V aed (/ a Aj. Whence ad : ae :: sin. (7 2 A z ) : sin. (180 7 2 ) :: pm : qn. In the fraction !?, - represents . qn' n 2 Dividing therefore sin * f-[J^_^) by _JL we shall sin. 180 /.). g' find the values of p and gs jp, representing the decre- ment in breadth on the angle a, proceeding along the plane a b, and q the number of molecules in height corresponding to the line a e. If the inclination on P, of any plane belonging to class c, which rests on the edge between P' and the 310 APPENDIX- CALCULATION OF THE REGULAR TETRAHEDRON. back plane of the figure, be known and called J 3 , the preceding formula will give the law of decrement producing that particular modification. Intermediary decrements. Fig. 333. The law of an intermediary decrement on an angle of the tetrahedron, is determined by the ratios of the defect or intercepted portions ae, af, a g, of the three primary edges a b, a c, a d. Let efg, fig. 333, be one of the six planes pro- duced by a modification of the tetrahedron belonging to class d. The general symbol representing a single plane belonging to this class is (B P B' q B"r), p representing the number of molecules contained in af, q the num- ber contained in a g, and r the number contained in ae. To determine the ratios of a e : af : a g, we re- quire the plane angles af e, afg; from which, as we know the angles e af, f a g, we may deduce the angles a ef, a gf. To obtain the plane angles afe, afg, we may have recourse to a spherical triangle. LAWS OF DECREMENT. 311 REGULAR TETRAHEDRON. The plane representing class d, on the figure in the tables of modifications, which corresponds to the plane e fg of fig. 333, is that which rests on the edge between P and P' and inclines on F'. Let the inclination of this plane on P be called / 4 . ........ V . . .1,. and let the plane angle afe be called A 5 . ....... afg . . . A 6 . we shall have sin. | A, == L O V - J 4 )+(180-/ 5 ) ]cos4[ I sin. /n sin. / 4 and sin. A 6 =z ^ / I/ sin. /, sin. 1 5 Having from these formula deduced the angles A 5 and A 6 , we have /: a g :: p : q :: sin, (120 A 6 ) : sin. A 6 sin. A 6 sin. (120 A 6 ) a f: a e :: p : r :: sin. (120 yl/ 5 ) : sin. A 5 .. I . sin.^ 5 * sin. (120" J 5 ) Hence the particular values of p, T 6 ). We may determine p and q, those being the lowest whole numbers which will express that ratio, by the means of either the natural sines, or their logarithms ; and when determined, they will express the law of decrement by which the new plane has been pro- duced. LAWS OF DECREMENT. 313 REGULAR OCTAHEDRON. Fig. 335. Its elements. The inclination of any two adjacent planes at their common edge 109 28' 16", may be called /, . Plane angles = 60, may be called A t . Edges all equal. Inclination of edge to edge measured over the solid angle 90. Inclination of plane to plane measured over the solid angle = 70 31' 44", and may be called J 2 . Ratio of a perpendicular a b : an edge af : : ^3 i 2. . . . an edge : | an axis :: ^2 : 1. Its unit of comparison Is 1, in reference to simple and mixed decrements on the angles, and also to decrements on the edges. Simple and mixed decrements on the angles. The law of a simple or mixed decrement on the angle of a regular octahedron may be determined by means of the perpendiculars a b, a f, drawn from the angle a upon the edges of the base : which perpen- 314 APPENDIX "CALCULATION OF THE REGULAR OCTAHEDRON. diculars are, from the nature of the figure, equal. We know the angle b a c, which we call J 2 , and we are supposed to know the angle b d e, which is the inclination of one of the planes of class b to the adja- cent primary planes. This inclination we shall call / 3 , and from these angles we may deduce the angle d e a, which we may call / 4 . Hence we have a d : a e : : sin. / 4 : sin. (180 / 3 ) : : p : -- |__1L sec. (90 | I.) r, , R COS But as -- _ , sec. R and tang. 90 a = cot. a, the equation becomes cot.i/ 3 R The second equation must evidently be simi- lar in its character to the first, but substituting J 3 and / a for / 2 and 7 3 . The terminal edges equal. Ratio of a terminal edge : a perpendicular on the base of plane P :: R : cos. \A,. Ratio of a terminal edge : a perpendicular on the base of plane M :: R : cos. |^ 2 . Ratio of a terminal edge : a greater edge of the base : : R : 2 sin. \ A , . Ratio of a terminal edge : a lesser edge of the base :: R : 2 sin. \ A z . Edge d b of the base : edge be:: cot. \ I 3 : cot.i/ 2 . For gi = %db :: gh~ b e :: tang. (90 M 3 ) : tang. (90- i 7.) :: cot. \ 1 3 : cot. | I z 332 APPENDIX CALCULATION OF THE OCTAHEDRON, RECTANGULAR BASE. The inclination of a terminal edge to the axis may be called A 3 , and may be thus found, If we suppose a spherical triangle to be represented by a segment kg bfof the octa- hedron, it would obviously be right angled at h ; and we know the side Whence we find cos. y g fb, which we call A 3 , by the known formula cos A cos.(90 1/ 2 ~R sin. | 7 2 cos. Ratio of I a diagonal of the base : ^ the axis :: sin. A 3 : cos. A 3 . The relation between 7,, 7 2 , and 7 3 may be thus discovered, cos. 7 zz ~ cos '2^2 cos. |7 3 If we suppose the angle h b i rr 90, to be the side of a spherical triangle, the angles of the same triangle would be 7,, i7 , and 1/3- A general equation to discover 7 , would be cos. 7, ~ cos.\/hb i. sin.i7 2 .sin.|7 3 cos.-7 2 . cos.^/ 3 * But as cos. V h b i ~ o, the equation be- comes that which has been given. From which formula, any two of the angles being known, the third angle may be found. LAWS OF DECREMENT. 333 OCTAHEDRON, RECTANGULAR BASE. Its units of comparison. The unit is z= 1 in reference to all decrements on the terminal angles, and to decrements on the edges of the base. For simple or nvxed decrements on the lateral angles, the unit will be the ratio of the perpendiculars b 7t, b /, drawn from the angle at b perpendicularly on the edges fd and e m. The ratio of those lines may be thus determined. fb : k b :: R : sin. V kfb = A, b m fb i b I :: R : sin. \J b m I zz A t whence . . . k b : b I :: sin. A , : sin. A a When the decrement is conceived to proceed along the plane P. the unit of comparison will be ?3- L sin. A % but when the decrement proceeds along the plane M, the decrement in breadth will evidently be measured by the line b I, and the unit will then become sin 3 sin. A t . Fig. 344. The laws of decrement on the terminal edges may be determined from the lines b a, b c, drawn perpen- dicular to the edge f 6, on which a decrement is sup- posed to have taken place, and meeting the edges 334 APPENDIX CALCULATION OF THE OCTAHEDRON, RECTANGULAR BASE. C) produced to a and c. These lines are to each other as tang. A t : tang. A 2 . When the decrement proceeds along the plane P, the unit of comparison is ^1 L, but when it is tang. A, conceived to proceed along the plane M, the unit becomes ?! - . tang. A , Knowing the elements of the primary form, the unit of comparison, and the symbol of any plane whose law of decrement we require, we are to mea- sure the inclination of that plane on one or more of the primary planes, and then to adapt such formulae to the particular case, as will give the particular values of the general indices of the plane in question. - LAWS OF DECREMENT. 33$ THE OCTAHEDRON WITH A RHOMBIC BASE. Fig. 345. Its dements. The inclination of the planes at the obtuse ter- minal edges may be called /, . Inclination of the planes at the acute terminal edges may be called /,. Inclination of the planes at the edges of the base may be called / 3 . Plane angles at the summit, being called A tj may be thus found, cot -^ cot. cos A = R A spherical triangle is marked on the upper part of the above figure, the angles of which are 90, \ 1 ^ and \ / 2 , and of which the required side b a c is the hypo- thenuse, which is found by the preceding formula. The two spherical triangles marked at the base of the figure, will give the follow- ing formulae; the required side being in each instance the hypothenuse of the as- sumed triangle. 336 APPENDIX CALCULATION OF THE OCTAHEDRON, RHOMBIC BASE. The greater lateral plane angle a c 6, may be called A zj and may be thus known, cos. A, = cot.U. cot.iJ 3 K Most acute of the lateral plane angles, a b c, may be called A^ and may be thus found, cos. A 3 = coMf._cot.jJ. Angles of the rhombic base may be thus deter- mined. Let the lesser angle be called A^ and cos.-M 4 = R cos " * L > sin.i/ 3 The required angle \ A^ is the angle e b c of the above figure, and is that side of the spherical triangle nearest to b, which is opposite the angle \ I 2 ; and hence the for- mula which is given. Ratio of \ the greater diagonal of the base : \ the lesser diagonal :: R : tang. \ A . Obtuse terminal edges are equal. Acute terminal edges are equal. Inclination of the obtuse terminal edge to the axis being called A 5 , may be thus found, A r, COS. \ I 9 cos. A, = R i i sin. /, Inclination of the acute terminal edge to the axis may be called A 6 , and may be thus found, . T, cos. \ I, cos. A 6 R -i- sin. i/ 2 The angles A 5 and A 6 , are those sides of the upper triangle marked on the figure, which subtend the angles \ I I and \ J 2 . And hence the formulae for their determination. LAWS OF DECREMENT. 337 OCTAHEDRON, RHOMBIC BASE. The ratio of an obtuse terminal edge : an acute terminal edge :: sin. A 3 : sin. A z . Ratio of \ the axis : | the greater diagonal of base :: R : tang. A 6 \ the axis : * the lesser diagonal of base :: R : tang. A 5 -| the axis : the obtuse terminal edge :: cos. A 5 : R. \ the axis : the acute terminal edge :: cos. A 6 : R. obtuse terminal edge : perpendicular a /':: R : sin. A 2 . acute terminal edge : perpendicular af:: R : sin.A 3 . Its units of comparison. The unit is rz 1, in reference to the following de- crements. 1. Simple and mixed on the terminal angles, pro- ducing mod. b. 2 obtuse lateral angles, producing mod. k. 3 acute lateral angles, producing mod. q. 4 . obtuse terminal edges. O 7 producing mod. u. 5 acute terminal edges, producing mod. x. 6. edges of the base, producing mod. z. 338 APPENDIX- CALCULATION OF THE OCTAHEDRON, RHOMBIC B3SE. The units of comparison for determining- the laws of intermediary decrements will be the ratios of those edges, of which portions are intercepted by the particular plane we are examining. Having determined the unit, and the symbol, and measured the inclination of the modifying plane on one or more of the primary planes, we may proceed to discover the law of decrement by some of the methods already described. LAWS OF DECREMENT 339 THE RIGHT SQUARE PRISM. Its elements. The inclination of any two adjacent planes at their common edge nr 90. Plane angles 90. Terminal edges equal. Lateral edges equal. Its units of comparison. Fig. 346. Let the terminal edge be to the lateral edge as mini this will be the unit of comparison for decre- ments on the terminal edges. For simple and mixed decrements on the angles of the terminal planes, the ratio of a b : af becomes the unit, which is V<2 m 2 m " ~~ nVo 9 an d f r similar decrements on the lateral angles the ratio of a e, being \ the diagonal of the lateral plane, to the terminal edge a d, may be regarded as the unit. But we cannot immediately determine the ratio of the intercepted portions of the lines a e, ad, from the inclination of the secondary plane on the lateral primary plane. It must be deduced from the ratio to the edge a d, of a perpen- dicular a c upon the diagonal mf. It is very obvious that this perpendicular possesses the character assigned to those lines from which the 340 APPENDIX CALCULATION OF THE RIGHT SQUARE PRISM. unit of comparison may be derived. For it would be wholly intercepted by a decrement by 1 row, when the edge a d is so intercepted. And the propriety of adopting this unit will be apparent, if we recollect that the inclination of two planes to each other is measured upon lines perpendicular to their common edge. Let the lines i h, h c, be perpendicular to the edge of the secondary plane at the point h. The angle i h c is all that can be known from actual measure- ment of the crystal ; but from this we know the angle i h a, and hence the ratio of a h to a i. The constant relation of a c to a d, may be readily found. We are supposed to know the ratio of a m to af', and a m : a f :: R : tang. \/ a m f call \J a mj] A^ ; And because a c is perpendicular upon fm, we have a c : a m or a d :: sin A , : R. Hence 55! ' becomes the unit of comparison in R reference to simple and mixed decrements on the lateral angles of the prism, the symbol representing which would be P A P . For decrements on the lateral edges the unit is = 1, and for intermediary decrements, the unit will be the ratios of the particular edges affected by each law of decrement. The general methods adopted for determining the laws of decrement on the cube, may be applied to this prism ; observing however the differences in the several units of comparison afforded by the two forms. LAWS OF DECREMENT. 341 THE RIGHT RECTANGULAR PRISM. Its elements. The inclination of any two adjacent planes at their common edge ~ 90. Plane angles zz 90. Parallel edges are always equal. Three adjacent edges always unequal. This inequality cannot be determined but by the means of secondary planes. Its units of comparison. Fig. 347. Let us suppose the edge o a : o b :: m : n o a : o c :: m : o will evidently be the unit of comparison for deter- o mining the laws of decrement producing the planes of class b. , for those of class c. o , for those of class d, whose symbol is G p P G. n " P G G p m It has been shewn in p. 267, that the planes com- prehended under class #, might be produced by four 342 APPENDIX CALCULATION OF THE RIGHT RECTANGULAR PRISM. different kinds of decrement, of which the general symbols would be A, P A, A p , and(Cp G q Br). The unit of comparison will be different in each of these cases. Let a b, a c, b c, fig. 347, be three diagonals, and o d, oj\ o e, three perpendiculars drawn from those diagonals to the angle o. The edges a b, a c, be, might be the edges of a plane produced by a decrement by 1 row of mole- cules, which plane would intercept the three edges o a, o by o c> together with the three perpendiculars drawn on the diagonals. The ratios of those perpen- diculars to the edges may therefore be assumed as the units of comparison for simple and mixed decre- ments on the angles. Thus for decrements on the p angle a o h, whose symbol is A, our unit will be o d : o c. For those on the angle a o c, whose symbol is P A, the unit is of : o b. And for those on the angle b o c, whose symbol is A p , the unit is o e : o a. To find a constant ratio between these perpen- diculars and the primary edges, let us suppose m, n, and o known, and from these quantities, the plane angles o a b being also known and called A v o a c A t o b c A 3 we have o a : o d : : R : sin . A l o a : o c :: R : tang. AI .-. o d : o c :: sin.A l : tang. A z We also find of : o b :: sin. A z : tang. A l o e : o a :: sin. A 3 : tang. (90 A^) The units for intermediary decrements are the ratios of ?w, w, and o, and the methods for deter- mining the several laws of decrement will be similar to some of those already employed. LAWS OF DECREMENT. 343 TJTE RIGHT RHOMBIC PRISM. Its elements. The inclination of terminal to lateral planes r= 90'. Inclination of lateral planes to each other varies in different minerals; call the greater angle at which they meet /, . Terminal plane angles being equal to the angles of the prism, are consequently /,, and (180 I t ). But being plane angles, we shall designate the greater by A t , and the lesser by A z . Half the greater diagonal of the terminal plane : half the lesser diagonal :: tang. \ A l : R. Lateral plane angles 90. Terminal edges equal. Lateral edges equal. Ratio of a lateral to a terminal edge is deter- minable from secondary planes only. Its units of comparison. Fig. 348. The unit of comparison for determining the dif- ferent laws of decrement by which this class of pri- mary forms may be affected, will be different for nearly all the different classes of modifications. 344 APPENDIX CALCULATION OF THE RIGHT RHOMBIC PRISM. Let the ratio of the terminal to the lateral edge o be known and expressed by the ratio of m : n or of \: n -. m The general expression of the unit of comparison will become s * T - in reference to simple or mixed decre- n R ments on the terminal angle A of the pri- mary form. This is evidently derived from the ratio " A > , it ef:fl:: 1 : !L m hence :: m cos. \A^ : n R. m sin. T^I^ j n reference to similar decrements on n R the angle E of the primary form. For let R e f : e n :: R : sin. \ ^ and ef: em :: 1 : m therefore sin. \A l e n e m i Z R m : n R. The ratios of the edges may be used as the units for determining the laws of decrement producing the planes belonging to the classes b and d of the modi- fications of this form. LAWS OF DECREMENT. 545 RIGHT RHOMBIC PRISM. Fig. 349. The unit of comparison for decrements on the terminal edges, is the ratio of the line fg, fig. 349, toy/; fg being perpendicular to i k, and conse- quently to a b. For the law of decrement is to be determined from the inclination of the plane abed, to one of the primary planes, let us suppose to JP, which inclination would be measured on the lines P* fi fi f[ t & * It is evident that the line fg falls within the de- scription already given of the lines from which the unit of comparison may be derived ; for, whatever the law of decrement may be which produces the new plane abed, we must have fh :Jb :: fg : fk. The ratio offg to f I may be thus discovered. We are supposed to have found fk : fl :: m : n .: 1 : - and knowing the angle efk, which we have called A,, we known the angle gf k =^4 l 90. hence fk:fg::R: cos. (^,90) 1 cos. (^,-90). ~TT~ wherefore fg:fl :: cos. (^,-90") . n :: m cos. (A , 90 n ) : n R. 346 APPENDIX- CALCULATION OF THE RIGHT RHOMBIC PRISM. The required unit is therefore * cos. (^ t 90') n R For decrements on the lateral edges of the prism, the unit is 1. The methods of applying these several units to the determination of the laws of decrement, will be similar to some of those already described. Fig. 350. - If we require the law of decrement producing the plane d e, fig. 350, and we know the angle a d e, which is the inclination of a primary plane on the plane d e, and may be called / 3 , we shall find the ratio of the edges of the defect, bd\be :: sin. (/ 3 ^J : sin. (180 7 3 ) :: p : q, which gives the required law of decrement. LAWS OF DECREMENT. 347 THE RfGHT OBLIQUE-ANGLED PRISM. Its elements. The inclination of terminal on lateral planes nz 90* Inclination of lateral planes to each other varies in the different individuals belonging to the class. Call the greater angle at which they incline to each other, /, . Terminal plane angles are consequently I l9 and (ISO /,), but they will be called A, and A z . Lateral plane angles ~ 90. The adjacent edges unequal. Their ratio, and that of the diagonals of the terminal planes, can be known from secondary planes only. Its units of comparison. Fig. 351. When the edges of any plane of class a or &, of the modifications of this form, which intersect the plane P, are parallel to a diagonal of that plane, the modi- fying plane results from a simple or mixed decre- ment. The unit of comparison for determining the laws of simple or mixed decrements belonging to class a, 348 APPENDIX CALCULATION OF THE RIGHT OBLIQUE-ANGLED PRISM. is the ratio of b d, a perpendicular on the diagonal, to the edge bf^ which may be thus found. Let the ratios of the edges be as follows, b c : b a :: m : n :: 1 :- m b c : b f : : m ; o : : I : ba: bf:: n: o ::l :- n The angle a b c has been called A t b c m A z From the ratios of the edges and the angles A l and A z , we may find the angles b a c m a c m, call this A 3 abm ~ b m c, . . ^ 4 thus . . b a : b d :: R : sin A 3 :: 1 : Sin ' A * R therefore b d : bf:: - 2 : :: n sin.^ 3 : o R. R n The unit for determinining the laws of simple and mixed decrements belonging to class b is the ratio of e c, a perpendicular on the diagonal, to an edge c w, or bf. We have m c ~ a b : e c :: R : sin.A 4 :: 1 : ^ R and because c n zz 6 /, we must have e c : b f:: - i : :: n sin. A. : oR. R n The ratios of the primary edges may be assumed as the units for determining the decrements producing the remainder of the planes belonging to class a and b ; and they should be so adapted to each particular LAWS OF DECREMENT. 349 RIGHT OBLIQUE-ANGLED PBISM. case, as to give the particular values of the indices of each individual plane. The following are the units of comparison for de- termining the laws of decrement on the edges. Fig. 352. For those which produce the planes 0, the unit is the ratio of the perpendicular g b, fig. 352, to an edge b /; and for those which produce the planes cf, the unit is the ratio of a perpendicular b h, to an edge To find b g : bf, we have ba : bg::R :sm.A 2 : H and as . . . . b a : b f : : I : n we must have b g : bf :: n sin. A z : o R. To find b h : bf> we have R be :bf:i 1 : - m b h\ bf :: m sin. A t : o R. 350 APPENDIX - CALCULATION OF THE RIGHT OBLIQUE-ANGLED PRISM. For decrements on the lateral edges, where the symbol is P G the unit is - G" . . - m m The application of these units to the determination of the several laws of decrement, will be similar to many of the examples already given. LAWS OF DECREMENT. 351 THE OBLIQUE RHOMBIC PRISM. Its elements. Fig. 353. The inclination of P on M, or M', fig. 317, varies in different minerals ; call it /,. Inclination of M on M' also varies in different minerals, and may be called / 2 . Plane angle/* a d, fig. 351, may be called A v . Let g a e, g a c?, d a e, be the three sides of a spherical triangle, whose angles would be 90, /,, and \ I 2 . The side g a d, which is \ A i9 may be thus found, sm. \ v Plane angley a e, or d a e, may be called A 2 . This angle is the hypothenuse of the tri- angle from which we have derived the pre- ceding formula, and may therefore be thus known, cot 7 cot 7 * cos A - CObya 2 Terminal edges equal. Lateral edges equal. 11 352 APPENDIX- -CALCULATION OF THE OBLIQUE RHOMBIC PRISM. The inclination of the oblique diagonal a g, to an edge a e, being called / 3 , may be thus found, cos. I = cos " sin. i The angle g a c is the third side of the triangle already used. Ratio of a terminal to a lateral edge can be known only by means of secondary planes. Its units of comparison. Let us suppose the ratio of y : ae :: m : n :: 1 : m The unit of comparison in reference to the decre- ments producing the planes of classes a and c, is the ratio of half an oblique diagonal a h to a lateral edge a e, which may be thus found. /_._T> T A i cos. -^ A , Ji Therefore a h : a e :: m cos. \ A v : n R. Fig. 354. The unit for determining the decrements producing those planes belonging to class e, of which the repre- sentative symbol is E, is the ratio of \ a horizontal diagonal^" h to the linear drawn from the solid angle aty perpendicularly on r m ; the line r m being paral- LAWS OF DECREMENT. 353 OBLIQUE RHOMBIC PRISM. lei to the diagonal g , and touching the solid angle at m. For the line fr is evidently in a plane per- pendicular tofd, and if the modifying planes we are considering were to be produced, they would cut fm and/r proportionally. The ratio of fh tofr, is thus found. fff fh R : sin. y A , 1 ^' II ^ jfji | J o J R fg-.fm 1 : *. m Therefore fh-.fm m sin. \ A i \ m sin. \A V nR. nR But . . fnfm cos. (/ 3 ~-90 cos. (/ 3 90 r ):R foil Therefore fh : fr : R m sin. f A v cos (/ 3 90") R m sin. | ^ t : cos. (/ 3 90"). Fig. 355. The unit for determining the laws of decrement producing the planes of classes /and g is the ratio of a b to a e, fig. 355, when the angle fa d is obtuse, or 2 Y 354 APPENDIX- IALCULATION OF THE OBLIQUE RHOMBIC PRISM. of a b' to a e, when that angle is acute ; a #, or a b 1 being perpendicular on the edge g d, and a e perpen- dicular on the edge e c. This ratio may be thus deduced. i 90") R ad: a e I n m ab: a e m cos. (A ,90) : nR a c : a e cc s. (A 90) : R a b : (i c m cos. (A t 90) . cos. (A 2 90) nR R m cos. (A 90) : wcos. (A, 90). The unit is equal to 1, in relation to decrements producing the planes i and /. The units for determining the several laws of de- crement producing the planes of classes b and d, and such of class e as are not represented by the symbol E, are the ratios of the primary edges ; and the methods of determining the laws of decrement producing any modification of this form, will be analogous to some of those already described. LAWS OF DECREMENT. 355 THE DOUBLY OBLIQUE PRISM. Its elements. The inclination of the primary planes is unequal at an?/ three adjacent edges, and is different in different minerals. Three adjacent plane angles unequal. Three adjacent edges unequal, and the ratios of these inequalities are to be deduced only from some secondary planes. The laws of decrement which produce the modify- ing planes of this class of primary forms, may be determined by the general methods already described at p. 295, and the units of comparison will then be the ratios of the primary edges. 356 APPENDIX CALCULATION OF THE THE HEXAGONAL PRISM. Its elements. The inclination of the adjacent lateral planes to each other =: 120. Inclination of the terminal on the lateral planes = 90. Plane angles of the summit = 120. Lateral plane angles z= 90. Ratio of the terminal to the lateral edge of each particular prism can be deduced only by means of some secondary plane. Let us suppose it known, and expressed by m : n~ or by 1 : m Its units of comparison. For decrements on the angles of the terminal plane, the unit is ; and for decrements on the terminal 2 n edges it is ^ 8 "L60, as will be ghewn below . wR For decrements on the lateral edges it is = 1. Fig. 356. The diagonals drawn on the terminal plane divide that plane into six equilateral triangles. The law of any simple or mixed decrement on an angle of the prism is deduced from the ratio of LAWS OF DECREMENT. 357 HEXAGONAL PRISM. ef\ eg, but efis half an edge of either of the equi- lateral triangles b i c or h i c, whence ef : e g : : \ m : n : : m : 2 n y which thus becomes the unit of comparison for simple or mixed decrements on the angles of this prism. The laws of intermediary decrements may be deter- mined by means of spherical triangles adapted in the manner already described. Decrements on the terminal edges. A decrement by I row on the edge b e, fig. 356, would intercept proportional parts of the edges b d, b c, and consequently if the whole of b d were inter- cepted by the new plane, the whole of b c, e g', and e //, would be intercepted also, and d h would be the edge of the new plane d h c g. And we observe that the entire of the line b , which is perpendicular to d h, would also be intercepted by the same plane. The ratio of b a : b c may therefore be taken as the unit of comparison for determining the laws of decre- ment on the terminal edges of the hexagonal prism. But b a is perpendicular to d i 9 the base of the equilateral triangle d i b ; whence d b : b a :: R : sin. 60 :: 1 : Sin * 60 R But . . db -.be :: 1 : - m Therefore b a : b c :: m sin. 60 : n R. The law of decrement on the lateral edges of the c5 prism, will be represented by the units contained in the ratio of the edges of the defect occasioned by such decrement. 358 APPENDIX CALCULATION OF THE HEXAGONAL PRISM. The individual or particular prisms belonging to the seven preceding classes are, as we have seen, distinguishable from each other by the comparative lengths of two or three of their adjacent edges, or by the particular values of some of their plane angles. These plane angles may be determined by means of spherical trigonometry from the inclination of the primary planes to each other; and this inclination may be ascertained by measurement with the gonio- meter. But the comparative lengths of the edges can be deduced in no other manner than from some second- ary plane, which, for that purpose, must be supposed to have been produced by a given law of decrement. If for example we assume that, any known second- ary plane has been produced by a decrement by 1 row of molecules, the ratio of the edges of the defect of the primary form would, as we have already seen, be equal to the ratio of those edges of the primary form of which they are respectively portions. If therefore we determine the ratio of the edges of the defect occasioned by the interference of the secondary plane, which we suppose to have been pro- duced by a decrement by 1 row of molecules, we shall, if our supposition be correct, evidently obtain the ratio of the corresponding primary edges. But it may happen that the plane which we have supposed to result from a decrement by 1 row of molecules, is really produced by some other law of decrement. The only method we possess of discovering whether we have determined the true dimensions of the prism, is to use those dimensions for ascertaining the laws of decrement producing other secondary planes ; and LAWS OF DECREMENT. 359 HEXAGONAL PRISM. if, when so applied, we find that the laws of decre- ment producing those planes are simple, we may conclude that we have determined the dimensions of our prism rightly ; but if the ratios we have assigned to the primary edges, suppose those planes produced by irregular and extraordinary laws of decrement, we may conclude that the plane which we have first observed, has resulted from some other law than by 1 row of molecules, and we must proceed to assign new dimensions to the prism, in conformity with the new law, by which we now suppose the plane first observed to have been produced. 360 APPENDIX CALCULATION OF THE THE RHOMBOID. In this fig. df is the oblique diagonal, a b . . . horizontal diagonal, d i ... axis. If we imagine a b, a c, b c. to be edges of a plane passing through the solid, that plane would be per- pendicular to the axis, and the line en e drawn upon it,, would consequently be perpendicular to the axis. But the point e at which this perpendicular touches the plane d afb, is the middle of the diagonal d f. If therefore we draw/o parallel to e n, we shall have o n HZ H d. ButjTo zz c w, and/ i d c ; therefore i o n d ~ on. Hence perpendiculars upon the axis of a rhomboid, drawn from two adjacent lateral solid angles, divide the axis into three equal parts. Its elements. The inclination of the adjacent planes at the su- perior edges will be different in different minerals, but may be designated generally by LAWS OF DECREMENT. 361 RHOMBOID. This angle is supposed generally to be measurable by the goniometer; but it may sometimes require to be deduced from the inclination of secondary planes to each other or to the primary. If two planes modifying the edge itself be used to determine the angle / we must know the inclination of those planes to each other, and that of one of them on the adjacent primary plane ; and hence the inclination of the primary planes is known. If we know the inclination of the plane a of the modification of the rhomboid, to the plane P, and call it /, we may determine /, from the following equation, , j j . _ cos. 30. sin. (ISO /) " ~ll~ It is apparent from the above fig. that if (180 /) and | I, be taken as two angles of a spherical triangle, the third angle must be 90, and the side subtending the angle | /, must be 30. and hence the given equa- tion is derived. If we know the inclination to the plane P, of that plane of mod. e which replaces the inferior angle of the plane P, we may still deduce /, from that inclination, by the preceding formula. For plane e on P is obviously 270 /, and consequently 1 270 (e on P). The inclination of the adjacent planes at the in- ferior edges will consequently be (180 / t ) and may be called J 2 , APPENDIX CALCULATION OF THE RHOMBOID. The superior plane angles may be called^,, and may be thus found, The angle f A l is one of the sides of the spherical triangle marked on fig. 357, whose angles are evidently 90, 60, and | /,. The general formula to determine \ A l is cos. i A. = "I^OL sin. * /, O but as cos. 60 ~ , the formula becomes 2 sin. |/ t Lateral plane angles will be (180 n A,) and may be called A a . Inclination of the superior edge to the axis may be called A^ and may be thus found, The angle b dn, fig. 357, is the inclination required, and is the hypothenuse of the tri- angle already used, whence nn. * - C0t - CQ1 C0t - J J . R Inclination of the oblique diagonal to the axis may be denoted by A 4 , and may be thus fo md, The angle e d n which we call A 4 , is the third side of the triangle marked on the figure, and may consequently be determined from the formula, cos. A . ' CQS> T- - sin. 60 Sum of the two preceding angles, being the inclination of an oblique diagonal to a superior edge, when measured over the summit, may be denoted bv A f . LAWS OF DECREMENT. 363 RHOMBOID. The inclination of an oblique diagonal to an adja- cent edge, measured over the inferior angle, would be 180 A 5 , and may be denoted by A* Ratio of a perpendicular upon the axis drawn from a lateral solid angle, to the axis itself, is as - tan.?. 4 3 : R. We have already seen that d n is y of the axis d , and d n : n b :: R : tang. A 3 Therefore d i 3dn : a b :: 3R : tang. A s Ratio off the oblique diagonal, to f the hori- zontal diagonal, as R : tang. \ A^. Ratio of y of the axis : f half the oblique dia- gonal :: cos. A 4 : R. Ratio of -3- of the axis : f the horizontal dia- gonal :: cot. A^ : tang. 60. For d n : n e :: tang. (90 1 Aj : R :: cot.^ 4 : R and e b : n e :: tang. 60 : R Therefore dn : e b :: cot. A^ : tang. 60'. Ratio of \ an oblique diagonal : an edge :: cos. \A, : R. Ratio of \ a horizontal diagonal : an edge :: sin. A : R. 2 2 APPENDIX CALCULATION OF THE RHOMBOID. Its units of comparison. Fig. 358. d The following are the units of comparison in rela- tion to the decrements producing the several classes of modifications contained in the tables. Class b. c, e, g, A*, unit cos ' J -, being the ratio K, of | an oblique diagonal : an edge. Class h* the unit is stn> T ^ ', when the rhomboid sin. A 5 is acute. sin. \ A l and , when it is obtuse. LAWS OF DECREMENT. RHOMBOID. Fig. 359. This unit is the ratio of e b : b n, fig. 398 and 399, b n being perpendicular on g w, which is parallel to df. Fig. 398 represents an acute rhomboid, in which e b : b f, or b g :: sin. \ A, : R bn : bg :: cos. (90 A 5 ) : R :: m\.A & : II therefore e b : b n :: sin \ A, : sin. A 5 . Fig. 399 represents an obtuse rhomboid, in which e b : b g :: sin. \A^ : II b n : b g :: cos. (A' 5 00) : R and . . . e b : b n :: sin. \A V \ cos. (A 5 90). If in fig. 358 we suppose cp a portion of the oblique diagonal produced, and in both 398 and 399, d p parallel to b w, the assumed value of b n will be readily perceived. The unit of comparison, in relation to decrements producing the classes e^/J z, and /, is the ratio of the edges and is consequently zn 1. In relation to class w, it is that of two equal per- pendiculars / , / c, on a superior edge, drawn from two lateral angles; and in relation to class p, it is that of two equal perpendiculars h a, h A, on an in- ferior edge, drawn from the parallel superior edges, and it is consequently in both cases ~ 1. The determination of the laws of decrement affect- ing the rhomboid, and the developement of the 366 APPENDIX CALCULATION OF THE RHOMBOID. various relations between that primary form and its numerous secondary forms, occupy a considerable portion of the Abbe H ally's crystallographical re- searches. Some, however, of the relations he has demonstrated, though curious in themselves, are not immediately useful to the mineralogist for deter- mining the mineral species to which a given crystal belongs. Little more will be attempted here than to give an outline of a method of determining the laws of decre- ment, similar to that which has been applied to the other classes of primary forms; and it is hoped that this will supply the mineralogist with as much assist- ance as his purposes will generally require. Simple and mixed decrements on the superior and inferior angles. The planes produced by these are contained in the classes b, c, e g, and k. Let us be supposed to have measured the incli- nation of one of the planes b on an adjacent primary plane, or of one of the planes c, over the summit on a primary plane, and if we call the measured angle / 3 , the ratio of the portions of the oblique diagonal and edge contained in the defect, would be as sin. (I 3 A & ) : sin. (180 / 3 ), and if we divide this ratio by c __fLj L, we shall R obtain the law of decrement producing the plane we have measured. Let us suppose the inclination of one of the planes belonging to the classes e, g-, or k 9 on that primary plane which i&. intersected by the modifying plane LAWS OF DECREMENT. 367 RHOMBOID. parallel to its diagonal, to be called / 4 , the ratio of the edges of the defect will be as sin. (/ 4 A & ) : *in. (ISO 1 7 4 ). This being divided as before by cos. \A l : R, will give the law of decrement producing the plane we have measured. Simple and mixed decrements on the lateral angles. Let us suppose the inclination known of the pri- mary plane P, to one of the adjacent planes of mod. A, and let this be called 1 5 . The angle measured, would be in the plane e b w, fig. 358. And as e b is perpendicular to b n, the ratio of the edges of the defect would be as radius to tang, of the supplement of the measured angle; and this beii g divided by cos. \Ai : R, will give the required law of decrement. Intermediary decrements on the terminal solid angles. The general symbol representing a single plane of mod. d is (B P B' q B"r). The values of the indices /?, q, and r may be dis- covered from the inclination of the particular plane represented by that symbol, on the two adjacent pri- mary planes, by means of a spherical triangle, and the plane triangles, adapted in the manner already described. Intermediary decrements on the lateral solid angles* These, as we have already seen in our account of the symbols representing the planes produced by them, may be referred to the angle at O, or to that at E. Let the plane from which we are about to 368 APPENDIX CALCULATION OF THE RHOMBOIT*. deduce our required law of decrement be one of those which appears on the angle at O. There are, as may be seen in the tables, three classes of modifications produced by these decre- ments, being classes f, z, and I. The distinction between these three classes has been already pointed out in the table of modifications of the rhomboid, and in p. 279, where their several relations to the theory of decrements are given. The general symbol representing one of the planes at the angle O, belonging to either of these modifi- cations, is (D' q b"r Dp), and the particular value of the indices may be discovered, in relation to any particular modification belonging to either class, by measuring the inclination to the two primary planes adjacent to the edge b", of the plane represented by the above symbol ; and finding the plane angles of the defect adjacent to the edge b", by means of a spherical triangle, and thence deducing the ratio of the edges of the defect in the manner already de- scribed. The indices of the individual modifications belong- ing to classy will be found in a constant ratio to each other. This results from the condition that the edge at which the modifying planes "intersect each other shall be parallel to the axis of the rhomboid. Let the index p be > 7, and q ^> r. The relations between p, q^ and r, may be thus stated, p = q (7-1) = q r. a = 1+ V *H-1 _ P 2 ' r' LAWS OF DECREMENT. 369 RHOMBOID. And the manner of deducing these may be given as an example of one of the methods* of analysis appli- cable to these investigations. The general equation of a plane in relation to three co-ordinate axes, is known to be A#+By-f C * + D = o. Let the distances from the origin at which the plane cuts the three axes, be represented by p, q, and r. We shall then have the following equations of the points where the axes are cut by the plane. For the point on the axis x we have x nz j?, y y To find the values of the co-efficients A B C D, in function of the quantities p, q, and r, with a view to substitute those quantities in the general equation for the co-efficients A, B, C, and D, Let y o, z zz o, and x x zz o, s = o, . . y A --D IT D C Therefore ~ p, whence A =z A p ~B~ ' q ' ~Y~ D c _ D C r * This method of determining the relations that may exist among crystals, has been used in a paper published by Mr. Levy in the Edinburgh Philosophical Journal, relative to another object which will be referred to in a later part of this Appendix. 3 A 370 APPENDIX CALCULATION OF THE RHOMBOID. The general equation may therefore be thus ex- pressed, P <1 or, dividing all the terms by D, it may be reduced to this general form, Fig. 360. Let the plane a d e, fig. 360, represent one of the planes belonging to class jfo whose indices are p, q, and r ; and let o e ~ p. o d = , o a =. r; the equation of this plane will then be, But as the line 0/at which the two planes of mod./ intersect each other, is parallel to the axis of the rhomboid, and passes through one of its superior edges, it might obviously be on the surface of some LAWS OF DECRE3JEM. 571 RHOMBOID. plane belonging to mod. e. The equation of this plane may be thus expressed, + y. + 1 = i. p' ^ pf r From the character of the planes of mod. e, the index p' is always =z 2 r. Knowing the relation of p' tor, we may discover the relations of/? and q to r, by finding their relations to p' ; and these relations may be known from the equations of the traces of the two planes on the plane of the x y, when referred to the points/I The equations of these traces are obtained by making s = o in the two preceding equations, whence the equation of the trace d e, is -\- &- 1 (1) P 9 and of b c ....... + = 1 (2) pf pf But as both traces pass through the point /^ tfce values ot\r and of^y must be equal in both equations. Hence from equation (1), y q ^ ' r P (2), y = pf jr Therefore . . q *LE = j/ x P x = 1 j ? 3 A ^? 372 APPENDIX CALCULATION OF THE RHOMBOID. In the like manner we may find 1 But at the point/, x y, therefore . . . f^^J. = P f ~ P 1-2. i -P p 1 whence . . . ' = ^ 1 p+q q = 2pp' As we know that p' 2 r, and that r cannot be less than 1, p' cannot be less than 2 ; and it must, in relation to any particular plane of mod. e, be either 2, 4, 6, 8, or some greater even number, according to the number of molecules supposed to be contained in the defect occasioned by that plane. It may be easily seen that when p' z= 2, we must have 2 But as the indices of planes produced by inter- mediary decrements must be whole numbers, it fol- lows that the planes a b c, and a d c, cannot both pass through the point f, unless p' be greater than 2. Let p' 4. and 4 = whence z= * ? -2 If we regard the figure 360, we may perceive that if o b, which we have called/?', be considered equal LAWS OF DECREMENT. 373 RHOMBOID. to 4, the line o d, which we have called g, must be greater than 2; for as the line b c is equally divided at the pointj^ if d o z= d />, the line d f would be parallel to o c. Therefore when p' zz: 4, we must have q zz: 3, and consequently p zn 6. And as r = . , -if we suppose the value of q to be successively increased to 4, 5, u, &c. we shall have the following series of indices to represent the series of planes of class /. = 6 q = 3 r 2 . 12 . 4 . . 3 . 20 . 5 . . . 4 . 30 . 6 . . 5 . 42 . 7 . . 6 &c. From what has been stated in the preceding pages, it will be readily perceived, that when, in addition to the inclination of the primary planes to each other, we know the unit of comparison, and the inclination of the secondary plane to the primary plane along which the decrement is conceived to proceed, we may immediately determine the law of decrement. For we can from these data directly deduce the ratio of the lines of the defect corresponding with those from whence we derive our unit; and if we divide this ratio by our assumed unit, we obtain, as we have before observed, the law of decrement producing the plane we have measured. 374 APPENDIX CALCULATION OF THE The preceding sketch of the methods of discovering the laws of decrement, will, it is hoped, be generally found sufficient for that purpose, wK" the angles at which the secondary planes incline on the primary are known, and where the ratios of the edges or other lines already described are also known. But it very frequently happens that the whole of the primary planes are obliterated by such an ex- tension of the secondary planes, as.produces an entire secondary crystal. In these cases we must recur to cleavage for determining the relative positions of the primal^ and secondary planes, and for measuring the angle at which they meet. The cleavage planes which we may adopt as the primary set, if more than one set be discoverable, should be those which are most compatible with the observed secondary forms. Having thus given an outline of the solution of our first problem, by shewing how the laws of decrement may be determined from certain data, we shall pro- ceed to examine the second, and to ascertain how the angles may be determined at which the secondary planes incline on the primary, the elements of the pri- mary form., and the law of decrement, being known. The methods used for determining these angles, will be nearly similar to those already described for determining the laws of decrement. The plane triangles which have been used for determining- the laws of decrement, have been both right-angled and oblique. Where a law of decrement is expressed by means of the ratio of the sides of a right-angled triangle, the angles are readily found by reducing the ratio to that of radius, and tangent of the required angle. LAWS OF DECREMENT, 375 Fig. 361. Let fig. 361 represent a section of any crystal whose planes af, a g', are perpendicular to each other, and let the lines b c, b d, b e^ be sections of planes modify- ing the edge or angle g a f. Thus let us suppose that we have the law of decre- ment given by which the plane b c has been produced ; and let the required angle g c b be called /. Let the ascertained ratio of the edges a g, a f, be as 5 : 4, and the law of decrement producing the plane b c, be 1 row of molecules. It follows that a c : a b :: 5 : 4 :: 1 : -. 5 But we also have a c : a b :: R : tang. (180 /) therefore - = '8 = tang. (ISO /) 5 and &, in the table of natural tangents, is the tang, of angle 38 40' nearly, = ISO /; therefore / ISO (38 40') = 141 20'. If we suppose b d the section of a plane resulting from a decrement by 2 rows in breadth, we should obviously have adiab :: 10:4 :: 5 : 2 :: 1 : -. 5 And if we call the angle g d b, /', we must have a d : a b :: R. tang. (180 /') 376 APPENDIX CALCULATION OF THE whence - = 4 = tan (ISO /') = 21 48' nearly, 5 hence /' = 180 (2 i 48') = 158 12'. By this method we may in all similar cases deter- mine any required angle, whatever may be the ratio ofag: af. Where the planes a c, a b, are not at right angles to each other, the angle c a 6, may be either acute or obtuse. In either case knowing the angle c a 6, and the particular values of a c and a b, deduced from the known ratio of m : n, and from the given law of decrement, we may obtain the angle a b c from the formula i , d'. tanff. \ s tang. \d -f~ where d =z difference of required angles, d' z= difference of given sides, s zz sum of required angles zz: 180 given angle. s 1 zz: sum of given sides. Where spherical triangles have been used for de- termining the law of decrement, they may also be used for determining the angles of the secondary planes with the primary, the law of decrement being known ; with this difference however, that where in the former examples we have sought the sides of those triangles, knowing the angles, we have now to determine the angles from the given sides : and the sides are known from the plane angles of the primary crystal, and from the ratio of the edges of the defect of the primary form, as deduced from the ratios of the corresponding primary edges, and the law of de- crement. LAWS OF DECREMENT. 377 The preceding parts of this section suppose the angles known at which the secondary plane whose law of decrement is required, inclines on one or more of the primary planes. But it may sometimes occur that the inclination of the secondary on the primary planes cannot be directly obtained. In cer- tain cases, where this happens, the laws of decrement may be deduced from the inclination of the secondary planes to each other. We shall suppose in the following examples of one or two particular and simple cases of this nature, that the unit of comparison is expressed by , and the n ratio of the edges or other lines of the defect by P m . Whence will express the law of decre- qn q ment by p molecules in breadth and q molecules in height. It has been already stated that where an edge is replaced by two similar planes, m will always be found to equal n^ and the fraction P y or its equi- q n valent , when reduced to its lowest terms in whole ? numbers, will express the ratio of the edges of the defect. 1. Let us suppose the edge of a cube replaced by 2 similar planes as in mod. f, or the lateral edge of a right square prism, as in mod. e. And let the inclination of the secondary planes to each other be called /. We shall find p_ _,.. R q ~ ~ tang. (| I&5) 378 APPENDIX - CALCULATION OF THE and the inclination of either of the new planes on the adjacent primary plane would be = 225 f /. 2. Let the lateral edges of the right or oblique rhombic prism, or any edge of the rhomboid be modified by 2 similar planes, and let the inclination of the primary planes to each other zr /, and that of the secondary planes to each other 1 '. Then = sn - - q sin. (I /'-i /) And the inclination of either of the new planes on the adjacent primary planes would be \r. The following application to a particular case of the proposed methods of calculation, will probably be sufficient to illustrate their general use. It will be found convenient, if we have to deter- mine the laws of decrement producing secondary planes upon any primary form, to determine in the first place the particular values of such of the elements of that form as we may require for ascer- taining those laws of decrement; and the values so determined may be reserved for any future occa- sion. Let it be required to determine the elements of the rhomboid of carbonate of lime : LAWS OF DECREMENT. 379 Here the angle - /, = 105 5' 7 2 = 74 55' A. 101 55'. We have seen that cos. * A =. 2 sin. i I I By means of the tables of logarithms we find the angle A l , thus, * ......... = 20* - log. 2 ..... = 0-3010300 log. sin. /, = log. sin. 52 32' 30" = 9-8997088 -- 10*2007388 Therefore . . . log. cos. f^, 9-7992612 Therefore f A t = 50' 57' 30", and consequently A? 101 55'. A 2 =78 5' A 3 =63 44' 45", which may be found thus from the formula. . cos. A = cot 60 cot - * 7 * R = cot - 60 cot. 52 32' 30' ; log. cot. 60 . . . = 9.7614394 log. cot. 52 32' 30' ; = 9.8843264 19.6457658 log. R .... 10. log. cos. A 3 . . . = 9.6457658 Therefore angle A 9 = 63 44' 45 /; . 380 APPENDIX CALCULATION OF THE A 4 rr 45 23' 26", which may be thus found. cos A = R ' cos ' 7> R ' cos> 5r 32/ 4 ~ sin. 60 sin. 60 log. R . . . . = 10- log. cos. 5V 32' 30" = 9-7840353 19-7840353 log. sin. 60" . . = 9-9375306 s . cos.^ 4 . . = 9-8465047 Therefore angle A 4 = 45 23' 26". A 5 109 8' 11". A 6 =i 70 51' 49". The following ratios may be known from the tables of natural sines, &c. when we require the nearest whole numbers by which they may be represented; or their logarithms may be taken from the tables, when we use logarithms only in our calculations. Perpendicular on axis drawn from lateral solid angle : axis :: tan.^ 3 : R :: i tang. 63 44' 45" : R :: i 2*0274279 : 1 :: -6758093 : 1. This may be reduced with sufficient accuracy to its lowest equivalent terms in whole numbers, by means of a common sliding rule, and will be found as 23 : 34 very nearly. Or if the logarithms be required, we have log. tang. 63 44 ; 45" = 10-3069454 log. 3 ...-. 0-4771213 9-2298241 log. R =10 0-0701759 LAWS OF DECREMENT. 381 \ oblique diagonal : \ horizontal diagonal :: R : tang. \A, :; R : tang. 50 51' 30" :: 1 : 1-2330626 :: 17 : 21 very nearly. or log. R . . . . = 10- log. tang-. 50-57' 30" =: 10-0909851 0-0909851 3- axis : \ oblique diagonal :: cos. A 3 : R :: cos. 63 44' 45" : R :: '4423539 : 1 :: 31 : 70 very nearly. or log. cos. 63 44' 45" = 9-6457698 log. R .... =10- - ""0-3542302 $ axis : \ horizontal diagonal :: cos. \ A 3 : tang. \A , :: cos. 31 52' 22" : tang. 50 37' 30" :: -849227 : 1*2330626 :: 11 : 16 nearly, or log. cos. 31 52' 22" = 9-9290216 log. tang. 50 57' 30" == 10-0909851 0-1619635 \ oblique diagonal : edge :: cos. \ A t : R :: cos. 50 57' 30" : R :: '6298254 : 1 :: 12 : 19 nearly, or log. cos. 50" 57' 30" = 9-7992615 log. R . . . . rz 10- 02007385 \ horizontal diagonal : edge :: sin. \ A^ : R :: sin. 50 57' 30" : R :: -7766881 : 1 :: 7 : 9 nearly, or log. sin. 50 57' 30" = 9-8902466 log.R .... = 10- 0-1097534 382 APPENDIX CALCULATION OF THE i horizontal diagonal : perpendicular b n (fig. 358 & 9) :: sin. \ A, : cos. (^ 5 90) :: sin. 50 57' 30" : cos. 19 8' 11" :: '7766881 : -9447409 :: 46 : 56 nearly. or log. sin. 50 57' 30" = 9-8902466 log. cos. 19 8' 11" = 9-9753128 0-0850662 Having thus determined the elements of the rhomboid of carbonate of lime, which we may re- mark are all deduced from the single angle /,, we may proceed to determine the laws of decrement producing any of its observed secondary planes. Let us now suppose that we have measured the inclination to the plane P, of a plane belonging to class b of the rhomboid, and that we have found it 143 28'. We have already seen in p. 366, that the ratio of those lines of the defect occasioned by the planes , from which the law of decrement is to be deduced, may be expressed by the fraction sin. (I a A.) sin. ( 180 / 3 ) In relation to the plane we have measured we find / 3 = 143 28' and we have found . A & = 109 8' 11" therefore I 3 A 5 . . = 34 19' 49" and 180 7 3 is evidently zn 36 32'. The ratio of those lines of the defect from which the law of decrement may be deduced, is therefore in this particular case sin. 34" 19' 49" sin. 36 32 ; LAWS OF DECREMENT. 383 If we recur to the tables of natural sines, we shall find the numbers constituting this ratio to be nearly 1 , which fraction being reduced to its lowest 18 terms will be This, as we have already shewn, is to be divided by the unit of comparison ; which in this instance is the ratio of \ the oblique diagonal to an edge, and 12 has been found equal to _. X. \J But to divide by , we must invert the terms of the latter fraction, and then multiply the first by it. 18 19 18 3 , . , . c rience x zz zz , which gives a law or decrement by 3 rows in breadth and 2 in height pro- ceeding along the plane P. If however instead of using the natural sines, &c. we use only logarithms, the law of decrement may be thus determined. log. sin. 34 19' 49" = 9-7512503 log. sin. 36 32' zz 9-7747288 - 0-0234783 To divide this by the unit of comparison, we must subtract the logarithm of that unit, which is given in p. 381, from the above logarithm of the ratio of the edges of the defect; this may be done' by the ordi- nary method of subtracting algebraic quantities, by changing the sign of the quantity to be subtracted and then adding ; . 384 APPENDIX CALCULATION OF THE hence if to 0-0234783 we add . . 0-2007385 we have -f 0' 1772602 as the logarithm of the natural number which is to determine the law of decrement. But the natural number corresponding to 0*1772602 is 1*504, and which, if we disregard the last figure, 4, may obviously be expressed by the fraction 15 3 - ~ , a result similar to that which has been already found by means of the natural sines. Fig. 362. Let us next require the law of an intermediary decrement producing a particular modification of the rhomboid belonging to class d. And let us suppose that we have measured the inclination to Pand P', of that plane with the letter d upon it, which rests on P', see fig* p. 204. Let d on P be found 132 13' d.. P' . . . . = 145" 57' Hence 180 132 13' = 47 47', which we may call / 6 and 180 145 57 34 3' 7 7 /, it will be recollected is 105*5'. LAWS OF DECREMENT. 385 We first require the plane angles e f d, dfg> fig. 362, which may be thus found. sn. R -J/-COS. I (/ I +/ ? + f y ) COS. I (J t +/ 6 _/ si. in sin. J t sin. I R I" cos> $ (/i+-*7+*6) cos. | (/.+/, / 6 ) sin. /! sin. 7 7 But /, = 105 5' 7 6 = 47 47' 152 52'+ J 7 , 34 3' z=186 55', i of which is 93 27' 30 and ... 152 52' 34 3'= 11 8 49', | of which is 59 24' 30 Again /, r= 105 5 ; I 7 = 3V 3' 139 8'+/ 6 , 47 3 47'i=186 55' \ of which is 93 27' 30" and .... 139 8' 47 47'= 91 2P,iof which is 45 40' 30". The preceding general formulae therefore become, sin. I V e fd R 4 /cos. 93 27 7 30" cos. 59' 24 7 30 77 ~" sin. 105 5' sin. 47 47' sin. J v <*fg = R 4/COS. 93 27' 30" cos. 45 40' 30^ sin 105 5' sin. 34 3' These equations may be resolved by the assistance i in 3c of the table of logarithms in the following manner. 386 APPENDIX CALCULATION OF THE log-. cos. 93 27'3G"zzl. cos. 86 32' 30" 8-7804792 log. cos. 59 24' 30" = 9-7066463 18-4871255 log. sin. 105 5'=1. sin.74 55'zz9- 9847740 log. sin. 47 47' .... =9-8695891 19.8543631 To extract the square root of this quotient, 1.3672376 divide it by 2, and f is 0-6836188 log. R 10- log. sin. |V e f d = 9-3163812 therefore \ y efd = 11 57' 28", and consequently V efd = 23 54' 56". log. cos. 93 27' 30"r=l. cos. 86 32' 30" = 8.7804792 log. cos. 45 40' 30" = 9-8443079 18-6247871 log. sin. 105 5 / =].sin.74 55'z=:9-9847740 log. sin. 34 3' .... =9-7481230 - 19-7328970 divide by 2 . . MOS1099 and J is ... 0-5540549 log. R ; =10- log. sin. | V d fS = 9-4459451 Therefore J V d fg = 16 12/ 49/ ', and consequently V dfg 32 25' 38". Now as we know the angle e df fdg = 101 55' we therefore know the angle d ef 45 10' 4" and dgf i=45 39' 22" and hence dfide :: sin. 54 10' 4" : sin. 23 54' 56" :: 8107 : 4054 :: 4 : 2 !LAWS OF DECREMENT. 387 and . . df: d g :: sin.45 39' 22" : sin. 32 25' 38" .: 7152 : 5360 :: 4 : 3 If we again dispense with the use of natural sines, we may still derive the same result by means of the logarithms of those sines. For log. sin. 54 10' 4" = 9-9088794 log. sin. 23 54' 56" = 9-6078695 0-3010099 The natural number corresponding to this result- ing log. is 2 very nearly, which may be represented 2 4 by the fraction zz And log. sin. 45 39' 22" = 9-8544077 log. sin. 32 25' SS" =: 9*7293493 0-1250584 The natural number corresponding to this resulting log. is 1-3337 =. ]l 3337 = i, which gives the same law of decrement as that already found. The general symbol to represent the plane e f g, would be (B'r B P B" q ), and its particular symbol will be found by substituting in this general symbol, for the letters p, q, and r, the particular values of the indices as we have just found them. We shall then have the symbol (B'2 B4 B"3), which repre- sents this particular plane, and signifies that the compound molecule abstracted in the production of this plane belongs to a treble plate, or is 3 molecules in height, 2 in the direction of the edge B', and 4 in the direction of the edge B. 3 c 2 388 APPENDIX - CALCULATION OF THE Let us now require the inclination to the primary planes, of the planes whose law of decrement we havejust determined. And first of the plane e i c, fig. 362, whose symbol is A. 4 The inclination of this plane to the primary is equal to the angle a b c, fig. 362; to obtain which we must first know the angle d b c. The law of decrement being 3 molecules in breadth and 2 in height, and the decrement in breadth being measured by f an oblique diagonal and an edge, it follows that the ratio of the lines of the defect may be thus expressed, db : dc :: 3 half oblique diagonals : Sedges. But we have before seen that \ oblique diagonal : edge :: 12 : 19 we have therefore db : dc :: 3x12 2x19 - 36 : 38. The sum of the sides d b, d c, of the triangle d b c is therefore 38 -j- 36 zz 74 ; and their difference is 38 36 = 2. The angle d b c which we require, is evidently the greater of the two angles d b c and d c b Now the sum of these two angles is 180 A 5 = 180 109 1 8' 11" = 70 51' 49" of which \ 35 25' 54". But to find the greater angle, we must also know their difference, which we may discover by means of the general formula given in p. 376. tang, \ d = d> ian , - H $ which formula in relation to this particular case be- comes tang, i d = LAWS OF DECREMENT, 389 From the tables of logarithms we find log. 2 . . . ; . 0-3010300 log. tang. 35 25' 54" = 9-8521719 10-1532019 log. 74 .... 1-8692317 log. tang. \ d . . 8-2839702 Therefore ^difference of the angles dbc and deb I 9 6' 6" and \ their sum being 35 25' 54" the greater angle dbc == 36 32' and v c b a is consequently 180 36* 32 / =143 28'. If we turn to p. 382, we may observe that this angle is the same we are supposed to have found by measurement, and from which we have deduced the law of decrement. We shall now deduce the inclination of the plane efg to each of the adjacent primary planes, from the known law of decrement producing it, and from the known angles'/, and A t . The symbol of this plane being as we have already seen (B'2 B4 B"3), the edges de, df, d g, of fig. 362, are as follows, de 2 df 4 dg - 3 The angle c d /, or f d g*, corresponds to A t , which has been found 101 55'. Consequently the sum of the unknown angles dfe and d ef, or d /"g and d gf, is zz 78 5'. 390 APPENDIX CALCULATION OF THE We require the plane angles dfe, and dfg, which are evidently less than d e^and d g f. 1. To find the angle dfe, from the formula tang, i d = d> tan - s we have df-\-de = 4 + 2 = 6 3 " 6 log. 2 . . . . = 0-3010300 log. tang. 39" 2' 30" = 9-9090149 102100449 loff. 6 = 0-7781513 long. tang. \d . = 9-4318936 Therefore | the difference of the unknown angles zn 15 7' 38" and consequently V4A = 39 ' 2/ SO 77 15 7 ; 38 /; =23 54' 52". 2. To find the angle dfg, we have df-\-fg 4 + 3 = 7 dffg =4 3 = 1 and } 5 as before = 39 2' 30'' therefore tang. -| rf ^ tang. 39 2' 30' 7 log. tang. 39 2' 30" = 9-9090149 log. 7 . . . . = 0-8450980 log. tang, i rf . . = 9-0639169 therefore | df = 6 36' 31" and v and//. From the middle of the edge d $, draw h b, hf, and the triangle b h f would represent the position on the primary form, of the plane e, fig. 363. The planes /' and e are thus observed to intersect each other at the points c and w, and consequently the line cw, would correspond with the common edge of the planes /, and e, if that edge were visible in fig. 363. The first parallelism observed by Mr. Monteiro was between the line of intersection of the planes o' and /. and the striae on the plane /. The plane o' on some of the crystals he examined, was so much broader than the plane o, as to exhibit this parallelism dis- tinctly. But as the striae are parallel to an edge of the pri- mary form, the edge at which the plane o 1 intersects the plane /, must be parallel to an edge of the pri- mary form, and evidently to the edge a b of fig. 64. The second parallelism observed was between those PARALLELISM OF EDGES. 397 edges of the plane o', which are produced by its in- tersection with tile planes /' and e. From this paral- lelism, as it has been already remarked, it is known that the plane o' is itself parallel to the edge at which the plane I and e meet. We have thus obtained two conditions, which en- able us to place the plane o' on the primary form. First an edge of that plane is parallel to the edge a b of fig*. 364, and secondly the plane itself is paral- lel to, and consequently may coincide with, the line c w, which represents the edge at which the planes / and c meet. If therefore we draw the line q r, parallel to a 6, and passing through the point c, and the line ojt?, parallel to q r, and passing through the point w, we shall, by joining o 7, and p r, obtain the position on the primary form of the plane o'. And the ratio of q d to d o, will evidently give the law of decrement by which the plane o 1 has been pro- duced. This ratio Mr. Monteiro says is easily deduced, but he does not point out the method of discovering it; it is however very obvious. Fig. 365. Let the plane dfs t be represented by fig. 365. Produce df, and from the point h, draw h D, parallel to I m, whence d v m If; and because the triangles vfh, Iffy are similar, nf is evidently T of hf\ and 398 APPENDIX ON THE if we draw n ,r, parallel d /*, the triangles xjn, dfh are similar, and x n = d o is of d h. But dh m dg- 3 fig. 364. The plane o' is produced therefore by a decrement, consisting of 3 rows in breadth, on the plane d b af. This demonstration, it may be remarked, is purely geometrical, and limited in its application, to this particular case. The same method might however be adapted to other cases; but the problem would frequently become extremely complicated, and diffi- cult of solution by the aid of geometry alone. Perceiving this difficulty, and the limited nature of the method itself, Mr. Levy has generalised the pro- blem by giving it an algebraical form, and has pub- lished an interesting paper on the subject in the 6th vol. of the Edinburgh Philosophical Journal, p. 227. In this paper, Mr. L. has given formulae for deter- mining the law of decrement, by which any secondary plane, modifying any parallelopiped, is produced, whenever two of the edges of that plane, not being parallel to each other, are parallel to two known edges of the crystal. The following brief abstract of Mr. Levy's paper is inserted here, for the purpose of affording the reader a more immediate reference to the formulae it supplies ; and as an additional example of a method of investigation, which may be advantageously applied to other points of crystallographical research. To derive these formulae, Mr. Levy has first sup- posed the edges of the primary form to be represented by three co-ordinate axes, and the primary planes, consequently, to correspond to the three co-ordinate planes. He has then found the equations of all the planes concerned in the solution of the problem; and by combining these equations, has obtained the PARALLELISM OF EDGES. 399 equations of the projections upon one of the co- ordinate planes, -of those intersections of the known planes, to which the edges of the new plane are respectively parallel. And from the necessary rela- tion subsisting among the co-efficients of some of the terms of these equations, the following equations are derived. Let ,, <7 r , r be the unknown indices of the new .1 5 " i 5' 5 ' plane, which we chall call plane 5. Let /?,, N DRAWING THE parallel upon our supposed transparent skreen, and not converging as they do in the above diagram.* The method of representing crystals in projection may be thus explained. Let us for a moment forget the abstract notion of the object being removed to an infinite distance from the eye, and let us imagine it distinctly within our view. Fig. 367. Let the figure to be represented, be a cube ; and let us imagine this cube to be resting upon a hori- zontal surface, and the eye to be placed opposite one of its planes, and in the direction of a line drawn perpendicularly through the centre of that plane. In these relative positions of the eye and the crys- tal, only that plane opposite to the eye will be visi- ble; and if a transparent skreen were interposed between the eye and the crystal, and held parallel to the plane which is seen, the only linear traces which could be marked on the skreen would be the edges of the observed plane, as represented in fig. 367. * This theoretical notion of the infinite distance of the object, is bor- rowed from mathematical considerations of the nature of infinite lines ; and may be taken here merely to imply what is stated in the text, that the edges, or other lines, which are parallel on the crystals, are to be represented by parallel lines in the drawing. FIGURES OF CRYSTALS. 405 Fig. 368. t. . J_ I If we now suppose the eye and the skreen to be moved horizontally toward the right of the spectator, the skreen retaining its parallelism to the plane F', the rays proceeding from the edges of that plane, may be conceived to pass obliquely towards the skreen in its new position, and the edges of the plane P" will now be visible, and may be traced on the skreen as in iig. 368. If we suppose the eye and the interposed skreen to move round the crystal, the skreen retaining its perpendicular position, but ceas- ing to be parallel to any plane of the cube, excepting at some particular points of its progress, it will be obvious, that while the eye and the skreen continue to move in the same horizontal plane^ the vertical planes of the crystal, and those only, will become visible in succession ; but the terminal plane will not be perceived. To see the terminal plane we must suppose the eye and the skreen to be raised ; or, if the eye retain its position, the back of the crys- tal must be elevated. 406 APPENDIX OX DRAWING THE Fig-. 369. It will be more consistent with most of the follow- ing explanations of the methods of drawing" the figures of crystals, to suppose the position of the eye fixed, and the back of the crystal to be elevated. In this new relative position of the crystal and the eye, the figure traced on the interposed skreen, would resemble that exhibited in fte. 369. O It is evident from the preceding explanation, that the relative positions of the eye, the object, and the skreen, may be varied at pleasure, so as to produce in the drawing, such a representation of the object as best suits the illustration it is intended to afford. And although, as it has been stated in p. J02, an advantage will generally attend the placing the figures of crystals belonging to the same class of primary forms always in the same position, there may nevertheless be exceptions to this rule when the position of the modifying planes on the secondary crystal, is such, as to require some new position for their more perfect exhibition. The position chosen by the Abbe Haiiy for the crystal of felspar, is per- haps the best that could be adopted for exhibiting advantageously the secondary planes of the crystals of that substance ; yet the front lateral planes of his figure correspond to the back planes of the doubly oblique prism as it is given in the tables of modi- fications. FIGURES OF CRYSTALS, 407 Having- thus given a brief outline of the theory of geometrical projection, we shall proceed to shew how the forms of crystals may be accurately deli- neated, without entering into any further general explanation of the means which will be employed for this purpose. To draw a Cube. , Fig. 370. On the line a 6, describe the square abed. Let the line e k be parallel to a b. From the points a b c d, draw the lines c e, a r, df> b #, and let these lines be more or less oblique, as the side h f m i is to be rendered more or less visible. Draw the perpendiculars e o, r c g 9 dh, b /, these being, as before, drawn more or less oblique, according as we wish to exhibit a greater or less difference between the two lateral planes of the prism shewn in the front of the figure. To elevate the back part of the prism in order to exhibit the terminal plane, take i i' equal toff; and draw h i', h f, and their parallels. Let the lateral edge h k be drawn in such proportion to the terminal edge a d, as it has been found by cal- culation on the particular crystal we are delineating ; and draw the upper terminal edges parallel to the lower ones. 3 F 410 APPENDIX ON DRAWING THE To draw a Right Oblique-angled Prism, Fig. 372. The right oblique-angled prism may he drawn in a similar manner, keeping the diagonal a b horizontal, and making the angle a d , and the ratios of the edges d a, d 6, and g m y such as they are found in the prism of which we propose to give the figure. ' The oblique rhombic prism may be drawn in a man- ner similar to the two preceding prisms, but it should have a little more elevation given to the back of the figure, in order to render the character of obliquity of the prism more conspicuous. From the necessity of elevating the back of right prisms for the purpose of shewing the terminal plane, it is apparent that the character of obliquity cannot be conferred on a figure so drawn, otherwise than by elevating the back of it rather more than that of the right prism. To draw the Doubly-oblique Prism. The double obliquity given to the figure of this primary form in the tables of modifications, is too slight to convey an accurate notion of its general FIGURES OF CRYSTALS. 411 character; that obliquity is therefore considerably increased in the" following figure. Fig. 373. As the lateral angles of this form are not right angles, its base obviously cannot rest on a horizontal plane, while its lateral edges are perpendicular. To obtain its horizontal projection therefore, while its lateral edges are perpendicular, we may suppose those edges produced until they touch the horizontal plane i m, over which the figures appears to stand. The area of the horizontal projection is clearly less than the base of the figure, and may be known from the ratio of the terminal edges, and from the plane angles of the lateral planes ; which elements are supposed to have been previously determined. SF2 412 APPENDIX OF DRAWING THE Fig. 374. Let as suppose the edge k 1 m', fig. 373, to be to the edge k' i'", as 7 to 4, and to the edge k' k" as 7 to 6. Let the line a b, fig, 374, be the length we may determine upon for the greater terminal edge of the prism ; divide this into 7 equal parts, and 4 of those parts will be the length of the edge k 1 i 1 ", fig. 373, and 6 of them will give the height of the prism. Let the inclination of P on M, fig. 373, be known and called /,, P..T ....... / M..T J 3 . And let the plane angle i'" k 1 k", be also known and called A t9 ki'k'm', A z . And let us suppose A v an acute, and AI an obtuse angle. It is evident that if the solid angle at wz', of such a figure, be supposed to touch the horizontal plane m, the lateral edges being kept perpendicular, the solid angle at k 1 must stand above the plane, and the solid angle at i 1 " still more above it. The elevation of the point at k' may be known by drawing the arc afj fig. 374, with a radius a b, and drawing a second radius bf, making the angle a bf= A 2 90. The perpendicular a g dropped on the line f b, will be the required height of the point k 1 above the horizontal plane, and the line g b will be the length of the horizontal projection of the greater terminal edge of the prism. FIGURES OF CRYSTALS. The increased elevation of the point at i'", fig. 373, may also be determined by drawing the arc c d, fig. 374, with a radius c b equal to of b, and making the angle c b d equal to 90 A t . The per- pendicular c e will be the increased elevation of the point at '", fig. 373, and the line e b will be the horizontal projection of the lesser terminal edge of the prism. Having thus obtained the horizontal projections of the terminal edges of the prism, we may find the vertical projections of the lateral edges in the fol- lowing manner. Let the horizontal projection of the greater diago- nal of the terminal plane be supposed parallel to the line i m, fig, 373, and the diagonal g e of the plan g b e h 9 must be parallel to the same line. Fig. 375. The edges of this plan are known from the figure 374. The length of the line g e may be determined by simply cutting a card so that the angle g b e, fig. 375, shall be equal to / 3 , and making b g, equal to b g-, fig. 374, and b e, equal to b e of the same figure. The point b of the card being laid on the point b of fig. 373, the edge ge may be made parallel to i ra, by means of a parallel ruler, and the lines g b, be, being traced by a pencil, their parallels h , h g, may be drawn to complete the plan. 414 APPENDIX ON DRAWING THE The length and position of the line g e may be also very easily determined geometrically. Make b p, fig. 375, equal to b g, fig. 374, and b n equal to b e of the same figure. From b as a centre, describe the semicircle n o, and the segment p g. Take n e equal to the arc of 90 / 3 , and p g equal to the same arc, and ge will obviously be the greater diagonal of the horizontal projection of the base of the prism. Knowing the horizontal projection g b e 7i, we may proceed to the projection of the prism ; elevating i i'j m m', sufficient to exhibit the terminal plane, and taking k k 1 equal to a f, fig. 374, and i" i'" equal to c e of the same figure. To draw a Rhomboid. Fig. 376. The ratio of the axis, to a .perpendicular drawn FIGURES OF CRYSTALS. 415 upon it from one of the lateral solid angles of the particular rhomboid we are about to delineate, is supposed to have been ascertained.* We should then determine the height our proposed figure is to be, which height will be the length of its axis. Our next step is to find a line which bears the same ratio to that which we have fixed on for the axis of our figure, as the perpendicular upon the axis of the crystal, does to the axis itself. This may generally be done with sufficient precision, by dividing the line we have assumed for our axis into such a number of equal parts, as will give the length of the required line in some other number of those parts. If, for example, we have found, that the per- pendicular upon the axis, is to the axis itself, in the ratio of 7 to 10, and if we determine that our figure shall be an inch high, the required line will be evi- dently & of an inch. If, however, we are desirous of still greater accu- racy, we may draw a perpendicular line equal to that which we have fixed on for the height of our rhomb- oid, and from the upper extremity of this line draw a second, inclining to it at the same angle that the axis does to a superior edge of the rhomboid we are about to represent; and if we now divide our first line into three equal parts, and from the upper point of section, draw a perpendicular to it which shall pass through the second line, the portion intercepted by the second line will be the required length of the perpendicular upon the axis. With a radius equal to this line, which is, in the case we are supposing, ^ of an inch, describe the circle abed ef> fig. 376. * The method of ascertaining this ratio has been already pointed out in p. 363. 416 APPENDIX ON DRAWING THE Divide the circumference of this circle into six equal parts, by the points a b c d f /, as in the figure, and draw the lines a d, / 6, g o, e c, within the circle. Draw k I parallel to f b. Draw the oblique lines shewn in the figure, from the several points on the circumference of the circle, and from its centre, to the line k /; and from the several points in that line where it is cut by the oblique lines, raise the per- pendiculars as they appear in the figure. On the middle perpendicular line, take a portion 3' 3', equal to the length we have determined on for the axis of the rhomboid, and after dividing this portion into three equal parts, draw the lines m n : p q^ r s, through the upper point 3, and through the points of division, and parallel to the line k I. The oblique lines are to be drawn more or less obliquely, according as we would have the rhomboid appear more or less turned round. To elevate the back of the rhomboid, so as to render a plane trun- cating its terminal solid angles visible, draw d i parallel to e c, and join a i. The line d i is the quan- tity of elevation intended to be given to the solid angle a 1 of the rhomboid; and the lines 12, 3, 4 5. are the proportional quantities which the other solid angles require to be elevated in order to preserve the symmetry of the figure. This imaginary elevation of the back of the figure, is thus produced in the drawing. On the perpendicular lines 4', i 1 , and 4', which pass through the line m n, take V a! equal to di\ and 4' jf, 4' b', each equal to the line 4 5. On the perpendiculars, I' and I', which pass through the line p , and b*, by tracing No. 3, fig. 384, .on a separate paper, as No. 5, fig. 385, and above it draw an entire primary form, as shewn by No. 4. To place the planes hi and * on the figure, we require their intersections with and 2 ; ob- serving that those planes intersect each other in a line parallel to the intersection of bi and P. The posi- tions of b'i and 2 may be obtained by a similar method of proceeding, and the other corresponding planes may be drawn by a similar process, or by parallel lines, or by finding the relation of their edges, or angles, to some known points on the crystal. We shall give our next illustrative example from the rhomboid. Let us suppose an obtuse rhomboid is to be represented, modified by planes belonging to classes g, ??z 3 o, and p, and whose symbol is iDi 2D2 iBi 6 P. o p m g P FIGURES OF CRYSTALS. Fig. 386. 431 Let P, P' P", represent the primary planes. The planes o maybe added, by merely lengthening the primary axis, drawing at its two extremities the three upper and three lower primary planes of the rhomboid, and joining their angles by six vertical lines, which will then constitute the vertical edges of the plane o. To add the planes m, take any points c c', on the two adjacent edges of the rhomboid, such that a line passing through both should be parallel to the hori- zontal diagonal of the plane P" ; and from these points draw lines on the planes P and P', parallel to their common edge. The intersection of the planes m with each other, is parallel to a line passing through two opposite solid angles, and through the axis. Fig. 387. This is apparent from fig. 387, in which if the 432 APPENDIX -ON DRAWING THE directing planes 1234, and 1654, be taken to re- present two planes w, their intersection will be the line 1 4, bisecting the axis at o. Having drawn the edges of the planes m on the primary planes in fig. 386, we may find their inter- sections with o and o'. by taking a' b', on the vertical edges of o, equal to the portion a b of the ajtis, cut off by the terminal edges of the planes m, and join- ;': : c //, c' b'. Fig. 388. X? / Our next step should be to add the planes/?. As wo shall have subsequently to add the planes ', it is not strictly necessary to find any other intersections of the plane/?, than those which correspond toff, fig. 388, J\o.2. But for the sake of an additional illustration, we shall find the intersections of/;, with /, and their adjacent intersection with each other. FIGURES OF CRYSTALS. 433 If we recollect that the planes p result from a decre- ment by 2 rows in breadth, it will be apparent that the directing plane/? ppf of No. 1, corresponds to p of No. 2, and p" p" p" f of No. 1, top" of No. 2. But the intersection of 'p p pf, and p" p" p" f, is the directing line//'. If therefore we take any point f in the vertical edge of No. 2, and draw//', parallel to//' No. 1, /' No. 2 being the intersection of the new edge with an oblique diagonal, and if from the points / and /' we draw lines parallel to the inferior primary edges, we shall obtain a representation of the planes p and p". The intersection ofp and m, No. 2, is shewn by the directing line p p, No. 1 ; and that of p 1 and m, No. 2, by p' h, No. 1 ; and the intersection ofp andp', No. 2, is parallel to the line pf, No. 1.* It now remains only to add the planes g to -our figure. For this purpose we may trace in pencil, as at No. 4, fig. 389, an accurate copy of No. 2, fig. 388 ; and above it draw the primary form, and the direct- ing planes shewn by No. 3. We observe here that g g' g", No. 3, corresponds to plane g 9 No. 4. The intersection of this plane with m is parallel to the horizontal diagonal of P", and its intersections with P and P' are parallel to g g', and g g", No, 3. Its intersections with p and ^/, are parallel to the directing lines i I and i V, No. 3; the points /, and /', being the only ones at which the edges of the planes p, p', and g, intersect each other, and the point i being common to the three planes. The intersections of g with o and o' are the lines i k, i k'. For the planes o, and o', might evidently * It may he observed that there are three dotted lines terminating at , No. 1, fig. 388; one of these has p at its other extremity, another has/, and the reader is requested to add/i' to the third. 3i 434 APPENDIX ON DRAWING THE intersect the primary planes P' and P, parallel to their oblique diagonals. If therefore two planes corres- ponding to o and o', No. 4, be conceived to pass through the oblique diagonals of No. 3, and to be produced until they cut the edges g g f and g g", it is obvious that they would cut those edges at the points k and &', and the point i would then be common to the three planes. Assuming any point in the front vertical edge of No. 4 3 we may draw those lines of plane g which intersect o and o', then those which intersect p and p', and P and P' 5 and finally the intersection with m, to complete the figure. Fig. 389. FIGURES OF CRYSTALS. 435 In drawing the secondary forms of crystals, it very frequently happens that the law of decrement will suggest some relation between the position of the secondary edges or angles, and some known points or lines of the primary form, which will supercede the necessity of any directing diagram. One instance of this will be seen if we turn to p. 420, where the rhombic dodecahedron is derived from the cube, through a previous knowledge of the relation of the two forms to each other. And many expedients will probably occur to those who are accustomed to draw crystals, which will greatly abridge the laborious processes just described. These will, however, form particular cases, and will depend on the degree of attention and ingenuity employed in framing the diagrams. The following figure will supply another example of the delineation of a secondary form, from ascer- taining its relation to the primary. Let it be proposed to circumscribe a cube with a figure contained within 24 trapezoidal planes, be- longing to mod. class b of the cube, the law of decre- 436 APPENDIX ON DRAWING THE 2 ment being expressed by 2 A a . The fig. No. 1, has the necessary directing planes drawn upon it, from which it appears that the lines e a, e b, e c, represent three intersections of the secondary planes with each other. If on No. 2 we draw the lines p p', q q', &c. through the middle of the diagonally opposite edges, and from the solid angle at e, draw lines parallel to e a, e b, e c, of No. 1, those lines will be the edges of the new figure, and they will cut the lines p p', &c. at a distance from the edges of the cube equal to -- of its diagonal. This will be apparent if we suppose the central point e of No. 1, to represent the solid angle e of No. 2 ; for the line e a evidently cuts a diagonal of the cube at a distance from its middle point equal to of its whole length. The plane a a' b, No. 1, represents one of the secondary planes, through the middle of which, if we draw the line a' a", that line will pass through the centre of the cube, and will consequently bisect its prismatic axes. A line drawn, therefore, from the solid angle at e, fig. 2, through the produced pris- matic axisyg*, will cut that axis at a distance from the surface of the cube equal to | the length of the axis itself, and will pass through the middle of the secondary plane. Having thus found the points at which the lines p p'-> q #'? & c> an dy*g*5 ^ ? ? an d k i) are cut y tne secondary edges, we may readily complete the figure. We shall give only one further illustration to com- plete this branch of our subject. In several of the preceding examples, the directing planes and lines have been drawn on separate, parallel, figures, for the purpose of exhibiting more distinctly the described methods of drawing. We may, however, in very FIGURES OF CRYSTALS, 437 many cases dispense with the second figure, and draw the directing- planes in pencil, on the same figure which we purpose either to build upon, or to trun- cate. If there should be many planes to be placed on the secondary form, it will be found expedient sometimes to draw the directing lines with the point of a needle only, as the thickness of even a fine pen- cil line may become a source of error. Fig. 391. Let us suppose an octahedron with a square base, derived from a square prism, mod. a, required to be drawn, so as to envelope the prism from which it is derived; this being the method, as it has been already explained, by which nature is supposed to build up secondary forms. And let the law of decrement pro- ducing the plane, be by 2 rows in breadth, and 3 in height, on the terminal angle. Its symbol would consequently be A. First draw a small prism of the proportional dimen- sions of the prism we propose to represent; and through the middle of the lateral planes draw the lines h k y i m, parallel to the terminal edges, and 438 APPENDIX ON DRAWING THE CRYSTALS. also draw the prismatic axis,//. The dimensions of the circumscribing- figure should be such, that its planes should touch the solid angles of the con- tained prism. The directing plane, b c d, evidently represents a plane derived from the modification we have sup- posed ; and the line e d passes through the middle of the plane. If, therefore, we draw gf parallel to d e, and touching the solid angle of the prism at , we obtain the pointy, at which the secondary plane cuts the prismatic axis. From the middle of the axis at o, take o I, equal to o/, and draw / ?z, touching another solid angle of the prism. The point n will evidently be on an edge of the secondary figure, which edge we know must be parallel to a diagonal of the ter- minal planes of the prism. Through the point n, therefore, draw h i parallel to one of those diagonals ; from the points h and i draw h m, i , and join k m to complete the base of the figure. And the terminal edges being drawn, the entire figure will result, as shewn by the dark lines in the engraving. ON MINERALOGICAL ARRANGEMENT. THERE appears to be a degree of difficulty felt by most collectors of minerals, with regard to the ar- rangement of their cabinets, and particularly when new minerals occur, concerning which little more is known than their names. This difficulty arises partly from the want of an accurate distribution of minerals into natural species, and partly from not attending sufficiently to a dis- tinction, which has been hitherto regarded with less notice than it deserves, between this distribution into species, which constitutes the basis of a natural clas- sification of the objects of any branch of natural history, and their artificial arrangement for some pur- pose of illustration, of convenience, or as objects of taste; which artificial arrangement may be regarded as analagous to the order in which words are placed in a dictionary for the convenience of reference. This distinction will be rendered sufficiently ap- parent if we refer to some other branches of natural history for its illustration. The botanist may perhaps place his specimens of dried plants in his portfolio, according to some pre- conceived notion of natural alliance; but when he arranges the plants themselves in his garden or his conservatory, their natural order is disregarded, the natural families are dispersed, and the situation assigned to each plant is determined by its habitudes, 440 APPENDIX ON its necessities, or its peculiar character in reference to the pleasure it is capable of affording to some of the organs of sense. Disparity in size also, among individuals belong- ing even to the same species of objects of natural history, will be another and a frequent cause of variance between their arrangement for purposes of amusement or use, and their natural classification. And examples will probably occur to the reader, of deviation from natural classification in the cabinet arrangement of minerals, where that arrangement has been intended to afford some particular illustration. The cabinet of Leske, described by Kirwan, con- tained several separate collections, arranged for the illustration of distinct objects. One among these, exhibited in a regular series the distinctive external characters of minerals as taught by Werner ; a second contained his systematic arrangement of most, if not of all, the mineral species then known ; and a third exhibited the mineral substances used in various arts and manufactures, and was thence denominated the economical collection. The collection of English minerals in the British Museum, is arranged according to counties^ the sub- ordinate arrangement of the minerals of each county being, however, systematic. When, therefore, the question relates to the ar- rangement of a mineral cabinet generally, we should enquire into the object of the collector in forming his cabinet. In some few instances, it is possible that specimens may have been collected merely as objects of taste, and their selection may have de- pended merely on their rarity, or on the beauty of their forms or colours. The arrangement best adapted to a cabinet of this description, must evidently be such as would best exhibit the forms and colours of MINERALOGICAL ARRANGEMENT. 441 the specimens, and must be, regulated by their size and character, according- to the taste of the possessor. But the views of those collectors by whom the greater number of cabinets are formed, are, probably, to derive from their specimens, an acquaintance with those general external characters of minerals , by which they are commonly discriminated from each other. I would recommend to this class of collectors, an arrangement of their cabinets in nearly an alphabetical order., which, as it will greatly facilitate the reference to particular specimens, will afford them more ready means of comparing different specimens with each other ; and every new substance that occurs, to which a name has been assigned, will also find an immediate place in the collection under its proper letter, if its precise station under any other leading name has not been previously determined. The alphabetical series here recommended, is that which is distinguished by roman capitals in the alpha- betical list which follows this section. In this list I have endeavoured to collect and arrange all the mineral species at present known, with such of their synonyms as are not merely trans- lations out of one language into another; and with the addition of such of the primary forms of those which are regularly crystallised, as appear to be accu- rately known. Most of these forms have been determined from an examination of the substances themselves, and their angles have been measured, principally by the reflec- tive goniometer, both by Mr. W. Phillips and myself ; but from Mr. Phillips's greater precision in the use of that instrument, I have generally relied on his measurements where they have differed from my own ; and in several instances, I have been indebted to 3 K 442 APPENDIX ON Mr. P. for the forms and measurements of crystals which I had not myself previously examined. Following this alphabetical list, will be found a second table of primary forms, arranged according to their classes. The synonyms have been collected chiefly from Leonhard's Handbuch der Oryktognosie, published in 1821, and corrected from such other sources as I have had an opportunity of consulting. The choice of a specific name among many synonyms, has been in some degree arbitrary, but I have generally been influenced in this choice by the previous adoption in this country of the name I have selected. On refer- ing to the list, the word Abrazite will be found at its head, with a reference to Zeagonite, that being the name under which the mineral, also called Abrazite and Gismondin, had been previously known here. For this reason I have retained many of the old names, as Chiastolite, for example, instead of Made, the name assigned to the same mineral by Hatty. In many instances, it will be perceived, I have adopted the names given by Haiiy, either because they have already become familiar to the English mineralogist, as Peridot, instead of Chrysolite, or because they have comprehended several of the older species under a single name, as Amphibole, which includes the Hornblende, Tremolite, and Actynolite of the Wer- nerian school. Although the basis of the proposed arrangement of minerals is alphabetical, it is to a certain extent founded on their chemical distinctions. But a difficulty presents itself when we attempt a purely chemical classification of minerals, which arises out of the uncertainty of our knowledge rela- tive to the essential constituents of many species. For however accurately these may have been analysed by MINERALOGICAL ARRANGEMENT. 443 the skill of modern chemistry, we are yet unable to determine which of their component parts are essen- tial to the composition of the substance analysed, and which are but accidental mixtures. In the in- stance of the crystallised sandstone from Fontain- bleau, no doubt can be entertained that either the carbonate of lime, or the grains of sand, must be regarded as accidental mixture, and foreign to the constitution of the species, accordingly as we chuse to consider the specimens, as arenaceous quartz agglu- tinated bj/ carbonate of lime, or as carbonate of lime inclosing grains of quartz. It would, however, present little difficulty to the chemist, to determine that the silex and lime are not chemically combined in the sandstone : but there are numerous other instances, in which even the sagacity of a Berzelius has probably failed in discriminating the matter accidentally present in several of the species of minerals which have been analysed, from that which is essential to the composition of each particular species. These doub'ts are suggested by the observed fact, that the crystalline form of the Fontainbleau sand- stone is similar to one of the secondary forms of car- bonate of lime ; and from remarking, that among the minerals which chemical analysis would raise into distinct species, there are several which appear to agree in their crystalline forms. Now if we regard the Fontainbleau sandstone as a variety of carbonate of lime, enveloping grains of quartz ; and as we observe that the crystalline form of the carbonate of lime is not altered by the presence of this siliceous mixture, we may infer that the crys- talline character of minerals is not affected by the accidental presence of foreign matter in their com- position, and consequently that minerals differing 3*2 444 APPENDIX ON widely in their chemical character, may really belong to one species. These considerations appear to confirm the pro- priety of the Abbe Haiiy's definition of a mineral species, as given in p. 6. For although it might have been sufficient, theoretically, to comprise within the terms of this definition, such individual minerals as are composed of similar particles united in equal proportions, yet in the present imperfect state of our knowledge of the true constituent elements of many minerals, it appeared practically necessary to super- add to this definition the condition that, if they be- long to the same species, the form of their molecules, or, which is the same thing in effect, their primary forms should also be similar. Hence when we find different minerals agreeing in their crystalline forms, and varying in their chemical composition, we shall probably determine their spe- cies more accurately from their crystalline than from their chemical characters. I say probably, for the 6rystallographical cha- racter has its uncertainties also. The natural planes of crystals are generally too imperfect to give mea- surements which may be said to agree very nearlv with each other; the differences among such as belong to the same species of mineral, amounting frequently to nearly a degree ; and the cleavage planes, which generally afford better corresponding results, cannot always be obtained; but if they could, the angles of mutual inclination even of those, are not always alike, owing probably to an interposition of foreign matter between the laminae of the crystal, and being there unequally dispersed. Nor do we know that the difference of the angles under which the primary planes of different species of minerals meet, is not Jess than our best goniometers can distinguish. I( MINERALOGICAL ARRANGEMENT. 445 demands great precision of hand and eye to obtain the true measurements of angles to a minute only, and we cannot say that a difference of species may not exist, with a difference of only a few seconds in the angles of inclination of their planes. We know that the greater angle of a right rhombic prism must lie somewhere between 90 and 180. If the angle were 90 , the prism would be square, and 180 would reduce the prism to a plane. But be- tween 90 and 180, we do not know how many different prisms may exist. If they differed by degrees only, their number could not exceed 89. If the difference consisted of minutes, there might be 5399 such prisms, all dis- tinguishable by the goniometer ; but if the differences consisted of seconds only, there might evidently be 323999 rhombic prisms, of which no more than 5399 could be distinguished by the ordinary goniometric instruments. But with all the uncertainties and difficulties at- tendant upon the crystallographical determination of a mineral species, the goniometer is probably the most accurate guide we at present possess to lead us to that determination. And it is almost the only one of whicfi the practical mineralogist can at all times avail himself. It appears almost unnecessary to state, that where a mineral is defective in crystalline character, or its chemical composition is unknown, it must be pro- visionally distinguished from other minerals by some other of its physical characters, as its specific gravity, hardness, fracture, &c. Instances have been already alluded to where chemistry would separate minerals from each other, which, crystallographically, belong to the same species ; of which the Amphiboles, and the Pyrox- 446 APPENDIX ON enes, afford examples. But there are a few cases also, in which minerals differing in their crystalline form, are similar in their chemical composition; as appears to be the case with the common, and white, iron pyrites. These anomalies will however, probably, be reconciled by the future investigations of science. Dr. Brewster has, with that attachment which we usually evince towards a favorite pursuit, given a preference to the optical characters of minerals, as the surest means of determining their species. See a memoir by Dr. Brewster, in the Edinb. Phil. Journ. vol. 7. p. 12. This memoir relates to a difference in the optical characters of the Apophyllites from different locali- ties, upon which Dr. Brewster proposes to erect a particular variety into a new species under the name of Tesselite. Berzelius, as it appears from a paper, preceding that by Dr. Brewster. in the same volume of the Journal, has, at Dr. Brewster's desire, ana- lysed the Tesselite, and found it agreeing perfectly in its chemical composition with the Apophyllites from other places. Chemically, therefore, the Tesselite does not appear a distinct species. A few days before Dr. Brewster's paper was pub- lished, it happened that I had been measuring the angles of the Apophyllites from most of the localities in which they occur, all of which I found to agree with each other more nearly than different minerals of the same species frequently do. The Tesselite is not therefore, crystallographically, a separate species.* But when chemistry and crystallography * I have found several crystals of this substance corresponding in a remarkable manner in their general form of flattened four-sided prisms, terminated by four-sided pyramids with truncated summits, but with their corresponding planes dissimilar. The planes which appear as the summits of some of these prisms, being only the lateral planes of very MINERALOGICAL ARRANGEMENT. 447 concur so perfectly as they do in this instance, in determining the species to which a mineral belongs,, it will be difficult to admit a variation of optical character, as a sufficient ground to alter that deter- mination. A paragraph published by Dr. Brewster in the 6th volume of the same Journal, p. 183, relative to the crystalline form of the sulphato-tri-carbonate of lead, furnishes an additional motive to believe that the connection between the optical characters of minerals and their crystalline forms, is not yet sufficiently understood. Dr. Brewster admits what I believe is not liable to question, that " the crystals of this substance are " acute rhomboids" But he adds, " Upon examining " their optical structure, I find that they have two " axes of double refraction, the principal one of " which is coincident with the axis of the rhomb. " The sulphato-tri-carbonate, therefore, cannot have " the acute rhomboid for its primitive form, but must " belong to the prismatic system of Mohs" But it appears from the " Outline of Professor Mohs's new system of Crystallography," published in vol. 3 of the same Journal, that a rhomboid cannot belong to his prismatic system. For it is stated in p. 173, that " The rhomboid^ and the four- sided oblique" " based pyramid^" (the fundamental form of tlje pris- matic system) " are forms which cannot by any means " be derived from each other / the (two) groups of short and otherwise disfiroportioned crystals ; so that a line passing through these, in the direction of their greatest length, would in fact be per- pendicular to the axis of the primary form. Sections perpendicular to the axes of these apparently similar prisms, would certainly present very different optical phenomena. But it is not probable that the prac- tised eye of Dr. Brewster should have been misled by their apparent similarity, and the differences he has observed will still remain to be explained. 448 APPENDIX ON a simple forms ) as well as their combinations, must " each be always distinct from the (other).''' If therefore in the hands of Dr. Brewster, the use Of optical characters cannot at present be relied upon for the determination of a mineral species, it may be doubted whether they can be successfully employed by less accurate and less intelligent observers. The proposed arrangement in the following alpha- betical list, is, as it has been already observed, to a certain extent, chemical ; several species, to which separate specific names have been given, being ar- ranged frequently under one head or genus, in the alphabet. And there are, probably, many other species which now stand singly in their alphabetical order, which in the opinion of some of my readers might, with equal propriety, be collected into other chemical groups. This collection of species into groups or genera, has not been regulated by any very precise rule. The leading principles, however, upon which they have been formed, are either the simplicity of com- position of the species of which they consist, or the apparent certainty with which that composition has been determined; some few species may, however, be considered as rather arbitrarily included under particular genera. In most of the genera, the first and second of these principles are apparent; an example of the third may be seen under the head of Cerium, where the Yttro- cerite is placed, although it contains a greater pro- portion of Yttria than it does of the oxide of Cerium. The species which are left in their alphabetical order, are generally those which are denominated earthy minerals, and are composed of Alumine, Lime, Magnesia, Silex, &c. in various proportions, which MINERALOGICAL, ARRANGEMENT. 449 are probably not as definite as they hav^ been some- times considered. The proposed alphabetical arrangement will appear to deviate the less from natural classification, if we recollect that there is not any one strictly exclusive natural order to supercede this arrangement, and re- quiring that Zircon should be placed before or after the siliceous genus; or that Lead should precede, or follow, Iron or Copper. There may be conceived to be as many natural classifications of minerals, as there are natural properties common to the sub- stances which are to be arranged. Thus, the metals (not including the bases of the alkalies and earths) might be arranged according to their fusibility, or their specific gravity, or their ductility, &c. Either of these characters might be adopted as the basis of a natural classification , and the order of the substances thus classed, would vary according to the generic character we might adopt. The primary forms of most of the crystallised minerals contained in the following alphabetical list, are indicated in italics. The measurements there given, are the most accurate I have been able to obtain ; but although they have been taken with much care, and probably do not vary much from the truth, they are to be regarded in strictness only as approxi- mations to the true angles at which the planes of the crystals incline to each other. I have added where I could, to the square, rectan- gular, and hexagonal prisms, the measurement of a primary plane on some modifying plane, which fre- quently occurs on the crystals ; and the class of modifications to which the modifying plane belongs, is indicated by its appropriate letter. 3 L 450 APPENDIX, &C. It does not fall within the scope of my plan to give more than a mere list of minerals and of their primary forms. Descriptions of the minerals themselves, and figures of their secondary forms, as they occur in nature, will form the substance of a volume on Mineralogy by Mr. W. Phillips, which is now in the press. AN ALPHABETICAL ARRANGEMENT OF MINERALS, WITH THEIR SYNONYMES AND PRIMARY FORMS. Abrazit, see Zeagonite. Achirite, see Copper, carbonate, siliceous. Actinolite, see Amphibole. Actinote, see Amphibole. Adamantine spar, see Corundum. Adularia, see Felspar, crystallised. Aehrenstein, see Barytes, sulphate. Aequinolite, see Spherulite. Aerolite, see Iron, native, meteoric. Aerosite, see Silver, sulphuret, antimonial. AGALMATOLITE ; Bildstein ; Figure-stone ; Koreite ; Lardite ; Pagodite. Agaphite, see Alumine, hydrate, compact. Agaric mineral, see Lime, carbonate, spongy. Agate, see Quartz. Agustite, see Emerald, var. Beryll. v Akanticone, see Epidote. Alabaster, see Lime, sulphate. Alalite, see Pyroxene. Albin, see Apophyllite. Albite, see Cleavelandite. Allagite, see Manganese, carbonate, siliceous. Allanite, see Cerium, oxide, ferriferous. Allochroite, see Garnet. Allophane, see Alumine, silicate. Almaudine, see Garnet. Alum, see Alumine, sulphate. 3 L 2 452 LIST OF MINERALS, THEIR ALUMINE. hydrate. crystallised ; Diaspore. A doubly oblique prism, P on M, 108 30'; P on T, 101 20' ; M on T, 65, as measured and described by W. P. stalactitic ; Gibbsite. compact ; Agaphite ; Calaite ; Johnite ; Turquoise, earthy, phosphate. crystallised ; Devonite ; Hydrargillite ; Lasionite ; Wavellite. A light rhombic prism, M on M', about 122 15'. silicate; Allophane. Cleavage parallel to the planes of a square or rectangular prism. sub-sulphate; Aluminite ; Hallite; Websterite. sulphate of, and potabh ; Alum, crystallised. fibrous. A regular octahedron. sulphate of Alumine and Potash ; Alum, crystallised, fibrous. A regular octahedron. , siliceous. Alum-stone. crystallised. An obtuse rhomboid, P on P', 92 50', as measured by W. P. amorphous. Aluminite, see Alumine, sub-sulphate. Amalgam, see Mercury, argentiferous. Amausite, see Felspar, compact. Amazon-stone, see Felspar, green. AMBER; Bernstein; Karase; Succin. AMBLYGONITE. Cleavage parallel to the lateral planes of a prism of about 105 45', with indistinct traces of cleavage oblique to the axis of the prism. I am in- debted to Mr. Heuland for the loan of the specimen I have measured. Amethyst, see Quartz. Amianthinite, see Amphibole, Amianthoide. Amianthoide, see Amphibole. Amianthus, see Asbestus. AMMONIA. muriate ; Sal ammoniac. crystallised. A regular octahedron. stalactitic. earthy. sulphate ; Mascagnin. stalactitic. earthy. SYNONYMES, AND PRIMARY FORMS. 453 AMPHIBOLE. crystallised. An oblique rhombic prism , P on M or M' 3 103 15'; M on M', 124 30'. fibrous, amorphous. The following varieties appear, from the measure- ment of their angles, to belong to this species. Common Hornblende, colour dark green or greenish black; Keniphyllite ; Keratophyllite. Carinthin, in colourless, yellowish, and greenish crys- tals. Basaltic Hornblende. The foliated Augite of Werner. The blue Hyperstene of Giesecke. The green Diallage of Haiiy ; Smaragdite ; Lotalite. Pargasite, in short green crystals. Actynolite; Actiuote; Strahlite ; the crystals green, slender, and sometimes radiating. Tremolite ; Grammatite ; the crystals colourless or green, or pink, or brownish and generally imbedded in Dolomite; frequently fibrous; and sometimes ra- diated. A transparent and colourless variety occurs with the white Pyroxene at New York. Several specimens sent me as white pyroxene were all amphibole except one, which contained two or three imbedded crystals of pyroxene. Amianthoide ; Amianthinite ; Asbestiiiite ; Byssolite ; two separate fibres of this substance have afforded the measurement of 124 30.' Amphigene, see Leucite. ANALCIME ; Cubicite. A cube. the crystals red ; Sarcolite. Anatase, see Titanium, oxide. ANDALUSITE; Micaphyllite ; Stanzaite. A right rhombic prism, M on M', 91 20', as measured by W. P. Andreasbergolite, see Harmotome. Anhydrite, see Lime, sulphate, anhydrous. ANTHOPHYLLITE. Cleavage parallel to the lateral planes of a rhombic prism of 125, and to both its diagonals ; and another cleavage apparently perpendicular to the axis of the prism. The bright plane which is generally visible in the specimens, is parallel to the greater diagonal of the prism. Anthracite, see Coal. Anthracolite, see Coal. 454 LIST OF MINERALS, THEIR Anthraconite, see Lime, carbonate, columnar. ANTIMONY. native. Cleavage parallel to the planes of an obtuse rhomboid-, P on P', about 117, but the measure- ment of different fragments does not agree within more than 2 degrees, owing to the dulness of the planes, and apparently to their being more or less curved. The rhombic planes are striated hori- zontally, and the bright planes, which are gene- rally conspicuous in the specimens of this substance, are perpendicular to the axis of the rhomboid. arseniferous. oxide ; White antimony. crystallised. A right rhombic prism ; M on M', 137; the broad bright planes of the crystals being parallel to the short diagonal of the prism. fibrous, radiating, earthy. ....... sulphuretted ; Red antimony. crystallised ; probably a right square prism ; Mr. Phillips having found one of the thin fibrous crystals measure on the lateral planes 90, and 135. fibrous. earthy ; Tinder ore. (Leonhard.) sulphuret ; Grey antimony. crystallised. A right rhombic prism , M on M', very neaily 90. Its secondary planes shew that the prism belongs to the rhombic class and is not square, fibrous, compact. Apatite, see Lime, phosphate. Aphrite, see Lime carbonate, nacreous. Aphrizite, see Tourmaline. APLOMB. A cube. APOPHYLLITE; Albin ; Fish-eye-stone; Icthyopthalmite. A right square prism ; M on a plane belonging to mod. class 0, 128 10'. Aquamarine, see Emerald, var. Beryll. Arendalite, see Epidote. ARFWEDSONITE; Ferriferous hornblende; see Annals of Philosophy for May 1823. Cleavage parallel to the lateral planes and to both the diagonals of a rhombic prism of 123 55'. SYNONYMES, AND PRIMARY FORMS. 455 Argentine, see Lime, carbonate, nacreous. Arkticite, see Scapolite. Armenite, is said to be either Quartz or Carbonate of lime coloured by blue Carbonate of copper. * Arragonite, see Lime, carbonate. ARSENIC. native. oxide. A regular octahedron. sulphuret. red ; Realgar. crystallised. An oblique rhombic prism ; P on M or M', 104 6' ; M on M', 74 14'. amorphous yellow ; Orpiment. crystallised. A right rhombic prism ; M on M', 100. Form determined and measured by W. P. foliated. Asbestinite, see Amphibole. Asbestoide, see Amphibole. ASBESTUS. common, the fibres parallel, lying in many directions, and as it were matted together, forming mountain paper leather cork wood. flexible; Amianthus. Asparagus-stone, see Lime, phosphate. Asphaltum, see Bitumen. Astrapyalite, see Quartz, sand-tubes. Atacamite, see Copper, muriate. Atlaserz, see Copper, carbonate, green. Atramentstein, see Iron, sulphate, decomposed. Augite, see Pyroxene. Augustite, see Lime, phosphate. Automolite, see Zinc, oxide, aluminous. Avanturine, see Quartz, amorphous. Axe-stone, see Jade. AXINITE; Thumerstone ; Thumite ; Yanolite. This sub- stance does not readily yield to cleavage, so as to afford a determination of its primary form from cleavage planes. The primary form best agreeing with those secondary forms under which it generally occurs, is a doubly oblique prism, P on M, 134 40' ; P on T, 115 17' ; M on T, 135 10', as measured by W. P. 456 LIST OF MIMERALS, THEIR Azabache, see Coal, Jet. AZURITE; Klaprothite ; Tyrol i te ; Voraulite. crystallised. A right rhombic prism ; M on M', m 30'. I am indebted to the kindness of Mr. Heuland for the very rare specimen which has enabled me to determine this form, compact. B Baikalite, see Pyroxene. Baldogee, see Green-earth. Bardiglione, see Lime, sulphate, anhydrous. BARYTES. carbonate ; Barolite ; Witherite. crystallised. A right rhombic, prism, M on M', 118 30", as measured by Mr. W. Phillips. The ordinary hexagonal crystals probably result from the intersection of threq of the primary crystals, fibrous, sulphate; Baroselenite. crystallised. A right rhombic prism, M on M', 101 42'. columnar. radiated ; Bolognian spar ; Litheosphore. granular, compact ; Ca\vk. acicular, diverging, and imbedded in some other substance ; Aehrenstein. earthy, hepatic. sulphate of Barytes and Strontian. Basanite, see Quartz. Baudisserite, see Magnesia, carbonate, siliceous, Beilstein, see Jade. Bell-metal ore, see Tin, sulphuret of Copper and Tin. BERGMANITE; Spreiistein. No crystalline form discoverable, nor any analysis, that I can find published. Is referred by Leonhard to Scapolite, but on what authority does not appear. JJERGMEHL, Mountain meal. Bernstein, see Amber. Beryll, see Emerald. Berzelite, see Petalite. Bildstein, see Agalmatolite. Bimstein, see Pumice. SYNONYMES, AND PRIMARY FORMS. 457 BISMUTH. carbonate.* native. oxide. sulphuret. cupriferous. plumbo-cupriferous ; Needle-ore. plumbo-argentiferous ; Bismuthic silver. Bitter-spar, see Lime, carbonate, magnesian. BITUMEN. liquid; Naphtha. viscid ; Petroleum. solid, elastic ; Elaterite. compact; Asphaltum. earthy ; Maltha. Dapeche, brought by Humboldt from South America, and probably does not beJong to the mineral king- dom. Fossil copal; Highgate resin. Retinasphaltum. Blattererz, see Tellurium, native, plumbo-auriferous. Bleiniere, see Lead arseniate. Bleischweif, see Lead, sulphuret, compact. Blende, see Zinc, sulphuret. Blizsinter, see Quartz, sand-tubes. Bloedit, see Magnesia, sulphate of and Soda. Bloodstone, see Quartz, calcedony. Bohnerz, see Iron, oxide, hydrous. BOLE ; Lemnian earth ; Terra de Siena ; Terra sigillata. Bolide, see Iron, native meteoric. Bolognian spar, see Barytes, sulphate, BoRAcicAciD; Sassolin. Boracite, see Magnesia, borate. Borax, see Soda, borate. Borech, see Soda, carbonate. Botryolite, see Lime, borate, siliceous. Bournonite, see Lead, triple sulphuret. BREISLAKITE. BREWSTERITE. si right oblique-angled prism, M on T, about 93 40'. Brongniartin, see Soda, sulphate of Soda and Lime. BRONZITE; fibrous Diallage metalloide. Cleavage parallel to the planes and to both the diagonals of a rhombic prism,) of about 93 30', with indications of another cleavage perpendicular to the axis of the prism. See Hypersthene. 3n- 458 LIST OF MINERALS, THEIR Brown spar; see Iron, carbonate. And see Pearl-spar. Brucite, see Condrodite. Brunon, see Titanium, oxide, siliceo-calcareous. BUCHOLZITE ; two different substances appear to have been included under this name, viz. a mineral from the Tyrol, called Fibrous Quartz by Werner, and a fibrous sub- stance frequently found accompanying Andaluzite. Byssolite, see Amphibole. C Cacholong, see Quartz, opal. CADMIUM, a metal found in combination with Zinc in seve- ral of its ores. Calaite, see Alumine, hydrate. Calamine, see Zinc, carbonate, oxide, silicate. Each of these species having passed under the common appellation of Calamine. Calcedony, see Quartz. Cantalite, see Quartz, yellowish-green. CARBONIC ACID. Carinthin ; see Amphibole. Carnelian, see Quartz, calcedony. Cascalhao, clay indurated by Iron and Quartz, and frequently inclosing grains of Quartz, found in rolled fragments at the diamond mines in Brazil. Catseye, see Quartz. Cawk, jsee Barytes, sulphate, compact. Celestine, see Strontian, sulphate. Cerauniansinter, see Quartz, sand-tubes. Ceraunite, see Jade, nephrite. CEREOLITE. Cerin, see Cerium. Cerite, see Cerium. CERIUM. fluate. sub-fluate. fluate of Yttria and Cerium. fluate of Yttria Cerium and Lime ; Yttro-cerite. Clea- vage parallel to the planes of a right rhombic prism of about 97, by the common goniometer. oxide, siliceous, red ; Cerite ; Ochroite. ferro siliceous, black ; Cerin ; Allanite. A right square prism,) as determined by W. P. CHABASIE. An obtuse rhomboid, P on P' ? 94 46'. SYNONYMES, AND PRIMARY FORMS. 459 Chalcolite, see Uranium, phosphate. Chalcosiderite,- see Iron, green earth, fibrous. Charcoal, mineral, see Coal. Chelmsfordite, possesses the external characters of Scapolite, and crystallises in square prisms. See Scapolite. CHIASTOLITE; Crucite ; Macle. Probably a right square or rectangular prism. Chlorite, see Talc. Chlorophane, see Lime, fluate. CHLOROPH^ITE, described by Dr. Mac Culloch. Chondrodite, see Condrodite. CHROME, oxide. CHRYSOBERIL; Cymophane. Aright rhombic prism, Mon M', 97 12'. The plane P is generally bright and striated. Chrysocolla, see Copper, carbonate, siliceous. Chrysolite, see Peridot. Chrysoprase, see Quartz, calcedony. Chusite, see Peridot, granular, decomposing. CIMOLITE. Cinnabar, see Mercury, sulphuret. CINNAMON-STONE; Essonite ; Hyacinth; Romanzovite. crystallised. A rhombic dodecahedron. The cleavage planes afford measurements of about 90 in one direction, and about 120 in one or two others, amorphous. CLEAVELANBITE ; Albite; Siliceous spar from Chesterfield in Massachusetts. See Annals of Philosophy for May 1823. crystallised. Cleavage parallel to the planes of a doubly oblique prism, P on M, 119 30' ; P on T, 115; M on T, 93 30'. laminar. COAL. carbon nearly pure ; Anthracite ; Anthracolite ; Gean- trace. compact, columnar, slaty, bituminous. compact ; Cannel coal, columnar. foliated ; Common coal, friable ; Mineral charcoal, ligniform. Wood coal. compact; Azabache ; Jet. fibrous ; Bovey coal ; Surturbrand. SL 2 460 LIST OF MINERALS, THEIR foliated ; Dysodile ; Paper coal, earthy, peat. COBALT. arseniate. Aright oblique-angled prism. M on T, 124. arsenical. grey. A cube with regular modifications. white. A cube with irregular modifications similar to those of Iron pyrites, oxide, black. ferriferous, brown, yellow, sulphate. stalactitic. sulphuret. botryoidal. amorphous. Coccolite, see Pyroxene, granular. Cockle, of the Cornish miners, see Tourmaline. COLLYRITE, or Kollyrite. Colophonite, see Garnet. Columbite, see Tantalite. COMPTONITE. A right rectangular prism. M on a plane belonging to mod. class . ....... siliceous, compact ; Baudisserite ; Magnesite. The silica probably not essential to the species, which may be merely a carbonate mixed with silex. pulverulent; Razoumoffskin. fluate ? hydrate, siliceous ; Meerschaum ; Myrsen ; Kil ; Kill- keffe. of Magnesia and Soda ; Bloedit. sulphate. crystallised. A right square prism ; M on a plane belonging to class c, 129. fibrous, earthy. of Magnesia and Soda. of Magnesia and Iron ; Hallotricum. Magnetic iron, see Iron, oxydulous. Malachite, see Copper, carbonate. Malacolite, see Pyroxene. Maltha, see Bitumen. MANGANESE. carbonate. crystallised. An obtuse rhomboid^ P on P', about 107* 20', but the planes of the only specimen 480 LIST OF MINERALS, THEIR I have seen are too much curved to admit of a very precise measurement, foliated ; Dialogite. compact ; Rhodochrosite. siliceous. anhydrous ; Allagite ; Photizite. hydrous ; Rhodonite, hydrate, oxide. crystallised. A right rhombic prism, M on M' , 100. compact, earthy. Wad. fibrous, frothy, earthy, silicate ; Red manganese ore. ..... foliated, compact. Helvin, which see. Hydropite. , ferriferous, in octahedrons from Piedmont. hydrous. The precise differences between the preceding varieties cannot be accurately stated, there being no exact descriptions of the different minerals analysed, except of the Helvin. phosphate of Iron and Manganese, sulphuret. Marble, see Lime, carbonate, compact. Marekanite, see Obsidian. Markasite, see Iron, sulphuret, arsenical. Marl, see Lime, aluminous. Mascagnin, see Ammonia, sulphate. Meerschaum, see Magnesia, hydrate, siliceous. METONITE. A right square prism^ mod. d. on a plane belonging to class , 122. This species nearly corres- ponds in measurement and chemical composition with Scapolite. Melanite, see Garnet, black. Melanteria, see Iron, sulphate. MELLILITE. A right square prism ; determined by W. P. . from measurement of lateral primary, and secondary planes. MELLITE ; Honeystone. An octahedron with a square base, P on P', 93 7; P on P", 118" 31'. SYN 7 ONYMES, AND Pit IM Ail IT FORMS. Menachanite, see Titanium, oxide. Menilite, see Quartz, opal. MERCURY. muriate. A right square prism, M on a plane belong- ing to mod. class c, 158. native. argentiferous; Native amalgam. A rhombic dodecahedron. sulphuret ; Cinnabar. crystallised. An acute rhomboid, P on P', 72". fibrous, pulverulent, compact, slaty. hepatic ; Corallenerz ; Liver ore. MESOLE. See Edinb. Phil. Jour. vol. 7. p. 7. MESOLINE. See Edinb. Phil. Jour. vol. 7. p. 7. MESOLITE. MESOTYPE. crystallised. A right rhombic prism-) M on M', 91* 10'. red, globular radiated ; Crocalite. .... earthy ; Edelite. yellow, globular radiated, or reddish orwbite ; Hogauite; Natrolite. Meteorite, see Iron, native. MIASZITE. MICA. The crystalline form of the brpwn Mica from Vesu- vius is an oblique rhombic prism, P on M or M', 98 40'; M on M', 100, as determined by W. P, from measure- ment of some brilliant crystals. From the analyses of different substances which have been denominated Mica, it appears probable that different species of minerals have been comprehended under that name, and that among these there may be different crystalline forms. One of these varieties appears, from the direction of some of its cleavages, to crystalline in right prisms, which are probably hexagonal. Micaphyllite, see Andalusite. Micarelle ; Finite and Scapolite have both passed under this name. Mieraite, see Lime, carbonate, magnesian. Mispickel, see Iron, sulphuret, arsenical. Misy, see Iron, sulphate, decomposed. Mocha-stone, see Quartz, Agate, dendritic. Molarite, see Quartz, buhrstone. 3P 482 LIST OF MINERALS, THEIR MOLYBDENUM, oxide, fibrous. pulverulent. sulphuret. The form of the only crystals I have seen is a regular hexagonal prism , which is probably the primary form. Moon-stone, see Felspar. Moroxite, see Lime, phosphate. Mountain, cork, leather, wood, see Asbestus. Mountain meal, see Bergmehl. MOUNTAIN SOAP. Miillers glass, see Quartz, hyalite. Muriacite, see Lime, sulphate, anhydrous. Muricalcite, see Lime, carbonate, magnesian. Mundic, a name given by the Cornish miners to Iron pyrites. MURIATIC ACID. Mussite, see Pyroxene. Myrsen, see Magnesia, hydrate, siliceous. N Naphtha, see Bitumen. Napoleonite, see felspar, globular. NAPOLITE. A blue mineral from Vesuvius, see Annals of Philosophy, vol. 7. p. 402. I have called it Napolite for the purpose of distinguishing it by name from Hauyne with which it has been classed, but to which species it appears not to belong. Natrolite, see Mesotype. Natron, see Soda. Necronite, is probably Felspar. It has two cleavages pro- ducing bright planes at right angles to each other, and an indistinct oblique cleavage, and has the same lustre and hardness as Felspar. Needle ore, see Bismuth, sulphuret. NEEDLE-STONE ; Scolezite. crystallised. A right rhombic prism , M on M', 91 20'. The Needle stone from Iceland, and that from Faroe, afford the same measurements by the re- flective goniometer. Dr. Brewster regards them however as distinct species, acieular. pulverulent ; Mealy stilbite. Neopetre, see Quartz, hornstone. NEPHELINE ; Sommite. A regular hexagonal prism^ M on a plane belonging to mod. class c, 134 The Nephelines from Monte Somma and from Capo.di Bove, afford an instance of chemical discordance in rela- tion to minerals having the same crystalline form. SYNONYMES, AND PRIMARY FORMS. 483 Nephrite, see Jade. NICKEL, arseniate. colouring clay or some other substances ; Pimelite, arsenical. , antimonial oxide ; black. sulphuret of Nickel, Arsenic and Iron. Nickel, Antimony and Arsenic, Nigrine, see Titanium. Nitre, see Potash, nitrate. Nosin, see Spinellane. NOVACUUTE ; Turkey stone, O OBSIDIAN ; Volcanic glass. in small rolled fragments ; Marekanite. fibrous. amorphous. Ochre, see Iron, oxide. Ochroite, see Cerium, oxide. Octahedrite, see Cerium, oxide. ODERIT ; probably Black mica. Oisanite, see Titanium, oxide, anatase. Oligiste Iron, see Iron, oxide. Olivenit, see Copper, arseniate. Olivin, see Peridot. Omphazit, appears from the specimens sent here, to be a mixture of Garnet, and that variety of Amphibole called by Haiiy Green diallage, and probably Cyanite. Onegite ; perhaps an ore of Titanium. Oolite, see Lime, carbonate. Opal, see Quartz. Ophite, see Serpentine. Orpiment, see Arsenic, yellow sulphuret. ORTHITE. Orthose, see Felspar. OSMIUM ; occurs alloyed with Indium, which see, Osteocolla, see Lime, carbonate, incrusting. Otrelite, see Schiller spar. P Pagodite, see Agalmatolite. PALLADIUM, native. Paran thine, see Scapolite. Pargasite, see Amphibole. Paulite, sec Hyperstene. OP 2 484 klST OF MINERALS, THEIR Pearl-spar, see Lime, carbonate, magnesian. There has been much uncertainty in the use of the terms Brown spar and Pearl spar ; the first of these having been applied to carbonate of Iron, and also to those varieties of Pearl spar, or Magnesian carbonate of Lime, which are of a brown colour, and probably to some other of the car- bonates, of Lime, and of Manganese. PEARL STONE. Peat, see Coal. Pechuran, see Uranium, oxide, ferriferous. Peliome, see Dichroite. Pentaclasite, see Pyroxene. PERIDOT. crystallised ; Chrysolite A right rectangular prism , M on a plane belonging to mod. class